philosophy of physics

The Everett Interpretation ———. (2010). A fair deal for Everettians. In Saunders, Barrett, Kent, and Wallac e (2010), 205– 226. Penrose, R. (1989). The emperor's new mind: Concerning computers, brains and the laws of physics. Oxford: Oxford University Press. Pric e, H. (2010). Dec isions, dec isions, dec isions: c an Savage salvage Everettian probability? In Saunders, Barrett, Kent, and Wallac e (2010), 369– 390. Rovelli, C. (2004). Quantum gravity. Cambridge: Cambridge University Press. Saunders, S. (1993). Dec oherenc e, relative states, and evolutionary adaptation. Foundations of Physics 23: 1553– 1585. ———. (1995). Time, dec oherenc e and quantum mec hanic s. Synthese 102: 235– 266. ———. (1997). Naturaliz ing metaphysic s. The Monist 80(1): 44–69. ———. (1998). Time, quantum mec hanic s, and probability. Synthese 114: 373– 404. ———. (2003). Physic s and Leibniz 's princ iples. In K. Brading and E. Castellani (Eds.), Symmetries in physics: Philosophical reflections, 289–308. Cambridge: Cambridge University Press. ———. (2010). Chanc e in the Everett interpretation. In Saunders, Barrett, Kent, and Wallac e (2010), 181– 205. Saunders, S., and D. Wallac e (2008a). Branc hing and unc ertainty. British Journal for the Philosophy of Science 59: 293–305. ———. (2008b). Saunders and Wallac e reply. British Journal for the Philosophy of Science 59: 315– 317. Saunders, S., J. Barrett, A. Kent, and D. Wallac e, eds. (2010). Many worlds? Everett, quantum theory, and reality. Oxford: Oxford University Press. Smolin, L. (1997). The life of the cosmos. New York: Oxford University Press. Struyve, W., and H. Westman (2006). A new pilot-wave model for quantum field theory. AIP Conference Proceedings 844: 321. Tappenden, P. (2008). Saunders and Wallac e on Everett and Lewis. British Journal for the Philosophy of Science 59: 307–314. Tegmark, M. (1998). The interpretation of quantum mec hanic s: Many worlds or many words? Fortschrift Fur Physik 46: 855–862. Available online at http://arxiv.o rg/abs/quant- ph/9709032. Timpson, C., and Brown, H. R. (2002). Entanglement and relativity. In R. Lupac c hini and V. Fano (Eds.), Understanding physical knowledge, Preprint no. 24, Departimento di Filosofia, Università di Bologna, CLUEB, 2002, 147–166. Available online at http://arxiv.o rg/abs/quant- ph/0212140. Tsvelik, A. M. (2003). Quantum field theory in condensed matter physics. 2d ed. Cambridge: Cambridge University Press. Tumulka, R. (2006). Collapse and relativity. In A. Bassi, T. Weber, and N. Zanghi (Eds.), Quantum mechanics: Are there quantum jumps? and on the present status of quantum mechanics , 340. Americ an Institute of Physic s Conferenc e Proc eedings. Available online at http://arxiv.o rg/abs/quant- ph/0602208. Vaidman, L. (2002). The many-worlds interpretation of quantum Mec hanic s. In the Stanford Encyclopedia of Philosophy (Summer 2002 edition), ed. Edward N. Zalta. Available online at http://plato .stanfo rd.edu/archives/sum2002/entries/qm- manywo rlds . Valentini, A. (2010). De Broglie-Bohm pilot wave theory: Many worlds in denial? In Saunders, Barrett, Kent, and Wallac e (2010), 476– 509. Page 18 of 21

The Everett Interpretation Wallac e, D. (2002). Quantum probability and dec ision theory, revisited. Available online at http://arxiv.o rg/abs/quant- ph/0211104. (This is a long (70-page) and not-formally-published paper, now entirely superseded by published work and inc luded only for historic al purposes.). ———. (2003a). Everett and struc ture. Studies in the History and Philosophy of Modern Physics 34: 87– 105. ———. (2003b). Everettian rationality: Defending Deutsc h's approac h to probability in the Everett interpretation. Studies in the History and Philosophy of Modern Physics 34: 415–439. ———. (2005). Language use in a branching universe. Unpublished manuscript; Available online from http://philsci-archive.pitt.edu. ———. (2006). Epistemology quantiz ed: Circ umstanc es in whic h we should c ome to believe in the Everett interpretation. British Journal for the Philosophy of Science 57: 655–689. ———. (2007). Quantum probability from subjec tive likelihood: Improving on Deutsc h's proof of the probability rule. Studies in the History and Philosophy of Modern Physics 38: 311–332. ———. (2008). The interpretation of quantum mec hanic s. In D. Ric kles (Ed.), The Ashgate companion to contemporary philosophy of physics, 197–261. Burlington, VT: Ashgate. ———. (2010). How to prove the Born rule. In Saunders, Barrett, Kent, and Wallac e (2010), 227– 263. Available online at http://arxiv.o rg/abs/0906.2718. ———. (2012). The emergent multiverse: Quantum theory according to the Everett interpretation. Oxford: Oxford University Press: 227–263. Wallac e, D., and C. Timpson (2007). Non-loc ality and gauge freedom in Deutsc h and Hayden's formulation of quantum mec hanic s. Foundations of Physics 37(6): 951– 955. ———. (2010). Quantum mec hanic s on spac etime I: Spac etime state realism. British Journal for the Philosophy of Science vol. 61: 697–727. Available online at http://philsci-archive.pitt.edu. Wilson, A. (2010a). Mac rosc opic ontology in Everettian quantum mec hanic s. Philosophical Quarterly 60: 1– 20. Wilson, A. (2010b). Modality Naturaliz ed: The Metaphysic s of the Everett Interpretation. Ph.D. diss., University of Oxford. Notes: (1) Arguably this has c hanged, but only in the last dec ade or so, and more so in the UK than elsewhere. (Students oc c asionally ask me how the Everett interpretation is perc eived outside Oxford; my flippant answer is that there is a signific ant divide between philosophers who do and do not take it seriously, and the divide is c alled the Atlantic Oc ean.) (2) This is largely anec dotal; see, however, Tegmark (1998). (3) This simplifies slightly: it is frequently c onvenient—notably in c ases involving symmetry—to define the spac e of states so that the mathematic s-to-physic s relation is many-to-one, and it is somewhat c ontroversial in some suc h c ases whether it is many-to-one (see, e.g., Saunders (2003) and referenc es therein.) Suc h c onc erns are orthogonal to the quantum measurement problem, though. (4) That is: what is the c orrec t view of sc ientific theories—semantic or syntac tic (c f. Ladyman and Ross (2007, 111–118) and references therein). (5) In the c ase of dynamic al-c ollapse theories, Tumulka (2006) has produc ed a relativistic ally c ovariant theory for non-interacting partic les, but to my knowledge there is no dynamic al-c ollapse theory empiric ally equivalent to any relativistic theory with interac tions. There has been rather more progress in the c ase of hidden variable theories (perhaps unsurprisingly, as these supplement but do not modify the already-known unitary dynamics); for three Page 19 of 21

The Everett Interpretation different rec ent approac hes, see Dürr et al. (2004, 2005) (hidden variables are partic le positions), Struyve and Westman (2006) (hidden variables are bosonic field strengths), and Colin (2003) and Colin and Struyve (2007) (hidden variables are loc al fermion numbers). As far as I know, no suc h approac h has yet been demonstrated to be empiric ally equivalent to the Standard Model to the satisfac tion of the wider physic s c ommunity. (6) Perhaps in some sense there are multiple interpretations of classical electromagnetism: perhaps realists could agree that the elec tromagnetic field is physic ally real but might disagree about its nature. Some might think that it was a property of spac etime points; others might regard it as an entity in its own right. I am deeply skeptic al as to whether this really expresses a distinc tion, but in any c ase, I take it this is not the problem that we have in mind when we talk about the measurement problem. (7) A move c ritic iz ed on tec hnic al grounds by Foster and Brown (1988). (8) Given that an “observer” is represented in the quantum theory by some Hilbert spac e many of whose states are not c onsc ious at all, and that c onversely almost any suffic iently large agglomeration of matter c an be formed into a human being, it would be more ac c urate to say that we have a c onsc iousness basis for all systems, but one with many elements that c orrespond to no c onsc ious experienc e at all. (9) In fac t many adherents of many-minds theories (e.g., Loc kwood and Donald) embrac e this c onc lusion, having been led to rejec t func tionalism on independent grounds. (10) It would ac tually be a c ase of disc arding all but one set of minds—one for eac h observer. (11) Barbour (1999) might be an exc eption; so might Allori et al. (2009), though it is unc lear if Allori et al. are ac tually advoc ating the interpretation rather than using it to illustrate broader metaphysic al themes. (12) For an elementary introduc tion, see, e.g., Kittel (1996); for a more systematic treatment see, e.g., Tsvelik (2003) or (old but c lassic ) Abrikosov, Gorkov, and Dz yalohinski (1963). (13) In exac tly what sense is c ontroversial, and the debate arguably overlaps(!) with others in mainstream metaphysic s; see Saunders (2010) and Wilson (2010a, 2010b) for further disc ussion. (14) It represents a departure from my position in Wallac e (2006). (15) See Wallac e (2012) for details; I learned the argument from David Deutsc h in c onversation. (16) For reasons of spac e I omit detailed disc ussion of the parallel tradition in Everettian quantum mec hanic s of identifying probability via long-run relative frequenc y (notably by Everett himself (1957) and by Farhi, Goldstone, and Gutmann(1989). I disc uss this program in detail in c hapter 4 of Wallac e(2012); my c onc lusion is that it works about as well, or as badly, as equivalent c lassic al attempts, though there is no direc t Everettian analogue to the best-systems approach. (17) A more prec ise way of stating both is that the program attempts to show that agents are rationally required to ac t as if mod-squared amplitude played the objec tive-probability role in David Lewis's Principal Principle; c f. Lewis (1980). (18) See, the disc ussions in e.g., Bell (1981a) or Maudlin (2002). (19) That the dynamic s are thereby required to violate Lorentz c ovarianc e does not unc ontroversially follow; c f. Myrvold (2002), Wallace and Timpson (2010), and Tumulka (2006). (20) For a more detailed analysis—whic h gives a slightly different ac c ount of why the Everett interpretation is an exc eption to Bell's result—see Timpson and Brown (2002). (21) Storrs Mc Call also explores these issues in developing his approac h to quantum mec hanic s (see, e.g., Mc Call (2000)); that approac h is related to, but not identic al to, the Everett interpretation (and, insofar as it relies on an explic it and prec ise c onc ept of branc hing without offering a dynamic al explic ation of when branc hing oc c urs, arguably fails to solve the measurement problem.) Page 20 of 21

The Everett Interpretation David Wallac e David Wallace studied physics at Oxford University before m oving into philosophy of physics. He is now Tutorial Fellow in Philosophy of Science at Balliol College, Oxford, and university lecturer in Philosophy at Oxford University. His research interests include the interpretation of quantum m echanics and the philosophical and conceptual problem s of quantum field theory, sym m etry, and statistical physics.

Unitary Equivalence and Physical Equivalence Laura Ruetsche The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter evaluates whether unitary equivalenc e is an appropriate c riterion of physic al equivalenc e for quantum theories that fall outside the scope of the comforting uniqueness results. It describes the Stone-von Neumann and Jordan-Wigner uniqueness results, provides examples of unitarily inequivalent representations, and proposes an alternative to unitary equivalenc e as a c riterion for physic al equivalenc e. The c hapter also offers an impressionistic introduc tion to the rudiments of operator algebra theory. K ey words: u n i tary equ i v al en ce, ph y si cal equ i v al en ce, u n i qu en ess resu l ts, S ton e-v on Neu man n , Jordan -Wi gn er, i n equ i v al en t represen tati on s, operator al gebra th eory 1. Intro ductio n By tradition, to quantiz e a theory of c lassic al mec hanic s, one c onstruc ts a Hilbert space representation of a set of magnitudes obeying canonical commutation relations (CCRs) c harac teristic of the theory in question. Similarly, to c onstruc t a quantum theory of spin systems, one finds a Hilbert spac e representation of c harac teristic canonical anticommutation relations (CARs) for spin systems. The representations in question take the form of symmetric Hilbert spac e operators satisfying the relevant c anonic al relations. Following standard prac tic e of eliding the distinc tion between operators and the physic al magnitudes (aka observables) they represent, I will c all suc h operators “c anonic al observables.” By tradition, other observables rec ogniz ed by the theory c an be obtained as polynomials of, and limits of sequenc es of polynomials of, the representation-bearing c anonic al observables. By tradition, a quantum state is an expec tation value assignment to this c ollec tion of magnitudes, whic h is normed, linear, and c ountably additive. Thus, a Hilbert spac e representation of the c anonic al relations c irc umsc ribing a quantum theory supplies a kinematics for that theory, that is, an ac c ount of the states it rec ogniz es as physic ally possible and the magnitudes it rec ogniz es as physic ally signific ant. In standard Hilbert spac e quantum mec hanic s, the Sc hrödinger equation equips suc h a theory with a dynamic s. Even very simple quantum theories, so realiz ed, harbor provoc ative diffic ulties. The quantum theory of two spin ½ systems features entangled states that c an be understood to predic t distant c orrelations, alarmingly suggestive of spooky ac tion at a distanc e. This is the diffic ulty of quantum nonloc ality. The quantum theory of a c at and a radioac tive atom, both modeled as bivalent systems, when equipped with what for all the world seems an appropriate model of purely unitary measurement, deposits the cat in a state eerily superposed between life and death. This diffic ulty is the quantum measurement problem. Foundational disc ussions of quantum non-loc ality and the measurement problem are legion. They are also largely immune to a foundational anxiety that is the mission of this c ontribution to induc e. The anxiety is whether the definite desc riptions earlier in the paragraph are appropriate: whether, that is, the quantum theory of two spin ½ systems is unique; more generally, whether, there might be multiple, physically inequivalent ways to c onc oc t a quantum theory from representations of the c anonic al relations c irc umsc ribing that theory. Page 1 of 23

Unitary Equivalence and Physical Equivalence The substantial literature on quantum nonlocality and the measurement problem is largely innocent of the uniqueness anxiety because it largely concerns quantum theories that fall within the scope of two results taken to silenc e that anxiety. These are the Stone-von Neumann and Jordan-Wigner theorems. Ac c ording to the former, every Hilbert spac e representation of the CCRs for a partic ular c lassic al Hamiltonian theory of finitely many partic les is unitarily equivalent to every other.1 For eac h finite n, the Jordan-Wigner theorem guarantees that representations of the CARs for n spin systems are unique up to unitary equivalenc e.2 These results are taken to silenc e the uniqueness anxiety bec ause unitary equivalenc e is widely supposed to explic ate physic al equivalenc e for (quantum theories obtained via) Hilbert spac e representations. Given the supposition, the results imply that superfic ially variant representations of the c anonic al relations c irc umsc ribing a quantum theory are merely variant means of expressing the same quantum kinematic s. That is, they are merely variants provided the quantum theory in question c onc erns a suitably “finite” system. But not all quantum theories do c onc ern suitably finite systems. Quantum field theory (QFT), and the thermodynamic limit of quantum statistic al mec hanic s (QSM), are quantum theories whose degrees of freedom are infinite in number. A typic al QFT c omes about as the quantiz ation of a c lassic al field theory, whic h assigns a field amplitude to every point of spacetime. To take QSM to the thermo-dynamic limit is to allow the number of mic roc onstituents of the system analyz ed, and the volume they oc c upy, go to infinity while the density remains finite. Conc erning infinite systems, QFT and the thermodynamic limit of QSM—quantum theories whic h I will lump together under the heading “QM∞”—fall outside the sc ope of the Jordan-Wigner and Stone-von Neumann uniqueness theorems. Indeed, the c anonic al relations c irc umsc ribing a theory of QM∞ c an admit c ontinuously many unitarily inequivalent Hilbert spac e representations. Ac c ording to very same c riterion of physic al equivalenc e, that warranted the reading of those theorems as results about physic al equivalenc e, theories of QM∞ c an admit infinitely many presumptively physic ally inequivalent Hilbert spac e representations. This c ontribution addresses quantum theories that fall outside the sc ope of the c omforting uniqueness results, with a view toward assessing whether unitary equivalenc e is an appropriate c riterion of physic al equivalenc e for these theories. More emphasis is put on means of assessment and what motivates them than on endorsing and defending a partic ular answer. Indeed, I will suggest that appropriate c riteria of physic al equivalenc e for quantum theories are sensitive to fac tors that are not obviously c riteria of identity for those theories, fac tors suc h as the uses to whic h those theories are being put and the scientific climates in which they find themselves. I proc eed as follows. The next sec tion sketc hes the Stone-von Neumann and Jordan-Wigner uniqueness results and explic ates the assumptions underlying their c onventional reading as results about the physic al equivalenc e of quantum theories. Sec tion 3 develops two ac c essible examples of unitarily inequivalent representations. With these examples in hand, sec tion 5 revisits the c ase that unitary equivalenc e is c riterial for physic al equivalenc e. Artic ulating a new-fangled alternative c riterion, sec tion 5 also identifies presuppositions favoring the traditional c riterion over the new-fangled one. These inc lude presuppositions about whic h relations between physic al observables serve to define other physic al observables, as well as presuppositions about what states are physic al. The concluding section 6 discusses strategies for securing these presuppositions, observes certain tensions within these strategies, and c omments on what to make of this state of play. Throughout, the exposition is informal, with referenc es to more thorough and rigorous disc ussions supplied. In an effort to keep the disc ussion self-c ontained and ac c essible, a tec hnic al interlude (sec tion 4) offers an inc omplete and impressionistic introduc tion to the rudiments of operator algebra theory. 2. The “Uniqueness” Results 2.1 Preliminaries This c ontribution derives its dramatic tension from the fac t that entrenc hed c riteria of physic al equivalenc e for quantum theories render a verdic t of ‘inequivalent!’ for what otherwise seem to be realiz ations of the same basic quantum theoretic al struc ture. So our first task will be to announc e a c riterion of individuation for quantum theories. With that c riterion in hand, we c an turn to the question of what it takes for superfic ially different realiz ations of the theory, so individuated, to be physic ally equivalent. What makes a quantum theory the theory it is? There is a c onsensus among the c ommunity of people who work Page 2 of 23

Unitary Equivalence and Physical Equivalence with suc h theories. Fulling reports that “most theoretic al physic ists, following Sc hwinger, regard the action principle as fundamental. Theories are defined by Lagrangians” (1989, 126). By fixing the symplec tic struc ture of a c lassic al theory, the Lagrangian fixes the c ommutation relations its quantiz er seeks to represent.3 But not all interesting quantum theories desc end from real or imagined c lassic al Lagrangians. Haag observes, The idea that one must first invent a c lassic al model and then apply to it a rec ipe c alled “quantiz ation” has been of great heuristic value. In the past two dec ades, the method of passing from a c lassic al Lagrangian to a c orresponding quantum theory has shifted more and more away from the c anonic al formalism to Feynman's path integral. This provides an alternative (equivalent?) rec ipe. There is, however, no fundamental reason why a quantum theory should not stand on its own legs, why the theory could not be c ompletely formulated without regardtoan underlying deterministic princ iple. (1992, 6) Thus, Fulling proposes a liberaliz ation of Sc hwinger's c onventional wisdom: There is, however, another point of view, more c onsistent with the spirit of axiomatic field theory. Any c ommutation or antic ommutation relations c onsistent with the dynamic s c an define a possible model. …In this approac h a formal theory c onsists of equations of motion plus c ommutation rules (or some more general algebraic relations). They need not determine eac h other, but it is a nontrivial requirement that they be mutually c onsistent. (Fulling 1989, 126) Agreeing that quantum theories are to be defined by their characteristic commutation relations and equations of motion, Fulling does not require that these emanate from the same sourc e, or that that sourc e be a Lagrangian. To keep the disc ussion trac table, we will foc us on the question of physic al equivalenc e for quantum theories spec ified up to their kinematics—that is, their ac c ounts of what states are physic ally signific ant, and (thinking of a state as a map from physic al magnitudes to their expec tation values) what physic al magnitudes lie in the sc ope of those states. For theories spec ified up to their kinematic s, Fulling's refinement of the c onventional wisdom identifies c anonic al c ommutation or antic ommutation relations as a princ iple of individuation. Where R is a set of suc h relations, let QR be the quantum theory c irc umsc ribed by those relations. We want to know: When are different kinematic al sc hemes for a theory QR physic ally equivalent? 2.2 Quantizing It will help us address these questions to review some basic strategies for c onstruc ting quantum theories. The state of a c lassic al Hamiltonian system is given by its position and momentum. Take the simplest c ase of a single system moving on the real line. The position and momentum variables, real numbers q and p, ac t as c oordinates for the phase spac e M of possible states of the system. M is just the real plane ℝ2 . For more c omplic ated systems—n partic les in Euc lidean three-spac e, say—the phase spac e of possible states is larger (ℝ2 n), but c onstruc ted along the same princ iples. A c lassic al Hamiltonian theory with phase spac e M represents physic al magnitudes (aka observables) by func tions from M to ℝ. The position and momentum observables for the simplest system are examples: they map points in phase spac e to their q and p c oordinate values, respec tively. All other observables pertaining to the system c an be expressed as func tions of these observables. The Hamiltonian observable H, whic h usually c oinc ides with the sum of the system's kinetic and potential energies, is of partic ular signific anc e. Fed into Hamilton's equations, H helps determine dynamic ally possible trajec tories—the system's position and momentum as func tions q(t), p(t) of a time variable t—through M. For more c omplic ated systems, the c lassic al observable set and the dynamic al presc ription are more c omplic ated, but the princ iples are the same. The c anonic al Hamiltonian quantization recipe exploits the fac t that the c ollec tion of c lassic al observables just desc ribed exhibits an algebraic structure. As smooth func tions on phase spac e, c lassic al observables form a set that is also a vec tor spac e over the real numbers. An algebra is just a linear vec tor spac e endowed with a (not nec essarily assoc iative) multiplic ative struc ture (see Kadison and Ringrose 1997 for an introduc tion). The vec tor spac e of c lassic al observables bec omes a Lie algebra upon being endowed with a multiplic ative struc ture supplied by the Poisson bracket. The Poisson bracket f, g of classical observablesf : M → ℝ and g : M → ℝ is (1) Page 3 of 23

Unitary Equivalence and Physical Equivalence For the c anonic al observables pi and qj (2) The Poisson brac ket also affords a partic ular expeditious expression of Hamilton's equations (3) A desc ription of the time-evolution of a general observable f : M → ℝ follows: df = {f, H} dt Given its c entrality both to the struc ture of c lassic al observables and to their dynamic s, it is tempting to think that the Poisson brac ket struc ture has a great deal to do with making a c lassic al Hamiltonian theory the theory it is.4 Ac c ording to the c anonic al Hamiltonian quantiz ation rec ipe, what it takes to quantiz e suc h a theory is to find a c harac teristic ally quantum mec hanic al analog of that theory's Poisson brac ket struc ture. In partic ular, one identifies canonical quantum observables with symmetric operators qˆi, pˆi acting on a separable Hilbert space H and obeying c ommutation relations c orresponding to the c lassic al Poisson brac kets of the c lassic al theory. When the c lassic al theory has phase spac e ℝ2 n and c anonic al observables qi and pi, these CCRs are (where [Aˆ, Bˆ] := AˆBˆ − BˆAˆ, Iˆ is the identity operator, and Planck's constant ℏ is set to one) (4) For a c lassic al theory with phase spac e M = ℝ2 n, the Hamiltonian quantiz ation rec ipe is typic ally realiz ed by the Schrödinger representation, set in the Hilbert spac e L2 (ℝn) of square integrable c omplex-valued func tions of ℝn. For n = 1, the Schrödinger representation defines qˆψ(x) = xψ(x) and pˆψ(x) = −i dψd(xx). A story similarly c entered on algebraic struc tures c an be told about quantum theories of spin systems. To build the quantum theory of a single spin system, find symmetric operators {ˆσ(x), σˆ(y), σˆ(z)} acting on a Hilbert space H to satisfy the Pauli Relations, whic h inc lude (5) Call the elements ˆσ(j) satisfying (5) the Pauli spin observables. The generaliz ation to n spin systems is straightforward. To build the quantum theory for n spin systems, find for eac h spin system k a Pauli spin σˆk = (σˆk(x), ˆσk(y), σˆk (z)) satisfying the Pauli Relations, expanded to include the requirement that spin observables for different systems c ommute. A set of operators satisfies the Pauli relations if and only if they satisfy the CARs (see Emc h 1972, 271– 272); thus (5) c an be taken to impose the algebraic struc ture of the CARs. 2.3 Unitary Equivalence as Physical Equivalence A Hilbert spac e representation of the c anonic al relations R c irc umsc ribing a quantum theory QR takes the form of operators Ĉi acting on a Hilbert space H to satisfy R. Having obtained such a representation, the aspiring quantum mec hanic c annot rest. She needs to build produc ts and linear c ombinations of her c anonic al observables if she is to have a viable theory of physics. Where pˆ and qˆ are her canonical observables, she will use pˆ2 /2m to desc ribe the kinetic energy of a partic le of mass m. Where V is the magnitude of some c onfiguration-dependent potential to which that particle is subject, she will use V qˆ to describe its potential energy and use a Hamiltonian func tion, whic h is a sum of these, to desc ribe its energy. In addition to suc h polynomials of the c anonic al observables, the aspiring quantum mec hanic also needs observables that are defined as limits of sequenc es of other observables—for instanc e, the unitary Sc hrödinger evolution operators Ȗ (t) = e−iĤ t are limits of the Taylor series of polynomials of Ĥ. Desc ribed in other terms, the aspiring quantum mec hanic needs to build an observable algebra using the c anonic al observables Ĉi as generators. This means building the algebra from polynomials, and limits of sequenc es of polynomials, of those observables. Hereinafter I will use “ordinary QM” to mean the tradition in whic h the observable algebra generated by representation-bearing canonical observables Ci acting on H will coincide with B(H ), the full set of bounded operators on H . In more technical terms (glossed immediately below), the aspiring quantum mechanic has begun with an irreducible representation of R, and used the weak operator topology to determine which sequences of Page 4 of 23

Unitary Equivalence and Physical Equivalence polynomials of the canonical observables converge. {Ĉi} acting on H is an irreducible representation if and only if the only subspaces of H invariant under the action of {Ĉi} are the 0 subspace and H itself. Unless otherwise announc ed, all representations disc ussed here will be irreduc ible. 5 A sequence Âi of operators on H converges to an operator  in H 's weak operator topology if and only if for all |ψ⟩, |ϕ⟩ ∈ H ,  ∣∣⟨ψ∣∣(Aˆ − Aˆi)∣∣ϕ⟩∣∣ goes to 0 as i goes to ∞. When I want to emphasize that the algebra B(H ) is generated by the set {Ĉi} of canonical observables representing R, I will use “B(H )Cˆi ” to designate it. In ordinary QM, quantum states are normalized, linear, positive, and countably additive maps from B(H ) to the complex numbers ℂ. Gleason's theorem tells us that states so conceived coincide with the set T+ (H ) of density operators on H , provided the dimension of H exceeds 2. Each Wˆ ∈ T+ (H ) determines a state via the trace prescription, which assigns each self-adjoint Aˆ ∈ B(H ) the expectation value Tr(ŴÂ). Call (B(H )Cˆi , T+ (H )) a kinematic pair for a theory of ordinary QM. (B(H )Cˆi , T+ (H )) is an instance of a general scheme (D, S ) for kinematic pairs. The first entry gives the algebra of physical magnitudes recognized by a theory, with the theory's observables c orresponding to that algebra's self-adjoint elements. The sec ond entry gives the theory's physic al states. The next sec tion motivates a c riterion of physic al equivalenc e applic able to generic kinematic pairs, then argues that for kinematic pairs of the form (B(H )Cˆi , T+ (H )) favored by ordinary QM, that c riterion reduc es to unitary equivalenc e. 2.4 Analyzing Physical Equivalence The basic type of physic al possibility rec ogniz ed by a quantum theory takes the form of an expec tation value assignment to the family of magnitudes rec ogniz ed by that theory. There is, of c ourse, a further interpretive question of how to understand the nontrivial probabilities implicit in such an expectation value assignment, and in partic ular of whether multiple, distinc t “value states” c orrespond to eac h given quantum state. Different interpretations of ordinary QM urge different understandings of the quantum state, and thereby eventuate in different pic tures of the set of physic al possibilities assoc iated with a quantum theory. But these disagreements oc c ur, as it were, “downstream” from the identific ation of a kinematic pair on behalf of the theory. A question we c an artic ulate and address without embroiling ourselves in these tendentious questions of interpretation is: When are c andidate realiz ations of a quantum theory QR, realiz ations spec ified up to kinematic pairs, physic ally equivalent? If the c ontent of a physic al theory c onsists in the set of physic al possibilities it rec ogniz es, then physic al theories have the same c ontent just in c ase they admit the same set of physic al possibilities. On this pic ture, a nec essary criterion for the physical equivalence of (D, S ) and (D′, S ′) is a one-to-one correspondence between the physic al possibilities admitted by the first pair and the physic al possibilities admitted by the sec ond. Claiming Glymour as an inspiration, Clifton and Halvorson (2001) analyz e this demand into two c onditions. The first is that this one-to-one c orrespondenc e ‘preserve expec tation values,’ in the sense spec ified by a c riterion I will c all PEV: PEV. There are bijections iobs :D → D′ and istate : S → S ′ such that (6) for all A  ∈  D and all ω  ∈  S . Whenever two kinematic pairs satisfy PEV, to eac h state ω of one pair there c orresponds a state istate(ω) in the other suc h that istate(ω)'s assignment of expec tation values to the observable iobs(A) exac tly duplic ates ω ‘s assignment of expec tation values to observable A. Part of how quantum theories c harac teriz e possibilities is as maps from physic al magnitudes to their expec tation values. Kinematic pairs satisfying PEV c harac teriz e exac tly the same set of possibilities thus c onc eived. So let us join Clifton and Halvorson in supposing that two theories Page 5 of 23

Unitary Equivalence and Physical Equivalence spec ified up to kinematic pairs are physic ally equivalent only if they satisfy PEV by admitting expec tation-value- preserving bijec tions of the sort it demands.6 PEV pays no obvious heed to the “algebraic structure” of the observable sets D and D′ . Nor does PEV engage the fac t that those sets are in some sense desc ended from realiz ations of the c anonic al relations R c irc umsc ribing the quantum theory QR. We might want a c riterion of physic al equivalenc e for kinematic pairs that is sensitive to suc h matters. After all, part of what makes a physic al theory the theory it is is the func tional relationships it posits between the physic al magnitudes it rec ogniz es. We have been rec ogniz ing this implic itly by identifying quantum theories by appeal to their constitutive CARs/CCRs, which express such relationships. Functional relationships between observables are moreover implicated in a theory's laws: Hˆ = pˆ2 /2m makes the energy of a free system of mass m a func tion of its momentum; the Sc hrödinger equation uses the energy of an isolated system to build a family Ȗ (t) = e−iĤt of operators desc ribing that system's time evolution, whic h implies (roughly speaking) that the operator Ĥ is a limit of a sequenc e of func tions of the evolution operators Ȗ (t). These c onsiderations suggest that c riteria of physic al equivalenc e for quantum theories spec ified up to kinematic pairs should inc lude a demand to the effec t that the algebraic struc tures of their observable families be suitably isomorphic . Reflec tions like the following might tempt one to hope that when one demands the preservation of expec tation values by imposing PEV, one gets a suitable isomorphism of algebraic struc ture for free: Where prime and unprimed commodities denote elements identified by the bijections satisfying PEV, suppose that iobs between observable algebras failed to preserve additive algebraic struc ture. Then there exists observables X, Y suc h that (X + Y′) ≠ (X′ + Y′). Let us also suppose that the observables (X + Y)′ and X′ + Y′ are different only if there is some state ω′ that separates them in the sense that ω′[(X + Y)′] ≠ ω′ (X′ + Y′). If PEV is satisfied, this separation c ondition implies ω(X+Y) ≠ ω(X+ Y). But that is impossible. So PEV c annot be satisfied by a bijec tion iobs that fails to preserve additive algebraic struc ture. Similar arguments invoking the separation c ondition establish that PEV is satisfied only by bijec tions between observable sets that are linear and preserve their identity elements (see Roberts and Roepstorff 1969, Prop. 3.1).7 The c omplete hope is that PEV on its own ensures a suitably “physic al” isomorphism between the observable algebras of kinematic pairs that satisfy it. Examples dashing this hope will be provided in sec tion 5. For now, let us take their existenc e to lend urgenc y to a (at present vague) demand that physic ally equivalent kinematic pairs enjoy suitably isomorphic observable algebras, and try to make that demand more prec ise. The algebras at issue are each supposed to be generated by elements satisfying the relations R circumscribing the quantum theory, relationships such as Where Ci satisfying R generate the observable algebra D and C ′ i satisfying R generate the observable algebra D′, I contend that a bijection iobs: D → D′ “preserves relevant algebraic structure” only if it enables the primed and unprimed theorists agree about what makes the c anonic al magnitudes c anonic al—only, that is, if iobs maps observables realiz ing R in the unprimed theory to observables realiz ing R in the primed theory. Continuing the c onvention that primed and unprimed c ommodities are those identified by i obs, this agreement requires that if the unprimed theorist's c anonic al relations are realiz ed by AB − BA = kI, then those observables’ primed c ounterparts also realiz e the c anonic al relations: (AB − BA)′ = kI′ = A′ B′ − B′ A′ (where the first and last term are different ways of taking primed c ounterparts of observables involved in the c anonic al relations). The linearity of i obs gives us (AB − BA)′ = (AB)′ − (BA)′. We are very c lose to drawing another c onc lusion about any i obs that satisfies PEV. Given other features already established for i obs, i obs will qualify as an isomorphism, if only i obs c ould be shown to be multiplic ative. Where A and A′ are algebras, a map α : A → A′ is a morphism if and only if α is linear, multiplicative (α(XY) = α(X)α(Y) for all X,  Y   ∈  A), and takes the identity to the identity. α is an isomorphism if it is one- to-one.8 Alas, we c annot use the separation c ondition tac tic to argue that if i obs satisfies PEV, then (XY)′ = X′Y′. The c atc h is that, unlike sums of self-adjoint elements of our observable algebras, produc ts of self-adjoint elements need not themselves be self-adjoint—and so need not themselves be observables, and so need not engage the gears of the separation c ondition. So shift attention to symmetriz ed produc ts, elements of the form XY + YX, whic h are self- Page 6 of 23

Unitary Equivalence and Physical Equivalence adjoint. Then a separation c ondition argument establishes that i obs satisfies PEV only if [* ] whic h makes iobs a Jordan homomorphism (Roberts and Roepstorff 1969, Prop. 6.1). But a Jordan homomorphism need not be an isomorphism. In partic ular, it need not be multiplic ative. [*] will be satisfied just in c ase (XY)′ = X′ Y′ + ZXY′ and (YX)′ = Y′X′ − ZXY′, even if ZXY′ is a nonzero element of the primed algebra. However, our requirement that iobs take c anonic al observables to c anonic al observables, along with the separation c ondition, implied that (AB − BA)′ = (AB)′ − (BA)′ = A′ B′ − B′ A′. That c annot be true unless ZAB′ = 0—unless, that is, iobs ac ts multiplic atively on c anonic al observables. Bec ause c anonic al observables generate the algebra, it follows that iobs ac t multiplic atively on the algebra, period. And that c ompletes the c ase that, supposing our observable set is ric h enough to separate states, an i obs satisfying PEV preserves relevant physic al struc ture only if it is an isomorphism between observables algebras D and D′. But i obs must ac c omplish one more task if it is to sec ure physic al equivalenc e. i obs is an isomorphism between the observable algebras of kinematic pairs. These kinematic pairs c annot be physic ally equivalent if they do not agree about what the theory's fundamental canonical magnitudes are. Thus, they are not physic ally equivalent unless the isomorphism i obs identifies the c anonic al magnitudes generating one algebra with the c anonic al magnitudes generating the other. Let us c onsolidate the foregoing reflec tions on what, beyond satisfying PEV, i obs must ac c omplish to establish physic al equivalenc e. Where Ĉi satisfying R generate the observable algebra D and Cˆ′ i satisfying R generate the observable algebra D′ , a map i obs preserves relevant physic al struc ture just in c ase it Preserves Algebraic Struc ture by satisfying PAS. i obs is an isomorphism between D and D′ suc h that (7) for all i. Our analysis of physic al equivalenc e for quantum theories spec ified up to generic kinematic pairs is thus: Kinematic pairs (D, S ) and (D′, S ′) for a quantum theory QR circumscribed by the relations R are physically equivalent if and only if there exist bijections iobs : D → D′ and istate : S → S ′ satisfying both PEV and PAS. This is essentially the analysis offered by Clifton and Halvorson, although the justific ation offered here uses words different from the words used in their justific ation.9 Now we are getting somewhere. It turns out that kinematic pairs of ordinary QM's form, kinematic pairs (B(H ), T+ (H )) and H ′ satisfy both PEV and PAS if and only if their collections Ci and {C′i} of canonical operators are unitarily equivalent in the following sense: A Hilbert space H , and a collection of operators {Oˆ′i} is unitarily equivalent to (H ′, {Oˆ′i}) if and only if there exists a one-to-one, linear, invertible, norm-preserving transformation (“unitary map”) U : H → H ′ such that U −1 Oˆ′i U = Oˆ′i for all i. Here is a sketch of an argument that (B(H ), T+ (H )) and (B(H ′), T+ (H ′)) satisfy both PEV and PAS if and only if they arise from unitarily equivalent representations of the c anonic al relations. To see that unitary equivalenc e is suffic ient, notic e that the U effec ting the unitary equivalenc e of primed and the unprimed representations of R furnishes both a bijection i (Ŵ) = UŴU from the unprimed kinematic pair's state space to the primed pair's state space and a bijection i (Â) = UÂU from the unprimed kinematic pair's observable algebra to the primed pair's algebra. The property of unitary maps that UU = U U = I along with the trace prescription guarantee that these bijections together satisfy PEV. i is moreover an isomorphism between the observable algebras, because of truths such as U(A + B)U = UAU + UBU . Finally, the isomorphism i , whic h is induc ed by a map identifying c anonic al elements of the unprimed algebra with c anonic al elements of the primed algebra, satisfies (7), and therefore satisfies (PAS). Notic e for future referenc e that bec ause a unitary map Page 7 of 23

Unitary Equivalence and Physical Equivalence between H and H ′ preserves inner products, a sequence Ai ∈  B(H ) converges weakly to an element A  ∈  B(H ) if and only if the sequence UAiU −1   ∈  B(H ) converges weakly to an element UAU −1   ∈  B(H ). To see that the unitary equivalenc e of the representations {Ci} and {C ′ i } underlying the kinematic pairs (B(H ), T+ (H )) and (B(H ′), T+ (H ′)) is necessary for those pairs to satisfy both PEV and PAS, note that B(H ) and B(H ′) are Type I von Neumann algebras. Because all isomorphisms between Type I von Neumann algebras are implemented unitarily, an isomorphism iobs satisfies PAS only if it is induc ed by a unitary map and takes Ci to {C ′ i }. But suc h an isomorphism is available only if the representations are unitarily equivalent. To summariz e: partic ulariz ed to kinematic pairs of ordinary QM's sort, the analysis of physic al equivalenc e in terms of PEV and PAS implies that suc h pairs are physic ally equivalent if and only if the representations of R generating them are unitarily equivalent. In 1931, von Neumann demonstrated what had been conjectured the previous year by Stone: the unitary equivalence (up to multiplic ity) of any pair of Hilbert spac e representations of the CCRs arising from a c lassic al theory with phase spac e ℝ2 n. The Jordan-Wigner theorem likewise establishes the uniqueness, up to unitary equivalenc e, of Hilbert spac e representations of the CARs for n degrees of freedom (n finite). Given our analysis of physic al equivalenc e, it follows that onc e we have settled on the CCRs or CARs c irc umsc ribing a quantum theory in the sc ope of these results, every ordinary QM-ish kinematic pair we c an c onstruc t on behalf of that theory is physic ally equivalent to every other. Disagree howsoever we might about the further interpretation of that theory, we c an at least agree about its c ore identity, an identity provided by a kinematic pair that is essentially unique. 3. Unitary Inequivalence: So me Examples This sec tion musters several examples of quantum theories falling outside the sc ope of the Stone-von Neumann and Jordan-Wigner theorems. The canonical relations circumscribing these theories admit unitarily inequivalent representations. Our working analysis of physic al equivalenc e implies that ordinary QM-ish kinematic pairs based on suc h representations are also physically inequivalent. We will raise some worries about whether ‘physic ally inequivalent!’ is the right verdict to reach about the representations sketched here. 3.1 The Infinite Spin Chain An exc eedingly simple quantum theory whose CARs admit unitarily inequivalent representations is the theory of infinitely many spin ½ systems in a linear array.10 As a warmup for our enc ounter with this theory, c onsider the quantum theory of a finite number n of spin ½ systems, arranged in a one-dimensional lattic e. To c onstruc t suc h a theory, we need only equip each location k in the lattice with a Pauli spina σˆk = (σˆk(x), σˆk(y), ˆσk(z)) in such a way that the c ollec tion of these Pauli spins satisfies the Pauli relations. One way to do this employs a vector space H spanned by a basis whose elements correspond to sequences sk, where eac h entry in the sequenc e takes one of the values ±1, and k ranges from 1 to n. (Notic e that there are finitely many distinc t suc h sequenc es, bec ause there are only finitely many ways to map a set of finite c ardinality into the set +1,−1 .) We introduce operators σˆj (z), j = 1 to n in such a way that sequences sk whose jth entry is ± 1 serve as ˆσj (z) eigenvectors associated with the eigenvalue ±1. Along with operators σˆj (y),  ˆσj (x), c onstruc ted by analogy to their single elec tron c ounterparts, these operators provide a representation of the Pauli relations, and thus the CARs, for n. Of c ourse, we c ould have c onstruc ted a representation of Pauli relations by many other, superfic ially c ompeting, means. But bec ause we are c onsidering only finitely many spin systems, the Jordan-Wigner theorem guarantees that any representation of the CARs for n is unitarily equivalent to any other. Let us belabor a c onsequenc e of that. Imagine that Werner and Erwin eac h build a representation of the Pauli relations for c hain of n spin system. Let σk (i)W be the operator on HW by whic h Werner represents the ith c omponent of spin for the kth partic le; let σk (i)E be the operator on HE by whic h Erwin represents the ith c omponent of spin for the kth partic le.11 If Werner's representation and Erwin's are unitarily equivalent, then there exists a unitary map U : HE → HW such that (8) Page 8 of 23

Unitary Equivalence and Physical Equivalence Bec ause unitary maps are linear and norm preserving, this unitary map not only identifies eac h Pauli spin operator in Erwin's representation with a Pauli spin operator in Werner's representation, it also extends in a way that respects the identifications between Pauli spins to a bijec tion between the full sets of bounded operators on eac h theorist's Hilbert spac e. The polarization of a system will be of partic ular interest. A system's polariz ation is desc ribed by a vec tor whose magnitude (∈ [0, 1]) gives the strength and whose orientation gives the direc tion of the system's net magnetization. On a single electron, it is represented by an observable mˆ whose three components correspond to three orthogonal c omponents of spin. Thus in the +1 eigenstate | +) of ˆσ(z) (understood as the z-c omponent of spin), the polariz ation has an expec tation value of + 1 along the z axis. For a finite c hain of spins, the polariz ation observable has components ˆm(i),  i  ∈  {x,  y,  z}, that are just the average, over links in the chain, of the n corresponding component of spin: ˆm(i) = 1 ˆσk (i). Let [sk]j ∊ {±1} denote the jth entry of the sequenc e sk. In n ∑ k=1 n the basis sequence sk, the z-component of polarization ˆm(z) takes an expectation value of magnitude 1n ∑ [sk]j . j=1 This quantity attains extreme values (of ±1) for sequenc es every term of whic h is the same. From their representations of the Pauli relations, Erwin and Werner both c onstruc t a kinematic pair of the ordinary QM form (B(H ),  T+ (H )). The Jordan-Wigner theorem implies that any other representation of the Pauli relations will be unitarily equivalent to Werner's. Werner's and Erwin's kinematic pairs thus satisfy both PEV and PAS. Suppose ρˆ is a state in Werner's state set assigning ˆm (z) the expec tation value +1. Then Erwin's state set must inc lude a state ρˆ′ (the image of ρˆ under the isomorphism induc ed by unitary map implementing the equivalenc e of the representations) and an observable ˆm (z)′ (the image of ˆm (z) under that map) suc h that the expec tation value of ˆm (z)′ in the state ρˆ′ is +1. Now let us leave the sc ope of the Jordan-Wigner theorem to c onsider a doubly infinite c hain of spins, its sites labeled by the positive and negative integers ℤ = …,−2, −1,0,1,2,… . We are after a representation of the CARs that assoc iates with eac h site k a Pauli spin satisfying the Pauli relations. But we c annot adapt the strategy adopted for the finite spin c hain to do so. Suc h an adaptation would build the representing Hilbert spac e from a basis consisting of all possible maps from ℤ to ± 1 . Because the set of such maps is nondenumerable, the Hilbert spac e envisioned would be nonseparable, breaking the staunc h tradition of using separable Hilbert spac es for physic s.12 One way to build a separable Hilbert spac e representation is to start with the “base” sequenc e [sk]j = +1 for j ∈ ℤ, and add all sequenc es that differ from the base one at only finitely many loc al sites. The c ollec tion of suc h sequenc es forms a basis; its members are sequenc es in whic h only finitely many + 1s appear. A Hilbert spac e spanned by the basis hosts a representation of the Pauli relations modeled on that of the finite spin c hain. In particular, it features operators σˆj (z)+ , j  ∈  Z. Sequences sk whose jth entry is ±1 are σˆj (z)+ eigenvectors associated with the eigenvalue ±1. Call this the H + representation for short. Much is lost in the abbreviation, for it matters to the algebraic structure of this representation which elements of B(H + ) play the role of which Pauli spins. A total polarization observable mˆ + can be defined in terms of the H + representation if it can be understood to be an element of the observable algebra B(H + ) generated by that representation. To be an element of B(H + ), the polariz ation observable must be a polynomial of Pauli spins, or the limit (in the appropriate sense) of a sequenc e of suc h polynomials. Consider the sequenc e of partial sums ˆm (z)N := 1 N σˆk(z)+ , which define 2N+1 ∑ the z c omponent of net polariz ation of finite stretc hes of the c hain. For eac h N, ˆm (z)Nk i=s −aN polynomial of Pauli spins and so a member of B(H + ). The z component of net polarization for the entire infinite chain would be given by the N→ ∞ limit of the sequences of partial sums ˆm(z)N, if that limit exists. In H + 's weak topology, it Page 9 of 23

Unitary Equivalence and Physical Equivalence does. Recall that a sequence Âi of operators on H converges to an operator  in H 's weak operator topology if and only if for all |ψ⟩, |ϕ⟩ ∈ H ,  ∣∣⟨ψ∣∣(Aˆ − Aˆi)∣∣ϕ⟩∣∣ goes to 0 as i goes to ∞. Considering the sequence ˆm(z)N, remark N ( [sk ]j + [s′ k ]j that ⟨sk ∣ˆm (z)N ∣s′ k ⟩ = 1 2 ) for basis sequences sk and s′k . Because − 1 occurs only finitely 2N+1 ∑ j=−N many times in eac h basis sequenc e, ⟨sk ∣ˆm (z)N ∣s′ k ⟩ will c onverge to 1 as N → ∞, no matter what sk and s′ k are. This shows that the sequence ˆm(z)N converges weakly. Its limit is an operator that has every vector in H + as an eigenvalue 1 eigenvector (because every element of H 's basis is such a vector, and that basis spans the spac e). In other words, in an ordinary QM-ish observable algebra generated by taking the weak c losure of the H + representation, the z-component of the global polarization just is the identity operator I+ on H + . The other c omponents of the total polariz ation c an also be defined as weak limits of polynomials of Pauli spins. Each component of the total polarization is an element of the algebra B(H + ), and so an observable in ordinary QM's sense. It is noteworthy that there are representations of the Pauli relations, modeled on the H + representation, for which the c onvergenc e c onstituting the global polariz ation as a bona fide observable fails to obtain. Consider, for instance, a representation whose “base state” sk is a sequence for which Nl→im∞∣⟨sk ∣ˆm(z)N∣sk⟩∣ does not converge (e.g., the sequence [sk]j = +1 for 2n 〈 |j|≤2n+1 and n odd; [sk]j = − 1 otherwise). The H + representation hosts a total polarization observable only because H 's weak topology facilitates the definition of suc h an observable. For other representations, this need not be so. Whether there is a global polariz ation observable hinges on our c hoic e of representation. Supposing kinematic pairs numbering polariz ation observables among their bona fide magnitudes differ physic ally from kinematic pairs that do not, this is a hint that when the spin c hains bec ome infinite, the c hoic e of representation c ould have physic al signific anc e. The Jordan-Wigner theorem applies to representations of the CARs for finitely many spin systems. Not applying to the infinite spin chain, the theorem is silent about whether the H + representation is unique up to unitary equivalenc e. Indeed, it is not. Consider, for c ontrast, a representation set in a Hilbert spac e whose basis elements c orrespond to the sequenc e [sk]j = − 1 for j ∈ ℤ, along with all sequenc es differing from this one in only finitely many places. Operators σˆk(i)− satisfying the Pauli relations are introduced in such a way that [sk]j, the jth entry in the basis sequence sk, gives the expectation value of ˆσj (z). Call this the H − representation. By parity of reasoning, the z-component of the total polarization mˆ (z)− is an observable generated by this representation. Enjoying every vector in H − as an eigenvalue −1 eigenvector, ˆm(z)− coincides with −I−. To establish that the H − and H + representations are not unitarily equivalent, suppose, for contradiction, that they were. Then, we know from the last sec tion, the unitary map between these representations would define bijec tions iobs and istate between the observable algebras and state sets of the kinematic pairs (B(H + ), T+ (H + )) and (B(H − ), T+ (H − )), bijections satisfying PEV and PAS. Any bijection iobs that preserves algebraic structure (as demanded by PAS) must map each σˆk(z)+ in the H + representation to σˆk(z)− in the H − . Preserving this correspondence through the sequence of polynomials of Pauli spins whose limit defines the z-component of global polarization, a unitarily implemented iobs obedient to PAS maps ˆm(z)+ on the H + representation to ˆm(z)− on the H − representation. But that is to say that iobs maps the identity operator on the first representation to − 1 times the identity operator on the sec ond. It follows that no bijec tion istate between states of the H + and H − representations can consort with iobs to preserve expectation values as demanded by PEV. Such an istate would have to identify a state ρ on the H + representation with a state istate(ρ) on the H − representation in such a way that istate(ρ) assigns iobs(ˆm(z)+ ) the same value ρ assigns ˆm(z)+ . But—keeping in mind that all states are linear and normed—ρ assigns ˆm(z)+ the value 1, because ˆm(z)+ is the identity operator, whereas any state on the H − assigns iobs(ˆm(z)+ ) the value −1, because iobs(ˆm(z)+ ) is −1 times the identity operator. Thus, no bijec tion iobs between the observable sets that preserves algebraic struc ture c an hope also to preserve expec tation values. The bijec tions that would have to exist, if the representations were unitarily Page 10 of 23

Unitary Equivalence and Physical Equivalence equivalent, c annot exist. We c onc lude that the representations fail to be unitarily equivalent. This argument that the H + and H − representations are not unitarily equivalent makes striking an apparent physic al differenc e between the kinematic pairs based on those representations. In an ordinary quantum theory built up from the H + representation, states whose polarizations differ from +1 in the z direction do not occur. In an ordinary quantum theory built up from the H − representation, only such states occur. Thus, the values of global polariz ation allowed distinguish physic ally between theories based on the inequivalent representations. Indeed, the rival quantum theories built on those representations c an be subjec t to a c ritic al test in the form of a measurement of the z-c omponent of global polariz ation. And this physic al differenc e would be expec ted to persist in finer-grained interpretations of the kinematic pairs corresponding to the representations, supposing those interpretations take systems in eigenstates of an observable to actually possess the corresponding eigenvalue of the observable. Sec tion 3.3 will raise the question of whether we c an with good c onsc ienc e regard the differenc e just eluc idated to be a genuine physical difference between rival quantum theories. 3.2 The Bead on a Circle To leave the sc ope of the Stone-von Neumann theorem, we need only c onsider the apparently simple system c onsisting of a single partic le c onstrained to move on the unit c irc le S1. The c anonic al variables of a c lassic al Hamiltonian treatment of this system are its position, given by an angular variable Φ∈ [0,2π], and its angular momentum ℓ ∈ ℝ. Thus its c onfiguration spac e is the c irc le S1 and its phase spac e is the c ylinder S1 × ℝ. Before quantiz ing13, a c hange of variables from the standard c ylindric al c oordinates (Φℓ) is in order. We are aiming in our quantiz ation to reproduc e the Poisson brac ket struc ture of the c lassic al theory (at least insofar as it applies to c anonic al observables) in c orresponding c ommutation relations between Hilbert spac e operators. The variable φ is not a c lassic al observable, bec ause c lassic al observables are c ontinuous func tions on phase spac e, but Φ is not. The disc ontinuity oc c urs as Φ approac hes 2π, and oc c urs bec ause the c onfiguration spac e of the system is a c irc le. So we will instead c harac teriz e the c lassic al algebraic struc ture in terms of the variables (9) whic h are c ontinuous on the c ylinder.14 Among these variables, the standard Poisson brac kets are given by (10) Following the Poisson brac ket goes to c ommutator rule, we build a quantum theory of the partic le on the c irc le, by finding a Hilbert spac e representation of the Circ ular Canonic al Commutation Relations (CCCRs) c orresponding to the Poisson brac kets (10): (11) These CCCRs (11) have a standard representation in terms of Hilbert spac e operators ac ting on L2 (S1), the spac e dϕ of func tions ψ(Φ) : S1 → ℂ that are square integrable with respec t to the measure 2π . (12) Happily, the spec trum of xˆ and ŷ is [−1, 1], whic h is exac tly as it should be for a system whose c onfiguration space is the unit circle S1. Also happily, ẑ's spectrum is 2π n, n ∈ 0,1,2,… . That is, it is the angular momentum spec trum for a partic le on a c irc le suggested by our boundary c onditions and the de Broglie relations. The Stone-von Neumann theorem c onc erns representations of CCRs expressing the quantiz ation of a c lassic al theory with phase spac e ℝ2 n. But the CCCRs are not the CCRs, and the phase spac e for a bead on the c irc le is not ℝ2 n for any n, but the c ylinder S1 × ℝ. Thus the Stone-von Neumann theorem does not imply that the standard representation (12) of the CCCRs is unique up to unitary equivalenc e. And it is not. Instead, there exists a family representations of the CCCRs labeled by θ ∈ [0, 1] (see Isham 1983, 1270–1272 for details). These representations are related to the standard one by: (13) Page 11 of 23

Unitary Equivalence and Physical Equivalence The θ = 0 member of this family is just the standard representation, whic h we will c all Q0 . For θ ≠ 0, the representation Qθ agrees with the Q0 representation about the spec tra of the c onfiguration observables x and y. But it disagrees about the angular momentum spec trum. In the standard representation Q0 , the angular momentum spectrum is 2π n for integer n; in Qθ, the angular momentum spectrum is 2π(n − θ) . Representations Qθ and Qθ′ are unitarily equivalent only if θ = θ. Thus we have a veritable host of unitarily inequivalent representations of the CCCRs. This indic ates a signature of the physic al differenc e between ordinary quantum theories based on unitarily inequivalent representations: their momentum spec tra.15 States possible ac c ording to an ordinary quantum theory built up from the Q0 representation inc lude ẑθ eigenstates in whic h the bead's angular momentum vanishes. States possible ac c ording to an ordinary quantum theory built up from the Qθ≠0 representation inc lude no ẑθ eigenstates in whic h the bead's angular momentum vanishes.16 The family Qθ of unitarily inequivalent representations of the CCCRs are not idle mathematic al c uriosities. They are related to the θ angles of Yang-Mills theory. With minor adjustments, they c an be applied to the Bohm-Aharonov effec t, wherein an elec tron translated around an (infinitely extended) solenoid experienc es a phase rotation determined by the flux through the solenoid.17 Exc luding the elec tron from the region of spac e oc c upied by the solenoid, we attribute it a c onfiguration spac e that is ℝ3 with a c ylinder removed. Like the c irc le, this topologic ally non-ℝn c onfiguration spac e frames a Hamiltonian theory that admits a family of inequivalent quantiz ations. Different members of this family c orrespond to different fluxes through the solenoid and henc e to different phase shifts for the transported electron (see Landsman 1990 for details). Thinking big, observe that mec hanic s set in a spatially c ompac t universe (suc h as, presumably, our own) will have a topologic ally non-ℝ2 n phase spac e, plac ing its quantiz ation outside the sc ope of the Stone-von Neumann theorem. 3.3 Other Unitarily Inequivalent Representations The foregoing examples hardly exhaust the field of unitarily inequivalent representations. Other examples drawn from QFT have elic ited the attention of philosophers of physic s. These inc lude: Foc k spac e representations quantiz ing the free Klein-Gordon field assoc iated with ‘inc ommensurable’ partic le notions (Clifton and Halvorson 2001; Arageorgis, Earman, and Ruetsc he 2003); the standard Minkowski vac uum representation and infrared c oherent representations (Baker 2009); representations assoc iated with different states of broken symmetry (Earman 2004; Liu and Emch 2005); representations generated from one another by time evolution in nonsta- tionary spac etimes (Arageorgis, Earman, and Ruetsc he 2002). Examples from the thermodynamic limit of QSM inc lude representations assoc iated with equilibrium states at different temperatures, or assoc iated with different phases at the same temperature (Ruetsc he 2003; Emc h 2007). Even the simplest quantum mec hanic al system—a single partic le c onfined to the real line—admits unitarily inequiva-lent representations in the form of the ‘position’ and ‘momentum’ representations (Halvorson 2001). Running roughshod over conventional expectations (see Teller 1979), the former makes available exac t position eigenstates; the latter does the same for momentum. Even on their own, the examples developed above should inspire us to interrogate the assumptions underlying our rec eption of unitary equivalenc e as c riterial for physic al equivalenc e. These inc lude assumptions about the proper c onfiguration of kinematic pairs for quantum theories, as well as the proper analysis of physic al equivalenc e for kinematic pairs generically conceived. When kinematic pairs of ordinary QM's form (B(H ),  T+ (H )) are subject to c riteria of physic al equivalenc e explic ated by PEV and PAS, it follows that kinematic pairs based on representations of c anonic al relations R are physic ally equivalent just in c ase those representations are unitarily equivalent. With respec t to the bead on the c irc le, embrac ing the c riterion limits the angular momentum spec trum for the system to the one given by a single member of the family Qθ of representations. Suc h a limitation hamstrings the quantum theory's c apac ity to model the Bohm-Aharonov effec t, as well as other phenomena admitting desc riptions in terms of “double-valued” wave functions, not to mention the applications catalogued in the penultimate Page 12 of 23

Unitary Equivalence and Physical Equivalence paragraph of sec tion 3.2. With respec t to the infinite spin c hain, embrac ing the c riterion requires us to deny that physic ally possible c onditions of the c hain inc lude both states in whic h its global polariz ation takes the value +1 in the z-direc tion and states in whic h its global polariz ation takes the value −1 in the z-direc tion. Suc h a denial is in tension with the behavior of ferromagnetic substanc es, whic h exhibit both kinds of spontaneous magnetiz ation.18 This exhibition is moreover an example of a c ritic al phenomenon engaging in what is known as universal behavior. The behavior of a wide variety of ferromagnetic substanc es is desc ribed by a phase diagram of a ferromagnet (see figure 14.1) c harac teriz ed by the same c ritic al exponents. The theory of universal phenomena is desc ribed elsewhere in the volume. The present point is that to use a phase diagram suc h as figure 14.1 to desc ribe a system, thereby bringing it within the ambit of this theory, is to allow it distinct possible states of spontaneous magnetization. Taking unitary equivalenc e to be c riterial for physic al equivalenc e c anc els this allowanc e. Figure 14.1 Phase diagram for a ferromagnet Resistance to the criterion also derives from a very different sort of attitude toward the infinite spin chain.19 It is a c ommonplac e among philosophers of spac e and time that solutions to the fundamental equations defining a spac etime theory, solutions c onnec ted by a spac etime symmetry of those equations, are physic ally equivalent.2 0 And it is easy to imagine embedding the infinite spin c hain into a spac etime theory in suc h a way that (for instanc e) the base state of the H + representation is connected to the base state of H − representation by the space time symmetry of flipping over the z-axis. This symmetry moreover extends to the Hilbert spac es in their entireties, identifying each vector in H + with its ‘flipflop’ in H − . Implying that each pair of symmetry-connected vectors represents the same physic al situation, the metaphysic al c ommonplac e implies that the c ollec tions of states implemented by density operators on H + and on H − represent the very same collection of physical situations. Ac c epting unitary equivalenc e as a c riterion of physic al equivalenc e means rejec ting the metaphysic al c ommonplac e.2 1 This suggests that widespread metaphysic al intuitions, as well as the use that prac tic ing physic ists make of unitarily inequivalent representations, stand intension with the established accounts of what quantum theories are and when they are physic ally equivalent. After a tec hnic al interlude, sec tion 5 explores reac tions to this distressing c irc umstanc e. 4. Technical Interlude For the sake of formulating and evaluating various responses to the prevalenc e in QMQ∞ of unitary inequivalent representations of the c anonic al relations identifying a quantum theory, it is time for us to take on board some formal notions. As previously announc ed, my exposition of these notions will be unrigorous.2 2 There is an abstrac t algebraic struc ture that all c onc rete Hilbert spac e representations of the c anonic al relations R c irc umsc ribing a quantum theory QR have in c ommon. This is the struc ture of the C* algebra AR generated by R. One way to c onstruc t AR is to start with a c onc rete Hilbert spac e representation of R, form polynomials of the c anonic al operators affording that representation, then c lose in the uniform (aka the norm) operator topology of the representing Hilbert spac e.2 3 The C* algebra AR results. (A c onc rete C* algebra is a uniformly c losed subset of R Page 13 of 23

Unitary Equivalence and Physical Equivalence the bounded operators on some Hilbert spac e.) If starting from another representation of R, we followed the same rec ipe to obtain a C* algebra A′ R , we would find that AR and A′ R are isomorphic : there exists a one-to-one map between the algebras preserving algebraic struc ture. As representation-independent, this is the algebraic struc ture shared by all of R′ s Hilbert spac e representations. Ordinary QM-ish observable algebras, algebras of the form B(H ), are weakly closed, and indeed can be regarded (as sec tion 2 presents them) as generated on behalf of a quantum theory QR by following the * rec ipe of the prec eding paragraph, but substituting weak for uniform c losure in the last step. (Bec ause weakly c losed sets are also uniformly closed, every B(H ) is also a C* algebra.) The criterion of convergence supplied by the uniform topology is harder to satisfy than the c riterion of c onvergenc e supplied by the weak topology. So there are sequenc es of Hilbert spac e operators that c onverge weakly, but not uniformly. For instanc e, the sequenc e of partial sums of n pairwise orthogonal projec tion operators on a separable infinite dimensional Hilbert spac e c onverges weakly to the identity operator as n goes to infinity. That same sequenc e of partial sums fails to converge uniformly. Given a concrete irreducible representation of canonical relations R on some Hilbert space H , its uniform closure, which is isomorphic to the canonical C* algebra AR, can turn out to be a proper subalgebra of B(H ), the observable algebra ordinary QM generates by weak closure from the concrete representation. (Rec all the example of the infinite spin c hain. The c onvergenc e of the limit that defines the global polariz ation observable is representation-dependent. That observable belongs to B(H + ), but has no counterpart in ACAR, the C* algebra generated as the uniform c losure of a representation of the Pauli relations.) One differenc e between unitarily inequivalent representations of R arises from observables that make it into the weak, but not the uniform, closures of concrete representations of R. The c anonic al C* algebra AR c an be c onsidered in abstrac tion from any c onc rete Hilbert spac e, as the algebraic struc ture all representations of R share. But an abstrac t C* algebra A will always admit a c onc rete Hilbert spac e representation, a morphism π from A into B(H ). Two representations (π,  H ) and (π′ ,  H ′) of the same algebra A are unitarily equivalent just in case there exists a unitary map U :  H → H ′ such that for each A  ∈  A,  Uπ(A)U −1 = π′ (A). Even if representations π and π′ are not unitarily inequivalent, there is still a isomorphism between π(A) ⊊ B(H )and π′ (A) ⊊ B(H ′)—just not one implemented unitarily, and so not one that extends to an isomorphism between B(H ) and B(H ′) in their entirety. The foregoing apparatus frames a C∗ algebraic approac h to quantum theories, whic h inc ludes the Hilbert spac e approac h of ordinary QM as a spec ial c ase. The C∗ algebraic approac h assoc iates the observables of a quantum theory with the self-adjoint elements of a C∗ algebra A appropriate to that theory. (In the spec ial c ase of ordinary QM, that algebra takes the form of B(H ) for some Hilbert space H .) The C∗ algebraic approach identifies quantum states on the observable algebra A with linear functionals ω :  A → C that are normed (ω)(I) = 1))and positive (ω(A∗ A) ≥ 0 for all A  ∈  A). ω(A) may be understood as the expectation value of (self-adjoint) A  ∈  A. These states are uniformly c ontinuous: if Ai is a sequenc e of elements of A that c onverges uniformly to A and ω is a state on A, then ω(Ai) c onverges in the good-old fashioned sense to ω(A). Ordinary QM adds an extra c odic il to its conception of states: admissible states on B(H ) need also be countably additive—a virtue that lacks a natural representation-independent account. Countably additive states on B(H ) are ultraweakly continuous: if Ai is a sequence of elements of B(H ) that converges in H 's ultraweak operator topology24 to A and ω is a c ountably additive state, then the expec tation values ω assigns Ai c onverge to the expec tation value ω assigns A. Like the states of ordinary QM, the set of states on a C∗ algebra A is c onvex. Its extremal elements—that is, states ω whic h c annot be expressed as nontrivial c onvex c ombinations of other states—are pure states; all other states are mixed. It is straightforward that a c ountably additive ordinary QM state implemented by a density operator Wˆ ac ting on a Hilbert space H carrying a representation π :  A → B(H ) of an algebra A defines a state ω on A. Simply set ω(A) = Tr(Wˆπ(A)) for all A  ∈  A. It is gratifying that we can travel in the other direction. Let ω be a state on a C∗ algebra A. Then there exists a Hilbert space Hω, a faithful25 representation πω :  A → B(Hω) of the algebra, and a cyclic26 vector |Ψω⟩ ∈  Hω such that, for all A  ∈  A, the expectation value the algebraic state ω assigns A is duplic ated by the expec tation value the Hilbert spac e state vec tor | ψ ω〉 assigns the Hilbert spac e observable π(A). The triple (Hω, πω , |Ψω⟩) is unique up to unitary equivalence, and known as the state's GNS representation. A Page 14 of 23

Unitary Equivalence and Physical Equivalence Clearly, other states φ on A c an be implemented as density operator states on ω's GNS representation. For example any Wˆ acting on Hω defines a state φ on A via ϕ(A) :   =  Tr(Wˆπω (A)) for all A  ∈  A. The set of states thus definable as density operators on ω's GNS representation c omprise what is known as ω's folium. If A admits unitarily inequivalent representations, not every state on A lies in ω's folium. Rec alling the infinite spin chain, consider, for example, the state on the algebra ACAR implemented by the base state of the H + representation. This is the state that assigns the z-c omponent of every Pauli spin, as well as the z-c omponent of every finite subchain polarization observable, the eigenvalue +1. Call this state ω+. We can regard the H + representation as ω+'s GNS representation. We have argued that in H + 's weak operator topology, the finite subc hain operators m(z)+N c onverge in the N→ ∞ limit to the identity operator I+. This implies that they c onverge ultraweakly to I+ as well. A state on ACAR that c annot be implemented by a density operator on ω+ 's GNS representation is the state, call it ω−, implemented by the base state of the H − representation. ω− is the state in whic h the spin at every site points in the negative z direc tion. So for eac h k, ω−(σk(z)+) = − 1. Henc e for eac h N, ω− (m(z)N+ ) = −1. But ω−(I+) had better be + 1, because ω− is a state and I+ is the identity operator. So m(z)+N converges ultraweakly to I+, but ω− (m(z)+N) does not converge to ω−(I+). But that means ω− fails to be ultraweakly continuous in H + . If ω− were implemented by a density operator on H + , it would be ultraweakly c ontinuous. ω− is not implementable by a density operator on ω+'s GNS representation. Every state in ω+'s folium is thus implementable. So ω− lies outside ω+'s folium. When the GNS representations πω and πφ of two algebraic states are unitarily equivalent, the folia of those algebraic states c oinc ide. If two pure algebraic states φ and ω have unitarily inequivalent GNS representations, their folia are disjoint: no algebraic state expressible as density matrix on φ 's GNS representation is so-expressible on ω's and vic e versa. (With mixed algebraic states, the situation is more delic ate. See Kadison and Ringrose 1997, c h. 4 for details.) Disjoint states are states whose folia are disjoint. The states ω+ and ω− on ACAR are disjoint. Sec tion 2.3 explic ated physic al equivalenc e for generic kinematic pairs by means of the pair of c onditions PEV and PAS, and promised an argument that kinematic pairs satisfying the expec tation-value preserving c ondition (PEV) will not in general thereby satisfy the algebraic -struc ture preserving c ondition PAS. We are now in a position to give that argument, whic h undersc ores the inc apac ity of PEV on its own to c apture a robust sense of physic al equivalenc e. Let AR be the c anonic al C* algebra for a quantum theory QR, and let ω and φ be disjoint pure states on it. QR's grip on whic h physic al possibilities are whic h begins with the c anonic al relations R and the expec tation values assigned observables satisfying those relations. States assigning different expec tation values to c anonic al observables c orrespond to different physic al possibilities. In order to be disjoint, ω and φ have to be different states, that is, there has to be some c anonic al observable regarding whose expec tation value ω and φ disagree. Now consider two ordinary QM-ish kinematic pairs for QR :  (B(Hω)T+ (Hω)) and (B(Hϕ ),  T+ (Hϕ )). Every separable infinite dimensional Hilbert spac e is isomorphic to every other, and in partic ular there is a unitary map U :  Hω → Hϕ . This U defines a pair of bijections iobs and istate between observable algebra and state sets of the kinematic pairs: for all A  ∈  B (Hω), iobs(A) = UAU −1 , for all W ∈  T+ (Hω), istate(W) = UWU −1 . It is trivial that these bijec tions together satisfy PEV. But it is mad to take states identified by the bijec tions to represent the same physic al possibility. No state in the first kinematic pair c an reproduc e the expec tation value assignment to c anonic al observables of any state in the sec ond kinematic pair. (That is just what it is for the representations to be disjoint.) istate therefore identifies states that ac t differently on AR , states c orresponding to different physic al possibilities. But istate was supposed to identify states c orresponding to the same physic al possibility. Something has gone wrong. A natural diagnosis is that the bijec tions defined by the unitary map above have gone astray by failing to respec t QR's algebraic struc ture. In partic ular, where Ci are the elements satisfying the c onstitutive relationships R and generating QR, iobs violates the (7) clause of the condition PAS, which requires: (We know this bec ause if iobs satisfied this c ondition, the representations would be unitarily equivalent, whic h by hypothesis they are not.) Violating (7), iobs satisfies PEV in a way that loses trac k of whic h observables are Page 15 of 23

Unitary Equivalence and Physical Equivalence c anonic al. That is why states identified by i state differ in the values they assign c anonic al observables. 5. Why Unitary Equivalence 5.1 Competing Criteria of Equivalence The disc losures of the tec hnic al interlude inspire an interpretive strategy Arageorgis (1995) has dubbed “Algebraic Imperialism.” The Algebraic Imperialist c onc eives of a theory of QM∞ not in terms of a partic ular c onc rete Hilbert spac e representation of the c anonic al relations R c irc umsc ribing that theory, but in terms of the abstrac t algebraic struc ture every suc h representation shares. That is, the Imperialist supposes the theory's physic al magnitudes to be given by the self-adjoint part of the C* algebra AR, and takes possible states to be the set SAR of states in the algebraic sense on AR. The Algebraic Imperialist equips a theory QR with a kinematic pair of the form (AR, SAR ). For a quantum theory that, like a typic al theory of QM∞, admits unitarily inequivalent representations, the Imperialist thereby rejec ts ordinary QM's conception of a kinematic pair as a double (B(H ),  T+ (H )) for some concrete Hilbert space H . Generated as the weak c losure of a c onc rete Hilbert spac e representation of R, the ordinary QM observable algebra B(H ) generally contains elements without counterpart in the Imperialist's observable algebra AR. And the Imperialist's state set AR generally includes states not implementable by members of the ordinary QM state set T+ (H ). The Imperialist takes more states to be physically possible than can be ratcheted into an ordinary QM ac c ount. She also takes fewer observables to be physic ally signific ant. Sec tion 2.3 anointed unitary equivalenc e a c riterion of physic al equivalenc e for kinematic pairs of the form (B(H )Cˆi ,  T+ (H )) on the grounds that (i) any two such pairs satisfied the demands PEV (preservation of expec tation values) and PAS (preservation of algebraic struc ture) just in c ase their generating representations were unitarily equivalent, and (ii) the demands PEV and PAS appropriately explic ated physic al equivalenc e for theories spec ified up to kinematic pairs. For the Imperialist, (i) is true, but irrelevant: she assigns theories of QM∞ kinematic pairs different in kind from those of ordinary QM. She should, however, take (ii) to be both true and relevant. If anything, more sensitive to the physic al import of algebraic struc ture than the advoc ate of ordinary QM, the Imperialist should welc ome the c riterion (PAS). Moreover the Imperialist has no spec ial reason to resist the preservation of expec tation values as a c riterion for physic al equivalenc e. So she should ac c ept sec tion 2.3's gloss of physic al equivalenc e in terms of the pair of c onditions (PEV and PAS), but deny that it be applied to kinematic pairs of ordinary QM's sort. Rather, the Imperialist would apply those c onditions to kinematic pairs (AR, SAR ) of the sort she recognizes. When the c riteria PAS and PEV for physic al equivalenc e are applied to kinematic pairs of the sort favored by Imperialists, unitary equivalenc e emerges as suffic ient, but not nec essary, for physic al equivalenc e. Let (thAe, SalgAe),b r(aAs′ ,A S,  AA′′) h baev ek inceamnoantiicc apla giresn feorra ato trhse Ror,y R Q′iR, criercspuemcstcivreiblye.d A bsys uthmee c tahneoren iicsa al nre ilsaotimonosrp Rhi,s man ad :s uApp→osAe ′ such that α(Ri) = R′i for all i. Because it is an isomorphism on the algebras’ generators, α extends to their products, linear combinations and uniform limits (Kadison and Ringrose 1997, Thm 4.1.8). Between A and A′ , α provides a bijec tion i obs(A) = α (A) that satisfies PAS. Conversely, if there is no suc h isomorphism, there is no bijec tion satisfying PAS. α also generates a bijec tion between states: i state : ω → ω ◦ α where ω ◦ α −1 assigns each A  ∈  A the value ω assigns α−1(A). Together, istate and iobs satisfy PEV. So kinematic pairs (A,  SA),  (A′,  SA′ ) satisfy PEV and PAS if and only if there is an isomorphism α :  A → A′ such that α(Ri) = R′i for all i. Unitary equivalenc e emerged as a c riterion for physic al equivalenc e for kinematic pairs cast in ordinary QM form (B(H ),  T+ (H )). It emerged from the analysis of physical equivalence for general kinematic pairs in terms of PAS and PEV. We have just seen that applying that same general analysis to kinematic pairs c ast in Algebraic Imperialist form (A,  SA) eventuates in algebraic isomorphism as a criterion of physical equivalence. Our overarc hing question is: Is unitary equivalenc e a suitable c riterion of physic al equivalenc e for kinematic pairs realiz ing quantum theories that fall outside the sc ope of the Stone-von Neumann and Jordan-Wigner theorems? We Page 16 of 23

Unitary Equivalence and Physical Equivalence now have a provisional answer. It depends. It depends on how we ought to c onstrue the kinematic pairs for suc h theories—in ordinary QM's form, in Algebraic Imperialist form, or some other way. But whic h form is the right one? The next, and c onc luding, sec tion presents some approac hes to this question. 6. Principles? The infinite spin c hain evoked an anxiety: its possible states should inc lude the states on ACAR we have c alled ω+ and ω−, but any theory of ordinary QM rec koning ω+ to be a possible state of the infinite spin c hain is physic ally inequivalent to any theory of ordinary QM rec koning ω− to be a possible state. The last sec tion c harts a strategy for alleviating this anxiety: ditc h ordinary QM's ac c ount of quantum kinematic s and embrac e instead the Imperialist's account. Limiting observables pertaining to the infinite spin chain to self-adjoint elements of ACAR , we c onstitute a quantum theory ac c ording to whic h both ω+ and ω− are possible states of the c hain. But at a c ost. Among the observables we have given up is the global polariz ation observable, the observable that distinguishes most c onspic uously and direc tly between the states ω + and ω −, the observable that labels an axis of the phase diagram distilling a ferromagnet's universal behavior. The Imperialist strategy for welc oming both ω+ and ω− as physic al undermines the physic s some people would like to do with those states. Of c ourse, other people—those with a partic ular, but widespread set of metaphysic al sc ruples about symmetry— would rejec t any observable that marked a physic al differenc e between ω+ and ω−, on the grounds that, as symmetry-c onnec ted, ω+ and ω− represent the same physic al situation. Those c ommitted to suc h sc ruples also have reason to resist the ordinary QM pic ture of quantum theories—ω+ and ω− c annot represent the same physic al situation if one's a possible state only if the other is not. The Imperialist's pic ture presents them with no suc h immediate absurdity. It does, however, require the symmetric ians to do some work to avert inanity. Symmetries like the flip-flop symmetry c onnec ting ω+ and ω− are implemented by automorphisms of ACAR . (As the name suggests, an automorphism α :  A → A of an algebra A is just an isomorphism that maps A to itself.) It turns out that for every pair of pure states on ACAR , there is some automorphism that c onnec ts them. So if every automorphism implements a symmetry, the symmetric ians’ c ommitment to identify symmetry-c onnec ted states c ollapses the state spac e of the theory to a single point. A physic al theory with suc h a state spac e is inane. To avert inanity, symmetricians need to distinguish automorphisms that implement symmetries from automorphisms that do not. Sec tion 5 determined that the explic ation of physic al equivalenc e for theories of QM∞ depends on the sorts of kinematic pairs it is appropriate to attribute suc h theories. We have just seen two princ ipled attitudes toward ω+ and ω− whic h c onc ur in rejec ting kinematic pairs of ordinary QM's sort as inappropriate. One attitude rec kons ω+ and ω− to be physic ally distinc t possible states of the infinite spin c hain, on the princ iple that this arrangement supports explanatory aspirations, exerc ises in unific ation, and programs of theory development. The other attitude rec kons ω+ and ω− to be physic ally identic al states, on the princ iple that they are c onnec ted by a symmetry. United in their departure from ordinary QM, these princ iples soon diverge. Those in the grip of the explanation- honoring princ iple c annot abide the Imperialist's ac c ount of quantum kinematic s, for that ac c ount obliterates what they take to be the physic ally fruitful distinc tion between ω+ and ω−. Those in the grip of the symmetry-honoring princ iple c an live with the Imperialist's ac c ount, provided they c an supplement their metaphysic al princ iple with another one, a princ iple able to identify those automorphisms of ACAR whic h implement symmetries. This illustrates a typic al predic ament: the question of what sort of kinematic pair it is appropriate to attribute theories of QM∞—the question on whose answer turns the explic ation of physic al equivalenc e for suc h theories—is a question to whic h there are many responses, guided by many princ iples, some of whic h are in mutual (or even internal) tension. We c an try to simplify the question about kinematic pairs by breaking it into c omponent questions: What observable algebras should we attribute theories of QM∞, and what state sets? Take the former question first. Suppose all hands agree that observables realiz ing the c anonic al relations R c irc umsc ribing a quantum theory QR are physic ally signific ant. Bec ause expressing the relations R requires forming linear c ombinations and produc ts of c anonic al observables, suppose that all hands agree as well that that the set of physic al observables is c losed under those algebraic operations. Then all we need to induc e agreement among all hands about QR's observable algebra is to get all hands to agree about what c riterion of c onvergenc e is appropriate to use in adding limit points to the algebra generated by the c anonic al observables. If they agree to uniform c onvergenc e, they agree as well * AR Page 17 of 23

Unitary Equivalence and Physical Equivalence that the C* algebra AR is the observable algebra for QR. If they agree to c onvergenc e in the weak operator topology of a Hilbert space H bearing a representation of R, they agree to B(H ) as the observable algebra. Segal (1959) struc k on an argument favoring uniform c onvergenc e. The c onc eptual c rux of Segal's argument is the c laim that if a sequenc e of observables Ai c onverges to another observable A, the world had better not be able to get itself into a state ω suc h that ω(Ai) fails to c onverge to ω(A). Segal used a broadly operationalist outlook to sec ure this c laim. Protoc ols for measuring A will inc lude ones where we measure many members of the sequenc e Ai and extrapolate to the limit of the sequenc e of outc omes obtained. This protoc ol would break down if ω(Ai) failed to c onverge to ω(A). Lest our experimental protoc ols lose their grip on the observables they are meant to measure, Segal c ontends, we should regard as observables only those quantities defined by limiting relationships Ai → A suc h that ω(Ai) c onverges to ω(A), no matter what ω is. To deal weak c onvergenc e the coup de grâce, Segal remarks that for any representation (π,  H ) of a canonical algebra A admitting unitarily inequivalent representations, there will be sequences Ai  ∈  A and states ω on A such that π(Ai) converges in H 's weak topology, but ω(Ai) fails to c onverge. (States ω on A not implementable by a density operator state on the representation (π,  H ) will have this feature.) If we added the weak limit of the sequence π(Ai) to our catalog of observables, we would rec ogniz e an observable whose value we c ould not gauge even by the most c areful sequenc e of measurements performed on systems in the state ω. To a good operationalist, a quantity whose value eludes experimental assessment is no real quantity. To build an observable algebra containing only genuine quantities, Segal urges, avoid weak limits. By his lights, uniform limits are fine. Bec ause all states on A are uniformly c ontinuous, if Ai c onverges uniformly to A, no matter what ω is, ω(Ai) c onverges to ω(A) as well.2 7 This is an admirably princ ipled argument for attributing QR the observable algebra AR generated by uniform closure, rather than any larger algebra B(H ) generated by weak closure. But as Segal develops it, the argument quite obviously makes c overt appeal to a partic ular answer to a question it was the aim of our simplifying strategy to set aside: What states are physic al? If QR's physic al states were required to reside in the folium of the representation whose weak operator topology was used to obtain B(H ), then for every weakly convergent sequenc e of observables Ai → A and every physic al state ω, ω(Ai) would c onverge to ω(A), and the limiting observables would survive Segal's test for signific anc e.2 8 Segal's c riterion for the signific anc e of an observable c annot operate in isolation from an ac c ount of whic h states are physic al. Operating in c onc ert with different ac c ounts of physic al states, Segal's c riterion makes different judgments about whic h observables are physic al. Operating in isolation from an ac c ount of physic al states, Segal's c riterion makes no judgment at all. We are after a principled way of settling the question of what sort of kinematic pair (Q,  S ) to attribute a theory QR. Even supposing we ac c ept Segal's princ iple that the c ontinuity properties of physic al states marc h in loc kstep with the c ontinuity properties of algebras of observables, we need a princ ipled way to fix either the state set S or the observable algebra Q of QR to prec ipitate a kinematic pair from that princ iple. The good news is that there are a variety of principles we might invoke to configure either S or Q. The bad news is that different principles eventuate in different kinematic pairs, which in turn are suited to different philosophical and physical projects. Adopting a single and uniform princ iple inevitably stymies some of those projec ts. Here are some examples of princ iples, and the projec ts they sustain and frustrate. • The Hadamard c ondition is a princ iple for identifying physic ally signific ant states on the c anonic al algebra A for a quantum field theory on c urved spac etime (see Wald 1994, c h. 4 for an ac c ount). States that fulfill the Hadamard c ondition are states for whic h a point-splitting presc ription for assigning the stress energy tensor an expec tation value suc c eeds. Suc h suc c ess is essential to the pursuit of semi-c lassic al quantum gravity, an approac h that marries quantum theory to the general theory of relativity by replac ing the stress energy tensor Tab in Einstein's field equations with its expec tation value 〈Tab〉φ. For a QFT QR set in a c losed spac etime, the Hadamard c ondition fosters signific ant progress on the question of what kinematic pair to assign QR. In suc h a spac etime setting, every Hadamard state is unitarily equivalent to every other. Thus, states satisfying the Hadamard c ondition are c onfined to those expressible as density operators on a partic ular c onc rete representation (π,  H ) of A. With states so confined, Segal's coordination principle delivers B(H )—perhaps with the stress-energy observable added ‘by hand’—as the observable algebra appropriate to QR. In open spac etimes the story is not so neat: unitarily inequivalent Hadamard states abound. Perhaps more distressingly, when it c omes to sustaining the aspirational pursuit of quantum gravity, the Hadamard c ondition is a mixed bag. One triumph of semi-c lassic al quantum gravity is its role in predic ting the “phenomenon” of blac k Page 18 of 23

Unitary Equivalence and Physical Equivalence hole evaporation. In blac k hole evaporation, the region of spac etime exterior to a blac k hole oc c upies a quantum state that is a thermal state at a temperature related to the surfac e gravity of the blac k hole, whic h eventually radiates its substanc e away. This Hawking radiation and the assoc iated model of blac k hole thermodynamic s serve fledgling projec ts in quantum gravity as something like a datum. To be c onsidered plausible, theories of quantum gravity should be able to ac c ommodate and predic t blac k hole evaporation. The c atc h is that the QFT state that in some respec ts best models the Hawking radiation exterior to a blac k hole during its evaporation is a state that violates the Hadamard c ondition (see Candelas 1980). Dismissing that state from physic al relevanc e frustrates the pursuit of quantum gravity by undermining one of our best models of a “phenomenon” quantum gravity is meant to save. • Let A be the c anonic al algebra for a QFT QR set in a spac etime M. Axiomatic approac hes to QFT inc lude an axiom demanding that M's spac etime symmetries be implemented unitarily (see Dimoc k 1980). We c an c ast this axiom as a princ iple that in order for a state ω to be physic al, its GNS representation must be one on whic h M's symmetries are unitarily implementable. Again, given the right sort of M, this princ iple for identifying physic al states helps identify a kinematic pair. Coupled with other axioms for Minkowski spac etime, the princ iple singles out a privileged vac uum state ω 0 on A. Taking the physic al states of the theory to be those in ω 0 's folium and applying Segal's coordination principle, we attribute QR the kinematic pair (B(Hω ),  T+ (Hω)). In more general spac etime settings, the story is not so neat. But even c onfining attention to Minkowski spac etime, the kinematic pair underwritten by the princ iple that symmetries be unitarily implementable stymies c ertain projec ts in physic s. For instanc e, so-c alled “infra-red states,” invoked in models of soft-photon sc attering (see Stroc c hi 1985, 87), fall outside the pale of physic al possibility limned by this kinematic pair. But these states are integral to our modeling and understanding a signific ant c lass of experimental partic le physic s phenomena. Rejec ting infrared states on princ iple limits the explanatory reac h of QR. This list c an be c ontinued: for instanc e, insisting that dynamic s be well defined mitigates in favor of abstrac t algebraic approac hes when dynamic s c annot be implemented unitarily on a fixed Hilbert spac e (Arageorgis et al. 2002), but in favor of c onc rete Hilbert spac e representations when the existenc e of a dynamic s is representation- dependent (see Emc h and Knops 1970). But even in its short form, the list suggests a moral that c omplic ates the searc h for overarc hing and univoc al c riteria of physic al equivalenc e appropriate to theories of QM∞. Those c riteria will depend on the sorts of kinematic pair suited to theories of QM∞. Their c apac ity for wide applic ation is a virtue of theories of QM∞. Any rec ipe that, given R, generated a kinematic pair for QR without looking for guidanc e to the details of QR's applications, threatens to erode that virtue. References Arageorgis, Aristidis (1995). Fields, partic les, and c urvature: Foundations and philosophic al aspec ts of quantum field theory in c urved spac etime. Ph.D. diss., University of Pittsburgh. Arageorgis., Aristidis, John Earman, and Laura Ruetsc he (2002). Weyling the time away: The non-unitary implementability of quantum field dynamic s on c urved spac etime and the algebraic approac h to QFT. Studies in the History and Philosophy of Modern Physics 33: 151–184. —— (2003). Fulling non-uniqueness, Rindler quanta, and the Unruh effect: A primer on some aspects of quantum field theory. Philosophy of Science 70: 164–202. Baker, David J. (2009). Against field interpretations of quantum field theory. British Journal for the Philosophy of Science 60: 585–609. —— (forthc oming). Spontaneous symmetry breaking in QFT. Philosophy of Science. Belot, Gordon (1998). Understanding elec tromagnetism. British Journal for the Philosophy of Science 49: 531– 555. —— (2007). The representation of time and c hange in c lassic al mec hanic s. In Butterfield and Earman (2007a), 133–227. Butterfield, Jeremy, (2007). On symplec tic reduc tion in c lassic al mec hanic s. In Butterfield and Earman (2007a), 1– 131. Page 19 of 23

Unitary Equivalence and Physical Equivalence Butterfield, Jeremy, and Earman, John, eds. (2007a). Handbook of philosophy of science: Philosophy of physics, Vol. 1. Amsterdam: Elsevier. —— (2007b). Handbook of philosophy of science: philosophy of Physics, Vol. 2. Amsterdam: Elsevier. Candelas, P. (1980). Vac uum polariz ation in Sc hwarz sc hild spac etime. Physical Review D 21: 2185–2202. Clifton, Robert and Hans Halvorson (2001). Are Rindler quanta real? inequivalent partic le c onc epts in quantum field theory. British Journal for the Philosophy of Science 52: 417–470. Dimoc k, J. (1980). Algebras of loc al observables on a manifold. Communications in Mathematical Physics 77: 219–228. Dürr, Detlef et al. (2006). Topologic al fac tors derived from Bohmian mec hanic s. Journal Annales Henri Poincaré 7: 791–807. Earman, John (2004). laws, symmetry, and symmetry breaking: Invarianc e, c onservation princ iples, and objec tivity. Philosophy of Science 71: 1227–1241. Earman, John and Laura Ruetsc he (2005). Relativistic invarianc e and modal interpretations. Philosophy of Science 72: 557–583. Emc h, Gérard (1972). Algebraic methods in statistical mechanics and quantum field theory. (New York: Wiley). Emc h, Gérard (2007). Quantum statistic al physic s. In Butterfield and Earman (2007b), 1075– 1182. Emc h, Gérard and H. J. F. Knops (1970). Pure thermodynamic phases as extremal KMS states. Journal of Mathematical Physics 11, 3008–3018. Emc h, Gérard and Liu, Chuang (2002). The logic of Thermostatistic al Physic s. (Berlin: springer). Fulling, Stephen A. (1989). Aspects of quantum field theory in curved space-time. Cambridge: Cambridge University Press. Glymour, Clark (1970). Theoretic al realism and theoretic al equivalenc e. In Boston Studies in Philosophy of Science, Vol. 7, ed R. Buc k and R. Cohen, 275– 288. Dordrec ht: Reidel. Gotay, M. J. (2000). Obstruc tions to quantiz ation. in Mechanics: From theory to computation, 171–216. New York: Springer. Haag, Rudolf (1992). Local quantum physics. New York: Springer-Verlag. Halvorson, Hans (2001). “On the nature of continuous physical quantities in classical and quantum mechanics. Journal of Philosophical Logic 30: 27–50. Healey, Ric hard (2007). Gauging what's real: The conceptual foundations of contemporary gauge theories Oxford: Oxford University Press. Isham, C. J. (1983). Topologic al and global aspec ts of quantum theory. In ed B. S. DeWitt and R. Stora. Les Houches, Session XL: Relativité, groupes et to pologie II Amsterdam: Elsevier. Kadison, R. V. and J. R. Ringrose (1997). Fundamentals of the theory of operator algebras, Vol. 1. New York: Ac ademic Press. Landsman, N. P. (1990). C∗-algebraic quantiz aton and the origin of topologic al quantum effec ts. Letters in Mathematical Physics 20: 11–18. Lèvy-Leblond, Jean-Marc (1976). Who is afraid of nonhermitian operators? A quantum desc ription of angle and phase. Annals of Physics 101: 319–341. Liu, Chuang and Emch, Gérard (2005). Explaining quantum spontaneous symmetry breaking. Studies in History Page 20 of 23

Unitary Equivalence and Physical Equivalence and Philosophy of Modern Physics 36: 137–163. Roberts, J. E. and G. Roepstorff (1969). Some basic c onc epts of algebraic quantum theory. Communications in Mathematical Physics 11: 321–338. Ruetsc he, Laura (2003). A matter of degree: Putting unitary inequivalenc e to work. Philosophy of Science 70 [Proc eedings]: 1329– 1342. —— (2011). Why be normal? Studies in History and Philosophy of Modern Physics , forthc oming. Segal, Irving E. (1959). The mathematic al meaning of operationalism in quantum mec hanic s. In L. Henkin, P. Suppes, and A. Tarski. Studies in Logic and the Foundations of Mathematics, ed., 341–352. Amsterdam: North- Holland. Sewell, G. (2002). Quantum mechanics and its emergent Metaphysics . Princ eton: Princ eton University Press. Simon, B. (1972). Topic s in func tional analysis. In, Mathematics of contemporary physics, ed. R.F. Streater. London: Ac ademic Press. Stroc c hi, F. (1985). Symmetry breaking Berlin: Springer. Summers, Stephen J. (1999). On the Stone-von Neumann uniqueness theorem and its ramifications. In John von Neumann and the foundations of quantum physics. 135–152. Budapest, 1999. Vienna Circle Institute Yearbook, 8. Dordrec ht: Kluwer. Teller, Paul (1979). Quantum mechanics and the nature of continuous physical quantities. Journal of Philosophy 7: 345–361. Wald, Robert M. (1994), Quantum field theory in curved spacetime and black hole thermodynamics . Chic ago: University of Chic ago Press Notes: (1) Subjec t to provisos beautifully explained by Summers 1999. (2) For an introduction, see Emch 1972, 269–275. (3) For an introduc tion, see Wald 1994, c h. 2. (4) See Butterfield 2007 and Belot 2007 for this idea elaborated. (5) This is a restric tion for the sake of simplic ity, and a drastic one. Almost generic ally, QM∞ systems are desc ribed by states assoc iated with reducible—that is, not irreduc ible—representations. See Earman and Ruetsc he 2005 for a disc ussion. (6) What if my theory differs from yours only in a trivial sc ale transformation? That is, we don't satisfy PEV, but there are bijections iobs: iobs :  Q → Q′ and istate : S → S ′ such that (say) istate(ω)(iobs(A)) = 2 × ω(A). Wouldn't it be mad to take this failure to satisfy PEV to disqualify our theories from physic al equivalenc e?! I am not sure it would be. Notic e that at least one of the theories entertains only states that fail to be normaliz ed. And notic e as well that we c an restore unitary equivalenc e by attributing the theorists the same observable algebra but different c onventions for c oordinating self-adjoint elements of that algebra and measurement proc edures. (Thanks to Dave Baker and Bryan Roberts, who independently raised this point.) (7) Here is one reason linearity is important. Since only the self-adjoint elements of Q and Q′ correspond to observables, PEV should take iobs to be a bijection between the self-adjoint parts Qsa and Q′sa of those algebras, rather than a bijection between the algebras in their entirety. But for any element Q of Q, there are A,  B ⊂  Qsa suc h that Q = A+ iB. So if an iobs restric ted to the self-adjoint parts of the observable algebras ac ts linearly—a supposition the result just c ited sec ures— it induc es a bijec tion between the entire algebras, whic h bijec tion we will c ontinue to c all iobs. * Page 21 of 23

Unitary Equivalence and Physical Equivalence (8) I am suppressing a further c ondition arising from the fac t that a quantum algebras has an adjoint operation * . It is that α (X*) = (α (X))*. We will want iobs to satisfy this c ondition bec ause we will want it to identify observables, self- adjoint elements, with observables. (9) Clifton and Halvorson (2001) follow Glymour's analysis of physic al equivalenc e, whic h supposes physic al theories to be interpreted as axiomatic systems. For Glymour, suc h theories are physic ally equivalent only if intertranslatable in suc h a way that axioms get translated as axioms and theorems get translated as theorems. Roughly speaking, Clifton and Halvorson assimilate the generators of an observable algebra to axioms and its other elements to theorems, thereby motivating (7) as the axiom-to-axiom demand and PEV as the theorem-to-theorem demand. I take my rec onstruc tion to agree in spirit with theirs. (10) Here I follow Sewell 2002, § 2.3, to whic h I refer the reader for details. (11) For the duration of this explic ation, I am dropping hats over operators to minimiz e notational c lutter. (12) Simon expresses a c ommitment to the tradition as he launc hes into an exposition of those aspec ts of func tional analysis he c onsiders most c entral to physic s: “Throughout, all our Hilbert spac es will be separable unless otherwise indic ated. Many of the results extend to non-separable spac es, but we c annot be bothered with suc h obsc urities” (1972, 18). Although there are some ways in whic h the mathematic s of separable Hilbert spac es are “nic er” (for instanc e, some operator topologies are first-c ountable), I am not aware of a c anonic al explanation of the tradition. (13) A projec t in whic h my impressionistic exposition tries to follow the c areful treatments of Isham (1983) and Gotay (2000). Thanks are owed to Gordon Belot for help with this. Blame for persisting misunderstandings is not. (14) Another approac h is to let the c onfiguration variable be unitary instead of self-adjoint. See Lèvy-Leblond 1976. (15) See Dürr et al. for an argument that Bohmian mec hanic s “provides a sharp mathematic al justific ation” (2006, 791) of the expec tation that c lassic al theories with topologic ally exotic c onfiguration spac es have unitarily inequivalent quantiz ations. (16) Appropriately weighted superpositions of ẑθ eigenstates might assign expec tation value 0 to angular momentum, but c an be empiric ally distinguished from ẑ θ eigenstates by repeated non-disturbing measurements of angular momentum. (17) For why this might be interesting, see Belot 1998 or Healey 2007. Dürr et al. 2006 c atalogs other topologic ally non-ℝ2 n phase spac es with physic al applic ations. (18) Another suppressed complication: the ferromagnetic-paramagnetic phase transition does not occur in the 1-d model provided by the infinite spin c hain, but does in models of higher dimension. See Emc h and Liu 2002 for further disc ussion. (19) It will not esc ape the reader's notic e that the morals drawn in this paragraph undermine the c onsiderations of the last paragraph. Sinc e I am here only trying to motivate disc ontent with unitary equivalenc e as a c riterion of physic al equivalenc e, that is fine with me: whether you are drawn by the c onsideration of the last paragraph or the present paragraph, you have a reason to be unhappy with the c riterion. Sinc e I myself am drawn by both sorts of c onsiderations, I fac e a real puz z le, whic h deserves more attention than it gets here. (20) This sounds simpler than it is; see Belot (this volume). (21) This problem is addressed systematic ally in Baker (forthc oming). (22) Kadison and Ringrose 1997, c h. 4, gives a more formal introduc tion. (23) ‘Closing in the uniform topology’ means adding to the algebra the limit points of all uniformly c onvergent sequenc es of elements that have made their way into the algebra by other means. (24) A near-relative of weak c onvergenc e: Ai c onverges ultraweakly to A just in c ase for eac h density operator W, | Tr(WAi) − Tr(WA) | c onverges to 0. Page 22 of 23

Unitary Equivalence and Physical Equivalence (25) π is faithful iff π (A) = 0 implies A = 0 for all A  ∈  A. (26) |ψ〉 is cyclic for πω (A) means {πω (A)|Ψ⟩} is dense in H . (27) Ruetsc he 2011 uses c onsiderations of Segal's sort to sketc h reasons, innoc ent of operationalism, for c oordinating a quantum theory's state spac e and its observable algebra. (28) The continuity claim follows from the fact that every density operator state on B(H ) is ultra-weakly c ontinuous. Laura Ruetsche Laura Ruetsche is Professor of Philosophy at the University of Michigan. Her Interpreting quantum theories: The art of the possible (Oxford, 2011) aim s to articulate questions about the foundations of quantum field theories whose answers m ight hold interest for philosophy m ore broadly construed.

Substantivalist and Relationalist Approaches to Spacetime Oliver Pooley The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter provides an up-to-date, c omprehensive disc ussion of substantivalist and relationalist approac hes to spac etime. It analyz es why Isaac Newton postulated absolute spac e before examining the so-c alled kinematic shift argument and evaluates orthodox spac etime substantivalism. The c hapter also desc ribes the strategies the relationalist c an pursue in the fac e of the c hallenge posed by Galilean c ovarianc e, and evaluates the relevanc e of the Einstein's Hole Argument in the substantivalist–relationalist debate. K ey words: spaceti me, Isaac Newton , absol u te space, k i n emati c sh i ft argu men t, spaceti me su bstan ti v al i sm, G al i l ean cov ari an ce, Ei n stei n 's Hol e Argu men t 1. Intro ductio n A signific ant c omponent of the philosophic al interpretation of physic s involves investigation of what fundamental kinds of things there are in the world if reality is as physic s desc ribes it to be. One c andidate entity has proven perennially c ontroversial: spacetime.1 The argument about whether spac etime is an entity in its own right goes by the name of the substantivalist– relationalist debate. Substantivalists maintain that a c omplete c atalog of the fundamental objec ts in the universe lists, in addition to the elementary c onstituents of material entities, the basic parts of spac e-time. Relationalists maintain that spac etime does not enjoy a basic , nonderivative existenc e. Ac c ording to the relationalist, c laims apparently about spac etime itself are ultimately to be understood as c laims about material entities and the possible patterns of spatiotemporal relations that they c an instantiate. In his Principia, Newton famously distinguishes absolute from relative spac e and states that the former “of its own nature without reference to anything external, always remains homogeneous and immovable” (Newton 1999, 408). Newton's desc ription of, and arguments for, absolute spac e are c ommonly (and rightly) taken to be a statement and defense of substantivalism. In sec tion 2, I c onsider Newton's reasons for postulating absolute spac e before examining, in sec tion 3, one of the strongest arguments against its existenc e, the so-c alled kinematic shift argument. These arguments highlight the c lose c onnec tion between the spatiotemporal symmetries of a dynamic al theory and the spac etime ontology that the theory is naturally interpreted as c ommitted to. It turns out that the Galilean c ovarianc e of Newtonian mec hanic s tells against both substantival spac e and the most obvious relationalist alternative. With hindsight, the natural substantivalist response to this predic ament is to jettison spac e for spac etime. Sec tion 4 reviews orthodox spac etime substantivalism. The most defensible substantivalist interpretation of Newtonian physic s has muc h in c ommon with a very natural interpretation of relativistic physic s. From this perspec tive, our c urrent best theory of spac e and time vindic ates Newton rather than his relationalist c ritic s, c ontrary to what early philosophic al interpreters c laimed (e.g., Reic henbac h, 1924). Sec tion 5 is a skeptic al review of two antisubstantivalist themes that motivate some c ontemporary relationalists. Its Page 1 of 48

Substantivalist and Relationalist Approaches to Spacetime c onc lusion is that the aspiring relationalist's best hope is Oc kham's raz or, so the foc us shifts onto the details of relationalists'proffered alternatives to substantivalism. The move to a four-dimensional perspec tive expands the range of possibilities available to the c lassic al relationalist. In sec tion 6 I distinguish three strategies that the relationalist c an pursue in the fac e of the c hallenge posed by Galilean c ovarianc e and c onsider how the c orresponding varieties of relationalism fare when one moves from c lassic al to relativistic physic s. It turns out that a number of well-known relationalist views find a natural home in this framework. A review of the substantivalist– relationalist debate c annot get away without mention of Earman and Norton's (in)famous adaptation of Einstein's Hole Argument. In the final sec tion, I highlight what many see as the most promising substantivalist response and relate it to so-c alled struc tural realist approac hes to spac etime. 2. Newto n's Bucket Newton's best-known disc ussion of absolute spac e c omes in a sc holium to the definitions at the start of the Principia. Ac c ording to the onc e-standard reading, Newton's purpose in the Sc holium is to argue for the existenc e of substantival spac e via the existenc e of absolute motion, whic h he supposedly takes to be established by his famous buc ket experiment and two-globes thought experiment (see, e.g., Sklar, 1974, 182– 184). While there is now widespread agreement that this account badly misrepresents Newton's arguments, there is less consensus over how they in fac t should be understood.2 Thanks to Koyré (1965) and Stein (1967), it is now rec ogniz ed that, in order to understand Newton's Sc holium, one has to apprec iate that Newton was in large part reac ting to Desc artes's c laims about the nature of motion. I briefly review the relevant Cartesian bac kground before giving an ac c ount of Newton's arguments that is essentially in agreement with that of Rynasiewic z (1995). Desc artes was one of the first natural philosophers to put the principle of inertia—the c laim that bodies unaffec ted by net external forc es remain at rest or move uniformly in a straight line—at the c enter of his physic s (Desc artes 1644, II 37, 39). At the same time, and apparently without rec ogniz ing the problem, he espoused an ac c ount of motion that is hopelessly inc ompatible with it. Desc artes distinguishes motion in an everyday sense of the term from motion in a stric t, philosophic al sense. Motion in the ordinary sense is said to be change of place (ibid., II: 24) and Desc artes gives a relational definition of a body's plac e in terms of that body's position relative to external referenc e bodies (ibid., II: 10, 13). Whic h bodies are to be treated as referenc e bodies is an arbitrary matter. Desc artes's ordinary notion of motion is therefore a relative one: a given body may be said to be moving uniformly, nonuniformly, or not moving at all, depending on whic h other bodies are taken to be at rest (ibid., II: 13). In c ontrast, Desc artes's definition of motion in the stric t sense, while still relational, was supposed to sec ure a unique proper motion for eac h body (ibid., II: 31). True motion is defined as “the transfer of one piece of matter, or of one body, from the vicinity of the other bodies which are in immediate contact with it, and which are regarded as being at rest, to the vicinity of other bodies” (ibid., II; 25, emphasis in the original).3 Newton gave a single definition of motion, as c hange of plac e, but he also rec ogniz ed two kinds of motion, depending on whether the plac es in question were the parts of a relative spac e (defined in terms of distanc es relative to material reference bodies) or the parts of substantival space. Newton's relative motion, therefore, c orresponds c losely to Desc artes's motion in the ordinary sense. It is the motion we most direc tly observe and, Newton agreed, it is what we mean by “motion” in everyday c ontexts. But, he insisted, when it c omes to doing physic s, we need to abstrac t from suc h observations and c onsider a body's true motion, whic h, he argued, has to be defined in terms of an independently existing absolute space. Newton's arguments appeal to alleged “properties, c auses, and effec ts” of true motion. His aim is to show that various spec ies of relative motion, inc luding Cartesian proper motion (though this is not targeted by name), fail to have the requisite c harac teristic s. If one assumes, as Newton tac itly did, that true motion c an only be some kind of privileged relative motion or else is motion with respect to an independently existing entity, Newton's preferred option wins by default. That eac h body has a unique, true motion and that suc h motion has the purported properties, c auses and effec ts, are unargued assumptions. Newton's c laims about the properties of true motion arguably beg the question against the Cartesian. The arguments from c auses and effec ts are more interesting, both bec ause they c onnec t with physic s and bec ause their premises were ac c epted by the Cartesians. The partic ular effec t of true motion that Newton c ites is almost an immediate c orollary of the princ iple of inertia: bodies that are undergoing genuine c irc ular motion “endeavour to Page 2 of 48

Substantivalist and Relationalist Approaches to Spacetime rec ede from the axis” bec ause, at eac h instant, their natural, inertial motion is along the tangent at that point; they only follow their c urved path bec ause of the applic ation of a c entripetal forc e. Desc artes fully endorsed these c laims about c irc ular motion. Indeed, they formed a c entral c omponent of his model of planetary motions and the c osmos (e.g., Desc artes, 1644, III: 58– 62). His definition of true motion, however, fails to fit these phenomena, as Newton's buc ket experiment was designed to illustrate. Newton asks us to c onsider a water-filled buc ket suspended by a wound c ord. Onc e released, as the c ord unwinds, the buc ket starts to rotate. Initially, the water is at rest and its surfac e is flat. As fric tion gradually transfers the buc ket's motion to the water, the water's surfac e bec omes ever more c onc ave until its rate of rotation reac hes a maximum and it is c omoving with the buc ket. The c onc avity of the water's surfac e reveals its endeavour to rec ede from the axis of rotation. Ac c ording to Desc artes's definition of true motion, however, the water is at rest both before the buc ket is released and at the end of the experiment, when the water and buc ket are onc e again at relative rest, even though the water now manifests an effec t of true rotation. It might also seem that Desc artes should c ount the water as truly moving just after the buc ket has been released, bec ause it is transferred from the vic inity of bodies in immediate c ontac t with it (viz ., the sides of the buc ket), even though, at this stage, the water's surfac e is flat. The effec ts of true motion are not c orrelated with true motion as defined by Desc artes in the way they are supposed to be. Newton c onc ludes that the effec t revealing true rotation “does not depend on the c hange of position of the water with respec t to [immediately] surrounding bodies, and thus true c irc ular motion c annot be [defined in terms of] suc h c hanges of position” (Newton 1999, 413).4 The Principia's Sc holium on spac e, time, and motion is no longer our only sourc e for Newton's views on these topic s. In the 1960s, a pre-Principia manusc ript, known after its first line as De Gravitatione, was published for the first time (Newton, 2004). In De Grav, Newton quite explic itly targets Desc artes, and one argument is partic ularly telling.5 Newton points out that, ac c ording to Desc artes's definition of motion, no body has a determinate veloc ity, and there is no definite trajec tory that it follows. From moment to moment a body's motion is defined with respec t to those bodies in immediate c ontac t with it, whic h (for any body in motion) c hange from moment to moment. There is nothing in this pic ture that allows us to identify at some time the exac t plac es through whic h a body has traveled and so a fortiori nothing that can tell us whether these places constitute a straight line which the body has traversed at a uniform rate. Desc artes's ac c ount of true motion, therefore, c annot sec ure a fac t of the matter about whether a body is moving uniformly, as the princ iple of inertia requires. Newton c onc ludes: “So it is nec essary that the definition of plac es, and henc e of loc al motion, be referred to some motionless being suc h as … spac e in so far as it is seen to be truly distinc t from bodies” (Newton, 2004, 20– 21). Talk of a “being” that is “truly distinc t from bodies” indic ates that Newton's alternative to Cartesian motion involves a variety of substantivalism. The waters are muddied, however, by Newton's explic it denial in De Grav that spac e is a substanc e. Newton's position does qualify as a version of substantivalism as defined above: ac c ording to Newton, spac e is a genuine entity of a fundamental kind. Newton's denial that spac e is a substanc e c omes in a passage where he also denies both that it is merely a property (“ac c ident”) and that it is “nothing at all.” In fac t, of the three c ategories—substanc e, ac c ident, or nothing—Newton states that spac e is c losest in nature to substanc e. His two reasons for denying that spac e is a substanc e relate only to how this c ategory was understood in the then- dominant Sc holastic tradition. In partic ular, spac e was disqualified from being a substanc e bec ause, on Newton's view, it does not ac t and bec ause, in a c ertain rather tec hnic al sense, Newton did not regard it as a self-subsistent entity.6 In postulating spac e as an entity with its own manner of existenc e, Newton was direc tly following a number of the early modern atomists, suc h as Patriz i (1943, 227, 240–241), Gassendi (see, e.g., Grant, 1981, 209) and Charleton (1654, 66). These authors all treated spac e as more substantial than traditional Aristotelian substanc es. And there are striking struc tural parallels between some of the arguments in De Grav and those in Charleton's book, whic h we know from one of Newton's early notebooks that Newton had studied. The c onc lusion must be that in postulating substantival spac e Newton adopted, albeit for truly original reasons, a metaphysic al pac kage already very muc h on the table. 3. The Puzzle o f Galilean Invariance For Newton's first two laws of motion to make sense, there needs to be a fac t of the matter about whether a body's Page 3 of 48

Substantivalist and Relationalist Approaches to Spacetime motion is uniform and, if it is not, a quantitative measure of how it is c hanging. Newton rec ogniz ed that Desc artes's definitions of motion failed to sec ure this and signed up to a metaphysic s that underwrites the required quantities. There is a sense, though, in whic h Newton's absolute spac e underwrites too muc h. The problem arises bec ause of the symmetries of Newtonian mec hanic s, in partic ular its Galilean invariance. 3.1 Spacetime and Dynamical Symmetries The relevant notions of symmetry c an be introduc ed in terms of c oordinate transformations. Given our topic , a little c are is needed bec ause the substantivalist and the relationalist do not share a c onc eption of a c oordinate system. Roughly speaking, a spac etime c oordinate system is a map from spac etime into ℝ4 but, of c ourse, only the substantivalist thinks of spac etime as an genuine entity.7 The relationalist thinks of a c oordinate system as assigning quadruples of real numbers (t, x, y, z)  =  (t,  x ⃗) to material events rather than to spacetime points. And, whereas the substantivalist will view every quadruple of a c oordinate system as assigned to something, the relationalist will view some sets of c oordinate values (those that the substantivalist thinks of as assigned to unoc c upied regions) as simply not assigned to anything at all. Despite this differenc e, both substantivalists and relationalists will view c ertain c oordinate systems as kinematically privileged in the sense of being optimally adapted to the partic ular spatiotemporal quantities that they eac h rec ogniz e. In the c ontext of c lassic al mec hanic s, the natural relationalist alternative to Newton's substantivalism is Leibnizian relationalism.8 Ac c ording to this view, a possible history of the universe is given by a sequenc e of relative particle configurations: the primitive spatiotemporal fac ts about the universe are c omposed solely of fac ts about the instantaneous relative distanc es between partic les (assumed to obey the c onstraints of Euc lidean geometry) and fac ts about the time intervals between the suc c essive instantaneous material c onfigurations. The ways in whic h a c oordinate system c an be adapted to these quantities is straightforward. The time c oordinate, t, is c hosen so that, for any material events e and e′, the differenc e, t(e) − t(e′), c orresponds to the temporal interval between e and e′, and is positive or negative ac c ording to whether e oc c urs later or earlier than e′. Finally, spatial c oordinates are c hosen so that, for all partic les i,j and for all times, √−(x−i− −−− −x−j )−2−+−−(−y−i −−−y−j−)−2−+−−(−zi−−−−zj−)−2   =  rij , where rij is the instantaneous inter-particle distance between i and j. Assuming that these distanc es evolve smoothly over time, one also requires that eac h partic le's spatial c oordinates are smooth func tions of the time c oordinate. Coordinate systems that enc ode the Leibniz ian relationalist quantities in this way are sometimes known as rigid Euclidean coordinate systems (Friedman, 1983, 82). If a particular coordinate system (t, x ⃗) satisfies these constraints then so will any (t′,  x ⃗′) related to it by a member of the Leibniz group9 of transformations: (Leib) R(t) is an orthogonal matrix that implements a time-dependent rotation. The c omponents of a ⃗ (t) are smooth func tions of time that implement an arbitrary time-dependent spatial translation and d is an arbitrary c onstant that c hanges the c hoic e of temporal origin. The manner in whic h a c oordinate system c an be adapted to the Newtonian's spatiotemporal quantities is very similar. Mutatis mutandis, the substantivalist imposes the same c onstraints as the relationalist, although now the spatial and temporal distanc e relations to whic h the c oordinate values are to be adapted hold between the points of spac etime rather than (only) between material events.10 There are also the substantivalists' trademark “same plac e over time” fac ts to enc ode. Here we simply require that the spatial c oordinates of eac h point of spac e remain c onstant. The transformations that relate c oordinate systems adapted to the full set of Newtonian spatiotemporal quantities form a proper subgroup of the Leibniz group (it might appropriately be labeled the Newton group11), sinc e the only rotations and spatial translations that preserve the extra c onstraint are time-independent: (New) Identified in this way, the Leibniz and Newton groups are examples of spacetime symmetry groups: they are Page 4 of 48

Substantivalist and Relationalist Approaches to Spacetime groups of transformations that preserve spatiotemporal structure (as enc oded in c oordinate systems). A c onc eptually distinc t route to identifying spec ial c lasses of c oordinate transformations goes via the dynamic al laws of a partic ular theory. A dynamical symmetry group is a group that preserves the form of the equations that express the dynamic al laws. Sinc e the Leibniz ian relationalist holds that every rigid Euc lidean c oordinate system is optimally adapted to all the real spatiotemporal quantities, they might expec t suc h c oordinate systems to be dynamic ally equivalent. In other words, it is natural for someone who thinks that the Leibniz group is a spac etime symmetry group to expec t it to be a dynamic al symmetry group as well. This might indeed be the c ase if, for example, the dynamic al laws dealt direc tly with the relative distanc es between bodies. But Newton's laws do not. Instead they presuppose that individual bodies have determinate motions independently of their relations to other bodies. If Newton's laws take their c anonic al form in a given c oordinate system K (whic h, we may imagine, is adapted to Newtonian spac e and time), then they will not take the same form in a c oordinate system, K′, related to K by an arbitrary member of the Leibniz group. The equations that hold relative to K′ will involve additional terms c orresponding to sourc e-free “pseudo forc es” that, the Newtonian maintains, are artefac ts of K′'s ac c eleration with respect to substantival space. 3.2 The Kinematic Shift Argument The mismatc h between dynamic al symmetries and (what Leibniz ian relationalists regard as) spac etime symmetries is a problem for the relationalist. But a similar problem afflic ts the Newtonian substantivalist. While the Newton group is a dynamic al symmetry group of c lassic al mec hanic s, it is not the full symmetry group. The equations that express Newton's three laws of motion and partic ular Newtonian forc e laws (suc h as the law of universal gravitation) are invariant under a wider range of coordinate transformations, namely those constituting the Galilei group: (Gal) As in (New), the rotation matrix is time-independent, but now uniform time-dependent translations of the spatial c oordinates (“boosts”) are allowed. Let's stipulate that two c oordinate systems are adapted to the same frame of reference if and only if they are related by an element of the Newton group.12 Two c oordinate systems related by a nontrivial Galilean boost are then adapted to different frames of referenc e. However, if Newton's laws hold with respec t to either frame, they hold with respec t to both of them. In partic ular, both frames might be inertial frames in that, with respec t to them, forc e-free bodies move uniformly in straight lines. This gives rise to the following epistemologic al embarrassment for the Newtonian substantivalist. Imagine a possible world W′ just like the ac tual world exc ept that, at every moment, the absolute veloc ity of eac h material objec t in W′ differs from its ac tual value by a fixed amount (say, by two meters per sec ond in a direc tion due North). W′ is an example of a world that is kinematically shifted relative to the ac tual universe. Two kinematic ally shifted worlds are observationally indistinguishable bec ause, by c onstruc tion, the histories of relative distanc es between material objec ts in eac h world are exac tly the same. The worlds differ only over how the material universe as a whole is moving with respec t to spac e. Sinc e substantival spac e is not direc tly detec tible, this is not an observable differenc e. Further, the Galilean invarianc e of Newtonian mec hanic s means that any two kinematic ally shifted worlds either both satisfy Newton's laws, or neither does. The upshot is that the Newtonian substantivalist is c ommitted to the physic al reality of c ertain quantities, absolute velocities, that are in princ iple undetec table given the symmetries of the dynamic al laws. Given how W′ was spec ified in the previous paragraph, we know it is not the ac tual world. But c onsider a world, W\", just like the ac tual world in terms of the relative distanc es between bodies at eac h moment but in whic h, at 12 a.m. on 1 January 2000, the absolute veloc ity of the Eiffel Tower is exac tly 527ms−1 due North. For all we know, Wʺ is the ac tual world.13 To paraphrase Maudlin (1993, 192), there may be no a priori reason why all physic ally real properties should be experimentally disc overable but one should at least be uneasy about empiric ally inac c essible physic al fac ts; ceteris paribus, one should prefer a theory that does without them.14 The c onc lusion of this sec tion is that the Galilean invarianc e of Newtonian physic s poses something of an Page 5 of 48

Substantivalist and Relationalist Approaches to Spacetime interpretative dilemma. On the one hand, to make sense of the suc c essful dynamic al laws it seems that we have to ac knowledge more spac etime struc ture than the Leibniz ian relationalist is prepared to c ountenanc e. On the other hand, Newton's manner of securing a sufficiently rich structure introduces more than is strictly required and therefore underwrites empirically undetectable yet allegedly genuine quantities. Absent an alternative way to make sense of Newtonian physic s, one might learn to live with Newton's metaphysic s. However, the irritant of absolute veloc ities motivates a searc h for an alternative. In the next sec tion, I c onsider a substantivalist way out of the dilemma. Relationalist strategies are explored in sec tion 6. 4. Spacetime Substantivalism 4.1 Neo-Newtonian Spacetime The substantivalist can do away with unwanted absolute velocities by adopting the essentially four-dimensional perspec tive afforded by spac etime. In this framework, there is an elegant way of c harac teriz ing a spatiotemporal struc ture that might seem to be neither too weak nor too strong for Newtonian physic s.15 The dynamic al quantities of c lassic al mec hanic s presuppose the simultaneity struc ture, instantaneous Euc lidean geometry and temporal metric c ommon to the Leibniz ian relationalist and the Newtonian substantivalist. They additionally require some extra transtemporal struc ture. Geometric ally, what is needed is a standard of straightness for paths in spac etime (provided in differential geometry by an affine connection). The possible trajectories of ideal force-free bodies c orrespond to those straight lines in spac etime that do not lie within surfac es of simultaneity. These straight lines fall into families of nonintersec ting lines that fill spac etime. Eac h family of lines are the trajec tories of the points of the “spac e” of some inertial frame. The resulting spac etime is known as Galilean or neo-Newtonian spacetime (see, e.g., Sklar 1974, 202–206; Earman, 1989, 33). Now, it is one thing to give a nonredundant c harac teriz ation of the spac etime struc ture that Newtonian mec hanic s assumes. It is another to provide a satisfac tory ac c ount of its metaphysic al foundations (that is, of what, in reality, underwrites this struc ture). The obvious option is to take the unitary notion of spac etime (rather than spac e and time separately) ontologic ally seriously. One regards spac etime as something that exists in its own right and whic h literally has the geometric struc ture that the affine c onnec tion, among other things, enc odes. In terms of suc h ontology, one c an provide a metaphysic al ac c ount of the distinc tion between absolute and relative motion in a way that respec ts the physic al equivalenc e of inertial frames. It will fac ilitate c omparison with relativistic theories to introduc e a formulation of Newton mec hanic s that makes explic it referenc e to this geometric al struc ture. In abstrac t terms, physic al theories often have the following general form. A spac e of kinematically possible models (KPMs) is first spec ified. The job of the theory's equations (whic h relate the quantities in terms of which the KPMs are characterized) is then to single out the subspace S of K c ontaining the dynamically possible models (DPMs).16 The KPMs c an be thought of as representing the range of metaphysic al possibilities c onsistent with the theory's basic ontologic al assumptions. The DPMs represent a narrower set of physic al possibilities. In a c oordinate-dependent formulation of Newtonian theory like that so far c onsidered, the KPMs might be sets of inextendible smooth curves in ℝ4 which are nowhere tangent to surfaces of constant t (where (t, x ⃗)  ∈ R4 ). The models assign to the c urves various parameters (m,…). Under the intended interpretation, the c urves represent possible trajec tories of material partic les, desc ribed with respec t to a c anonic al c oordinate system, and the parameters represent various dynamic ally relevant partic le properties, suc h as mass. The spac e of DPMs c onsists of those sets of c urves that satisfy the standard form of Newton's equations. In the local spacetime formulation of the theory, one takes the KPMs to be n-tuples of the form M, tab,hab, ∇a, 1, 2, …).17 M is a four-dimensional differentiable manifold and tab, hab and ∇a are geometric-object fields on M that enc ode, respec tively, the temporal struc ture (both the simultaneity surfac es and the temporal metric ), the Euc lidean geometry of instantaneous spac e and the inertial struc ture (that is, whic h paths in spac etime are straight). Together they represent neo-Newtonian spacetime. The “matter fields” i, which represent the material c ontent of the model, c an be c urves (maps from the real line into M, representing partic le trajec tories) or fields (maps from M into some spac e of possible values, either enc oding forc e potentials or c ontinuous matter distributions). Page 6 of 48

Substantivalist and Relationalist Approaches to Spacetime The theory's DPMs are pic ked out by a set of equations that relate these various objec ts. Some of these will involve both the spac etime struc ture and the matter fields. For example, Newton's sec ond law bec omes: (N2) For those not familiar with the tensor notation, the essential points to note about this equation are the following. Fa and ξa are spac etime four-vectors. Fa stands for the four-forc e on the partic le. As in more traditional formulations of Newtonian theories, it will be spec ified by one or more additional equations. ξa is the four-veloc ity of the partic le; it is the tangent vec tor to the partic le's spac etime trajec tory if that trajec tory has been parameteriz ed by absolute time.18 Note, in partic ular, the explic it appearanc e of ∇a in the equation. ξn ∇nξa is the four-ac c eleration of the partic le; it c harac teriz es how the partic le's trajec tory deviates from the adjac ent tangential straight line in spac etime (that is, from the relevant inertial trajec tory). 4.2 Symmetries Revisited In sec tion 3 the Galilei group and the Leibniz group were introduc ed as sets of c oordinate transformations, and the dynamic al symmetries of Newtonian mec hanic s were c harac teriz ed in terms of the form invarianc e of its equations. We now see that the implementation of this approac h is not c ompletely straightforward: various formulations of Newtonian mec hanic s involve different sets of equations, and these c an have different invarianc e properties. In partic ular, when Newton's laws are rec ast so as to make explic it referenc e to the geometric al struc ture of neo- Newtonian spac etime, the resulting equations are either generally covariant (they hold with respec t to a set c oordinate systems related by smooth but otherwise arbitrary c oordinate transformations) or they are coordinate- independent (they direc tly equate c ertain geometric al objec ts, rather than the values of the objec ts' c omponents in some c oordinate system). There is an alternative way to c harac teriz e the symmetries of a spac etime theory. Rather than foc using on the theory's equations, one c onsiders their solutions. Suppose, now, that a group G of maps from spac etime to itself has a natural ac tion on the spac e of KPMs. G is a symmetry of the theory if and only if it fixes the solution spac e S .19 Note the need to relativize this characterization of symmetry to groups of maps from spacetime to itself. If we were allowed to consider any action of any group on the space of KPMs, the requirement that S be fixed would be too easily satisfied. In particular, for any two points s1,s2 in S , we could find a group action on the space of KPMs such that S is fixed and s1 is mapped to s2. That is, every point in S would be mapped to every other by some symmetry transformation or other.20 If the KPMs and DPMs of Newtonian mec hanic s are defined ac c ording to the first formulation above (as c ertain c lasses of c urves in ℝ4 ), the symmetry group of the theory turns out, as one might have expec ted, to be the Galilei group. This is not so if the loc al spac etime formulation of the theory is adopted. The spac e of KPMs then c arries an action of the diffeomorphism group Diff (M): for any ⟨M , tab,  hab,  ∇a,  Φi⟩  ∈ K and for any d ∈ Diff(M), ⟨M ,  d *  tab,  d *  hab,  d * ∇a,  d * Φi⟩  ∈  K .21 It follows from the tensorial nature of the equations that pick out the solution subspace that, if M,tab,haa,∇a, i satisfies the equations then so does M,d* tab,d* h,d* ∇a,d* i) (Earman and Norton, 1987, 520). In other words, the full group Diff (M) fixes S and therefore counts as a symmetry group of this formulation of the theory. At this point it might look as if the formulation-dependenc e that afflic ts a definition of dynamic al symmetries in terms of the invarianc e of equations has simply been reproduc ed at the level of models. We c an, however, reintroduc e the distinc tion between spac etime and dynamic al symmetries in model-theoretic terms. Our c harac teriz ation of the models of a loc al spac etime formulation of Newtonian mec hanic s involved distinguishing those geometric -objec t fields on M that represent spac etime struc ture from those that represent the material c ontent of spac etime. Let's write M,A1,…,An,P1,…, Pm for a generic spacetime model, where the Ais stand for the fields that represent spac etime struc ture and the Pjs stand for the fields that represent the matter c ontent. Rec all that the c oordinate- dependent definition of spacetime symmetries given in section 3 required that the encoding of spacetime structure by the c oordinate system was preserved. Analogously, we c an identify a theory's spacetime symmetry group as the set of elements of Diff (M) that are automorphisms of the spac etime struc ture; that is, we require that d* Ai = Ai for eac h field Ai .2 2 In the c ase of our Newtonian theory, the proper subgroup of Diff (M) that leaves eac h of tab, hab and ∇a invariant is the Galilei group. Page 7 of 48

Substantivalist and Relationalist Approaches to Spacetime An alternative method of singling out a subgroup of Diff (M) foc uses on the matter fields rather than the spac etime struc ture fields. Typic ally the matter c ontent of a solution will have no nontrivial automorphisms and, if it does, there will be other solutions whose matter c ontent does not share these symmetries. We c an, however, ask whether a diffeomorphism ac ting solely on the matter fields maps solutions to solutions. In other words, we pic k out a subgroup of Diff (M) via the requirement that (for a given c hoic e of Ai) for all ⟨M , Ai,  Pj ⟩  ∈  S ,  ⟨M ,  Ai,  d *  Pj ⟩ should also be in S .23 Note that, in this definition, d acts only on the matter fields and not on the spac etime struc ture fields. The subgroup of diffeomorphisms with this property is sometimes identified as a theory's dynamical symmetry group (Earman, 1989, 45– 46). Again, in the spac etime formulation of Newtonian theory set in neo-Newtonian spac etime, if this group turns out to be the Galilei group one has a perfec t matc h between the spac etime symmetry group and the dynamic al symmetry group.2 4 The model-theoretic perspec tive supports the idea that the problems fac ed by both the Leibniz ian relationalist and the Newtonian substantivalist involve mismatches between these two symmetry groups. Whenever the spacetime symmetry group is a proper subset of the dynamic al symmetry group, the theory will admit nonisomorphic models whose material submodels are nevertheless isomorphic . (Suc h models will be related by dynamic al symmetries that are not also spac etime symmetries.) This will give rise to supposedly meaningful yet physic ally undisc overable quantities: models will differ over some quantities in virtue of different relations of matter to spac etime struc ture and yet (assuming the material c ontent of the models enc ompasses all that is observable) suc h differenc es will be undetec table. This is exac tly the situation of the Newtonian substantivalist, who postulates a ric her spac etime struc ture than that of neo-Newtonian spac etime. Models related by Galilean boosts of their material c ontent differ over undetec table absolute veloc ities. The mismatc h fac ed by the Leibniz ian relationalist is of the opposite kind: the Galilei group is a proper subset of the Leibniz group. Stric tly speaking, this is not a c ase where the dynamic al symmetry group of some theory is smaller than the spac etime symmetry group. For that we would need a spac etime formulation of a theory that (i) was set in so-c alled Leibniz ian spac etime and (ii) had the Galilei group as its dynamic al symmetry group (e.g., in virtue of an isomorphism between the set of its matter submodels and those of standard Newtonian theory). But the way in whic h the equations of the spac etime formulation of standard Newtonian theory single out its matter submodels uses the full struc ture of neo-Newtonian spac etime. The mismatc h between the two groups is prec isely what stands in the way of c onstruc ting suc h a relational theory. Before c onsidering the spac etime substantivalist interpretation of relativistic physic s, a brief c omment on the relation between the two formulations of Newtonian mec hanic s that I have been disc ussing: I first c harac teriz ed the privileged c oordinate systems of the c oordinate-dependent form of Newtonian physic s as those adapted to the spatiotemporal quantities rec ogniz ed by the Newtonian. From this perspec tive, the spac etime formulation of the theory simply makes explic it struc ture that, while implic it, is no less present in the c oordinate-dependent, Galilean- c ovariant formulation of the theory. Suppose one starts with (the c oordinate expressions of) the equations of the spac etime formulation of a Newtonian theory. These equations will be generally c ovariant. However, one c an use the symmetries of the theory's spac etime struc ture to pic k out a spec ial c lass of c oordinate systems in whic h the values of the c omponents of the fields representing the spac e-time struc ture take on c onstant or vanishing values. In these c oordinate systems the otherwise generally c ovariant equations apparently simplify. In other words, the Galilean c ovariant equations just are the generally c ovariant equations, written with respec t to c oordinate systems that “hide” the objec ts that represent spac etime struc ture.2 5 4.3 Relativistic Spacetimes In the previous sec tion, I maintained that the geometric al struc ture of neo-Newtonian spac etime featured, implic itly or otherwise, in the various formulations of c lassic al mec hanic s. A similar c laim holds true for spec ial relativity. This is most obvious in generally c ovariant or c oordinate-free formulations of the equations of any spec ially relativistic theory, where the Minkowski metric struc ture, enc oded by the tensor field ηab, figures explic itly. But it is equally true of the “standard” formulations of the equations that hold true only relative to privileged inertial c oordinate systems related by Lorentz transformations. One c an think of these c oordinate systems as the spac etime analogues of Cartesian c oordinates on Euc lidean spac e: the c oordinate intervals enc ode the spac etime distanc es via the condition (tp − tq)2 − ∣x ⃗p − x ⃗q∣2   =  Δs2 . Minkowski geometry is thus implicit in the standard formulation of the laws. Page 8 of 48

Substantivalist and Relationalist Approaches to Spacetime Bec ause of the manner in whic h spac etime geometry features in the formulation of the laws, substantivalists hold that it explains c ertain features of the phenomena c overed by those laws.2 6 Consider, for example, the “twin paradox” sc enario. Of two initially sync hroniz ed c loc ks, one remains on Earth while the other performs a round trip at near the speed of light. On its return the traveling c loc k has tic ked away less time than the stay-at-home c loc k. The geometric al fac ts behind this phenomenon are straightforward: the inertial trajec tory of the stay-at-home c loc k is simply a longer timelike path than the trajec tory of the traveling c loc k. Note that the substantivalist does not simply assert that the number of a c loc k's tic ks is proportional to the spac etime distanc e along its trajec tory. Cloc ks are c omplic ated systems the parts of whic h obey various (relativistic ) laws. One should look to these laws for a c omplete understanding of why the “tic ks” of suc h a system will indeed c orrespond to equal temporal intervals of the system's trajec tory. But sinc e, for the substantivalist, these laws make (implic it or explic it) referenc e to independently real geometric struc ture, an explanation that appeals to the laws will, in part, be an explanation in terms of suc h geometric struc ture, postulated as a fundamental feature of reality. Soon after formulating his spec ial theory of relativity, Einstein began a dec adelong quest for a theory of gravitation that was c ompatible with the new notions of spac e and time. The triumphant c ulmination of this effort was his public ation of the field equations of his general theory of relativity (GR) in 1915.2 7 These equations, known as the Einstein Field Equations (EFEs), relate c ertain aspec ts of the curvature of spac etime, as enc oded in the Einstein tensor, Gab, to the matter c ontent of spac etime, as enc oded in the energy momentum tensor, Tab: 2 8 (EFE) One goal above all others guided Einstein's searc h: the generaliz ation of the “spec ial” princ iple of relativity to a princ iple that upheld the equivalenc e of frames of referenc e in arbitrary states of relative motion. In this, Einstein was motivated by the belief that the role of primitive inertial struc ture in explaining phenomena in both Newtonian physic s and in spec ial relativity was “epistemologic ally suspec t”: real effec ts, he believed, should be trac eable to observable, material causes. In the early stages of his searc h, Einstein had a c ruc ial insight, whic h he thought would play a key role in the implementation of a generaliz ed relativity princ iple. Ac c ording to his Principle of Equivalence, an inertial frame in whic h there is a uniform gravitational field is physic ally equivalent to a uniformly ac c elerating frame in whic h there is no gravitational field. It is not hard to see why one might take this princ iple to extend the princ iple of relativity from uniform motion to uniform ac c eleration.2 9 When Einstein first published the field equations of GR, he believed that their general covariance ensured that they implemented a general princ iple of relativity. Sinc e smooth but otherwise arbitrary c oordinate transformations inc lude transformations between c oordinate systems adapted to frames in arbitrary relative motion, it might seem that there c an be no privileged frames of referenc e in a generally c ovariant theory. Almost immediately, Kretsc hmann (1917) pointed out that this c annot be c orrec t, arguing that it should be possible to rec ast any physic al theory in generally c ovariant form. Compared to its predec essors, GR is without doubt a very spec ial theory. But one will do more justic e to its c onc eptual novelty, not less, by seeking as muc h c ommon ground with previous theories as possible. We have already met generally c ovariant formulations of Newtonian and spec ially relativistic physic s. As these examples attest, Kretsc hmann's instinc ts were sound and it is now well understood that the general c ovarianc e of GR does not implement a general princ iple of relativity. In fac t, GR arguably inc ludes privileged frames of referenc e in muc h the same way as pre-relativistic theories.3 0 It is illuminating to c onsider a c onc rete example. Compare the loc al spac etime formulation of spec ially relativistic electromagnetism to its generally relativistic version. Models of the former are of the form M,ηab, Fab, Ja), where ηab enc odes the Minkowski spac etime distanc es between the points of M, the c ovariant tensor field Fab represents the elec tromagnetic field and the vec tor field Ja represents the c harge c urrent density. The dynamic ally possible models c an be pic ked out via the c oordinate-free form of Maxwell's equations: (Maxwell) The only differenc e between this theory and the generally relativistic theory is that, in the latter, the Minkowski metric ηab is replac ed by a variably c urved Lorentz ian metric field, gab. Maxwell's equations remain as c onstraints Page 9 of 48

Substantivalist and Relationalist Approaches to Spacetime on the DPMs of the GR version, and the same relativistic version of Newton's sec ond law (whic h is formally identic al to the nonrelativistic version: equation (N2) of sec tion 4.1) holds in both theories.3 1 But, whereas the spec ially relativistic metric was stipulated to be flat, to be physic ally possible ac c ording to GR, c ombinations of gab, Fab and Ja must now obey the EFEs. In the c ase of SR, the standard, Lorentz -invariant form of Maxwell's equations is rec overed by c hoosing inertial c oordinate systems adapted to the spac etime distanc es.3 2 The only reason that the same c annot be said of GR is that the pattern of spac etime distanc es c atalogued by gab does not allow for global c oordinate systems that enc ode them. One c an, however, always c hose a c oordinate system that is optimally adapted the spac etime distanc es in the infinitesimal neighborhood of any point p ∈ M. In any c oordinate system in whic h, at p, gμν = diag(1, −1, −1, −1) and gμνγ = 0, the laws governing matter fields will take their standard Lorentz -invariant form at p. It is in terms of this strong equivalence principle that the phenomena that figure in Einstein's original principle are nowadays understood.3 3 From a modern perspec tive, apparent “gravitational fields” have prec isely the same status as the potentials that give rise to the c entrifugal and c oriolis forc es in Newtonian physic s: they c orrespond to pseudo forc es that are artifac ts of the failure of the relevant c oordinate systems to be fully adapted to the true geometry of spac etime. The “forc e” that holds us on the surfac e of the Earth and the “forc e” pinning the astronaut to the floor of the ac c elerating roc ket ship are literally of one and the same kind. In GR, gravitational phenomena are not understood as resulting from the ac tion of forc es.3 4 5. Reaso ns to be a Relatio nalist? We have seen that substantivalism is rec ommended by a rather straightforward realist interpretation of our best physic s. This physic s presupposes geometric al struc ture that it is natural to interpret as primitive and as physic ally instantiated in an entity ontologic ally independent of matter. Although I have only c onsidered nonquantum physic s explic itly, the c laim is equally true of both nonrelativistic and relativistic quantum theories (Weinstein, 2001). One might therefore wonder: why even try to be a relationalist? For some, the Hole Argument, disc ussed below in sec tion 7, is the major reason to seek an alternative to substantivalism. Here I wish to review some other antisubstantivalist themes that have motivated relationalists. The conclusion will be that the only strong c onsideration in favor of relationalism is Oc kham's raz or: if a plausible relational interpretation of empiric ally adequate physic s c an be devised, then the standard reasons for postulating the substantivalist's additional ontology are undermined. 5.1 A Failure of Rationality? In Sec tion 6.2 I review Julian Barbour's approac h to dynamic s. Ac c ording to Barbour, there is something “irrational” about standard Newtonian mec hanic s. Barbour sets up the problem in terms of the data required for the Newtonian initial value problem: given the equations of a Newtonian theory, what quantities must be spec ified at an instant in order to fix a solution? Following Poinc aré, Barbour emphasiz es that the natural relational data—instantaneous relative distanc es and their first derivatives— are almost but not quite enough. Inaddition, three further parameters, whic h spec ify the magnitude and direc tion of the angular momentum of the entire system, are needed.3 5 Barbour c laims that there is something odd about this fac t (see, e.g., Barbour, 2011, § 2.2). Fixing the Euc lidean relative distanc es between N partic les and their rates of c hange requires the spec ific ation of 6N − 12 numbers; 6N numbers are required to spec ify their positions and veloc ities in absolute spac e. The Newtonian initial value problem, however, requires 6N − 9 numbers. Six of the 6N numbers that spec ify the partic les' absolute positions and veloc ities c an be thought of as spec ifying the orientation of the system as a whole and the position of its c enter of mass. States that differ solely in terms of these quantities only differ nonqualitatively, in terms of which particular points of space are related to the material system in (qualitative) ways c ommon to both situations; the struc tural pattern of relations between spac e and matter is shared by both states. Even if suc h differenc es are to be regarded as real differenc es (something to be questioned in sec tion 7), Newtonian physic s should not be expec ted to take them into ac c ount (pace Barbour, 1999, 83). It does not, after all, single out partic ular points of spac e by name. The remaining three parameters c an be thought of as spec ifying the absolute veloc ity of the c enter of mass. As we have seen, while the Newtonian substantivalist regards this as a Page 10 of 48

Substantivalist and Relationalist Approaches to Spacetime genuinely qualitative matter, the spac etime substantivalist does away with these quantities by embrac ing neo- Newtonian spac etime. Arguably, 6N − 9 numbers is prec isely what the spac etime substantivalist should expec t to be needed as Newtonian initial data. What, then, is “irrational” about Newtonian mec hanic s? In Barbour's view it fails to be maximally predic tive: relative to the Leibnizian data, there is an apparent breakdown of determinism, whic h is only restored by spec ifying the global angular momentum. At one level, this simply amounts to a prejudic e in favor of the relational quantities over the Newtonian ones. For Barbour, this preferenc e is based on the fac t that it is the relational quantities that are “direc tly observable” (see, e.g., Barbour, 2010, 1280) but (i) direc t observability is an extreme c riterion for determining ontologic al c ommitment and (ii) instantaneous relative distanc es are not direc tly observable.3 6 One c an still agree with Barbour's less ambitious point. Without ac c epting that there is something inherently objec tionable about standard Newtonian theory, one might nevertheless prefer a theory that does as well (is at least as explanatory etc .) but with fewer resourc es. Barbour's observations about initial data highlight that, if an adequate relational theory of the envisaged type c an be found, it will be more predic tive, bec ause its initial data form a proper subset of those of the Newtonian theory. 5.2 The Spacetime Explanation of Inertia In sec tion 6.3 I c onsider the dynamical approach to spec ial relativity, defended in Brown (2005) and Brown and Pooley (2006). I noted above that substantivalists view the postulated spac etime geometry as explanatory. Brown is suspic ious of this doc trine. In assessing it, the putative explanatory roles of affine struc ture and of metric struc ture should be treated separately. Here I only c onsider the former; the latter is disc ussed in sec tion 6.3. The idea that affine struc ture plays a quasic ausal role in explaining the motions of bodies figures signific antly in Einstein's c ritic ism of Newtonian mec hanic s and SR and in his subsequent understanding of GR. Consider Einstein's example from his early review paper on GR, of two fluid bodies separated by a great distanc e and in relative rotation about the line joining their c entres. The two are of the same siz e, shape, and nature exc ept that one body is spheric al whereas as the other is oblate. In Newtonian mec hanic s the explanation of this differenc e is that the oblate body, but not the spheric al body, is rotating with respec t to the inertial frames: the absolute rotation of the oblate body c auses its oblateness. Einstein labels the Newtonian spac etime struc ture with respec t to whic h suc h rotation is defined as a “merely fac titious c ause” of the differenc e; he held that a genuine explanation should instead c ite another “observable fac t of experienc e” (Einstein, 1916, 112– 113). He initially maintained that this requirement was met in GR because he believed that the theory satisfied what he later called Mach's Principle (Einstein, 1918, 241– 242): if the metric field gab were fully determined by the distribution of matter throughout the universe, then the difference between the inertial behavior of the two bodies would be traceable to differences in their relations to distant (observable) masses. As Einstein soon rec ogniz ed, GR does not satisfy Mac h's Princ iple so defined.3 7 Inertial struc ture, as enc oded in gab, is influenced but not determined by the matter c ontent of spac etime.3 8 Einstein c eased to regard the field desc ribing inertial struc ture as having a sec ondary status relative to ponderable matter. By 1921, the objec tion to Newtonian absolute spac e was no longer that it was invisible. Instead the fac t that it acted without being acted upon was held up as problematic ; a “defec t” not shared by the spac etimes of GR (Einstein, 1922, 61–62).3 9 At around the same time, Weyl advoc ated a similar c onc eption of the role of inertial struc ture (he c alled it the “guiding field”), whic h he regarded as “physic ally real” in both pre-relativistic physic s and in GR (Weyl, 1922, § 27). What does Brown objec t to in this pic ture? Consider the “c onspirac y of inertia”: the relatively simple c ase of the forc e-free motions of a c ollec tion of Newtonian partic les (Brown, 2005, 15, 141). Despite the fac t that, ex hypothesi, the partic les are not influenc ing one another, they move in a highly c oordinated way: there are spac etime c oordinate systems with respec t to whic h all the trajec tories are straight lines.4 0 As we have seen, the c oordinate-free geometric al desc ription of this state of affairs regards the trajec tories as c oinc iding with the straight lines of an affine c onnec tion on spac etime, the flatness of whic h allows for the global inertial c oordinate systems in terms of whic h the phenomenon was stated. For the substantivalist, inertial struc ture is an element of reality that exists independently of the partic les and their motions. On this view, aspec ts of the geometry play a role in explaining the phenomenon bec ause, as stated in the substantivalist version of Newton's first law, the partic les' trajec tories are c onstrained as a matter of physic al nec essity to be aligned with features of this realistic ally c onstrued geometry. For Brown, this is merely verbal pseudo-explanation. His preferred point of view reverses the Page 11 of 48

Substantivalist and Relationalist Approaches to Spacetime arrow of explanation: the geometry is just a codification of the phenomenon, whic h (in pre-relativistic physic s) must be taken as primitive. If the only role of inertial struc ture was to explain pure inertial motion, Brown's c omplaint against the substantivalist would have some intuitive forc e; a flat affine c onnec tion c ould be thought of as a rather direc t c odific ation of the regularities manifest in the phenomena via the “c oordinative definition” of spac etime geodesic s as the trajec tories of forc e-free bodies.4 1 Explaining inertial motion, though, is not the real purpose of inertial struc ture. Inertial struc ture figures c entrally in the explanation of noninertial motion. In c ontrast to the forc e-free c ase, the sequenc e of relative distanc es between interac ting partic les manifest over time in a Newtonian universe displays no obvious regularity. It is a rather remarkable fac t that, by postulating a highly symmetric geometric al struc ture in terms of whic h the motions of individual partic les are to be understood, one c an provide an elegant explanation, in terms of simple forc e laws, of the c omplic ated and irregular history of relational quantities. Anyone who is not amaz ed by this c onspirac y has not understood it. Sinc e the postulated deep struc ture is not manifest in the surfac e phenomena it seems genuinely explanatory and not a mere c odific ation. It is from an applic ation of standard inferenc e-to-the-best-explanation reasoning to this type of sc enario that substantivalism gets its real support.4 2 Brown suggests that, around 1927, Einstein c eased to assign a quasi-c asual role to spac etime in determining the inertial trajec tories of bodies (Brown 2005, 161). The alleged reason is that, at this time, Einstein c ame to rec ogniz e that the princ iple of inertia does not need to be postulated as a basic law in GR; it is instead a theorem.4 3 In Brown's view this fac t undermines the idea that spac etime struc ture “in and of itself” ac ts “direc tly” on forc e-free bodies bec ause it shows that, in GR, when suc h bodies undergo geodesic motion,“suc h motion is ultimately due to the way the Einstein field gμν c ouples to matter, as determined by the field equations” (Brown, 2005, 162– 163). Brown's pic ture is, then, that the relationship of the motions of forc e-free bodies to inertial struc ture in GR is radic ally different to this relationship in Newtonian physic s and SR. In the latter theories, inertial struc ture is a mere c odific ation of basic , mysterious inertial behavior. In the former, it rec eives a dynamical explanation via the c oupling of matter fields to the metric , as desc ribed by the EFEs. The substantivalist should not be espec ially troubled by Brown's c laim, for it c onc edes that, in our best (c lassic al) theory of spac etime, the metric struc ture of spac etime is a primitive element of reality that plays a role in determining the inertial behavior of bodies. Even so, there are reasons to doubt that the c ontrast between GR and pre-relativistic theories c an really bear the weight Brown demands of it. The derivation of geodesic motion from the EFEs basic ally involves two steps. First, one notes that the EFEs imply the vanishing of the c ovariant divergenc e of the stress energy tensor: ∇a Tab = 0. Sec ond, one makes various assumptions about the nature of the stress- energy tensor to be assoc iated with a forc e-free partic le that, together with the vanishing of the divergenc e of stress-energy, c an be shown to entail that the partic le's trajec tory is a geodesic . Now, the sec ond step of this derivation is as applic able in SR as in GR.4 4 What differenc e there is between the theories must therefore c onc ern the status of the c onservation princ iple, ∇a Tab = 0. This equation, of c ourse, also holds in SR. Further, while in SR it c annot be derived from the gravitational field equations (there are none), it is a c onsequenc e of the matter field equations (as it is in GR also).45 In sum, geodesic motion is arguably as muc h a theorem in SR as it is in GR. An alternative perspec tive to Brown's is that the “dynamic al c oupling” of matter fields to inertial struc ture is in essential respec ts the same in GR and pre- relativistic theories. It is true that the geodesic theorem also demonstrates that it has limited validity in GR but not in SR (rec all footnote 41). But the reason why, for example, rotating bodies do not deviate from geodesic s in SR is not that the relationship between realistic ally c onstrued inertial struc ture and matter is radic ally different in this theory to that in GR. The reason is simply that, in SR, there is no c urvature to whic h rotating bodies might c ouple.4 6 We have yet to meet a dec isive reason to look for an alternative to substantivalism. Huggett, whose “regularity ac c ount” of relational spac etime I c onsider in sec tion 6.3, states that his reasons for advoc ating relationalism are the usual ones: “worries about ‘Leibniz shifts’ and c onsiderations of ontologic al parsimony that militate against the introduc tion of biz arre non-material substanc es” (Huggett, 2006, 41). The kinematic shift has already been dealt with; other Leibniz shifts are disc ussed in sec tion 7. It is not c lear why Huggett regards spac etime as “biz arre.” In a pre-relativistic c ontext, one feature of spac etime that might seem odd is its failure, emphasiz ed by Einstein, to obey the ac tion– reac tion princ iple. As we have seen, however, if this is a failing, it is not one shared by the spac etime of GR. That leaves c onsiderations of ontologic al parsimony: other things being equal, a relationalist interpretation of physic s might seem preferable to substantivalism bec ause it makes do with fewer metaphysic al c ommitments. The Page 12 of 48

Substantivalist and Relationalist Approaches to Spacetime question is: Are other things equal? It is time to examine some c onc rete relationalist proposals. 6. Three Varieties o f Relatio nalism Call the objec ts to whose existenc e a theory is c ommitted the ontology of the theory. Call the range of (primitive) distinc tions that a theory is able to express via its (primitive) predic ates and terms—roughly, the set of (primitive) properties and relations to whic h the theory is c ommitted—the theory's (primitive) ideology.4 7 In these terms, the problem fac ed by the Leibniz ian relationalist is that c lassic al mec hanic s employs an ideology that appears to presuppose a substantivalist ontology. Inertial struc ture is naturally understood in terms of relations that hold of spac etime points; it c annot be understood (straightforwardly) in terms of properties and relations that are instantiated only by material objec ts. On the other hand, bona fide relationalist ideology appears to be too impoverished a basis for an empiric ally suc c essful alternative to Newtonian theory. This means that there are two obvious strategies open to relationalists. On the first, they c an attempt to expand their ideology so that it underwrites the same physic al distinc tions as substantivalist inertial struc ture but in a way c ompatible with a relationalist ontology. To c omplete this program, the relationalist then needs to rec onstrue standard Newtonian theory in terms of these new relationalist quantities.4 8 Variants of this strategy are the topic of sec tion 6.1. On the sec ond strategy, the relationalist seeks an alternative theory to Newtonian mec hanic s that employs only traditional relationalist quantities but is, although empiric ally distinc t from Newtonian theory, nonetheless empiric ally adequate. Sinc e the inertial frames are empiric ally determinable (in our neighborhood of this universe), suc h a theory still needs to ac c ount for them, at least as a feature of solutions that c ould serve as models of the ac tual world. Unlike theories that result from the first strategy, however, it will not c onstrue them as enc oding primitive spatiotemporal properties and relations. The most promising version of this approac h is reviewed in sec tion 6.2. These c ourses of ac tion c orrespond c losely to the first two of three options identified by Nic k Huggett (1999). He sees Newton's globes thought experiment as illustrating that no theory has the following three c harac teristic s. (i) Its spatiotemporal ideology is restric ted to Leibniz ian relations; (ii) its dynamic ally allowed histories of suc h relations are exac tly those predic ted by Newtonian theory and; (iii) inertial effec ts supervene on the spec ified spatiotemporal relations between bodies. Strategies (1) and (2) c orrespond to relinquishing (i) and (ii) respec tively. But, Huggett observes, the relationalist might try to retain (i) and (ii) by dropping requirement (iii) (Huggett 1999, 22– 23). This amounts to a third, non-obvious strategy: do not c hange the theory and do not add to the ideology and yet somehow be a relationalist. This type of “have-it-all” relationalism is the topic of sec tion 6.3. 6.1 Enriched Relationalism 6.1.1 Classical Mechanics Part of the substantivalist's response to the kinematic shift argument involved replac ing persisting spac e with spac etime. If relationalists likewise adopt a four-dimensional perspec tive, a number of options ric her than Leibniz ian relationalism bec ome available. The most straightforward is Newtonian relationalism (Maudlin, 1993, 187). Whereas the Leibniz ian relationalist posits spatial relations that hold only between simultaneous material events, the Newtonian relationalist simply posits that all material point events stand in spatial distanc e relations.4 9 If we impose the natural c onstraints, the embedding of a Newtonian relational history into neo-Newtonian spac etime is fixed up to Galilean transformations.50 Whether this by itself entitles the relationalist to exploit the full resourc es of Newtonian dynamic s (as Maudlin thinks; 1993, 192– 193) need not be resolved, for the Newtonian relationalist c an also interpret the dynamic al laws direc tly in terms of relationalist quantities. For example, the absolute veloc ity of partic le i at time t is just the limit, as δt goes to z ero, of the direc ted distanc e between the instantaneous stage of i at t and its instantaneous stage at t + δt divided by δt. The Newtonian relationalist therefore has available a relational understanding of the very quantities that feature in the standard form of Newtonian laws expressed with respec t to an inertial frame. Unfortunately, Newtonian relationalism, like the Newtonian substantivalism from whic h its ideology is plundered, is vulnerable to the kinematic shift argument.51 Sinc e absolute veloc ities are unobservable, the Newtonian relationalist is c ommitted to physic ally real distinc tions that are in princ iple empiric ally inac c essible. The Newtonian Page 13 of 48

Substantivalist and Relationalist Approaches to Spacetime relationalist must ac c ept that, while only one inertial frame disc loses the true Newtonian relational distanc es, that frame is forever beyond our grasp. The spac etime substantivalist solves the kinematic shift problem by replac ing Newtonian with neo-Newtonian spac etime. There is an obvious relational analogue of this move: sinc e neo-Newtonian spac etime's inertial struc ture is equivalent to a relation of c ollinearity between triples of spac etime points, the relationalist c an add to their ideology a three-place relation of collinearity between material events. The neo-Newtonian relationalist c laims that, for three nonsimultaneous events e1, e2 , e3 , the relation col(e1, e2 , e3 ) holds just if, from the substantivalist perspec tive, e1, e2 and e3 lie on a single inertial trajec tory.52 The move solves the kinematic shift problem, but only at the c ost of leaving the relationalist's ideology too impoverished to fix the embedding of a relational history into neo-Newtonian spac etime. An example of Maudlin's nic ely illustrates the point: [C]onsider two partic les in a neo-Newtonian spac etime that are uniformly rotating about their c ommon c enter of mass. Until the first rotation is c omplete, no triple of oc c upied event loc ations are c ollinear. Even after any number of rotations, the c ollinearity relations among oc c upied points will be c onsistent with any periodic rotation, uniform or nonuniform. (Maudlin 1993, 194) One c annot, therefore, interpret the spac etime c oordinates in whic h the dynamic al laws take their standard form as just those c oordinates adapted to the neo-Newtonian relationalist's ideology; if the relationalist ontology is suffic iently sparse, this ideology underdetermines the inertial frames.53 This kinematic al underdetermination is not nec essarily fatal to neo-Newtonian relationalism. It means that the neo- Newtonian relationalist c annot simply lay c laim to standard Galilean-invariant dynamic s. On the strategy we are c onsidering, however, the relationalist suc c eeds so long as they c an identify, in a relationally respec table manner, a set of relational DPMs that c orrespond to the full set of Newtonian DPMs. Can the neo-Newtonian relationalist find dynamic al laws expressed direc tly in terms of neo-Newtonian relations that ac hieve this? As far as I know, no one has seriously attempted to c onstruc t suc h laws. Even so, one knows that any suc h laws will exhibit a partic ularly unattrac tive feature: they will not be expressible as differential equations that admit an initial value formulation.54 In standard Newtonian theory the spec ific ation of the instantaneous positions and veloc ities of the partic les with respec t to some inertial frame suffic es, via the laws, to determine the partic les' relative positions and motions at all times. Stric tly speaking, what needs to be spec ified at an instant transc ends the spec ific ation of the intrinsic state of that instant: in spec ifying veloc ities one is spec ifying quantities that are ultimately grounded in the pattern of the instantiation of collinearity relations between nonsimultaneous spacetime points. But the relevant points lie in the infinitesimal neighborhood of eac h instant and so c an be used, via the usual limiting proc edure, to define derivative quantities that are possessed at that instant. What Maudlin's example illustrates is that there c an be finite stretc hes of time in a neo-Newtonian relational world suc h that no nonsimultaneous triples of material events occurring during that time instantiate the collinearity relation. Indeed, this is a generic feature of neo-Newtonian relational worlds of point partic les. The c ollinearity relation, therefore, c annot be used to define derivative quantities that c an supplement the instantaneous data definable in terms of Leibniz ian relations. The problem fac ed by Newtonian relationalism suggested neo-Newtonian relationalism; the trouble fac ed by neo- Newtonian relationalism suggests another, rather desperate, relationalist maneuver. If the problem is the lac k of appropriate instantaneous quantities, why not simply c o-opt as primitive c ertain derivative instantaneous quantities available to the substantivalist? This, in essenc e, is how Sklar proposes the relationalist treat absolute ac c elerations; as intrinsic , primitive, time-varying properties possessed by partic les at every instant of their trajec tories (Sklar 1974, 230). Huggett usefully dubs them Sklarations (Huggett, 1999, 27). It is not obvious how the relationalist is supposed to use this additional ideology. Skow notes that simply supplementing Leibniz ian relational initial data with Sklarations fails to fix a Newtonian history. Consider, for example, the following pair of two-partic le solutions to Newton's theory of gravitation (Skow 2007, 783– 784). In one solution the two bodies follow c irc ular orbits about their c ommon c enter of mass. In the sec ond solution the partic les travel in on parabolic paths from spatial infinity to slingshot past eac h other before heading bac k out to infinity. Suppose that the distanc e of c losest approac h of the partic les in the sec ond solution matc hes the c onstant Page 14 of 48

Substantivalist and Relationalist Approaches to Spacetime separation of the partic les in the first. Then, at the moment of c losest approac h, the sec ond solution matc hes the first in terms of its Leibniz ian initial data and absolute ac c elerations: the separation between the partic les is the same, its rate of c hange is z ero (it is the moment of c losest approac h) and, bec ause ac c elerations due to gravity depend only on the masses of partic les and the relative separation between them, the Sklarations are the same too. In fac t, this problem is generic . Prec isely bec ause, ac c ording to any Newtonian theory satisfying Newton's third law of motion, forc es and henc e absolute ac c elerations are func tions only of the relative distanc es, they are effec tively already inc luded in the Leibniz ian initial data. Thus every set of Leibniz ian initial data “supplemented” with Sklarations will radic ally underdetermine the future evolution of any system of interac ting Newtonian partic les. As we saw in sec tion 5.1, this evolution depends on the overall angular momentum and the Leibniz ian initial data, with or without Sklarations, does not tell us what this is. Skow's assumption about the appropriate initial data for a theory employing Sklarations c ould be questioned. Why should the Sklar relationalist not inc lude, say, the first time derivatives of Sklarations?55 What the relationist really needs to provide are some relatively natural equations involving Sklarations that fix their theory's DPMs, for these will determine what the appropriate initial data are. Sklar himself did not flesh out his proposal. Both Friedman (1983, 234) and Huggett (1999, 27) suggest that the Sklar relationist c an simply utiliz e Newton's sec ond law as expressed in arbitrary rigid Euc lidean c oordinate systems, that is, c oordinate systems adapted to the Leibniz ian relational ideology. However, it is not at all straightforward how Sklarations are supposed to feature in suc h a formulation of Newton's sec ond law.56 More signific antly, Friedman's and Huggett's attempt to reinterpret the standard equations in terms of Sklarations does not even get off the ground unless Sklarations are additionally c onstrained to be embeddable in spac etime as four- ac c elerations. But why should the instantiation of an allegedly primitive monadic property be c onstrained in this way as a matter of metaphysic al nec essity? Regarded as a kinematic al c onstraint, the requirement is very fishy from a relationalist perspec tive. The alternative is to view the c onstraint only as a restric tion on the physic ally possible, that is, as an additional “law of motion” governing the evolution of Sklarations (and c onstraining admissible initial data).57 Either way, a strong suspic ion must remain that, in this guise, Sklarations do not c onstitute a genuine alternative to accelerations and the attendant substantivalist c ommitments they require. The relationalist needs ideology weaker than Newtonian relations but richer than the neo-Newtonian's collinearity relation. In partic ular, ideology that is suffic iently ric hly instantiated in the neighborhood of any instant is needed in order to avoid the initial value problem fac ed by the neo-Newtonian relationalist. Sklarations might have been expec ted to provide what was needed but, bec ause the quantities are treated as primitive, their nec essary c onnec tions with the relational states of the world at earlier and later times is severed. Putting these c onnec tions bac k in by hand looks like substantivalism by another name. This suggests that to avoid the pitfalls of Sklarations the relationalist should look for ideology that is instantiated by some n-tuples of nonsimultaneous events. And to avoid the pitfalls of neo-Newtonian relationalism, this ideology should be instantiated by n-tuples of nonsimultaneous events in the infinitesimal neighborhood of any instant. Further, any kinematic constraints on the possible instantiation of this ideology should be comprehensible independently of its interpretation, once appropriately embedded, in neo-Newtonian spacetime. It is c ertainly possible to spec ify relational ideology that meets these require-ments58 but at this point we should take a step bac k and rec all the struc ture of the original dilemma posed by Galilean invarianc e. A dynamic al symmetry group that was larger than the spac etime symmetry group leads to in-princ iple unobservable quantities; a spac etime symmetry group larger than the dynamic al symmetry group requires a nonstandard story about the privileged status of some dynamic ally preferred frames of referenc e (for they form a proper subset of those maximally adapted to the spac etime quantities). I have given the impression that, in the c ontext of c lassic al mec hanic s, the struc tures of neo-Newtonian spac etime get things just right, but in fac t this is not the c ase: the dynamic al symmetry group of Newtonian physic s is in fac t larger than the Galilei group. In inertial frame c oordinates, the field-theoretic form of Newton's law of gravitation is expressed by the following equations: (Grav Force) Page 15 of 48

Substantivalist and Relationalist Approaches to Spacetime (Poisson) where ϕ is the gravitational potential, G is Newton's c onstant and p is the mass density. In the c oordinate-free notation of Sec tion 4, these equations bec ome: and the theory has models of the form M,tab, ha, ∇a,ρ,ϕ). The coordinate-dependent equations are invariant under Galilean transformations, whic h are also dynamic al symmetries in the model-theoretic sense, but these transformations do not exhaust the symmetries. Consider the Maxwell group59 of c oordinate transformations: (Max) Like those of the Leibniz group, they involve an arbitrary, time-dependent translation term, a ⃗ (t). Like those of the Newton and Galilei groups, the rotation matrix R is not time-dependent. They therefore c orrespond to a spac etime struc ture that embodies an absolute standard of rotation but no general standard of ac c eleration.6 0 These transformations will also preserve the coordinate-dependent form of the equations of Newtonian gravitation, and map solutions to solutions, so long as the gravitational potential field is also transformed appropriately: where f(t) is an arbitrary time-dependent func tion that is c onstant on surfac es of simultaneity. It follows that, if d is a diffeomorphism corresponding to such a transformation and M,tab, hab,∇a,ρ,ϕ) is a model of the spacetime formulation of Newtonian gravity we are c onsidering, then so is (M,tab,hab,∇a, d*, ρϕ ′). This means that the neo- Newtonian substantivalist is in prec isely the same kind of predic ament as the substantivalist who advoc ated absolute spac e: their metaphysic s grounds physic al quantities (in this c ase absolute ac c elerations, rather than absolute veloc ities) that it is impossible in princ iple to detec t. Here is another way to see the problem. Sinc e, for the type of diffeomorphism under consideration, d* tab = tab and d* hab = hab, if M1   =  ⟨M , tab , hab,  ∇a,  ρ,  ϕ⟩ is a model of the theory, then so is M2   =  ⟨M , tab, hab,  ∇′a,  ρ,  ϕ′⟩, where ∇′a  =  (d−1 ) *  ∇a . That is, the laws and a given matter distribution ρ fix the temporal and spatial metric struc tures, but they leave it underdetermined whether the combination of inertial structure and gravitational force is that given by (∇a,ϕ) or by (∇a,  ϕ) or by (∇′a,  ϕ′). And if we take the postulated inertial structures ontologically seriously, these differences correspond to qualitative differenc es. For example, in one model a given partic le might be forc e-free and moving inertially; in the other it might be ac c elerated under a gravitational forc e.6 1 A natural thought at this point is that M1 and M2 are merely mathematically distinct representations of the same physic al possibility and that ϕ and ∇a are gauge-dependent quantities. But one c annot simply dec lare this so by fiat. One should also provide a c harac teriz ation of a gauge-invariant reality in terms of whic h the gauge dependent quantities c an be understood. It turns out that the substantivalist c an, indeed, do this. The solution is Newton– Cartan theory, a formulation of Newtonian gravitation first developed by Cartan and Friedrichs.62 In this theory, just as in GR, gravitational phenomena are not the effec ts of forc es. The flat inertial c onnec tion ∇a is replac ed by dynamical inertial structure ∇aNC (in part) governed by the following generalization of the coordinate-free form of Poisson's equation: (PoissonNC) whic h relates the Ric c i tensor RaNbC defined by ∇NabC to the mass density. Our two models of Newtonian gravity set in neo-Newtonian spacetime, M1 and M2 , correspond to a unique model of Newton-Cartan theory (up to isomorphism). Any given (∇a,ϕ) pair that solves nongeometriz ed Newtonian gravity determines a unique dynamic al c onnec tion but the c onverse is not true: a given Newton-Cartan c onnec tion c an always be dec omposed into a flat c onnec tion and a gravitational potential, but this dec omposition is nonunique in a way that c orresponds exac tly to the underdetermination of gravitational theory set in neo-Newtonian spac etime.6 3 Page 16 of 48

Substantivalist and Relationalist Approaches to Spacetime While the problem of the symmetries of Newtonian gravity and its substantivalist solution are relatively well-known, the fac t that an enric hed-ideology relationalist strategy c an also be fruitfully pursued is far less apprec iated. When c anvassing enric hed relationalist options earlier in the sec tion, the operative assumption was that Newtonian dynamic s was Galilean invariant. Now that the larger Maxwell group has been rec ogniz ed as a symmetry group, a reevaluation is needed. The equations of any N-body Newtonian system whose forc e laws obey Newton's third law c an be re-expressed as: (1) where r ⃗ij   :   =  (x ⃗i − x ⃗j ) is the directed relative distance between particles i and j and F i⃗ j is the force exerted by particle i on particle j (Hood, 1970; see Earman, 1989, 81, for discussion). For the time derivatives of r ⃗ij to be well defined, the full inertial struc ture of neo-Newtonian spac etime is not required. All that is needed is a standard of rotation, that is, exac tly the spatiotemporal struc ture invariant under the Maxwell group. Sinc e we are assuming Newton's third law is also satisfied, the only spatial dependence of F i⃗ j will be on r ⃗ij . It follows that Equation (1) is invariant under the Maxwell group. The only ideology, in addition to Leibniz ian-relational quantities, needed to ground a standard of rotation is the transtemporal c omparison of the direc tions of the direc ted distanc es between material bodies.6 4 For example, the Maxwellian relationalist c an postulate a primitive four-plac e relation A on vmaaluteer ibael etwveeennts 0 s uacnhd t2hπa,t ,t ow bhee nin (tee1rp,er2e)t eadn da s( eth3e, ea4n)g alere b peatwires eonf −se−i1m→eu2 ltaannde −oe−3u→es4 e. vents, A(e1, e2, e3, e3) takes a What this shows is that the full set of Newtonian solutions for a finite system of interacting particles c an be given a bona fide relationalist interpretation (with or without Newtonian gravitation). With Earman (1989, 81), we should now ask whether the basic idea c an generaliz e to field theory. Maxwellian relationalism for field c onfigurations c an easily be implemented using Barbour's best-matc hing mac hinery, disc ussed in sec tion 6.2, so field theory per se is not an obstac le.6 5 Barbour's mac hinery, however, is only applic able to spatially finite systems (or systems with appropriate spatial boundary conditions). In such “island universe” scenarios, the Maxwellian invariance of dynamic s does not trouble the neo-Newtonian sub-stantivalist, for a preferred inertial c onnec tion c an be identified via the c ondition that the total three-momentum of the whole system is c onstant. Underdetermination only genuinely arises for the neo-Newtonian substantivalist in Newtonian cosmology when one considers, for example, infinite homogeneous matter distributions. As far as I know, no relationalist theory for suc h situations has been devised. The Maxwellian relationalist seems to be in the unfortunate position of having a solution applic able to those c ases that are not genuine problems and no solution for the truly troubling c ases. In the c ontext of field theory, there is one relatively easy way out for the relationalist.6 6 Rec all that the troubles fac ed by the neo-Newtonian relationalist arose bec ause, in a world of point partic les, the three-plac e c ollinearity relation typic ally will not be instantiated by material events in the infinitesimal neighborhood of a given material point. If the relationalist embrac es a plenum, this problem goes away. In the c ontext of Newtonian gravity, the relationalist c an c ombine a material plenum with the insight of Newton– Cartan theory and postulate a primitive three-plac e c ollinearity relation on material events that holds of triples of material events in a physic ally possible world just if, in the c orresponding substantivalist model, they lie on a geodesic of the substantivalist's dynamic al affine c onnec tion. Suc h a Newton–Cartan relationalist still has work to do. The c harac teriz ation of the position just given made c ruc ial referenc e to substantivalist models. Can the standard mathematic al formalism of Newton– Cartan theory be independently understood in terms of suc h relational ideology? What are the material fields and why must they c onstitute a plenum? Similar questions rec ur in the c ontext of relativistic physic s, where fields are no longer optional extras. It is to relativity we now turn. 6.1.2 Relativity In the c ontext of c lassic al mec hanic s, the relationalist who pursues the enric hed ideology strategy is forc ed to be c reative. Simply c o-opting substantivalist ideology (by restric ting the domain of possible instantiation to the material events) fails, primarily bec ause of the relative sparseness of the relationalist's ontology in c omparison to the substantivalist's plenum of spac etime points. In the c ontext of SR, however, the flat-footed move works. Restric ting the substantivalist's ideology—Minkowski spac etime distanc es—to material events fixes the embedding of a relational history into Minkowski spac etime (up toisomorphism). Further, Minkowski distanc es c onc eived of as relationalist ideology c an be used to frame dynamic al princ iples direc tly in relationalist terms.6 7 Page 17 of 48

Substantivalist and Relationalist Approaches to Spacetime Central to relativistic mec hanic s (even if not to relativistic physic s in general) is the idea that unac c elerated motion is default behavior and that ac c elerations are due to forc es. In order to lay c laim to this pic ture, the Minkowski relationalist needs ac c ounts of both ac c elerations and forc es. In c lassic al mec hanic s, forc es were unproblematic for the relationalist (bec ause they are func tions of Leibniz ian relational quantities); it was ac c eleration that proved troublesome. In relativity, the difficulties are reversed. Consider the standard, c oordinate-dependent forms of relativistic laws. The privileged c lass of c oordinate systems relative to whic h these equations hold are simply those adapted to the Minkowski distanc e relations between material events (c f. sec tion 4.3). Dynamic ally signific ant absolute ac c eleration, therefore, is simply ac c eleration relative to the c oordinate systems adapted to the relationalist's spatiotemporal distanc es. In fac t, the Minkowski relationalist c an do better and give an intrinsic c harac teriz ation of ac c eleration. Rec all that, in Minkowski spac etime, the inertial trajec tories are not struc ture over and above the spatiotemporal distanc es; the straight line in spac etime between two temporally separated events is the path of maximal temporal distanc e. This means that a partic le will be unac c elerated just if, for any temporally ordered points p, q, r of its trajec tory, I(p, r) = I(p,q) + I(q, r) and, conversely, if I(p, r) I(p, q) + I(q, r), we know that the particle is accelerated between the points p and r (Earman 1989, 129). The four-ac c eleration of a trajec tory at a point just is the intrinsic c urvature of the trajec tory at that point and so, as for c urves in Euc lidean spac e, one c an define the ac c eleration of the partic le (both its magnitude and its direc tion) in terms of suc h distanc es.6 8 The Minkowski relationalist treatment of forc es is less straightforward. The c oordinate-free statement of the sec ond law, Fa = mξn∇ nξa, is formally the same in c lassic al mec hanic s and SR and, in both c ases, the four-forc e, Fa, must be a spac elike vec tor. This formal identity, however, hides a c ruc ial differenc e. In the neo-Newtonian c ase, spac elike vec tors lie in (that is, are tangent to) surfac es of simultaneity. As a result neo-Newtonian four-forc es c an be defined in terms of Leibniz ian spatial distanc es, whic h are intrinsic to suc h surfac es. In the Minkowski c ase, if one c onsiders an arbitrary spac elike hyperplane and the ac c elerations of a number of interac ting partic les at the points where their trajec tories intersec t this plane, then, in general, none of these ac c elerations will be tangent to the hyperplane. It is no ac c ident that in relativistic theories Fa is standardly given as a loc al func tion of fields.6 9 Some think that the need to invoke fields is a problem for the relationalist. On one (natural) interpretation, fields are simply assignments of properties or states to the points of spac etime (Field, 1985, 40). Suc h a view does indeed presuppose substantivalism, but there is an alternative available, and it is one that arguably fits more naturally the language employed by physic ists. On this other view, the field itself is reified as a vast, spatiotemporally extended objec t in its own right.7 0 Adopting this sec ond c onc eption of fields does not, by itself, amount to relationalism; many substantivalists will agree that at least some fields are extended objec ts in spac etime (rather than properties of spac etime).7 1 Taking a “relational” view of a field also does not by itself c ommit one to the view that suc h a field c ould exist in the absenc e of spac etime, or, without spac etime, have the very properties one's theory c harac teriz es it as having (cf. footnote 38). The devil will be in the details. Consider the simple c ase of a field, ϕ , with just one degree of freedom per spac etime point. The relationalist wishes to view ϕ as an extended, physic al entity rather than as an assignment of properties to spac etime. Sinc e spac etime itself is supposed not to exist, this extended objec t c annot be c harac teriz ed in terms of the spatiotemporal loc ations of the various field intensities.7 2 Instead, the relationalist should view the field as c harac teriz ed by the infinite number of fac ts about the Minkowski distanc es between its pointlike parts; together these fully c harac teriz e the pattern of field intensities. These distanc es c annot (in prac tic e) be spec ified direc tly. But there is nothing relationally improper about desc ribing the field relative to a Lorentz ian c oordinate system so long as suc h a c hart is thought of as a map direc tly from the field itself into ℝ4 that enc odes the Minkowski distances. Consider, now, the substantivalist's presentation of a theory of such a field. The KPMs will be of the form M,ηab,ϕ and the DPMs will be pic ked out via an equation relating ϕ and ηab. Suppose the relationalist's only way to identify dynamic ally possible field c onfigurations was to use this mac hinery. Would they then be in the embarrassing position of relying on a substantivalist “fairy tale” without a proper explanation of why it works (Earman 1989, 172)? It does not seem so. That ϕ is the only field in the model reified by the relationalist does not mean that ηab is a fic tion. The substantivalist suggests that one understands fields as assigning various properties and relations. In this c ase, the relationalist agrees. They just disagree about the subjec t of predic ation: for the substantivalist it is spac etime itself, for the relationalist it is the one substantival field of the model, ϕ . Equations relating ϕ to the other Page 18 of 48

Substantivalist and Relationalist Approaches to Spacetime fields then have a straightforward relationalist reading as c laims about the allowed (geometric al) properties of ϕ itself. So far I have only c onsidered sc alar fields. More c omplex fields c an pose additional problems for the relationalist. Standard vec tor and tensor fields, for example, are not obviously c onc eptually independent of the struc ture of the manifold on whic h they are defined. Their degrees of freedom at a point are normally understood as taking values in the tangent space at that point (or in more c omplex spac es c onstruc ted in terms of it), whic h might appear to presuppose the differentiable struc ture of the manifold on whic h the fields are defined. In fac t, even c harac teriz ing sc alar fields normally involves this manifold struc ture, for one is normally interested in smooth fields. In this c ase, however, it is c lear how one c an do away with referenc e to an independent manifold. What one requires (roughly speaking) is that the field's values vary smoothly as a func tion of the distanc es between its parts: fields themselves c an have the struc ture of a differentiable manifold in virtue of these Minkowski distanc es. Vec tor and tensor fields, c onc eived of as substantival entities in their own right, will likewise have a manifold struc ture, but there is something suspic iously c irc ular about taking the spac es in terms of whic h a field's degrees of freedom are defined to be themselves defined in terms of that field's own spatiotemporal extension. An alternative is to try to understand the degrees of freedom of some material fields in terms of their interac tions with other fields whose relational c redentials are not in doubt.7 3 The upshot is that the c ombination of Minkowski relationalism and a relational interpretation of fields is at least a going c onc ern as an interpretation of SR. The final task for this sec tion is to c onsider whether the pic ture c an be adapted to GR. The strong similarities between SR and GR stressed in sec tion 4.3 might lead the relationalist to be optimistic . In fac t, the move from flat to c urved geometric struc ture, and the manner in whic h it features in GR, presents a formidable obstac le. Rec all that the Minkowski relationalist does not reify the metric field ηab. Instead this field is regarded as c ataloging primitive spatiotemporal distanc es that hold between the parts of bona fide material fields. At the level of kinematic s, the generaliz ation of this to GR is straightforward. Minkowski distanc es are simply replac ed by those of a c urved semi-Riemannian geometry. A c ruc ial c onsequenc e of this move is that the distanc es instantiated between material events need no longer fix (independently of the dynamic al laws) all the fac ts about the geometry of spac etime. In partic ular, c onsider an “island universe” involving a matter distribution of finite spatial extent. The spatiotemporal distanc es instantiated in the history of the material world will not fix the geometry of the empty spacetime regions beyond it. This is not a problem of princ iple. After all, the relationalist will c laim that there is literally nothing beyond the boundary of the material universe to instantiate one geometry rather than another. There is also no diffic ulty, in princ iple, with this type of relationalist regarding geometry as dynamic al and as influenc ed by matter. For example, the laws of a relational theory c ould lay down how the network of spatiotemporal relations instantiated in some temporally thic k slic e through the material world determine (together with other dynamic ally relevant properties of matter) the pattern of spatiotemporal relations instantiated in earlier and later regions of the material universe. The partic ular diffic ulty GR poses for the envisaged relationalist involves the c ombination of these two fac tors. In GR the geometric al properties of the supposed nonentity beyond the material universe do make a dynamic al differenc e. For example, the entire history of spatiotemporal distanc es instantiated in our island universe up to some time will not rec ord whether a “gravitational wave” (that is, a propagating ripple in the fabric of spac etime itself) is approac hing from outside the system and will thus underdetermine the system's future evolution (Earman, 1989, 130; Maudlin, 1993, 199). In response, the relationalist c ould rule out by fiat models with empty regions of spac etime. To do so, however, is not only to give up on the goal of empiric al equivalenc e with standard theory; it is to impose a restric tion that is arbitrary by the lights of the relationalist's own theoretic al apparatus. The relationalist does not have problems with empty regions per se. What they have problems with is those regions having a determinate geometry that need not supervene on the properties of and relations between matter-filled regions and with the geometry of empty regions playing the dynamic al role that GR assigns it. The better “relationalist” move is to treat the metric tensor as a “material” field in its own right, but then, sinc e all parties affirm the existenc e of a substantival entity whose properties are c harac teriz ed by gab, it is not c lear what substantive issue remains.7 4 6.2 Barbour's Machian Relationalism There are two straightforward relationalist responses to the mismatch between relationalist spacetime symmetries Page 19 of 48

Substantivalist and Relationalist Approaches to Spacetime and the dynamic al symmetries of Newtonian mec hanic s. The previous sec tion c overed one of these: enric h relationalist ideology in order to bring spac etime symmetries into line. This sec tion investigates the other: c hange the dynamic s in order to bring the dynamic al symmetries into line. The most thorough and suc c essful development of this strategy is that of Julian Barbour and c ollaborators. The label Machian relationalism is appropriate for three reasons. First, it ac c ords with Barbour's own terminology. He sees the requirement that a theory be maximally predic tive with respec t to relational initial data (in the sense disc ussed in Sec tion 5.1) as a prec ise version of Mach's Principle and he takes his approac h to dynamic s to reveal that GR is in fac t a Mac hian theory. Sec ond, in the c ontext of pre-relativistic partic le dynamic s, the spac etime quantities Barbour takes as fundamental are even sparser than those of the Leibniz ian relationalist: the Euc lidean nature of the instantaneous relative distanc es between partic les is ac c epted as primitive, but the temporal intervals between suc c essive instantaneous c onfigurations are not. In his c ritique of Newton, Mac h (1901, 222– 226) c laimed that the question of whether a motion is in itself uniform is senseless, on the grounds that a motion c an (allegedly) only be judged uniform relative to some other motion or material proc ess.7 5 Finally, Barbour's partic le theories provide a c onc rete implementation of Mac h's idea that the inertial properties of a body might be understood in terms of that body's relations to the rest of the bodies in the universe, rather than with respec t to substantival spac etime struc ture (Mac h 1901, 231– 235). Up to this point I have presented the DPMs of a theory as singled out in terms of differential equations that must be everywhere satisfied within a model by its c onstituent fields and partic le trajec tories. In some formulations of dynamic s, the DPMs are singled out in terms of their relations to other KPMs. Mac hian relational theories are most illuminatingly developed in this type of framework. Consider, in partic ular, the Lagrangian formulation of Newtonian mec hanic s. Central to this framework is a system's configuration space, Q, the points of whic h represent possible instantaneous states of the system. Ac c ording to the substantivalist, suc h a state for an N-partic le system c orresponds to a set of positions for eac h partic le relative to some inertial frame. Q is then 3N-dimensional. As the system evolves, the point in Q representing the system's instantaneous state traces out a continuous curve. In Lagrangian mec hanic s, KPMs (that is, metaphysic ally possible histories) are (monotonic ally rising) c urves in the produc t spac e formed from Q and a one-dimensional spac e, T, representing time. The DPMs are those c urves that extremiz e a partic ular func tional of suc h histories (the action). This framework c an be adapted to Leibniz ian and Mac hian relationalism in a straightforward way. First, sinc e the relationalist's possible instantaneous states c orrespond to sets of inter-partic le distanc es (rather than positions defined with respect to spacetime structure), the relationalist replaces Q with the relative configuration space QRC S . For N partic les, QRC S is (3N − 6)-dimensional. In fac t, the relationalist might be tempted to go further. Formulating dynamic s in terms of QRC S involves treating transtemporal c omparisons of length as primitive. Distinc t c urves in QRC S c an c orrespond to exac tly similar sequenc es of Euc lidean c onfigurations if some of the c orresponding c onfigurations represented in the two c urves differ in overall siz e. This is not true in shape space (QS S ), a c onfiguration spac e of one less dimension that treats only the ratios of distanc es within the same c onfiguration as physic ally meaningful. Sec ond, standard theory distinguishes histories that c orrespond to a single path in c onfiguration spac e being trac ed out at different rates with respec t to the primitive temporal metric . The Mac hian relationalist, in c ontrast, will view eac h path in c onfiguration spac e as c orresponding to exac tly one possible history. They therefore dispense with T, the spac e enc oding primitive temporal separations, in favor of a “timeless” formulation of dynamic s in terms of c onfiguration spac e alone. One way to do this is to equip the spac e with a metric . The DPMs are then pic ked out via a geodesic principle: physic ally possible histories c orrespond to paths in c onfiguration spac e of extremal length relative to the metric. The implementation of this sec ond step c an be ac hieved via a reinterpretation of Jacobi's Principle, part of the standard toolkit of Newtonian dynamic s.7 6 The metric struc ture of three-dimensional physic al spac e c an be used to dFoefE fp ina=ert  ia(c Elmese− tmrVioc(v oxi nn⃗1g ,Q  i…n kenro,ti wax ln⃗Nly )a. )sD. ytThnheae mk giinceesot diisce simnicce otprrripcino: crdaipstlek2eidn i  sb= yth  me∑nu:l t(iip2ml)2yi i ndgx ⃗id⋅s k2dinx ⃗ib⋅y. Iats c goenofodremsiacls f accotrorre spond to histories (3) Page 20 of 48

Substantivalist and Relationalist Approaches to Spacetime Its solutions c orrespond to Newtonian histories of a system of N partic les with a total energy E interac ting ac c ording to the potential V. Tkin looks like the standard Newtonian kinetic energy but note that λ represents an arbitrary parameteriz ation of paths in Q: the path length I is invariant under reparameteriz ations: λ ↦ λ′ = f(λ), where df/dλ 0 but f is otherwise arbitrary. The equations of motion corresponding to (2) are: (4) These simplify dramatic ally, reduc ing to the standard form of Newton's sec ond law, if the freedom in the c hoic e of λ is exploited to set FE = Tkin, that is, E = Tkin + V. The substantivalist sees imposing this requirement as a way to determine the rate at whic h the system trac es out its path in Q relative to a primitive temporal metric . The Mac hian sees the equation as defining an emergent temporal metric in terms of the temporal parameter that simplifies the dynamic s of the system as a whole (Barbour 1994, 2008).7 7 Jac obi's princ iple involves a metric on Q. To c onstruc t a relational theory, we need a metric on QRCS. One c an be obtained by replac ing Tkin (a func tion of veloc ities defined with respec t to inertial struc ture) with a func tion of the relative velocities ṙij. Theories of this kind were independently disc overed on a number of oc c asions during the twentieth c entury. They predic t mass anisotropy effec ts (how easy it is to ac c elerate a body bec omes direc tion dependent) that are ruled out by experiment.7 8 It is also not c lear how they might generaliz e to field theory, where analogues of the transtemporal partic le identities used in the definition of the ṙijs are absent. Barbour and Bertotti (1982) found a way to surmount both problems. Figure 15.1 Best matching. The curves C1 and C2 in Q correspond to the same sequence of relative c onfigurations. q2 is the point on the orbit c ontaining q1 that minimizes the distanc e along the c urve from p1. r2 is similarly related to q2 . C2 is the best-matc hed c urve; the length along it gives the length along C, the c orresponding c urve in QSS. The (ambitious) relationalist thinks of instantaneous c onfigurations as c ompletely c harac teriz ed by the ratios of inter-partic le separations. A three-dimensional c oordinate system enc odes suc h data just if ∣x ⃗i − x ⃗j ∣/∣x ⃗m − x ⃗n ∣  =  rij /rmn for all particles i,j, m, n. If one coordinate system satisfies this constraint, so will any other related to it by a rigid rotation, translation, or a dilation (an overall c hange of sc ale). The relationalist therefore regards the points of Q, not as spec ific ations of positions in some inertial frame, but as natural representations of relational c onfigurations. The representation involves some redundanc y: points of Q c onnec ted by an element of the similarity group (the group of rigid translations, rotations, and dilations) c orrespond to the same relative c onfiguration. Q is partitioned by the group into sets of suc h points (the group orbits). Consider, now, two paths in Q that c orrespond to the same sequenc e of relative c onfigurations. A metric on Q will, in general, assign them different lengths. However, starting from any given point p in Q, one c an use the ac tion of the similarity group on Q to define a unique length, by shifting the points of any curve through p along the corresponding group orbits so as to extremiz e the length assigned to the c urve. This is the proc ess Barbour and Bertotti c alled best 79 Page 21 of 48

Substantivalist and Relationalist Approaches to Spacetime matching.7 9 It is depic ted in Figure 15.1. If best matc hing is to define a metric on QS S (the quotient of Q by the similarity group), the metric on Q must have the right properties. In partic ular, suppose p1 and p2 are points on the same group orbit that are widely separated in Q. Consider two paths through p1 and p2 , respec tively, that c orrespond to the same sequenc e of relative c onfigurations. Suppose one now best matc hes these paths, keeping the points p1 and p2 fixed. Best matc hing only leads to a well-defined metric on QSS if the same result is obtained in each case. The metric ds2   =  FEds2kin satisfies this requirement provided FE meets c ertain c onditions. If one first c onsiders best matc hing just with respec t to the Euc lidean group (translations and rotations), V must be a func tion only of the relative distanc es, ṙij . This requirement is satisfied by familiar Newtonian potentials. The c orresponding best-matc hed theories, whic h take DPMs to be geodesic s of the metric induc ed on QR CS, have as solutions sequenc es of relative c onfigurations that c orrespond to the standard Newtonian solutions with z ero overall angular momentum (relative to the c enter-of-mass frame). The fac t that a subset of standard Newtonian solutions is rec overable by this method highlights the fac t that the theories provide a relational interpretation of inertial struc ture: best matc hing establishes a nonprimitive “equiloc ality relation,” c orresponding to the spac e of the inertial frame in whic h the system's total linear and angular momenta vanish. Note, also, that the rec overy of only a proper subset of the solutions of standard dynamic s is arguably a strength of the best-matc hing theory (assuming solutions c apable of modeling the ac tual world fall within this set). This is bec ause the theory predic ts and explains a feature of the world (the vanishing of its overall angular momentum) that is a c ontingent fac t on the orthodox Newtonian view (Pooley and Brown, 2002; Pooley, 2004).80 Best matc hing with respec t to dilations imposes a more severe requirement: FE must be a homogeneous func tion of the x ⃗ i s of degree −2, in order to c ompensate for the sc aling behavior of ds2kin . Standard Newtonian potentials do not have this property, but they c an nevertheless be inc orporated as effec tive potentials in sc ale-invariant theories if a weak, epoc h-dependent universal forc e is also inc luded (Barbour, 2003, 1556– 7). Barbour's framework for nonrelativistic partic le dynamic s, therefore, c onstitutes a genuinely relationalist (and potentially fruitful) alternative to Newtonian physic s as standardly c onc eived. What one is really interested in, though, is how the program transfers to relativistic physic s. The best matc hing idea c an be applied in the c ontext of SR (Barbour and Bertotti 1982, 302– 303), but I move straight to a c onsideration of GR, where the results are truly surprising. The first step is to c onsider how one might generaliz e the framework to c onfigurations manifesting a variably c urved Riemannian geometry. One c onfronts the issue, raised in sec tion 6.1.2, of how to deal with the possibility that the geometry of empty spac e might be both nontrivial and nonreduc ible to relations between material bodies. Barbour bites the bullet. In the c ontext of GR, the Mac hian “relationalist” takes the geometry of substantival (instantaneous) space as primitive. Assume that instantaneous spac e has the determinate topology of some c losed 3-manifold without boundary, Σ. The obvious analogue of Q is then Riem(Σ), the spac e of Riemannian 3-metric s on Σ. An analogue of QRC S is superspace: the space of 3-geometries. Two points (Σ,hab) and (∑,  h′ab ) of Riem(Σ) correspond to the same 3- geometry just if they are isometric, that is, just if, for some diffeomorphism d of ∑,  h′ab  =  d * hab . Superspace is therefore Riem(σ)/Diff(σ), the quotient of Riem(σ) by the group of diffeomorphisms of σ. Proc eeding as before, one seeks an ac tion princ iple on superspac e defined, via best matc hing, in terms of a metric on Riem(σ). In this c ase, best matc hing is implemented by diffeomorphisms of σ. Seeking as direc t a parallel as possible with Jac obi's Princ iple (2) leads to a Riem(σ) geodesic princ iple of the form: (5) The first integral inside the square root is the analogue of the c onformal fac tor FE in (2); 8 1 the sec ond is the maneatrloicg vueel oocf itthiees ((pwairtha mreestepreizcet dto) tkhinee atircb ietrnaerryg pya. tInh tphaisra cmaestee,r T λ)= aGndaA btchde h˙ gaeb n˙hecrda, lw fohremre o h˙f athb  e= su dpheramb /edtλri ca rGe aAtbhced is hac hbd + Ahabhc d, where A is an arbitrary c onstant. Best matc hing with respec t to 3-diffeomorphisms is ac hieved by replacing T with TBM  =  GAabcd (˙hab  −  Lξ˙  hab)(h˙cd − Lξ˙  hcd) and extremizing with respect to variations in ˙a 82 Page 22 of 48

Substantivalist and Relationalist Approaches to Spacetime ξ˙a .8 2 Theories of this kind make good sense but they do not provide direc t analogues of GR.8 3 For these, one needs to c onsider a Riem(σ) ac tion princ iple that involves a subtle but radic al differenc e. It has the form: (6) with W and T defined as before. The differenc e between Princ iples (5) and (6) is that, in the former, integration over 3-spac e oc c urs within a global square root, but in the latter the square root is taken at eac h point of spac e and oc c urs within the spatial integration.8 4 Whereas the reparameteriz ation invarianc e of (5) gives rise to a single c onstraint, the position of the square root in (6) leads to an infinity of c onstraints, one assoc iated with eac h point of spac e. These must be propagated by the equations of motion if the theory is to be c onsistent. This happens only if A = − 1 in GAabcd and W = Λ + αR, where R is the scalar curvature tensor of hab, and α is 0 or ± 1. The choice of α = 1 and the imposition of best matc hing with respec t to 3-diffeomorphisms transforms (6) into the ac tion princ iple for GR found by Baier-lein, Sharp, and Wheeler (1962). (Or, stric tly speaking, a time-reparameteriz ation analogue of the BSW ac tion, bec ause the BSW ac tion involves Lie derivatives defined with respec t to the shift-vec tor field, rather than with respec t to the veloc ity of 3-vec tor field. Thanks to Edward Anderson here.) This is dynamic ally equivalent to the standard spac etime ac tion restric ted to globally hyperbolic spac etimes. In other words, without any spacetime presuppositions and starting with a family of “timeless” ac tion princ iples for the evolution of 3- geometries of the general form (6), the requirement of mathematical consistency alone (almost) uniquely singles out an ac tion princ iple c orresponding to GR.8 5 The BSW ac tion princ iple for GR, whic h formally singles out c urves in Riem(σ), is degenerate: a point and a direc tion in Riem(σ) fail to pic k out a unique solution. By itself, this is not a problem for the Mac hian relationalist. An analogous property holds of the best-matc hed ac tion princ iples for partic le dynamic s: given a point and direc tion in Q, a c ontinuous infinity of c urves solve the equations. The reason this is not a drawbac k in the partic le c ase is that eac h of these c urves c orresponds to the same sequenc e of relative c onfigurations: they projec t down to a single c urve in the quotient of Q by the relevant group. The same is not true for the BSW ac tion. After projec ting down from Riem(σ), one still has a c ontinuum of c urves for eac h point and direc tion in superspac e. Sinc e these c urves c orrespond to non-isometric sequenc es of 3-geometries, and sinc e suc h 3-geometries are the Mac hian's fundamental ontology, the Mac hian is c ompelled to regard these c urves as c orresponding to physic ally distinc t histories. The theory is therefore radic ally indeterministic . The indeterminism is only removed if one c an find a way to regard all c urves with the same initial data as representations of a single physic al history. As we shall see in sec tion 7, the spac etime substantivalist, who regards spatio temporal geometry as primitive, c an do this, bec ause the different sequences of 3-geometries correspond to different foliations of a single 4-dimensional spacetime. The Mac hian, however, who regards spac etime geometry as sec ondary to, and defined in terms of, the dynamic al evolution of spatial geometry, has no suc h option (Pooley, 2001, 16– 18). Fortunately for the Machian, this otherwise devastating underdetermination can be resolved in strictly 3- dimensional terms. In the partic le c ase, the ambitious relationalist esc hewed transtemporal sc ale c omparisons and regarded only the shapes of c onfigurations as fundamental. Analogous moves are possible in the c ontext of GR. In partic ular, in a conformal 3-geometry only angles and the ratios of (infinitesimal) distanc es are regarded as physic ally fundamental. In terms of Riem(σ), one regards any two 3-metric s related by a (spatially varying) sc ale transformation as physically equivalent: hab ∼ ϕhab, ϕ 0. Conformal superspace is the quotient of Riem(σ) by suc h sc ale transformations (in addition to 3-diffeomorphisms). It c an be viewed as analogous to QSS. Solutions to the BSW ac tion that share initial data in superspac e (that is, sequenc es of 3-geometries c orresponding to different foliations of the same spac e-time) do not projec t down to a unique c urve in c onformal superspac e. However, the equations of GR c an be rec ast so as to determine a unique suc h c urve (Barbour and Ó Murc hadha, 2010). Its points c orrespond to the foliation of the c orresponding spac etime by spac elike hypersurfac es of constant mean extrinsic curvature. The fac t that these geometric ally privileged hypersurfac es simplify and make trac table the initial value problem in GR has been known sinc e the work of James York in the 1970s. What the Mac hian perspec tive provides is an (alternative) understanding of the relevant equations in terms of a generaliz ation of best matc hing to (volume preserving) c onformal transformations (Anderson et al., 2003, 2005). The Mac hian perspec tive on GR is not relationalist in the sense of this c hapter, but it does offer a mathematic ally Page 23 of 48