philosophy of physics

Unification in Physics nature and laws of whic h “we do not even attempt to define, bec ause we c an eliminate [them] from the equations of motion by the method given by Lagrange for any c onnec ted system” sec t. 552). Displac ement, magnetic induc tion and elec tric and magnetic forc es were all defined in the Treatise as vec tor quantities (Maxwell 1873, sec t. 11, 12), together with the elec trostatic state, whic h was termed the vec tor potential. All were fundamental quantities for expression of the energy of the field and were seen as replac ing the lines of forc e. (10) For an extended disc ussion of unific ation in Spec ial Relativity see Morrison (2000). (11) There are two different types of SU(3) symmetry: the one that ac ts on the different c olors of quarks, whic h is an exac t gauge symmetry mediated by gluons, and the flavor SU(3) symmetry, whic h rotates different flavors of quarks to eac h other. The latter is an approximate symmetry of the QCD vac uum and henc e is not fundamental. It arises as a c onsequenc e of the small mass of the three lightest quarks. (12) For a more c omprehensive disc ussion of symmetry and its uses in physic s, see Bangu (this volume), as well as the edited c ollec tion by Brading and Castellani (2003) and Morrison (1995; 2000). (13) These c an also be thought of as phase transformations where the phase is c onsidered a matrix quantity. See Aitc hinson and Hey (1989) for a disc ussion of this topic . (14) Isospin ac tually refers to similar kinds of partic les c onsidered as two states of the same partic le in partic ular types of interac tions. For example, the strong interac tions between two protons and two neutrons are the same, whic h suggests that for strong interac tions they may be thought of as two states of the same partic le. So, hadrons with similar masses, but differing in terms of c harge, c an be c ombined into groups c alled multiplets and regarded as different states of the same objec t. The mathematic al treatment of this c harac teristic is identic al with that used for spin (angular momentum). The SU(2) group is the isospin group and is also the symmetry group of spatial rotations that give rise to angular momentum. (15) In order to satisfy the symmetry demands assoc iated with the SU(2) group and in order to have a unified theory (i.e., have the proper c oupling strengths for a c onserved elec tric c urrent and two c harged W fields), the existenc e of a new gauge field was required, a field that Weinberg assoc iated with a neutral c urrent interac tion that was later disc overed in 1973. For a disc ussion of the diffic ulties surrounding the neutral c urrent experiments, see Galison (1987) and Pic kering (1984). (16) Indeed, despite its disc overy, the properties of the Higgs boson and whether it is a single partic le of a family or particles remains largely unknown. Further data from CERN will hopefully reveal these features and what their impact will be on the Standard Model. (17) CP is a symmetry that states that the laws of physic s should be the same if a partic le were interc hanged with its antipartic le (C symmetry, or c harge c onjugation symmetry), and left and right were swapped (P symmetry, or parity symmetry). In addition to its role in weak interac tions, it also plays an important role in the attempts of c osmology to explain the dominanc e of matter over antimatter in the Universe. (18) The Yukawa interac tion desc ribes the c oupling between the Higgs field and massless quark and elec tron fields. Through spontaneous symmetry breaking, the fermions ac quire a mass proportional to the vac uum expectation value of the Higgs field. (19) For an extended disc ussion of these and other problems fac ing the elec troweak theory, see Quigg (2009). My discussion borrows from his exposition. (20) Of c ourse, the notion of naturalness here is not something that c an be given a prec ise definition, sinc e it is relative to the gaps in our theoretic al knowledge of physic s at high energies. Sinc e the quantum c orrec tion inc ludes effec ts from high energy, there is an unc ertainty about their extent and validity. At energies beyond that for whic h our theories are valid, new physic s may emerge making the quantum c orrec tions depend entirely on the energy sc ale. Henc e, the notion of naturalness c an be thought of as sc ale relative. (21) A potential problem for SUSY breaking is whether it c an be ac c omplished in a “natural” way. Bec ause there seems to be no obvious way to break supersymmetry far below the grand unific ation energy, this problem, in some sense, is simply a reinc arnation of the hierarc hy problem. Page 23 of 24

Unification in Physics (22) A string is an objec t with a finite spatial extent that has an intrinsic tension in the same way that a partic le has intrinsic mass. The presenc e of an intrinsic tension means that string theory possesses an inherent mass sc ale, a fundamental parameter with the dimensions of mass that defines the energy sc ale at whic h “stringy” effec ts (effec ts assoc iated to the osc illation of the string) bec ome important. The various osc illation modes of the string are effec tively loc aliz ed in its immediate neighborhood and behave like elementary partic les with different masses related to the osc illation frequenc y of the string. Bec ause a string is like a c ollec tion of infinitely many point partic les, c onstrained to fit together to form a c ontinuous objec t, it has infinitely many degrees of freedom. Consequently, its assoc iated quantum theory required the existenc e of several spatial dimensions (26). The invention of superstring theory—a string with extra degrees of freedom that make it supersymmetric —has reduc ed that number to 11. (23) LQG inc orporates many of the important aspec ts of general relativity, but differs from the latter in its quantiz ation of spac e and time at the Planc k sc ale, as in quantum mec hanic s. In other words, the spac e c ontaining all physic al phenomena is itself quantiz ed. Lee Smolin, one of the originators of LQG, has proposed that a loop quantum gravity theory inc orporating either supersymmetry or extra dimensions, or both, be c alled “loop quantum gravity II.” (24) For a general disc ussion of RG in the c ontext of explanation more generally, see Batterman (2002). (25) Zinn-Justin (1998) disc usses some of the c onnec tions between the use of RG in statistic al physic s and quantum field theory. (26) My discussion of these issues borrows from Weinberg (1983). (27) See Georgi (1993). The point I want to stress here is that we do not need the dec oupling theorem to establish exac t results to see why reduc tionism is problematic . Instead we foc us on what the theorem does show: that the physic s at short distanc es is not only unimportant at longer length sc ales but that it is immune from c hanges that take plac e there in muc h the same way that atomic physic s is irrelevant to understanding turbulenc e and the Navier Stokes equations at high Reynolds numbers. In other words, it simply does not matter for these types of problems whether matter bec omes disc rete at Fermis rather than Angstroms, and it is that fac t that c auses diffic ulties for the reduc tionist pic ture. Marg aret Morris on Margaret Morrison is Professor of Philosophy at the University of Toronto. She is the author of several articles on various aspects of philosophy of science including physics and biology. She is also the author of Unifying scientific theories: Physical concepts and m athem atical structures (Cam bridge, 2000) and the editor (with Mary Morgan) of Models as m ediators: Essays on the philosophy of natural and social science (Cam bridge, 1999).

Measurement and Classical Regime in Quantum Mechanics Guido Bacciagaluppi The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter provides an up-to-date disc ussion of work on two distinc t problems in the foundations of quantum mec hanic s: the problem of the c lassic al regime and the measurement problem. It explains that c ontemporary work has foc used on the role of environmental dec oherenc e in the emergenc e of c lassic al kinetic s and dynamic s, and argues that the suc c ess of appeals to dec oherenc e to solve the problem depends on the interpretation of the quantum theory. The c hapter also c onsiders the c ollapse postulate, the Born rule, and the apparatus of positive operator value measures. K ey words: qu an tu m mech an i cs, cl assi cal regi me, measu remen t probl em, en v i ron men tal decoh eren ce, cl assi cal k i n eti cs, dy n ami cs, col l apse postu l ate, B orn ru l e, posi ti v e operator v al u e In this essay, I shall foc us on two of the main problems raising interpretational issues in quantum mec hanic s, namely the notorious measurement problem (disc ussed together with the theory of measurement in sec tion 4) and the equally important but not quite as widely disc ussed problem of the c lassic al regime (disc ussed together with dec oherenc e in sec tion 3). The two problems are distinc t, but they are both intimately related to some of the issues arising from entanglement and density operators, whic h are thus briefly reviewed in sec tion 2. A few fundamentals are rehearsed in sec tion 1. The essay will aim to be fairly nontec hnic al in language, but modern in outlook and c overing the c hosen topic s in more depth than most introduc tory treatments. The philosophy and foundations of quantum mec hanic s offer many more examples of live researc h issues, and muc h progress has been ac hieved rec ently in suc h traditional approac hes as c ollapse theories, pilot-wave theories and Everett interpretations, and in the (time-honored but rec ently revived) area of axiomatic rec onstruc tions of the theory. Rec ent years have seen fasc inating advanc es also in the study of the other great puz z le raised by entanglement, namely quantum mec hanic al nonloc ality. No in-depth c overage of these other topics will be attempted. 1. A Few Fundamentals 1.1 Phenomenology of Measurements In c lassic al mec hanic s, measurements are idealiz ed as testing whether a system lies in a c ertain subset of its phase spac e. This c an be done in princ iple without disturbing the system, and the result of the test is in princ iple fully determined by the state of the system. In quantum mec hanic s, none of these idealiz ations c an be made. Instead: (i) measurements are idealiz ed as testing whether the system lies in a c ertain (norm-c losed) subspac e of its Hilbert spac e; 1 (ii) a measurement in general disturbs a system: more prec isely (and in the ideal c ase), unless the state of the system is either c ontained in or orthogonal to the tested subspac e, the state is projec ted onto either the tested subspac e or its orthogonal c omplement (this is known as the “c ollapse” of the quantum state, or Page 1 of 32

Measurement and Classical Regime in Quantum Mechanics the “projec tion postulate”); (iii) this proc ess is indeterministic , with a probability given by the squared norm of the projec tion of the state on the given subspac e (the “Born rule” or “statistic al algorithm” of quantum mec hanic s).2 For instance, take a spin-1/2 system initially in the state (1) where |+x〉 and |−x〉 are the states of x-spin up and down. If we test for x-spin-up (for the subspace P+x ), the final state will be either |+x〉 with probability |α|2, or |−x〉 with probability |β|2. Often, one c onsiders testing together a family of mutually orthogonal sub-spac es.3 Suc h a measurement is usually described as measuring a “self-adjoint (linear) operator” (or “observable”) (2) where the (real) numbers ai are c alled the eigenvalues of the operator A and are assoc iated with the outc omes of the measurement. The Pi are the projec tors onto the given subspac es.4 These subspac es are c alled the eigenspac es of A and are the subspac es of all vec tors | ψ i) (the eigenvec tors of the operator) suc h that (3) or equivalently (4) This is the origin of the traditional identific ation of quantum mec hanic al observables with (self-adjoint) operators.5 The c ollapse postulate then states that upon measurement of A a state | ψ 〉 will c ollapse onto Pi| ψ 〉 (suitably renormaliz ed), with probability pi = 〈ψ| Pi| ψ〉. The quantity (5) is then the average value or expec tation value of the operator A in the state | ψ 〉. Note that unless the state is an eigenstate of the operator measured, there is a statistic al spread of results, that is, the dispersion of A in the state | ψ〉, (6) is nonz ero. The assoc iation between self-adjoint operators and families of mutually c ompatible tests may seem purely c onventional from the above desc ription. This is not quite so. Self-adjoint operators play a further role in quantum mec hanic s, namely as (mathematic al) generators of the unitary Sc hrödinger evolution. Now, think of a Stern– Gerlac h spin experiment. A Stern– Gerlac h magnet produc es (approximately) a magnetic field that is inhomogeneous in just one spatial direc tion. Classic ally, what suc h a magnetic field c an do is deflec t along this direc tion a partic le with nonz ero magnetic moment, the amount of the deflec tion being proportional to the magnetic moment itself. In quantum mec hanic s, spin operators of the form (7) (with P+ and P− the projec tion operators onto the “up” and “down” spin states in some direc tion) will appear in the Sc hrödinger evolution that c ouples the spin of the partic le to its position degrees of freedom, and the deflec tion experienc ed by the partic le will in fac t be proportional to the eigenvalue + ℏ or − ℏ . In this sense, the 2 2 measurement is indeed sensitive to the eigenvalues of the c orresponding spin operator, and not just to the projec tions of the state on the mutually orthogonal eigenspac es.6 This c loser relation between a measurement and a single self-adjoint operator will be lost in the c ase of the generaliz ed measurements disc ussed in sec tion 4.4. 1.2 Minimal Interpretation and Standard Interpretation The above phenomenologic al rules yield a minimal interpretation of the formalism: some laboratory proc edures are taken to be state preparations, and others are taken to be tests. Quantum mec hanic s yields probabilistic relations between states and outc omes of tests (Born rule). And, depending on their outc ome, tests are assoc iated with further (preparatory) transformations of the state (c ollapse postulate). To be sure, the terms “preparation” and Page 2 of 32

Measurement and Classical Regime in Quantum Mechanics “test” (or “measurement”) are phenomenologic al, but in the c ases in whic h we (or the working physic ist) would normally apply them, any fundamental approach to quantum mechanics must allow us to recover the usual predic tions of the theory, inc luding in partic ular the fac t that future predic tions will depend on the previous outc omes in the way spec ified by the c ollapse postulate. A c ommon alternative interpretation of the formalism (often c alled the “standard” or “orthodox” or “quantum logic al” or “Dirac – von Neumann” interpretation: we shall adopt the first of these terms) takes it that a quantum system has c ertain properties also independently of measurements, namely properties c orresponding to tests that the system passes with probability 1. These properties, whic h are uniquely fixed by the quantum state, c an be further identified either with the state itself (or rather the one-dimensional subspace spanned by the vector state)— as is standardly done in the quantum logic literature, most explic itly by Jauc h and Piron (1969)—or with an eigenvalue assoc iated with that vec tor (henc e also the name “eigenstate-eigenvalue link,” due to Fine (1973), for this interpretational rule)7 . For instanc e, an elec tron in a state of spin up in the x-direc tion will have a property c orresponding to the vec tor |+x〉, or, simply, a value + ℏ for spin in the x-direc tion. Ac c ording to the standard 2 interpretation, a c ollapse of the quantum state is thus an ac tual c hange in the properties of the quantum system. Assuming that quantum mechanics is meant to apply to any physical system whatsoever, and that there should not be a fundamental differenc e in the way it is interpreted ac ross different domains, intuitions from the mic rosc opic and the mac rosc opic domains of applic ation of the theory will pull in different direc tions. Applying the minimal interpretation to mac rosc opic systems would mean that suc h systems will merely appear to have c ertain properties if measured (the Moon is not there until we look). In this domain, something like the standard interpretation would seem more natural (at least prima fac ie). On the other hand, applying the standard interpretation to the mic rosc opic domain would mean that measurements appear to induce a discontinuous change in the properties of a mic rosc opic system, in a way that is not nec essarily c ompatible with the Sc hrödinger equation. This tension is the origin of the measurement problem of quantum mechanics (which we shall eventually discuss in section 4.6). Obviously, the minimal interpretation is an instrumentalist interpretation, while the standard interpretation involves an ontologic al c ommitment to the quantum state. The former c ould be seen as a stripped-down version of some historic ally more ac c urate reading of the “Copenhagen interpretation”. Note also that, while Sc hrödinger c learly had an ontologic al c ommitment to the wave func tion, it is not c lear that it c ould be phrased in the abstrac t terms of the standard interpretation. He appears to have rather been interested in the 3-dimensional manifestation of his wave func tions, in partic ular in terms of c harge density (see also sec tion 3 below). Something like the standard interpretation instead may have been adopted by both Dirac and von Neumann. 2. Density Operato rs and Reduced States 2.1 Density Operators Vec tors in Hilbert spac e, as we have seen, define probability measures over the results of measurements of quantum mec hanic al observables. Indeed, up to phase fac tors, the assoc iation between unit vec tors and suc h probability measures is one-to-one, sinc e it is c lear that if two unit vec tors differ by other than an overall phase fac tor, there will be at least one test (the projec tion onto the subspac e spanned by one of them), for whic h they will define different probabilities.8 To get rid of overall phase fac tors, we c an also identify a quantum state defined by the vec tor | ψ〉 with the one- dimensional projec tion operator onto | ψ 〉, denoted by | ψ 〉 〈ψ | , i.e. the linear mapping that takes any vec tor state | ψ〉 to the state 〈ψ | φ〉| ψ 〉 (the state | ψ〉 multiplied by the c omplex number 〈ψ | φ 〉). This c an be suggestively written as (8) This identific ation is partic ularly useful if one wishes to generaliz e the notion of a quantum state further. Indeed, it is c lear that the probability measures defined by vec tors in Hilbert spac e will not be the most general suc h probability measures. The set of these measures ought to be a convex set, that is, c losed under c onvex sums. One c an write a c onvex sum of two states c orresponding to projec tion operators, say onto | ψ1〉 and | ψ2 〉 as the operator (9) Page 3 of 32

Measurement and Classical Regime in Quantum Mechanics that maps any vec tor | φ〉 to the superposition (10) with p1+p2 = 1. We c an now write the c orresponding probability for the system passing a c ertain test represented by the projec tion P as (11) Here Tr(ρP) is the symbol for the so-c alled trac e of the operator ρP, defined for any operator A as (12) with the | ψi〉 forming a basis of the Hilbert spac e.9 As already mentioned in sec tion 1.1, operators of the form A| ψ 〉 = ai| ψ i〉 c an be used to c lassify simultaneous experimental tests for families of mutually orthogonal subspac es. A system will test positively to only one of these tests, and to this test c an be assoc iated an eigenvalue of the c orresponding operator. Sinc e Tr(ρPi) is the probability for the outc ome i in a test of Pi, the expression (13) is equal to the expec tation value of the self-adjoint operator A. The operator ρ is known as a density operator, bec ause in the expression (11) it plays a role similar to that of a probability density. Note that the one-dimensional projec tion operators are the extremal elements of the c onvex set of density operators, those that c annot be dec omposed further in terms of c onvex c ombinations of other density operators. Now, it is a deep theorem due to Gleason (1957) that the states defined by density operators are the most general probability measures that c an be defined over the possible tests that c an be (ideally) performed on a quantum system. A probability measure in Gleason's sense, as one would expec t, is a positive, normaliz ed mapping that in the finite-dimensional c ase is additive and in the infinite-dimensional c ase σ-additive for families of mutually orthogonal projec tors.10 Quantum mec hanic al states in the sense of density operators c an be alternatively c harac teriz ed as the most general (linear) expectation value functionals on the self-adjoint operators. This is actually what von Neumann shows in what has c ome to be known as his no-hidden-variables theorem (von Neumann 1932, pp. 305– 324 of the English translation). More prec isely, von Neumann takes a state s to be an assignment of an expec tation value to eac h self-adjoint operator A, subjec t to a c ontinuity requirement (whic h is vac uous in finite dimensions), a trivial normaliz ation requirement s(1) = 1, a positivity requirement and a linearity requirement (14) for any two observables A and B and real numbers a and b. He then proves that the only suc h expec tation func tionals on the self-adjoint operators are of the form Tr(ρA), with ρ a density operator. That is, the most general states in this sense are indeed the quantum mechanical states. Von Neumann took this result as showing that there c ould be no more prec ise desc ription of ensembles of quantum mec hanic al systems (in partic ular no states with z ero dispersion for all observables), and thus as ruling out “hidden variables.” Note, however, that von Neumann himself explic itly points out that assumption (14) is natural in the c ontext of c ommuting observables (where we see it is analogous to Gleason's additivity requirement), but is a very nontrivial assumption in the c ase of nonc ommuting ones (pp. 308– 309). As noted forc ibly by Grete Hermann (1935), this vitiates his c onc lusion about hidden variables.11 Avery simple geometric al intuition for the c onvex struc ture of density operators in the c ase of spin-1/2 systems c an be gained as follows. Imagine mapping eac h state of spin-up in the direc tion r to the c orresponding unit vec tor in three spatial dimensions, (16) Page 4 of 32

Measurement and Classical Regime in Quantum Mechanics This mapping between the vec tor states of a spin-1/2 system and the unit sphere is a bijec tion (one-to-one and onto). It turns out that it c an be extended to an affine isomorphism (i.e., a map that preserves c onvex c ombinations). What this means in partic ular is that for any two vec tor states | ψ〉 and | φ〉, whic h are mapped onto unit vec tors r and s on the sphere, we c an map the density operator (17) to the point λr + (1 − λ)s in the interior of the unit ball in three dimensions. This representation is known as the Bloc h sphere or the Poinc aré sphere. We c an use it to establish geometric ally many propositions about density operators. Here are a few examples. Density operators c an be dec omposed nonuniquely as c onvex c ombinations of vec tor states, in fac t in infinitely many ways, and as c ombinations of arbitrarily many vec tor states (even c ontinuously many). On the other hand, for eac h density operator, there is generally a unique dec omposition as a c ombination of spin-up and spin-down in a single direc tion (as a c ombination of antipodal points on the sphere).12 The only exc eption is the state that lies at the c enter of the ball, whic h is the equal-weight c ombination of up and down states in any direc tion (“maximally mixed” state). We also see that the only states that are extremal (also c alled pure states) in the c onvex set of density operators are indeed the vec tor states that map to the unit vec tors on the sphere. 2.2 Proper and Improper Mixtures The nonuniqueness in general of c onvex dec ompositions of a density operator is one of their most striking features, and a major differenc e between probability measures in quantum and c lassic al mec hanic s. Also in c lassic al mec hanic s one c an introduc e states that are c onvex c ombinations of the pure states defined by points in the phase spac e (whic h c orrespond to trivial—or “dispersion-free”—probability distributions). These general states are simply probability measures over phase spac e. But it is always possible to dec ompose a c lassic al probability measure uniquely as a c onvex c ombination of extremal states (a c onvex set with this property is known as a “simplex”). Indeed, both mathematic ally and physic ally, when we deal with a probabilistic state in c lassic al mec hanic s, we are always dealing with a statistical mixture of nonprobabilistic states, that is, probabilities arise through our ignoranc e of the ac tual pure state of the system, and any statistic al distributions of measurement results are attributable to this same ignoranc e. There is no possible ambiguity, sinc e the spac e of c lassic al probability measures is a simplex. In quantum mec hanic s, things are different. Even though formally density operators c an always be written as “mixtures” (i.e., as c onvex c ombinations of pure states), at the very least their nonunique dec omposability will introduc e an ambiguity in their interpretation. Assuming that in some c ase a density operator has arisen through our ignoranc e of the ac tual pure state of the system, this is not manifest in the form of the density operator. We might know that the spread of results observed in our tests is partly due to our ignoranc e of what the quantum state ac tually is, and partly due to the probabilistic nature of the vec tor states themselves, but knowledge of how to thus “apportion the blame” is knowledge in exc ess of that enc oded in the density operator itself. It c orresponds formally not just to the density operator, but to a partic ular c onvex dec omposition. Unlike the c lassic al c ase, this dec omposition c annot be uniquely retrieved from the state alone. This feature of quantum mechanical “mixtures” is essential to the question of how they should be understood, espec ially in the c ontext of our distinc tion between the minimal and standard interpretations of the theory. There is, however, an even more essential issue for the question of how to understand density operators. Of c ourse, density operators c an arise as genuine statistic al mixtures of pure quantum states (for instanc e a state obtained by randomly mixing systems prepared in different pure states). This is generally referred to as a proper mixture. So, for instanc e, if we know that a measurement of spin-x on an elec tron has been ac tually c arried out, but we are ignorant of the result, then we should apply the c ollapse postulate, but average over the results (so-c alled nonselec tive measurement). In this c ase we will have a proper mixture of the states | +x〉 and | −x〉 due to ignoranc e (we do not know whic h state we should ac tually best use for further predic tions).13 However, there are other c ases in whic h density operators arise that are not thus related to our ignoranc e, namely as so-c alled reduced states, states of subsystems of a larger system desc ribed by an entangled pure state. Indeed, the phenomenologic al rules sketc hed in sec tion 1.1 (c ollapse postulate and Born rule) turn out to have Page 5 of 32

Measurement and Classical Regime in Quantum Mechanics surprising c onsequenc es when applied to the c ase of entangled states. Take a singlet state of two spin-1/2 systems (18) ait nlide ste inst afo sru Pb+s1px  a⊗ce P o+2rtxh. oTghoen tael stto w thille c toemstee do uotn nee. gNaotwiv ete wsti tfho rp Pro+1bxa  b⊗ili tPy− 21x,. aTnhde trhees ustlta wtei lwl bilel b∣+e 1xu⟩n d⊗ist u∣−rb2xe⟩d o, rs ince ∣−x1 ⟩  ⊗  ∣+x2 ⟩, each with probability 1/2. In this case, we see that the results of the spin measurements performed on the two elec trons are perfec tly (anti-)c orrelated. Correlations, albeit weaker, will be observed quite in general if spin is measured along two different direc tions on the two subsystems (as c an be easily c hec ked explic itly). Entanglement thus introduces what appear to be irreducible correlations between results of measurements (even c arried out at a distanc e), and this for a generic pair of tests. This is the origin of nonloc ality in quantum mec hanic s. On the other hand, performing a measurement (or any other manipulation) on one of a pair of entangled partic les does not affec t the probability distributions for results of measurements on the other. This is the so-c alled no- signaling theorem. (That is, while c onditionaliz ing on the outc omes of one measurement in general affec ts the probabilities for the other, c onditionaliz ing on performing the measurement does not.) It is easy to see this in the example: we have perfect anti-correlations between outcomes on the two sides, but averaging over the outcomes on one side yields bac k the usual 50– 50 distribution on the other side. By explic it c alc ulation, one c an c hec k the c laim in the general c ase, that is, for measurements along different spin direc tions on the two sides. The no-signaling theorem is c ruc ial to our purposes, sinc e it allows us to generaliz e the desc ription of quantum systems to subsystems of entangled systems. Indeed, although such subsystems cannot be associated with any vec tor in their Hilbert spac e, we c an assign them a suitable probability measure for eac h test we may want to c arry out on them, bec ause the no-signaling result guarantees that the probability of suc h a test is well-defined independently of whether any test (or whic h one) is c arried out on the rest of the system. So, we c an define a probability measure for a test on a subsystem by simply taking the marginal of the probability measure assoc iated with the entangled state of the total system when the relevant test is paired with an arbitrary test on the rest of the system. But now, bec ause of Gleason's theorem, we know that suc h a state must be given by a density operator. Let us see this in a c onc rete example. Suppose we wish to define the probability for a measurement of spin-x on one of a pair of spin-1/2 systems in some arbitrary entangled state. We c an write the state of the pair as (19) If we were to measure spin-x on both elec trons of the pair, the resulting Born-rule probabilities would be (20) and averaging over the results for the sec ond elec tron, we obtain (21) In this way, one c an determine the probabilities for arbitrary tests on the first (and similarly on the sec ond) elec tron, and so assoc iate with it a state in Gleason's sense (a probability measure for any family of mutually orthogonal projec tions), even though it is not desc ribed by a vec tor in Hilbert spac e. A more c ompac t way of thinking of suc h a state is in terms of a c onvex c ombination of the states that one would obtain through the c ollapse postulate were one to perform a measurement on the other elec tron. So, for instanc e, if one were to perform a measurement of spin- x on the first elec tron, one would obtain the two (normaliz ed) states: (22) and (23) Page 6 of 32

Measurement and Classical Regime in Quantum Mechanics Writing (24) and (25) we see that the state of the sec ond elec tron would c ollapse to | ψ+〉 or | ψ −〉 with the probabilities p+ and p− (defined by (21)), respectively. We c an now determine the probabilities for any tests on the sec ond elec tron by taking the weighted average of the probabilities defined by | ψ +〉 and | ψ −〉, with the weights p+ and p−, respec tively. We c all this the reduced state of the sec ond elec tron, and write it formally as (26) This representation makes it explic it that a reduc ed state is a density operator. Furthermore, the no-signaling theorem shows us explic itly that the representation (26) c annot be unique. If a different measurement were to be c arried out on the first elec tron, then the states (24) and (25) would have to be different, if the total state is entangled, and the c orresponding probabilities would generally also be different. As a simple example, take the singlet state (18). Measuring spin in direc tion r on the first elec tron will c ollapse the sec ond elec tron into a state of spin in the same direc tion r, whatever this might be, due to the rotational symmetry of the state. Thus, the reduc ed state of an elec tron from a pair in the singlet state will have the form (27) (in the c ase of the singlet the probabilities for the different results will always be equal to 1/2) and will be independent of r.14 How are we to interpret density operators arising as reduc ed states of entangled systems? Certainly not as proper mixtures! Indeed, if a c omposite quantum system is in a pure entangled state, this state c annot be further dec omposed as a weighted average of other quantum states, so c annot be interpreted in terms of ignoranc e. But then, neither c an the states of the subsystems be interpreted in terms of ignoranc e, despite the fac t that the subsystems are necessarily described by density operators. Contrapositively, were the subsystems themselves in pure states (and we ignorant of whic h pure states they were in), then the c omposite would be in a mixed state, bec ause it would ac tually be in a produc t state (but we ignorant of whic h produc t state it was in). A mixed state arising as the reduc ed state of a subsystem, where the total system is in a pure state, is generally referred to as an improper mixture. The reduc ed state of an elec tron from an entangled pair is a paradigm example of an improper mixture, so that a dec omposition suc h as (26) should not be taken as indic ating that the system is indeed either in the state |ψ+〉 or in the state |ψ−〉. At least from the point of view of the minimal interpretation, there is nothing espec ially problematic about this. Quantum systems have dispositional properties to elicit certain outcomes under certain test circumstances, irrespec tively of whether we seek to explain them further. If we do seek to explain these further, the c ase of subsystems of entangled systems will turn out to be partic ularly tric ky, but from the point of view of the minimal interpretation it is perfec tly natural for subsystems of entangled systems to have suc h dispositional properties. The only aspec t of note is that in the c ase of suc h subsystems we explain the distributions over outc omes purely in dispositional terms (just as in the c ase of systems in pure states), while in other c ases, we may have reason to analyz e the distributions over outc omes partially in terms of ignoranc e. Instead, the existence of entanglement and reduced states has rather disquieting consequences for the standard interpretation. Indeed, if the system is neither in the state |ψ+〉 nor in the state |ψ−〉 (nor in any other state that might appear in a c onvex dec omposition of the density operator of the system), then the system simply lac ks the Page 7 of 32

Measurement and Classical Regime in Quantum Mechanics properties that in the standard interpretation are assoc iated with these states. We c an still apply the standard interpretation and assoc iate properties of the system with tests that the system will pass with probability 1. In general, however, these properties will no longer c orrespond to one-dimensional subspac es of the Hilbert spac e, but only to higher-dimensional ones (the name “eigenstate-eigenvalue link” becomes a bit of a misnomer in this c ase).15 In extreme c ases, suc h as with two entangled elec trons (where eac h elec tron's spin spac e is itself only two-dimensional), the individual elec trons will have no nontrivial spin properties: the only test they pass with probability 1 is the trivial one testing the projec tion onto the whole of the Hilbert spac e! 2.3 The Bit Commitment Problem We shall c onc lude this sec tion with an example illustrating both the notion of density operators and some of the mystery surrounding entangled states. Bec ause a mixed state c harac teriz es all statistic al predic tions of quantum mec hanic s for measurements on a system, it is impossible, by means of measurements performed on that system, to distinguish whether a density operator c orresponds to a proper mixture or an improper mixture, or whic h proper mixture (if any) it c orresponds to. This c an be illustrated with an example from quantum information theory, the so- c alled bit c ommitment problem. The problem is as follows: Alic e c ommits herself to sending Bob a definite bit of information (0 or 1). She then sends it, and Bob rec eives it. How c an he make sure that what she sends is, indeed, what she had c ommitted herself to? (In whatever sc heme we devise we must additionally ensure that Bob does not infer the ac tual bit of information any sooner than when Alic e in fac t sends it.) Example: Alic e and Bob are on the phone, and they dec ide to bet on something. First Alic e tosses a c oin. Then Bob c hooses heads or tails. Finally, Alic e tells him whether it was heads or tails that had c ome up. The protoc ol is fair (or safe) if Bob is sure that Alic e does not lie and if Alic e is sure that Bob did not know the outc ome of her toss before he c hose heads or tails. There is an obvious c lassic al solution to this problem (assuming Bob is not an expert loc k-pic ker): Alic e writes the result of her toss on a piec e of paper (1 for “heads,” 0 for “tails”), puts it into a safe, sends the safe to Bob but keeps the key. After Bob has c hosen heads or tails, Alic e sends the key as well. The question is now whether there is a quantum solution to this problem that is rigorously fair (and c ould be implemented by sending just a few elec trons instead of keys and safes). Here is an attempt to realiz e this. (One c ould also phrase it in terms of polariz ation states of photons, in whic h c ase Alic e c ould send them along a more or less standard optic al fiber as used in telec ommunic ations.) Alic e takes some random sequenc e of z eros and ones, say 1100010101110010…, and prepares a c ollec tion of elec trons as follows. If the result of her c oin toss (her “bit c ommitment”) is “heads,” she prepares the elec trons one after the other as spin-up (for 1) and spin-down (for 0) in the x-direc tion; if her result is “tails,” she does exac tly the same, but with spin states in the y-direc tion. She then sends the elec trons, in sequenc e, to Bob. At this point, Bob has an ensemble of elec trons. We assume he knows that Alic e has prepared them either in x-spin states or in y-spin states, but sinc e the sequenc e is random, there are as many up states as down states on average. Sinc e further (28) the ensemble is c harac teriz ed by the maximally mixed state, irrespec tive of whether Alic e had got heads or tails. As this c harac teriz ation gives the maximal information Bob c an extrac t from the ensemble by making measurements, he has no way of telling whether Alic e has prepared the elec trons in x- or y-spin states. At a later stage, Alic e tells Bob whic h way she had prepared the elec trons, together with the random sequenc e she used. Now Bob c an ac tually c hec k whether Alic e is telling the truth. Indeed, if he makes a sequenc e of measurements on the elec trons, in the order they were sent, then, if the direc tion of his measurements is the same as the one in whic h they have been prepared, say x, he will reproduc e the random sequenc e told him by Alic e; but if the direc tion of his measurements is the other one, say y, then he will obtain a c ompletely new random sequenc e whic h is unrelated to the first (and whic h Alic e c ould thus not have antic ipated). Thus, the fac t that no information on top of that provided by the density operator is available, in partic ular about how a proper mixture has been prepared, provides a “safe” in whic h the ac tual information about the result of Alic e's toss is inac c essible without a “key.” Page 8 of 32

Measurement and Classical Regime in Quantum Mechanics But the same fac t gives Alic e also the possibility of c heating. Instead of sending Bob one of the two above proper mixtures, Alic e c an send him, say, the right-hand elec trons from an ensemble of pairs prepared in the singlet state. Sinc e it is impossible for Bob to tell whether the state he rec eives, namely again the maximally mixed state, is a proper or improper mixture, he sees no differenc e between this c ase and the previous c ase. But in this situation, Alic e c an wait for Bob to c hoose heads or tails and then perform a sequenc e of, respec tively, spin-x or spin-y measurements, tell Bob she had done that before sending him the elec trons (as a way of preparing the c orresponding proper mixture by way of c ollapsing the state), and tell him the sequenc e of results she obtains (exc hanging ones and z eros). Sinc e results of spin measurements on pairs of elec trons in the singlet state are perfec tly anti-c orrelated, when Bob measures his elec trons, he obtains, indeed, the sequenc e Alic e has told him, not suspec ting that Alic e has just then c ollapsed the elec trons into the states he measures. Thus, the indistinguishability of proper and improper mixtures prevents Bob from finding out that Alic e is c heating, while the objec tive difference between proper and improper mixtures (namely, in terms of the state of the c omposite system) makes all the differenc e for Alic e in enabling her to c heat in the first plac e! This situation turns out to be extremely general. If a protoc ol for bit c ommitment is based on the idea that a density operator c ould be one of two different proper mixtures, whic h information is then disc losed later on, then there always exists a c heating strategy based on the fac t that this same density operator c ould be an improper mixture. This result is c alled the no-go theorem for safe bit c ommitment protoc ols (Lo and Chau 1997; Mayers 1997). 3. Classical Regime and Deco herence The problem of the c lassic al regime is the question of whether and how the sweeping suc c ess of c lassic al physic s (in partic ular on the mac rosc opic sc ale) c an be explained in quantum mec hanic al terms. While in the philosophic al literature it is the measurement problem that usually takes pride of plac e, the problem of the c lassic al regime is equally important in assessing the empiric al adequac y of quantum theory and its interpretations. In this sec tion we shall look at this problem as it is generally viewed today, through the eyes of dec oherenc e theory. To fix the ideas, however, we start with a c ouple of early examples of work related to this problem. Sc hrödinger (1926) c ontributed a seminal paper on the c lassic al regime, in whic h he showed that Gaussian wave pac kets for the harmonic osc illator maintain their shape and siz e (narrow in both position and momentum) and follow the trajec tories predic ted by Newtonian mec hanic s. He believed this provided the model for the relation between “mic romec hanic s” and “mac romec hanic s.” Another early treatment of “c lassic al” trajec tories was given by Heisenberg (1927) in his analysis of α -partic le trac ks as emerging through repeated c ollapse of the wave func tion in a bubble c hamber. An alternative treatment of α -partic le trac ks was given by Mott (1929), who showed that the wave func tion of the c ombined system of α -partic le and gas was c onc entrated on c onfigurations in whic h the gas was ioniz ed along straight lines. These examples (at least in hindsight) represent rather different approaches to understanding the problem of the c lassic al regime, c harac teriz ed by different (or potentially different) interpretational approac hes. Sc hrödinger had an ontologic al c ommitment to the wave func tion. At the time, he thought of it as representing (or manifesting itself as) a c harge density in 3-dimensional spac e. Thus, in order to rec over a c lassic al regime, it is essential in a Sc hrödinger-like approac h to identify quantum states that are both kinematic ally and dynamic ally like c lassic al states, that is, for whic h the c lassic al quantities suc h as position and momentum are both approximately well- defined and evolve in an approximately c lassic al manner.16 As for Heisenberg, it appears that at the time he did not even believe in the existenc e of wave func tions, but only in the transition probabilities between values of (measured) quantum mec hanic al observables.17 For suc h a view, it is essential that the transition probabilities defined by the Born rule reduc e approximately to 0 or 1 for results of measurements performed along classical trajectories. Thus, such an approach (if applied consistently throughout, in partic ular up to the mac rosc opic sc ale) arguably aims at an instrumental rec overy of the predic tions of c lassic al mec hanic s. The standard interpretation and the minimal interpretation of quantum mechanics that we have introduced in sec tion 1.2 c an be seen as sanitiz ed versions of the approac hes by Sc hrödinger and Heisenberg, respec tively. Instead, Mott's treatment is an early example of a dec oherenc e analysis, in whic h no c ollapse need be invoked to Page 9 of 32

Measurement and Classical Regime in Quantum Mechanics destroy the interference between the wave components corresponding to the different trajectories. As I see it, a dec oherenc e-based approac h is best viewed as interpretationally neutral, but as providing a very powerful tool for any approac h to the problem of the c lassic al regime. A beautiful example of the importanc e of the problem of the c lassic al regime for foundational issues is given by Einstein's (1953) c ontribution to the Edinburgh Festschrift for Max Born. Einstein desc ribes a mac rosc opic ball (of 1 mm diameter), bounc ing elastic ally to and fro inside a box along the direc tion x. The wave func tion of the ball is given by a standing wave, whic h fills the entire box, and has a similarly spread-out distribution in momentum. Ac c ording to Einstein, Born's statistic al interpretation provides an adequate desc ription of the situation for an ensemble of systems (at least ac c ording to his own reading of Born). However, an individual ball must have a well- defined mac rosc opic state, and that is not desc ribed by the wave func tion. To the objec tion that the Sc hrödinger equation has other solutions, that are suffic iently loc aliz ed in position and momentum, Einstein replies that these solutions will spread out in time. Einstein c onsiders also two attempts at interpretation of the wave func tion alternative to Born's. One is de Broglie– Bohm theory, in whic h a partic le will have a well-defined trajec tory guided by the wave func tion.18 In Einstein's example, however, the veloc ity of the ball will be equal to z ero, so that, in Einstein's view, de Broglie– Bohm theory fails to provide the c orrec t mac rosc opic desc ription of the ball as bounc ing to and fro inside the box. The other one is Sc hrödinger's idea of the wave func tion literally desc ribing a wavelike nature of material partic les (whic h in some form Sc hrödinger had rec ently returned to). In this c ase, however, even the mac rosc opic ball is a wavelike objec t filling the whole box, and Einstein's verdic t is that also Sc hrödinger's reading of the wave func tion fails to do justic e to the c lassic al regime. Einstein's own c onc lusion is that a statistic al interpretation of the wave func tion in the sense he attributes to Born is the appropriate interpretation to give to the theory. 3.1 Coherent States and Ehrenfest's Theorem The first question we shall disc uss now is the sense in whic h one c ould talk of a quantum state as being approximately c lassic al and behaving approximately c lassic ally, in the spec ial c ase of pure quantum states. The obvious c andidates are wave func tions with a small spread both in position and in momentum (small c ompared to some mac rosc opic sc ale). This was Sc hrödinger's initial guess as to the appropriate c andidates for the desc ription of c lassic al partic les in quantum mec hanic s. The Heisenberg unc ertainty relations give a lower bound for the produc t of the spreads in position and in momentum, but for suffic iently massive (“mac rosc opic ”) systems, this in itself is a very small limitation. For instanc e, it is c ompatible with the unc ertainty relations that a system has a spread in position of 10−13 c m and a spread in momentum of 10−13 gc m/s. If the system has a (mac rosc opic ) mass of 1 g, the latter c orresponds to a spread in veloc ity of 10−13 c m/s. If we are merely interested in desc ribing our system on suc h a mac rosc opic sc ale, we c an reasonably say that the system has both a well-defined position and a well-defined momentum. Note that suc h a wave func tion will typic ally be nonz ero everywhere both in position spac e and in momentum spac e. “Small spread” means that the “bulk” of the wave func tion is loc aliz ed. Indeed, it is well-known that those wave func tions that attain the lower bound given by the unc ertainty princ iple are Gaussian wave pac kets (i.e., they have the shape of Gaussian bell curves when represented either as functions of position or as functions of momentum), and as suc h have infinite “tails.”19 It is obvious, on the other hand (as in Einstein's example), that even for very massive systems there are states with mac rosc opic ally large spreads. For instanc e, take ψ1 and ψ2 to be two quantum states of a mac rosc opic system with very small spreads, but with mac rosc opic ally different average values of position and momentum, say x and p in one case, x′ and p′ in the other. Then the state 1/√2(ψ1   + ψ2  ) will have spreads of the order of |x − x′| and |p − p′|. An obvious question is thus whether states with small spreads in both position and momentum remain such under the quantum evolution. With regard to this, as already mentioned, Sc hrödinger (1926) made the following disc overy. Gaussian wave func tions for a harmonic osc illator (i.e., with the potential proportional to the square of position, e.g., an ideal spring) keep exac tly the same shape and move exac tly along the c lassic al trajec tories. These states, whic h are both kinematic ally and dynamic ally “c lassic al” are c alled the coherent states of the harmonic osc illator.2 0 Sc hrödinger was led by this result to think that all c lassic al behavior c ould be explained in these terms by quantum mec hanic s and, indeed, the result c an be generaliz ed in various ways. But we shall see Page 10 of 32

Measurement and Classical Regime in Quantum Mechanics that this hope was misplac ed. A simple way of generaliz ing these results, at least in part, is as follows. For short, write 〈A〉 to mean 〈ψ(t)| A| ψ(t)〉, that is, the average value of an operator A in the state |ψ(t)〉 (see equation (5)). Then, with m the mass of the partic le, Q and P the position and momentum operators, and V(Q) the operator representing the potential (whic h is a func tion of position), one c an derive the two parts of Ehrenfest's theorem: (29) (the average momentum is mass times the time derivative of the average position), and (30) (the time derivative of the average momentum is equal to the average forc e). Thus, the average position and momentum almost obey Newton's sec ond law, with the qualific ation that the c lassic al value of the forc e at the average position is replac ed by the average value of the forc e. This holds for all quantum states, but if the state has a small spread in position, the average value of the forc e is approximately equal to the value of the c lassic al forc e. Thus, a state with a small spread in position will follow an approximately c lassic al trajec tory as long as its spread remains small (at least if the external potential in whic h it moves is uniform enough on the sc ale over whic h the state is spread).21 Do position spreads remain small? In the c ase of a Gaussian, any inc rease in the position spread leads to a dec rease in momentum spread and vic e versa. Typic ally, under the unitary evolution, the spread in position inc reases.2 2 In the simple c ase of no potentials (“free Gaussian”), if the system has mac rosc opic mass, the spread of the state will remain small for a very long time. For a system with mass 1 g, starting off in a Gaussian state with position spread 10−13 c m, it will take 600 years for the spread in position to inc rease to 10−4 c m, and it will take another 6,000 million million years for it to further inc rease to twic e that siz e. If potentials are present, the spreading c an be enhanc ed or c ounterac ted, for example, if the wave func tion is in a potential well it may stay trapped there. In the c ase of the hydrogen atom, the spreading of wave func tions was pointed out to Sc hrödinger by Lorentz in their well-known c orrespondenc e of 1926 (published in Prz ibram 1963). In partic ular, Lorentz showed that elec trons in the hydrogen atom would be spread out over their entire orbits, even for the c ase of high-energy orbits. The examples so far are somewhat mixed, and one might think that Sc hrödinger's intuition might yet prove sound at least for suffic iently mac rosc opic systems. That is prec isely what Sc hrödinger replied to Einstein upon rec eipt of his draft for the Born Festschrift, to whic h Einstein replied that one c ould repeat the c alc ulation taking not a 1 mm ball but a dust partic le, and get a spread-out state within 24 hours!2 3 Regardless of the quantitative details, the disc ussion so far has presupposed that the state of our system always remain a pure state. That is, the time evolution equation of the system (the Sc hrödinger equation) may inc lude external potential terms, but it inc ludes no interac tion terms. If quantum interac tions are inc luded, however, the pic ture c hanges dramatic ally. And that is the c ase we really need to disc uss. 3.2 Entanglement with the Environment If two quantum systems do not interac t, the state of eac h system will evolve (unitarily) within the Hilbert spac e that desc ribes that system, and the state of the c omposite system (if initially a produc t state!) will always retain its produc t form, | ψ (t)〉| φ (t)〉. If the two systems interac t, instead, the state of the c omposite system will evolve (unitarily) within the produc t Hilbert spac e, and in general the state of the c omposite system will have the entangled form (31) We c an use this state in the standard way to make predic tions for the c omposite system, as well as for either subsystem (in partic ular to c alc ulate the spread in position or in momentum of either subsystem). Indeed, a measurement on a subsystem is just a spec ial kind of measurement on the c omposite system, so the usual formalism applies. Equivalently, as disc ussed already in sec tion 2.1, we c an make predic tions for measurements on Page 11 of 32

Measurement and Classical Regime in Quantum Mechanics a subsystem using the reduc ed state of that subsystem, whic h is an improper mixture that takes the form, say, (32) In some c ases, it maybe more c onvenient to write this as an integral, for instanc e over Gaussian wave pac kets c entered at different positions (although as mentioned in sec tion 2.1 a dec omposition of the form (32) always exists). If the c omponent states are Gaussians with mac rosc opic ally different average positions (and/or momenta), the spreads of the state now c an be mac rosc opic ally large, just as with pure states that are sums of suc h Gaussian wave pac kets. Rec all that improper mixtures are not ignoranc e-interpretable, so that a mac rosc opic ally large spread in position or momentum that arises in this way through quantum interactions cannot be discussed away simply by applying an ignoranc e interpretation to the mixed state. Suc h a state appears to be genuinely nonc lassic al. Thus, we have to ask whether interac tions typic ally lead to mixed states with large spreads, or whether we c an find a regime in whic h these spreads remain small. Now, however, it is c learly the c ase that quite c ommon interac tions do in fac t lead to suc h apparently nonc lassic al states. One c lass of interac tions that lead to mixtures of mac rosc opic ally different states are measurement interac tions, as with Sc hrödinger's (1935) own example of the c at. Although the sc enario is well-known, here is the desc ription of the thought experiment, as given by Schrödinger himself: A c at is penned up in a steel c hamber, along with the following diabolic al devic e (whic h must be sec ured against direc t interferenc e by the c at): in a Geiger c ounter there is a tiny bit of radioac tive substanc e, so small, that perhaps in the c ourse of one hour one of the atoms dec ays, but also, with equal probability, perhaps none; if it happens, the c ounter tube disc harges and through a relay releases a hammer whic h shatters a small flask of hydroc yanic ac id. If one has left this entire system to itself for an hour, one would say that the c at still lives if meanwhile no atom has dec ayed. The first atomic dec ay would have poisoned it. The ψ – func tion of the entire system would express this by having in it the living and the dead c at (pardon the expression) mixed or smeared out in equal parts. Suc h an example c learly provides a link between the problem of the c lassic al regime and the problem of measurement. We shall postpone disc ussion of the latter, however, sinc e the two problems are distinc t. In the c ase of the measurement problem, we have a spec ial c ase of failure or apparent failure of c lassic ality at the kinematic al level, but the observed behavior of a measuring apparatus (when c oupled to the measured system) is ac tually far from c lassic al (thus we need not worry about rec overing c lassic al dynamic s). The spec ial twist of the measurement problem is that preparations and measurements are what is needed to apply quantum mec hanic s in the first plac e: if it turned out that these c ould not be analyz ed theoretic ally, the theory would in some sense be undermining itself.24 From the point of view of the c lassic al regime, however, something perhaps even more startling happens, namely that very c ommon and spontaneous interac tions of a system with its environment lead to the same kind of states with large spreads. To fix the ideas, think at first of a pair of c oupled harmonic osc illators and start them off in the nonentangled state (33) Both c lassic ally and quantum mec hanic ally, two c oupled osc illators will rec urringly exc hange energy, that is, evolve to and fro between this state and the nonentangled state | first exc ited〉 | ground〉. But quantum mec hanic ally, this will happen through intervening stages of the form (34) whic h are entangled; and the single osc illators will be c orrespondingly in mixtures of their ground and first exc ited states. As above, these mixed states arise from quantum interac tions and the ensuing entanglement. Thus, they do not allow for an ignorance interpretation. Now imagine a harmonic osc illator c oupled to a thermal bath of harmonic osc illators. It will be taking energy from Page 12 of 32

Measurement and Classical Regime in Quantum Mechanics and giving energy to all of them. If initially the osc illator and the bath are unentangled, the rec urrenc e time for disentangling again c ould be arbitrarily long (or infinite), and in general the state of the osc illator maybe a mixture of any of its energy states. Indeed, if the osc illator is assumed to be in thermal equilibrium with its environment, its quantum mec hanic al desc ription is a mixture of all its energy states. The spread in position and momentum c an be c alc ulated in various ways. One rather suggestive way uses the fac t that for high temperatures one c an rewrite the equilibrium state as a mixture of all possible c oherent states of the osc illator, with weights depending on their energy. One c an thus pic ture the osc illator as roughly spread out over the c lassic al trajec tories c orresponding to the most probable energies (see, e.g., Donald 1998). This example illustrates very well the following general idea, whic h I owe to Matthew Donald. While in c lassic al statistic al physic s we may think of equilibrium states, at least intuitively, as desc ribing our ignoranc e of the ac tual mic rostate of a system, quantum equilibrium states should generally be thought of as improper mixtures: there is no matter of fac t about whic h pure state desc ribes the system, and any mac rosc opic spreads resulting from the weighted average in the mixture are genuine nonclassical features. A mac rosc opic osc illator will c learly not draw in enough energy to be spread out over mac rosc opic sc ales, if the environment is, say, at room temperature (a c lassic al osc illator will not start jittering on a mac rosc opic sc ale); but as a matter of fac t, one c an easily think of systems that are muc h more sensitive to the influenc e of a thermal environment, and are thus highly problematic from the point of view of justifying an approximate desc ription in terms of c lassic al physic s. One example is a molec ule of gas in equilibrium in a box. Every suc h molec ule will be spread out over the entire volume of the box (Donald 1998). Thus, deriving c lassic al statistic al physic s from quantum mec hanic s is part of the problem of the c lassic al regime (c f. also Wallac e 2001). Another possible example is that of a Brownian partic le suspended in a fluid. Our c lassic al intuition is that it is tossed around by the molec ules of the fluid, whic h influenc e the partic le's motion in a very irregular way. If, however, the interaction of the Brownian particle with its environment is treated quantum mechanically, it would seem that its state will be an improper mixture spread over all its c lassic ally possible positions. Radioac tive dec ay always involves entanglement with the environment, and if the emitted radiation c auses a c arc inogenic mutation that kills a c at, this is only one c omponent in a c omplic ated entangled state (that inc ludes not only the undec ayed c omponent, but also c omponents desc ribing dec ays at different times). The similarity with Sc hrödinger's c at is not ac c idental: this is prec isely a Sc hrödinger c at, but arising spontaneously, without the need for the experimenter's “diabolical device.” A little thought will multiply the examples. “Environmental” interac tions suc h as these are c learly ubiquitous. And if this is what they lead to, then it is c lear that, at least in its original form, Sc hrödinger's approac h to the problem of the c lassic al regime is doomed to failure. 3.3 Decoherence and the Classical Regime Luc kily, the same interac tions that lead to entanglement with the environment also provide at least a c ruc ial ingredient for the resolution of the problem, bec ause they also induc e decoherence between the various c lassic al c omponents they superpose.2 5 To explain the c onc ept of dec oherenc e, let us first look at a very elementary example, namely the two-slit experiment. One repeatedly sends elec trons or other partic les through a sc reen with two narrow slits, the partic les impinge upon a sec ond sc reen, and we ask for the probability distribution of detec tions on the surfac e of the sc reen. In order to c alc ulate this, one c annot just take the probabilities of passage through the slits, multiply with the probabilities of detec tion at the sc reen c onditional on passage through either slit, and sum over the c ontributions of the two slits. There is an additional “interferenc e term” in the c orrec t expression for the probability, and this term depends on both of the wave c omponents passing through one or the other slit. There are, however, situations in whic h this interferenc e term (for detec tions at the sc reen) is not observed, that is, in whic h the c lassic al probability formula applies. This happens for instanc e when we perform a detec tion at the slits, whic h at least phenomenologic ally induc es a c ollapse of the wave func tion. The disappearanc e of the interferenc e term, however, c an happen also spontaneously, when no detec tion at the sc reen is performed, for instanc e if suffic iently many “stray partic les” sc atter off the elec tron between the slits and the sc reen. In this c ase, Page 13 of 32

Measurement and Classical Regime in Quantum Mechanics the reason why the interference term is not observed is because the electron has become entangled with the stray partic les, and the results of any observation on the elec tron are determined by its reduc ed state alone. As in our disc ussion of reduc ed states in sec tion 2.2, the probabilities for results of measurements performed only on the elec tron are c alc ulated as if the wave func tion had c ollapsed to one or the other of its two c omponents. The intuitive pic ture is one in whic h the environment monitors the system of interest by c ontinuously “measuring” some quantity c harac teriz ed by a set of “preferred” states (“eigenstates of the dec ohering variable”). Interac tion potentials are func tions of position, so the preferred states will tend to be related to position, or to be in fac t joint approximate eigenstates of position and momentum (sinc e information about the time of flight is also rec orded in the environment), that is, c oherent states. The loc aliz ation thus ac hieved c an be on a very short length sc ale, that is, the c harac teristic length above whic h c oherenc e is dispersed (c oherenc e length) c an be very short. A spec k of dust of radius a = 10−5 c m floating in air will have interferenc e suppressed between (position) c omponents with a width of 10−13 c m. Even more startlingly, the timesc ales for this proc ess are minute. The above c oherenc e length is reac hed after a mic rosec ond of exposure to air. One c an thus argue that generic ally the states privileged by dec oherenc e at the level of c omponents of the quantum state are loc aliz ed in position or both position and momentum, and therefore kinematic ally c lassic al. (One should be wary of overgeneraliz ations, but this is c ertainly a feature of many c onc rete examples that have been investigated.) What about c lassic al dynamic al behavior? Interferenc e is a dynamic al proc ess that is distinc tively quantum, so, intuitively, lac k of interferenc e might be assoc iated with c lassic al-like dynamic al behavior. To make the intuition more prec ise, think of the two c omponents of the wave going through the slits. If there is an interferenc e term in the probability for detec tion at the sc reen, it must be the c ase that both c omponents are indeed c ontributing to the partic le manifesting itself on the sc reen. But if the interferenc e term is suppressed, one c an at least formally imagine that eac h detec tion at the sc reen is a manifestation of only one of the two c omponents of the wave func tion, either the one that went through the upper slit, or the one that went through the lower slit. Thus, there is a sense in whic h one c an rec over at least one dynamic al aspec t of a c lassic al desc ription, a trajec tory of sorts: from the sourc e to either slit (with a c ertain probability), and from the slit to the sc reen (also with a c ertain probability). That is, one rec overs a “c lassic al” trajec tory at least in the sense that formally the probabilities reduc e to those of a classical stochastic process. In the c ase of c ontinuous models of dec oherenc e with interac tions based on the analogy of approximate joint measurements of position and momentum, one c an do even better. In this c ase, the trajec tories at the level of the c omponents (the trajec tories of the preferred states) will approximate surprisingly well the c orresponding c lassic al (Newtonian) trajec tories. Intuitively, one c an explain this by noting that the preferred states are the states that themselves tend to get least entangled with the environment, so they will tend to follow the Sc hrödinger equation more or less undisturbed. But in fac t, as we have seen from Ehrenfest's theorem, narrow wave pac kets follow approximately Newtonian trajec tories. Thus, the resulting “histories” will be c lose to Newtonian ones on the relevant sc ales.2 6 The most intuitive physic al examples for this are the observed trajec tories of α -partic les in a bubble c hamber, whic h are indeed extremely c lose to Newtonian ones, exc ept for additional tiny “kinks.” Indeed, one should expec t slight deviations from Newtonian behavior. These are due both to the tendenc y of the individual c omponents to spread, and to the detection-like nature of the interaction with the environment, which further enhances the c ollec tive spreading of the c omponents (a narrowing in position c orresponds to a widening in momentum). These deviations appear as noise, that is, partic les being kic ked slightly off c ourse.2 7 Other examples will inc lude trajec tories of a harmonic osc illator in equilibrium with a thermal bath (so the dec omposition we mentioned above is not just suggestive, but in fac t quite ac c urate), and trajec tories of partic les in a gas (whic h are a prec ondition for then applying c lassic al derivations of thermodynamic s from c lassic al statistic al mec hanic s). Thus we see that dec oherenc e provides us with tantaliz ingly c lassic al struc ture, both kinematic al and dynamic al, at the level of c omponents of the wave func tion. It is thus natural to assume that it will play a c ruc ial role in any resolution of the problem of the c lassic al regime. Whether it c an play suc h a role and how, however, will depend on the interpretational approach one adopts toward quantum mechanics. Let us take first the minimal interpretation of the theory, ac c ording to whic h quantum mec hanic s is about the results Page 14 of 32

Measurement and Classical Regime in Quantum Mechanics of preparations and measurements, and merely provides a probabilistic link between these two. If one adopts this view, the problem of the c lassic al regime is a question about the results of measurements performed on c ertain “c lassic al,” generally mac rosc opic systems (or possibly c ertain elements of their environment). Dec oherenc e tells us that it is indeed possible to isolate a c lassic al regime (at least one2 8 ) for whic h appropriate measurements will reveal either ac tual quasi-c lassic al trajec tories, or the appearanc e thereof. What we mean by this is the following: (a) if the measurements along a quasi-c lassic al trajec tory are ac tually c arried out (as in Heisenberg's treatment of α -partic le trac ks), then the results obtained will “line up” along the quasi-c lassic al trajec tories provided by dec oherenc e; but even if (b) the intermediate measurements are not c arried out, and only the final measurement is, one c an c onsistently assign retrospec tively the whole trajec tory to the system (sometimes merely guessing what the trajec tories “must have been”). This distinc tion is related to what is known as the movability of the Heisenberg “c ut” between observer and observed (whic h we disc uss in the next subsection). One will thus rec over the predic tions of c lassic al mec hanic s, but only instrumentally. Indeed, measurements will need to be regarded as primitive even in c lassic al mec hanic s, and it will be out of measurements that we will rec onstruc t “objec ts” that “look” and “behave” c lassic ally (the Moon is not there if we do not look). What of the standard interpretation? In a sense, the problem of the c lassic al regime is more interesting if one adopts this view, bec ause if one manages to derive a c lassic al regime within quantum mec hanic s in the standard interpretation, then this would recapture also the standard interpretation of classical mechanics (with measurements being derived notions). However, as we have seen, if one rejec ts a minimal interpretation of the formalism, but has some fuller ontologic al c ommitment to the wave func tion as desc ribing a quantum system itself, then dec oherenc e appears to exac erbate the problematic nature of the c lassic al regime. Indeed, quantum interac tions tend to c reate improper mixtures at the level of the c omponent systems. Therefore, it would appear that they destroy c lassic ality, as in the c ase of Sc hrödinger's c at. As in Einstein's disc ussion, if one wishes to keep a fuller ontologic al c ommitment to the wave func tion, or to provide a desc ription of individual quasi-c lassic al systems within quantum mec hanic s, one will have to replac e the standard interpretation (or quantum theory itself) with some alternative approac h. The same broad frameworks that are usually proposed as relevant to the measurement problem appear to be useful (but note our c onc luding qualific ations in sec tion 5). Today's Everett interpretations are intimately c onnec ted with dec oherenc e. Indeed, the revival of Everettian ideas c an be trac ed bac k to Zeh's work on dec oherenc e from the early 1970s, and was taken up in the philosophy literature arguably starting in the early 1990s with the work of Saunders, and later of Wallac e and others. In these modern versions of Everett, either the “many worlds” or the physic al c orrelate of the “many minds” are explic itly identified with the stable struc tures c reated by dec oherenc e at the level of c omponents of the universal wave func tion.2 9 Pilot-wave theories along the lines of de Broglie and Bohm also need to address explic itly the problem of the c lassic al regime, sinc e in general the trajec tories defined in the theory are highly nonc lassic al (see, e.g., Holland 1995, c h. 6, and, for a different point of view, Allori and Zanghì 2009). At least in the nonrelativistic partic le theory, it would seem that the c omponents preferred by dec oherenc e c orrespond nic ely with the “full” and “empty” waves of the theory. In Einstein's example, the mac rosc opic ball or dust partic le will be dec ohered by the environment inside the box, and the system will be effec tively guided by only one of the c omponents running in opposite direc tions and that form the standing wave when superposed. However, it is less c lear whether similar results are available in the c ase of quantum field theoretic generaliz ations (see, e.g., Wallac e 2008). Finally, spontaneous c ollapse theories might also be able to take advantage of the struc tures provided by dec oherenc e (whic h generally operates on a muc h faster timesc ale than spontaneous c ollapse), but explic it studies c ombining c ollapse models and dec oherenc e are notably lac king from the literature. 3.4 Heisenberg's “Cut” We c onc lude this sec tion by expanding on the remarks in the last subsec tion on the Heisenberg “c ut.” In partic ular, we wish to make prec ise in what sense dec oherenc e is relevant to Heisenberg's disc ussion of the movable c ut between observer and observed (or to von Neumann's disc ussion of measurement c hains), and in what sense it is not. This will provide also a good entry into the topic of measurement, treated in the next sec tion. Espec ially in the early 1930s, Heisenberg used to emphasiz e the importanc e of the movability of the “c ut” between Page 15 of 32

Measurement and Classical Regime in Quantum Mechanics the quantum and the classical domains in ensuring the consistency of quantum mechanics (cf. Heisenberg 1930, c h. 4; 1949, pp. 7– 21 and 35– 46; and espec ially 1985). Neither Heisenberg nor any of the other founding fathers of quantum mec hanic s believed in a rigid boundary between a quantum world, to whic h one c ould apply quantum mec hanic s, and a c lassic al world, to whic h one c ould apply only c lassic al mec hanic s and to whic h the apparatus and the observer belonged. Any parts of the world (inc luding ostensibly “c lassic al” ones) c ould be treated quantum mec hanic ally if one so wished.3 0 Consistenc y of the theory had to be ensured, ac c ording to Heisenberg, in the sense that applying quantum mec hanic s to a “c lassic al” part of the world should produc e the same predic tions as if c lassic al mec hanic s had been used. At the risk of pre-empting somewhat our disc ussion of measurements in the next sec tion, let us c onsider a so- c alled “measurement c hain,” say the example disc ussed by von Neumann (1932) in his c hapter on quantum measurement: we measure the temperature of a (quantum) gas using a (c lassic al) thermometer, or we treat the interac tion between the gas and the thermometer quantum mec hanic ally, and we observe (c lassic ally) the height of the merc ury c olumn, or we treat also the interac tion between the thermometer and the human retina quantum mec hanic ally, and our brain registers (c lassic ally) the image on the retina, or we treat the whole physic al proc ess quantum mec hanic ally, and it is only our c onsc iousness that bec omes (“c lassic ally”) aware of the outc ome (and c ollapses the physic al state). Now, there are two senses in whic h we c an establish the c onsistenc y of these desc riptions. First, if the suc c essive (quantum or c lassic al) interac tions are suc h as to c orrelate perfec tly the values of the temperature and the values of the quantities that are meant to rec ord the temperature, then it follows straightforwardly that, irrespec tive of where the c ollapse postulate and Born rule are applied, one will obtain the same final results with the same probabilities. This is ac tually the sense in whic h both Heisenberg and von Neumann are interested in establishing c onsistenc y.3 1 Sec ond, we c an c onsider the influenc e of dec oherenc e. Note that if no dec oherenc e were present, then performing some other measurement on the thermometer (i.e., a measurement inc ompatible with that of the length of the merc ury c olumn), or somewhere further along the measurement c hain, would reveal interferenc e terms between the c omponents of the state c orresponding to different measured temperatures. The plac ing of the “c ut” would influenc e the final statistic s, just as the timing of the c ollapse does in the c ase of the two-slit experiment (c ollapse behind the slits or at the sc reen). Conversely, onc e dec oherenc e has kic ked in at the level of the thermometer, there is no further measurement we would be able to perform in practice on the thermometer that c ould distinguish whether the thermometer is a c lassic al or a quantum system. And similarly for the retina and for the brain of the observer. It is in this stronger sense that dec oherenc e establishes that the loc ation of the c ut between the quantum and the classical domain (where the collapse postulate is applied along the measurement c hain) is arbitrary.3 2 4. Theo ry and Pro blem o f Measurement We now turn to disc ussing the theory and problem of measurement. We shall start by disc ussing measurements in some detail, using Stern–Gerlac h measurements as our exemplar, and generaliz ing the phenomenologic al notion of a measurement (and of a measurable quantity or “observable”) using the tools provided by the so-c alled POV measures. This theoretic al disc ussion will then provide the basis for disc ussing the measurement problem in sec tion 4.6. 4.1 Discretized Position Measurements Quite surprisingly, there is no perfec t analogue for the c ollapse postulate in the c ase of measurements of c ontinuous quantities, suc h as position. Naively one would expec t a wave func tion ψ (x) to c ollapse to a (renormaliz ed) Dirac δ-func tion c entered at some point q, that is, to ψ(x)δ(x − q), with a Born probability density given by | ψ(q)| 2 . The problem with this is the mathematic al fac t that any func tion that is nonz ero at a single point has square integral 0, and is thus identified with the z ero vec tor. Dirac 's famous δ-func tions are thus not ac tually quantum states, so, trivially, one c annot c ollapse a state to a δ-func tion. This was rec ogniz ed already by von Neumann (1932), who used disc retiz ation proc edures to desc ribe measurements of position. For instanc e, we c an (ideally) test for whether a wave func tion lies in the subspac e of all Page 16 of 32

Measurement and Classical Regime in Quantum Mechanics square-integrable func tions that are nonz ero in the interval [x1,x2 ]. If the test is positive, the original wave func tion ψ(x) will collapse to χ[x1 , x2 ](x) ψ (x) (suitably renormalized),33 with probability (35) (This suggestion is so obvious that the problematic nature of the collapse postulate for continuous quantities often goes unnotic ed.) Suc h disc retiz ed measurements of position are all we need to analyz e explic itly how a spin measurement works, and in fac t to generaliz e it to inc lude more realistic kinds of spin “measurements.” 4.2 Ideal Spin Measurements Let us first desc ribe the c ase of an ideal measurement of spin. Note that a system with both spin and position degrees of freedom is desc ribed using the tensor produc t of the Hilbert spac es used to desc ribe a “pure” spin-1/2 system (a two-dimensional c omplex Hilbert spac e) and a spinless partic le (the Hilbert spac e of Sc hrödinger's wave func tions), just as if one were c omposing two separate systems. Take an elec tron that we assume to be initially in a state (36) What this means is that the elec tron is desc ribed as having a spin state, given by the vec tor | φ〉 in the two- dimensional spin spac e of the elec tron, as well as a wave func tion, ψ . Now suppose we want to perform a measurement of spin in some given direc tion, and that with respec t to this spin basis, |φ〉 = α|+〉 + β|−〉, so that (36) equals (37) If we pass the elec tron through an ideal Stern– Gerlac h magnet, the evolution of the state will be desc ribed by the appropriate Sc hrödinger equation, whic h is unitary. Therefore, we c an c onsider separately the deflec tion of the two c omponents and superpose the results. We obtain (38) (where ψ + and ψ − are suitably deflec ted versions of ψ ). We see that the spin degree of freedom of the elec tron is now entangled with its position degrees of freedom. We now detec t the elec tron on a sc reen, that is, perform a position measurement. Indeed, we perform a disc retiz ed measurement of position, bec ause we only need to distinguish whether the elec tron hits the half of the sc reen assoc iated with the up or down c omponent of the spin (whic h, as mentioned in footnote 6, depending on the experimental setup might or might not c oinc ide with the upper or lower half of the sc reen, respec tively). Assuming that ψ + and ψ − do not overlap, the standard c ollapse postulate and Born rule, applied to the detec tion of the elec tron on the sc reen, will yield either (39) or (40) and thus we c an ac tually derive the c ollapse postulate and Born rule for the spin measurement from the c ollapse postulate and Born rule for the disc retiz ed measurement of position. 4.3 “Unsharp” Spin Measurements Real experiments, however, will not yield exac tly the above result. Let us return to the disc ussion of our Stern– Gerlac h example. It is a fac t that wave func tions, even if at any one time they c an be z ero outside of a given interval, will (typic ally) spread instantaneously out to infinity, so that while we c ould expec t the bulk of ψ+ and ψ− to be c onc entrated eac h on one half of the sc reen, they will have “tails” spreading out to the “wrong” half of the sc reen, say (41) Page 17 of 32

Measurement and Classical Regime in Quantum Mechanics Here, ψ ++ is meant to represent that part of ψ+ that is distributed over the half-sc reen assoc iated with the up result, and ψ +− the part that is distributed over the half-sc reen assoc iated with the down result; and similarly for ψ−− and ψ −+. We shall assume for simplic ity that (42) and thus also (43) In this c ase, applying the c ollapse postulate and Born rule to detec ting the elec tron on the sc reen yields either (44) or (45) (both to be suitably renormaliz ed), with probabilities (46) and (47) respec tively. We see that the effec t of the measurement on the spin state of the elec tron is no longer simply given by the standard c ollapse postulate. Indeed, the two possible states of the elec tron after the measurement are not even produc t states, so that the spin of the elec tron is still entangled with its spatial degrees of freedom, and the spin part of the elec tron is c ollapsed to an improper mixture: either (48) or (49) with the same probabilities (46) and (47) (also here, we need to suitably renormaliz e, sinc e the weights in eac h dec omposition need to sum to 1). Note that these are the states we obtain if we, indeed, know the result of the spin measurement and c an selec t one of these two final states on the basis of the measurement result (thus performing a so-c alled selec tive measurement). If we do not know the outc ome of the spin measurement, then future predic tions for spin measurements on the elec tron will use a state that is itself a (proper) mixture of the two c orresponding density operators. We c an obtain this by simply adding the two (unnormaliz ed) states (48) and (49), to yield (50) This is now a c ase of nonselec tive measurement, in whic h we obtain a mixed state that is partially ignoranc e- interpretable. But—as in the c ase of the bit c ommitment problem of sec tion 2.3—we need to know the past history of the system (how the state has been prepared), in order to know how and how far to interpret this mixed state in terms of ignoranc e. If the measurement is ideal, then the c orrec t dec omposition of the state is in terms of spin-up or spin-down; if the measurement is c orrec tly modeled by the above, the c orrec t dec omposition is given in terms of (48) and (49). In this example (where we have c ombined an ideal Stern– Gerlac h magnet with a more realistic position state), we see that the probabilities in (50) are independent of the shape of the position state (and indeed, of whether it is “ideal” or “realistic ”). One easily realiz es that even more general transformations on the spin state of the elec tron c an be induc ed by a detec tion on the sc reen, if one c onsiders that the Stern– Gerlac h magnetic field itself is not “ideal” (in order to satisfy the Maxwell equations, it c annot be perfec tly homogeneous in the direc tions perpendic ular to that of measurement). Or, indeed, if one c onsiders that one c ould have c hosen, at least in princ iple, any other unitary c oupling between the spin and position degrees of freedom of the elec tron before proc eeding to the detec tion on the sc reen. Page 18 of 32

Measurement and Classical Regime in Quantum Mechanics 4.4 General Phenomenology of Measurements The above examples of various kinds of spin measurements serve as perfec t illustrations of the general phenomenology and theory of measurements in quantum mechanics. As discussed in section 1.1, measurements are phenomenologically captured by the collapse postulate, which describes transformations on the state of the measured system, and the Born rule, whic h gives the probabilities for suc h transformations. Both rules need to be generaliz ed. We shall sketc h this generaliz ation here, but only in the disc rete, finite-dimensional c ase. Let us first slightly redesc ribe the c ollapse postulate and Born's rule for the c ase of the standard measurements of sec tion 1.1. Take a family of mutually c ompatible quantum mec hanic al tests, c orresponding to a family of mutually orthogonal sub-spac es of the Hilbert spac e. (If they do not span already the whole Hilbert spac e, we c an add to the family the orthogonal c omplement of their span, c orresponding to the system testing negatively to all the tests.) The c orresponding projec tion operators form a so-c alled (PV, or projec tion-valued) resolution of the identity: (51) where 1 is the identity operator on the Hilbert spac e. In this c ase, we also talk of a PV-observable.3 4 In the c ase of a selec tive measurement of this PV-observable, with outc ome i, the state of the system c ollapses as: (52) or more generally, writing ρ for the initial state to c over also the c ase when it might not be pure: (53) (in both c ases with suitable renormaliz ation). The probabilities for the c ollapses (52) and (53) are, respec tively, 〈ψ|Pi|〉 and (54) (Note that the trace is cyclic, that is, Tr(AB) = Tr(BA) for any two operators, and that Pi2   =  Pi for projections.) In the c ase of a nonselec tive measurement, the c ollapse takes the form (55) (already normaliz ed, bec ause of (51)). In the c ase of the more realistic spin measurements just disc ussed, instead of the transformation (52), we have a transformation to an (improper) mixture, whic h we c an write as (56) or (57) depending on the outc ome, with probabilities given by the trac e of (56) or of (57), respec tively. In the most general c ase, the transformation (52) or (53) takes the form of a so-c alled operation, or completely positive map: (58) with suitable operators Aij for each outcome i. (If there is only one Aij corresponding to the outcome i, the operation is said to be “pure,” bec ause it maps pure states to pure states.) The c orresponding probabilities are given by (59) where we have defined the so-c alled effect3 5 Ei as (60) Page 19 of 32

Measurement and Classical Regime in Quantum Mechanics In the nonselec tive c ase, the transformation (55) bec omes (61) And the normaliz ation of the probabilities (59) yields the analogue of (51), namely (62) that is, the Ei form an effec t-valued (or POV, or positive-operator-valued) resolution of the identity (or POV- observable). Interesting spec ial c ases are obtained when, as in the c ase of the spin measurements above, the operations are c ombinations of the projec tions from a PV-observable (so-c alled “unsharp” measurements of the c orresponding PV-observable); or when the effec ts Ei are in fac t mutually orthogonal projec tions, but the c orresponding operations are not simple projec tions, but have a more general form (“disturbing” measurement of the c orresponding PV-observable). Other c ases of POV-observables c an be interpreted as c orresponding to sequenc es of measurements of PV-observables (or of other POV-observables), yet others as c orresponding (in c ertain spec ific senses) to joint unsharp measurements of inc ompatible PV-observables. The c loser relation between a measurement and a single self-adjoint operator mentioned in sec tion 1.1 is c learly lost in the general case. These transformations provide the general form of the phenomenologic al c ollapse postulate, and the c orresponding probabilities the general form of the phenomenologic al Born rule. The above disc ussion of spin measurements, however, illustrates also the general theoretical description of such measurements. Indeed, one c an show (this is known as the Naimark dilation theorem) that any c ompletely positive map on the states of the measured system c an always be obtained by suitable interac tion with some other system, followed by a PV- measurement on this other system (i.e., a transformation of the form (53) or (55), where it should again be emphasiz ed that the Pi need not be one-dimensional projec tions). This other system c an be thought of either as a generally mic rosc opic anc illary system or degree of freedom (e.g. the position of the elec tron in a Stern– Gerlac h measurement, or the photon in the Heisenberg mic rosc ope), or as an “indic ator variable” or “pointer variable” of a generally mac rosc opic measuring devic e. We shall see this in detail (for the c ase of ideal measurements) in sec tion 4.5. POV-observables provide a very powerful tool for desc ribing the phenomenology of quantum mec hanic al measurements. And they have bec ome a c ompletely standard tool in various branc hes of quantum physic s (e.g., quantum information theory). For instanc e, it is well-known that using a measurement of a single PV-observable it is impossible to rec onstruc t c ompletely the quantum state desc ribing an ensemble of systems. (If the PV-observable is spin in some direc tion, and if the state is pure, say α | +〉 + β| −) in that basis, the measurement statistic s will determine only the absolute values of the c oeffic ients α and β, not their relative phases.) But there are single POV-observables (so-c alled informationally c omplete observables) that allow suc h a rec onstruc tion. A simple example is given by the resolution of the identity (63) whic h intuitively pools together the information provided by measurements of spin in the three direc tions x, y, and z (and c an be seen as one sense of a joint unsharp measurement of the three PV-observables (Cattaneo et al. 1997)). Indeed, suc h a POV-measurement c an be performed simply by throwing a die and measuring spin in the direc tion x, y, or z depending on whether the die shows up 1, 2, or 3 (mod 3). Note, however, that from the Naimark dilation theorem we also know that there is a single interac tion with an anc illa or measuring devic e that will implement on the electron any set of six operations needed to measure the POV-observable (63). Note that Gleason's theorem c an be formulated also in terms of probability measures over the outc omes of all Page 20 of 32

Measurement and Classical Regime in Quantum Mechanics possible POV experiments, yielding again the quantum mec hanic al mixed states as the most general states defining probabilities for the outc omes of suc h experiments (Busc h 2003).3 6 We c onc lude by mentioning one example of c ontinuous POV measurements, the so-c alled “unsharp” measurements of position, whic h provide a c ontinuous alternative to von Neumann's disc retiz ation proc edures. The wave func tion ψ (x) c ollapses upon measurement to a wave func tion (64) (suitably renormaliz ed), where α(x − q) is not a δ-func tion but a normaliz ed Gaussian c entered at q. The probability density for the c ollapse is given by (65) This POV-measurement has the intuitive properties of a measurement of position, in that after the c ollapse the wave func tion is c onc entrated around the point q. Readers may rec ogniz e this as the family of operations that take plac e spontaneously in the spontaneous c ollapse theory by Ghirardi, Rimini, and Weber (1986).3 7 4.5 The Standard Model of Measurement The model of measurement that underlies standard discussions of the measurement problem (although usually phrased mainly in terms of ideal measurements) is direc tly related to the theoretic al desc ription based on the dilation theorem, as follows. In the ideal c ase, one takes a basis of eigenvec tors of the observable one wishes to measure on the system of interest, {| φi〉}, and c ouples it one-to-one to an orthonormal family of states {| ψj〉} of the apparatus, in the sense that for some “ready state” |ψ0〉 of the apparatus, (66) for all i. This is indeed possible through a single unitary evolution, bec ause it is simply a requirement that orthonormal states be mapped into orthonormal states. The outc omes of the measurement are assumed to c orrespond to orthogonal subspac es (not nec essarily one- dimensional), or their c orresponding projec tions Pk, eac h c ontaining one or more of the | ψi〉 (depending on the “resolution” of the measurement). If eac h outc ome c orresponds to a single | φi〉, the measurement is said to be maximal.38 Under this c oupling, an arbitrary state of the system will interac t with the apparatus in the ready state as (67) If the measurement is maximal, applying the standard c ollapse postulate to the pointer observable will now yield any one of the states (68) with probability | αi| 2 . If the measurement is non-maximal, more than one | ψi〉 will lie in the subspac e assoc iated with the measurement outc ome, and the c ollapse will yield some superposition of the states (68).3 9 More generally, a measurement will involve an arbitrary c oupling between the system of interest and the apparatus, so that the final state of the c omposite will have the form (69) or performing the sum over i first, (70) Defining βj as the norm of ∑i αij ∣φi ⟩, and ∣∣φ∗j ⟩ as 1   ∑i αij ∣φi⟩, (70) can be rewritten as (71) βj Page 21 of 32

Measurement and Classical Regime in Quantum Mechanics Applying the standard c ollapse postulate to the pointer observable will now yield any one of the states (72) with the corresponding probability βj2 (or some superposition thereof if more than one |ψj〉 lies in the subspace assoc iated with the measurement outc ome). 4.6 The Measurement Problem In this sec tion, we shall build on the theory of measurement we have just sketc hed, and desc ribe the measurement problem of quantum mechanics. The phrase “measurement problem” denotes a complex of interrelated questions, but we shall take the following to be its c ore: whether the prac tic al rules of quantum mec hanic s (c ollapse postulate and Born rule) are derivable from first princ iples, by applying the theory (in partic ular the dynamic s of the theory, as given by the deterministic Sc hrödinger equation) to a measurement situation, i.e. a situation in whic h we have an appropriate interaction between a system and a measuring apparatus. As we have seen from generaliz ing the example of the Stern–Gerlac h measurement, a theoretic al desc ription of a measurement c an indeed be given by c oupling the system of interest (the spin of the elec tron) to some “indic ator” variable (the position of the elec tron on the upper or lower half of the sc reen). And we have also seen that the c ollapse postulate and Born rule for the system (in their most general form) c an be obtained by applying the c ollapse and Born rule (in their more restric ted form) to the indic ator variable itself. Suppose that from an appropriate applic ation of the Sc hrödinger equation, and without explic itly invoking the c ollapse postulate and the Born rule for the indic ator variable, one c ould derive that in the c orrec t frac tion of c ases the final state after a measurement is given by (68) or (72), rather than by (67) or (71). Then the c ollapse and the Born rule would be derivable from first princ iples, irrespec tive of whether one adopts the minimal or standard interpretation of the theory. Indeed, under the standard interpretation, states suc h as (68) or (72)—or even appropriate superpositions of suc h states—c orrespond to the apparatus possessing an intrinsic property indic ating a definite outc ome (a subspac e Pk representing an appropriate mac rostate of the apparatus). And under the minimal interpretation, these same states mean that the apparatus has a surefire dispositional property to be seen as indic ating a definite outc ome if somebody looks. However, these final states do not obtain if the desc ription just given in sec tion 4.5 is c orrec t. Just like in the bit c ommitment problem of sec tion 2.3, where there is an objec tive differenc e between the c ase in whic h Alic e sends a statistic al mixture of elec trons in various spin states, and the c ase in whic h she sends elec trons from entangled pairs (a differenc e enabling her to c heat), so in the c ase of the standard model of measurement there is an objec tive differenc e between the c ase of a statistic al mixture of states assoc iated with different measurement results and states in whic h the mac rosc opic outc ome is entangled with the mic rosc opic value of the measured observable. The theoretic al desc ription of measurement in terms of a unitary interac tion has merely shifted the plac e of applic ation of the phenomenologic al rules. Thus, if it is a c orrec t desc ription of the proc ess of measurement, it does not provide a derivation of the c ollapse postulate and Born rule, whether the interpretation of c hoic e is the minimal interpretation or the standard interpretation. Before disc ussing this further, we should pause to c onsider whether we have been misled by the power of the dilation theorem and been overly rash in adopting this model of measurement. That is, we should see whether the negative result just described is merely an artifact of the model of measurement adopted. What c ould c ount as a derivation of the c ollapse postulate and Born rule from the Sc hrödinger equation? At first, it might seem c onc eptually mistaken even to pose suc h a question. How c an a probabilistic proc ess ever be derived from a deterministic one? Ac c ording to von Neumann, the differenc es run even deeper, in that the former is a thermodynamic ally irreversible proc ess, while the latter is reversible.4 0 On the other hand, in the c ase of c lassic al thermodynamic s and statistic al mec hanic s, we are familiar with the c laim that a phenomenologic ally irreversible theory c an be reduc ed (in some appropriate sense) to an underlying reversible one. The obvious first attempt at Page 22 of 32

Measurement and Classical Regime in Quantum Mechanics answering the problem is thus to point out that it is perfec tly possible for a deterministic evolution to underpin statistic al results, if we c onsider statistic al states (that is, genuinely statistic al states, whic h are proper mixtures) rather than pure states. That is, while the Sc hrödinger evolution maps pure states into pure states, it is perfec tly possible to obtain a final state that is a proper mixture of different readings of the apparatus, if the initial state is not pure but itself a proper mixture. The intuition here is that the initial state of system and apparatus should be given not by a produc t of pure states, but more realistic ally by a state of the form | ψ0 〉 ⊗ ρ0 , where ρ0 is a suitable statistic al state of the apparatus. Indeed, any realistic apparatus will arguably be a mac rosc opic objec t, and thus its exac t mic rostate will not be spec ifiable. Instead, the state of the apparatus will be given only in terms of c ertain mac rosc opic parameters, analogously to the mac rostates of statistic al mec hanic s, and thus for instanc e to be desc ribed as lying in some subspac e P0 or as some appropriate proper mixture of mic rostates.4 1 For simplic ity, let us stic k to our Stern– Gerlac h example, even though the “indic ator” variable (the position of the elec tron) is not itself mac rosc opic .4 2 Imagine that initially we are not able to prepare the wave func tion ψ for the spatial degrees of freedom of the elec tron, but only a proper mixture ρ  =   ∫ βλ ρλdλ , where eac h ρλ c orresponds Λ itself to a wave func tion ψλ (a pure state) loc aliz ed within the spread of the original ψ. Imagine further that ρ dec omposes into (73) where the weights μλ are related to the c oeffic ients in the dec omposition (37) as (74) Here Λ+ is the set of indic es for whic h | ψ〉 ⊗ ρλ evolves to some final state entirely c ontained in the subspac e c orresponding to 1 ⊗ P+ (i.e., the projec tion onto the half of the sc reen assoc iated with “up”). And c orrespondingly for Λ−. Sinc e the initial mixture was by assumption ignoranc e-interpretable, also the final state is a proper mixture suc h that in a frac tion | α | 2 of c ases, the elec tron has ended up on the half of the sc reen assoc iated with “up, | and in a frac tion | β| 2 of c ases, the elec tron has ended up on half of the sc reen assoc iated with “down,” as desired. The problem with this obvious strategy is that it does not work in general. Indeed, if the initial state of the elec tron is not (37) but, say, (75) then, even assuming that eac h ρλ still ends up on one or the other half of the sc reen, it is not c lear why the new sets Λ′+ and Λ′− into which the set Δ splits should again satisfy (74) with γ and δ substituted for α and β. Indeed, this c onstraint is impossible to satisfy for all initial spin states | φ〉 if the temporal evolution is unitary.4 3 The apparatus would have to c onspire to know in advanc e what spin state it is meant to measure in order to be in the appropriate statistic al mixture of mic rostates that will produc e the desired outc omes with the desired frequenc ies. 4 4 For the c ase of ideal measurements, the fac t that the measurement problem c annot be solved by invoking an initial mixed state of the apparatus was already disc ussed by von Neumann (1932, sec tion VI.3), who remarks that this idea was often proposed as a solution to the measurement problem.4 5 (It is periodic ally “redisc overed,” whic h only shows that von Neumann's book is often referred to but still not widely read.) Von Neumann used this to support his c laim that one needs indeed two different kinds of proc esses (namely c ollapse and unitary evolution) to desc ribe the behavior of quantum systems with and without measurements. This “insolubility theorem,” as it is now known, has sinc e been widely generaliz ed, in partic ular to inc lude also measurements of POV observables.4 6 Thus, we are left with our original c onc lusion, irrespec tive of the model of measurement we c hoose. In a nutshell, measurements understood as quantum interactions magnify quantum superpositions to the macroscopic level (bec ause of the linearity of the dynamic s), and thus do not lead to the phenomenologic ally c orrec t behavior (c ollapse postulate and Born rule). If we apply the Sc hrödinger equation to desc ribe the measurement proc ess, then we do not obtain states that would seem to inc lude definite measurement results, but superpositions of suc h Page 23 of 32

Measurement and Classical Regime in Quantum Mechanics states, nor do we obtain any kind of probabilistic distributions over final states. Under the minimal interpretation, this might not be very satisfac tory, but need not be partic ularly troubling, sinc e the interpretation only seeks to provide an instrumentalist reading of the theory. And in various variants of the Copenhagen interpretation, one c an argue that one should in fac t expec t measurements—or the quantum-c lassic al interfac e—to display a pec uliar behavior. What is essential, on these interpretations, is c onsistenc y between different c hoic es of when and where to apply the c ollapse postulate and the Born rule. And, as we disc ussed in sec tion 3.3, this c onsistenc y is ensured by dec oherenc e, or in a weaker sense by the existenc e of perfec t c orrelations along a measurement c hain.4 7 As a matter of fac t, von Neumann c onsidered the measurement problem to be purely the question of whether suc h c onsistenc y (in the weaker sense) c ould be ensured, and his treatment of the measurement problem c onsists prec isely in showing that unitary evolutions exist that will produc e the perfec t c orrelations (i.e., essentially, in showing that the standard model of measurement is well-defined). Collapse c ould oc c ur when the thermometer rec ords the temperature of the gas, or when the length of the merc ury c olumn is rec orded in the photons traveling to the eye, or in our retina, or along the optic nerve, or when ultimately c onsc iousness is involved. If all of these possibilities are equivalent as far as the final predic tions are c onc erned, von Neumann c an maintain that c ollapse is related to c onsc iousness while in prac tic e applying the c ollapse postulate at a muc h earlier (and more prac tic al) stage in the desc ription. From the point of view of the standard interpretation, however, the problem is serious, bec ause in the state (67) or (71) the system and the apparatus are entangled, and the mixed state resulting for the apparatus is not ignoranc e- interpretable. Thus, the apparatus does not have a reading under the standard interpretation. As we have also seen, in the c ase of the standard interpretation dec oherenc e does not help; if anything it makes the situation even worse, bec ause it will produc e suc h mac rosc opic improper mixtures even independently of observer-engineered measurement situations. Insofar as the standard interpretation is meant as an approac h to quantum mec hanic s that treats it as a fundamental theory, rather than as a phenomenologic al theory, we see that the standard interpretation fails. In partic ular, it fails to support a theoretic al analysis of the proc ess of measurement that ensures that measurements have definite outcomes, let alone one that enables one to rederive the phenomenological rules for the description of measurements (the c ollapse postulate and the Born rule). Everett theories, pilot-wave theories and spontaneous c ollapse theories are again the options of c hoic e if one wishes to provide a solution to the measurement problem rather than a minimalist or (neo)-Copenhagen dissolution,4 8 but a detailed disc ussion of these goes beyond the scope of this essay. 5. Co nclusio n We have disc ussed two of the main interpretational problems of quantum mec hanic s, both engendered by the nature of quantum mec hanic al entanglement, and the c onsequent failure of the ignoranc e interpretation of reduc ed states. The two problems are equally important if one wishes to give a foundationally adequate reading of quantum mec hanic s. We are not here in the business of disc ussing what a foundationally adequate reading of quantum mec hanic s might be. The minimal interpretation, while not being entirely satisfac tory, will arguably c ount as adequate if one has an instrumentalist picture of science. More sophisticated Copenhagen or neo-Copenhagen views may also find it easier to negotiate these two problems.4 9 Yet more robust ontologic al requirements will prompt one to seek a more suc c essful replac ement for the standard interpretation of quantum mec hanic s, with the help of dec oherenc e and usually along the lines of de Broglie– Bohm, c ollapse or Everett. A solution to the problem of the c lassic al regime, however, will not automatic ally be also a solution to the measurement problem (and vic e versa). While piec es of apparatus are generally mac rosc opic systems or arguably at least kinematic ally c lassic al systems, their dynamic al behavior in probing the quantum world is dec idedly nonc lassic al, and solving the dynamic al aspec ts of the measurement problem is thus distinc t from deriving approximately Newtonian trajec tories. For instanc e, modern-day Everettians c an use the results of dec oherenc e in an extremely effec tive way, both toward the solution of the problem of the c lassic al regime and toward that of the Page 24 of 32

Measurement and Classical Regime in Quantum Mechanics measurement problem. But the Everettian solution to the measurement problem relies more heavily on a successful derivation of the Born rule (e.g., the dec ision-theoretic approac h proposed by Deutsc h (1999) and Wallac e (2007)). Should the c ritic s of the Deutsc h– Wallac e approac h prove c orrec t (e.g., Lewis 2010), Everettians might still be lac king a derivation of the Born rule from first princ iples, and thus a full solution to the measurement problem. Conversely, a solution to the measurement problem will not automatic ally be a solution to the problem of the c lassic al regime. For instanc e, rec ent developments in de Broglie– Bohm theory have inc luded various proposals for desc ribing at least large portions of the standard model of partic le theory (see, e.g., Colin and Struyve 2007; Dürr et al. 2005; Struyve and Westman 2007; and the review in Struyve 2011). Critic s, however, argue that the c onfiguration variables in these models (whic h are guided by the relevant wave func tional) are not nec essarily dec ohering variables (Wallac e 2008). It may well be, as argued in partic ular in the “minimalist” model of Struyve and Westman (2007), that there are c hoic es for c onfiguration variables that will ensure that measurement results (suitably c onstrued) will always be well-defined. But should the c ritic s prove c orrec t, the “measured” c lassic al trajec tories will be no more real than those of the minimal interpretation, and pilot-wave theorists would still lac k a fully satisfac tory solution to the problem of the c lassic al regime. Muc h progress has been ac hieved in rec ent years on the resolution of these two problems, and generally in the philosophy and foundations of quantum mec hanic s. One should expec t to see more in years to c ome. References Albert, D. (1992). Quantum Mechanics and Experience. Cambridge, MA: Harvard University Press. Allori, V., and Zanghì, N. (2009). On the c lassic al limit of quantum mec hanic s. Foundations of Physics 39(1): 20– 32. Bac c iagaluppi, G. (2000). Deloc alised properties in the modal interpretation of a c ontinuous model of dec oherenc e. Foundations of Physics 30: 1431–1444. ———. (2003). The role of dec oherenc e in quantum mec hanic s. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2008 edition, http://plato .stanfo rd.edu/archives/fall2008/entries/qm- deco herence/ ). ———. (2008). The statistic al interpretation ac c ording to born and Heisenberg. In C. Joas, C. Lehner, and J. Renn (eds.), HQ-1: Conference on the History of Quantum Physics (Vols. I and II), MPIWG preprint series, Vol. 350 (Berlin: MPIWG), (Vol. II) c hapter 14, pp. 269– 288 (available at http://www.m piwg- berlin.mpg.de/en/reso urces/preprints.html). ———. (2010). Collapse theories as beable theories. Manuscrito 33(1) (spec ial issue edited by D. Krause and O. Bueno): 19–54. ———. (2012). Insolubility theorems and EPR argument. European Journal for Philosophy of Science, forthc oming, DOI: 10.1007/s13194-012-0057-7. Bac c iagaluppi, G., and Crull, E. (2009). Heisenberg (and Sc hrödinger, and Pauli) on hidden variables. Studies in History and Philosophy of Modern Physics 40: 374–382. Bac c iagaluppi, G., and Valentini, A. (2009). Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay conference. Cambridge: Cambridge University Press. Barbour, J. B. (1999). The end of time. London: Weidenfeld and Nic olson. Bassi, A., and Ghirardi, G.C. (2000). A general argument against the universal validity of the superposition princ iple. Physics Letters A 275: 373–381. Bell, J. S. (1966). On the problem of hidden variables in quantum mec hanic s. Reviews of Modern Physics 38: 447– 452. Reprinted in Speakable and Unspeakable in Quantum Mechanics. Cambridge: Cambridge University Press, 1– 13. Page 25 of 32

Measurement and Classical Regime in Quantum Mechanics Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden” variables, I and II. Physical Review 85, 166–179 and 180–193. Bohr, N. (1949). Disc ussion with Einstein on epistemologic al problems in atomic physic s. In P. A. Sc hilpp (ed.), Albert Einstein: Philosopher-scientist, The Library of Living Philosophers, Vol. VII (La Salle: Open Court), 201–241. Born, M., ed. (1969). Albert Einstein, Hedwig und Max Born: Briefwechsel 1916–1955 (Munich: Nymphenburger Verlagshandlung). Translated by I. Born as The Born–Einstein letters: Correspondence between Albert Einstein and Max and Hedwig Born from 1916– 1955 (London: Mac millan, 1971). Born, M., and Heisenberg, W. (1928). La méc anique des quanta [Quantenmec hanik]. In Lorentz (1928), 143–184. Translated (from the original German) in Bac c iagaluppi and Valentini (2009), 408– 447. Broglie, L. de (1928). La nouvelle dynamique des quanta. In Lorentz (1928), 105–141. Translated in Bac c iagaluppi and Valentini (2009), 374–407. Brown, H. R. (1986). The insolubility proof of the quantum measurement problem. Foundations of Physics 16: 857– 870. Busc h, P. (2003). Quantum states and generaliz ed observables: A simple proof of Gleason's theorem. Physical Review Letters 91: 120403/1–4. Busc h, P., and Shimony, A. (1996). Insolubility of the quantum measurement problem for unsharp observables. Studies in History and Philosophy of Modern Physics 27B: 397–404. Busc h, P., Grabowski, M., and Lahti, P. (1995). Operational Quantum Physics. Berlin and Heidelberg: Springer. Cattaneo, G., Marsic o, T., Nistic ò, G. and Bac c iagaluppi, G. (1997). A c onc rete proc edure for obtaining sharp rec onstruc tions of unsharp observables in finite-dimensional quantum mec hanic s. Foundations of Physics 27: 1323–1343. Colin, S., and Struyve, W. (2007). A Dirac sea pilot-wave model for quantum field theory. Journal of Physics A 40: 7309–7342. Deutsc h, D. (1999). Quantum theory of probability and dec isions. Proceedings of the Royal Society of London A 455: 3129–3137. Donald, M. (1998). Disc ontinuity and c ontinuity of definite properties in the modal interpretation. In D. Dieks and P. E. Vermaas (eds.), The modal interpretation of quantum mechanics (Dordrecht: Kluwer), 213–222. Dürr, D., Goldstein, S., Tumulka, R., and Zanghì, N. (2005). Bell-type quantum field theories. Journal of Physics A 38: R1–R43. Einstein, A. (1953). Elementare Überlegungen z ur Interpretation der Grundlagen der Quanten-Mec hanik. In Scientific Papers Presented to Max Born (Edinburgh: Oliver and Boyd), 33–40. Translated by R. Deltete in Nature 356(2 April 1992): 393–395. Fine, A. (1970). Insolubility of the quantum measurement problem. Physical Review D 2: 2783–2787. ———. (1973). Probability and the interpretation of quantum mec hanic s. British Journal for the Philosophy of Science 24: 1–37. Fuc hs, C. A. (2010). QBism, the perimeter of quantum bayesianism. http://arxiv.o rg/abs/1003.5209. Ghirardi, G.C., Rimini, A., and Weber, T. (1986). Unified dynamic s for mic rosc opic and mac rosc opic systems. Physical Review D 34: 470–491. Gleason, A. M. (1957). Measures on the c losed subspac es of a Hilbert spac e. Journal of Mathematics and Mechanics 6(6): 885–893. Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Page 26 of 32

Measurement and Classical Regime in Quantum Mechanics Zeitschrift für Physik 43: 172–198. ——— (1930), Die physikalischen Prinzipien der Quantentheorie. Leipz ig: Hirz el. Translated by C. Ec kart and F. C. Hoyt as The Physical Principles of the Quantum Theory Chic ago: University of Chic ago Press, 1930. ——— (1949). Wandlungen in den Grundlagen der Naturwissenschaft. Züric h: Hirz el. ——— (1985). ‘Ist eine deterministisc he Ergänz ung der Quantenmec hanik möglic h? In W. Pauli, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a., Band II: 1930–1939, ed. by K. v. Meyenn, A. Hermann, and V. F. Weisskopf (Berlin and Heidelberg: Springer), 407– 418. Translated by E. Crull and G. Bac c iagaluppi (with a brief introduc tion), http://philsci- archive.pitt.edu/8590/. Hermann, G. (1935). Die naturphilosophischen Grundlagen der Quantenmechanik. Abhandlungen der Fries'schen Schule 6: 75– 152. Sec tion 7, Der Zirkel in Neumanns Beweis, translated by M. Seevinc k (http://mpseevinck.ruho sting.nl/seevinck/trans.pdf). Holland, P. R. (1995). The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge: Cambridge University Press. Jauc h, J. M., and Piron, C. (1969). On the struc ture of quantal proposition systems. Helvetica Physica Acta 42: 842–848. Lewis, P. J. (2010). ‘Probability in Everettian quantum mec hanic s. Manuscrito 33(1) (spec ial issue edited by D. Krause and O. Bueno): 285–306. Lo, H.-K., and Chau, H. F. (1997). Is quantum bit c ommitment really possible? Physical Review Letters 78: 3410– 3413. Lorentz , H. A., ed. (1928). Electrons et photons: Rapports et Discussions du Cinquième Conseil de Physique Solvay. Paris: Gauthier-Villars. Mayers, D. (1997) Unc onditionally sec ure quantum bit c ommitment is impossible. Physical Review Letters 78: 3414–3417. Mott, N. F. (1929). The wave mec hanic s of α -ray trac ks. Proceedings of the Royal Society of London A 126 (1930, No. 800 of 2 Dec ember 1929): 79– 84. Neumann, J. von (1932), Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. Translated by R. T. Beyer as Mathematical Foundations of Quantum Mechanics (Princ eton: Princ eton University Press, 1955). Pric e, H. (1996). Time's Arrow and Archimedes’ Point. New York: Oxford University Press. Prz ibram, K., ed. (1963). Briefe zur Wellenmechanik. Vienna: Springer. Translated by M. J. Klein as Letters on Wave Mechanics (New York: Philosophic al Library, 1967). Saunders, S., Barrett, J., Kent, A., and Wallac e, D., eds. (2010). Many Worlds? Everett, Quantum Theory, and Reality. Oxford: Oxford University Press. Sc hrödinger, E. (1926). Der stetige Übergang von der Mikro- z ur Makromec hanik. Naturwissenschaften 14: 664– 666. ———. (1928). Méc anique des ondes [Wellenmec hanik]. In Lorentz (1928), 185–213. Translated (from the original German) in Bac c iagaluppi and Valentini (2009), 448– 476. ———. (1935). Die gegenwärtige Situation in der Quantenmec hanik. Naturwissenschaften 23: 807– 812, 823– 828, 844–849. Translated in J. A. Wheeler and W. H. Zurek (eds.), Quantum Theory and Measurement (Princeton: Princ eton University Press, 1983), 152– 167. Sc hulman, L. S. (1997). Time's Arrows and Quantum Measurement. Cambridge: Cambridge University Press. Struyve, W. (2011). Pilot-wave approac hes to quantum field theory. Journal of Physics: Conference Series 306: Page 27 of 32

Measurement and Classical Regime in Quantum Mechanics 012047/1–10. Struyve, W., and Westman, H. (2007). A minimalist pilot-wave model for quantum elec trodynamic s. Proceedings of the Royal Society of London A 463: 3115–3129. Wallac e, D. (2001). Implic ations of quantum theory in the foundations of statistic al mec hanic s. http://philsci- archive.pitt.edu/410/. ———. (2007). Quantum probability from subjec tive likelihood: Improving on Deutsc h's proof of the probability rule. Studies in History and Philosophy of Modern Physics 38: 311–332. ———. (2008). Philosophy of quantum mec hanic s. In D. Ric kles (ed.), The Ashgate Companion to Contemporary Philosophy of Physics (Aldershot: Ashgate), 16–98. Preliminary version available as “The quantum measurement problem: State of play (Dec ember 2007),” http://philsci- archive.pitt.edu/3420/. Zurek, W. H. (2003). Dec oherenc e, einselec tion, and the quantum origins of the c lassic al. Reviews of Modern Physics 75: 715–775. Zurek, W. H., and Paz , J.-P. (1994). Dec oherenc e, c haos, and the sec ond law. Physical Review Letters 72: 2508– 2511. Notes: (1) A subspac e is a subset that is c losed under linear c ombinations. We shall assume familiarity with the basic c onc epts of Hilbert spac es. (2) Terminology varies, and sometimes the terms “c ollapse postulate” or “projec tion postulate” inc lude also the Born rule. (3) Note onc e and for all that we are not nec essarily assuming that these subspac es are one-dimensional. Alternatively, one c an think of testing them in suc c ession, in any order. Explic it applic ation of the c ollapse postulate and the Born rule will show that one will obtain the same results with the same probabilities and the same final state, irrespec tively of the order in whic h the tests are performed. (4) Linear operators are mappings on the Hilbert spac e (or a subspac e thereof) that map superpositions into the c orresponding superpositions. The adjoint of a linear operator A is a linear operator A* suc h that 〈A*ψ | φ 〉 = 〈ψ | Aφ 〉 for all vec tors | ψ〉, | φ 〉 for whic h the two expressions are well-defined. An operator is self-adjoint iff A = A*. A projec tion operator P is a self-adjoint operator suc h that P2 = P. For ease of exposition, we shall mostly c onfine ourselves to the c ase of operators with “disc rete spec trum” (the sum in (2) is disc rete), or even to finite- dimensional Hilbert spac es. (5) Note that any self-adjoint operator c an be dec omposed uniquely into a sum (or more generally an integral) of projec tors onto a family of mutually orthogonal subspac es. This is the so-c alled spec tral theorem, whic h in elementary linear algebra is just the diagonaliz ability of self-adjoint matric es. (6) Inc identally, note that whether a (c lassic al or quantum) partic le moves up or down in a Stern– Gerlac h magnetic field will depend also on whether the inhomogeneous magnetic field is stronger at the north pole or at the south pole. Inverting either the gradient or the polarity of the field will invert the direc tion of deflec tion of a partic le. (Sinc e rotating the apparatus by 180 degrees corresponds to inverting both the gradient and the polarity it has no net effec t on the deflec tion.) Thus the c hoic e of the words “up” and “down” for labeling the results is rather c onventional. (The existenc e of these two different set-ups for measuring spin in the same direc tion is c ruc ial in disc ussing c ontextuality and nonloc ality in pilot-wave theory.) (7) Note that already ac c ording to the minimal interpretation, a quantum system desc ribed by a vec tor in Hilbert spac e has a set of dispositional properties to elic it spec ific responses with given probabilities in measurement situations (and these are fixed uniquely by the sure-fire disposition to elic it a c ertain response with probability 1 in a suitable measurement). The standard interpretation further identifies this set of dispositions with an intrinsic property of the system. Page 28 of 32

Measurement and Classical Regime in Quantum Mechanics (8) Note that thinking of Hilbert-spac e vec tors in terms of their assoc iated probability measures also makes readily intelligible why one c onsiders only unit vec tors. Indeed, normaliz ation of the vec tor ensures that the probabilities are c orrec tly normaliz ed (i.e., add up to 1). (9) One c an c hec k that the definition of the trac e is indeed independent of the basis. In finite dimensions and given a matrix representation of A, the trac e is simply the sum of the diagonal elements of the c orresponding matrix. (10) Normaliz ation means p(1) = 1, with 1 the identity operator (i.e., the projec tion onto the whole of the Hilbert spac e). The theorem holds for quantum systems with Hilbert spac e of dimension at least 3 (but see the remark at the end of sec tion 4.4 below). (11) The relevant sec tion 7 in Hermann's essay has been translated into English by M. Seevinc k (see http: //mpseevinc k.ruhosting.nl/seevinc k/trans.pdf). The same point was famously made by Bell (1966), who further pointed out the absurdity of requiring linearity of the hypothetical “dispersion-free states” (which would have to assign an eigenvalue to eac h observable as a definite value). Bell uses the following example: c onsider the operators σx,σy and σx + σy. For a linear, dispersion-free state λ, ((15)) But the left-hand side takes the possible values ±√2, while the right-hand side takes the possible values −2,0,+2, so that (15) c annot be satisfied. (12) Tec hnic ally, a density operator (in arbitrary dimensions) is a “c ompac t operator,” and for suc h operators a disc rete (if not nec essarily finite) dec omposition as a sum of mutually orthogonal projec tors always exists. (13) Of c ourse the c ollapse postulate is a phenomenologic al rule, so if one does not believe that c ollapse is fundamental, there is a sense in whic h proper mixtures c annot be prepared in this way. Nevertheless, any fundamental approac h to quantum mec hanic s, even if it denies the reality of c ollapse, will have to explain the appearanc e of the possibility of preparing proper mixtures, just as it will have to explain the appearanc e of c ollapse. (14) Geometric ally, this is the maximally mixed state at the c enter of the Bloc h sphere. (15) Tec hnic ally, these are all those subspac es that c ontain the (norm-c losed) range of the density operator, i.e. the (norm-c losure of) the subspac e of all vec tors that are images of vec tors under the linear mapping defined by the density operator. (16) It is important to add that, at least by 1927, Sc hrödinger was well aware that this “c harge density” was not simply a c lassic al c harge density, but a quantity that would (approximately) behave as a c lassic al c harge density only in c ertain respec ts and/or in the appropriate regime. See in partic ular his c ontribution to the 1927 Solvay c onferenc e (Sc hrödinger 1928), and also the disc ussions in Bac c iagaluppi and Valentini (2009, c h. 4, esp. sec tion 4.4) and Bac c iagaluppi (2010, esp. sec tion 4). (17) See in partic ular Born and Heisenberg's c ontribution to the 1927 Solvay c onferenc e (Born and Heisenberg 1928), and the disc ussions in Bac c iagaluppi (2008), and Bac c iagaluppi and Valentini (2009, esp. c hs. 3 and 6). (18) Rec all that Bohm (1952) had rec ently redisc overed and extended the pilot-wave theory by de Broglie (1928). (19) This need not be a problem in itself, say if one interprets the wave func tion along Sc hrödinger's lines as manifesting itself in 3-dimensional spac e as a c harge (or mass) density. It may bec ome a problem if the “tails” are themselves highly structured, as will happen in spontaneous collapse theories in the case of measurements or Sc hrödinger c ats, as this allows for an Everettian-style c ritic ism of the idea that suc h a wave func tion represents a single c opy of a quasi-c lassic al system (i.e., the tail is itself a “tiny” live or dead c at). (20) They are very important also in quantum optic s, bec ause eac h mode of the elec tromagnetic field is a harmonic osc illator. (21) Thus, while we might want to identify kinematic ally a c lassic al state with one with small spreads in both position and momentum, it is spec ific ally the smallness of the spread in position that determines whether this state will behave c lassic ally also in terms of its dynamic s (in the sense of Ehrenfest's theorem). Page 29 of 32

Measurement and Classical Regime in Quantum Mechanics (22) Note that this is not a stric t result, but only a phenomenological arrow of time, sinc e the Sc hrödinger equation is time-symmetric . (23) See Sc hrödinger to Einstein, no date (but between 18 and 31 January 1953), AHQP mic rofilm 37, sec tion 005– 012 (draft ms.) and 005– 013 (c arbon c opy), and Einstein to Sc hrödinger, 31 January 1953, AHQP mic rofilm 37, sec tion 005– 014 (both in German). (24) See sec tions 4.1– 4.5 for the theoretic al disc ussion of measurements, and sec tion 4.6 for the measurement problem. (25) This subsec tion is mostly based on my entry for the Stanford Encyclopedia of Philosophy (Bac c iagaluppi 2003). (26) For a review of more rigorous arguments, see, e.g., Zurek (2003, pp. 28– 30). Suc h arguments c an be obtained in partic ular from the Wigner func tion formalism, as done, e.g., by Zurek and Paz (1994), who apply these results to derive c haotic trajec tories in quantum mec hanic s. (27) For a very ac c essible disc ussion of α -partic le trac ks roughly along these lines, see Barbour (1999, c h. 20). (28) The question of uniqueness of a c lassic al or “quasi-c lassic al” regime has been quite hotly debated espec ially in the “dec oherent histories” literature, and it appears that explic it definitions of quasi-c lassic ality always remain too permissive to identify it uniquely. But maybe uniqueness is not stric tly nec essary (as nowadays often argued in the c ontext of the Everett interpretation). For these issues, see, e.g., Wallac e (2008). Attempts to enforc e uniqueness in other ways appear to overshoot the mark. Indeed, various “modal” interpretations based on the biorthogonal dec omposition theorem, the polar dec omposition theorem, or the spec tral dec omposition theorem for density operators, selec t histories uniquely, but end up agreeing with the results of dec oherenc e only in spec ial c ases, failing to ensure c lassic ality in general (Donald 1998; Bac c iagaluppi 2000). (29) For a c omprehensive c ollec tion of rec ent papers on the Everett interpretation, in partic ular c overing the more modern developments referred to here, see Saunders et al. (2010). (30) While this point is espec ially c lear in Heisenberg's writings, it is c lear that it was espoused also by other main exponents of what is known c ollec tively as the Copenhagen interpretation. For instanc e, Bohr often applies the unc ertainty relations to mac rosc opic piec es of apparatus in his replies to Einstein's c ritic al thought experiments of the period 1927– 1935 (Bohr 1949). And Pauli, c ommenting to Born on Einstein's views, is adamant that under the appropriate experimental c onditions also mac rosc opic objec ts would display interferenc e effec ts (Pauli to Born, 31 Marc h 1954, reprinted in Born 1969). (31) Indeed, von Neumann's aim was simply to show that there always exist unitary evolutions that will produc e suc h perfec t c orrelations, in order to establish c onsistenc y in this first sense. Heisen-berg's disc ussion, although tec hnic ally somewhat defec tive (see the analysis in Bac c iagaluppi and Crull 2009), is along similar lines. Note, however, that Heisenberg is particularly interested in the case of the Heisenberg microscope, where the electron interac ts with a mic rosc opic anc illa (the photon), and one c onsiders alternative measurements on the anc illa. For Heisenberg's purposes it is thus important that interferenc e is still present and that dec oherenc e does not kic k in until later. (32) The same point is valid if we are talking about the empiric al determination of when and where c ollapse oc c urs in spontaneous c ollapse theories. See the nic e disc ussion in Albert (1992, c h. 5). (33) The function χ[x1,x2] (x) is the so-called characteristic function of the interval [x1,x2], that is, the function that is 1 on the interval and 0 outside. (34) Instead of talking of resolutions of the identity, one c an also talk of PV “measures,” in the sense that (analogously to a probability measure), one c an assign to eac h “event” (subset I of the indic es labeling the results) a corresponding projection ∑i∈I Pi. One will talk similarly of POV measures when the requirement that the elements of the resolution of the identity be projections is relaxed. Page 30 of 32

Measurement and Classical Regime in Quantum Mechanics (35) Tec hnic ally, an effec t is a positive operator with spec trum c ontained in the interval [0, 1]. (36) In this formulation, the theorem holds in all dimensions. (37) For further details of POV observables, we refer the reader to standard referenc es, e.g. Busc h, Grabowski, and Lahti (1995). (38) Note that the c orresponding subspac e in the apparatus Hilbert spac e need not be one-dimensional: in the c ase of the spin measurements of sec tion 4.2, we had infinite-dimensional projec tions onto the upper or lower half of the detec tion sc reen. Given that the “apparatus” will usually be a mac rosc opic system, the idea that a reading should c orrespond to a large subspac e of its state spac e rather than to a single state is quite appealing. A reading ought to c orrespond rather to a mac rosc opic state of the apparatus than to a mic rosc opic state, and a mac rosc opic state c ould well be represented by an appropriate subspac e Pk. (39) Note that in this c ase the system is c ollapsed to an improper mixture of the states | φi〉. (40) Von Neumann's c harac teriz ation is based on an extensive thermodynamic analysis, whic h we shall not enter into, but it should be immediately c lear that the transformation (61) is not time-reversible. (41) We shall assume this for the sake of argument, even though we have suggested in sec tion 3.2 that these mixtures might be improper in the first plac e. (42) Rec all that on the minimal interpretation the indic ator variable is merely a variable that if measured will produc e the result that the apparatus reads either up or down. The position of the elec tron fulfills this role perfec tly. (43) Note also that even if this were possible, suc h a solution to the measurement problem would run into trouble when trying to reproduc e the experimental violations of the Bell inequalities, at least unless the mic rostates of the apparatuses are correlated before the measurements. For a related point see Bacciagaluppi (2012). (44) Arguably, the only loophole is if one c onsiders models in whic h the initial c orrelations between the mic rostate of the apparatus and the state of the system (and between the mic rostates of different apparatuses) are explained in retroc ausal terms, and thus are no longer c onspiratorial. The models of measurement by Sc hulman (1997) are probably best understood in this way. For a more general disc ussion of retroc ausal models in quantum mec hanic s, see Pric e (1996). (45) Historic al puz z le: who is von Neumann referring to? Someone like Sc hrödinger who suggested matter should be literally desc ribed by wave func tions? Or something like the early Copenhagen “disturbanc e” theory of measurement? (46) See e.g. Fine (1970), Brown (1986), Busc h and Shimony (1996), Bassi and Ghirardi (2000) and Bac c iagaluppi (2012). (47) I would suggest, however, that if one c onsiders the in-princ iple possibility of performing arbitrary measurements unimpeded by decoherence, then problems of consistency arise again in the context of thought experiments of the type of Wigner's friend. (48) As a prominent example of a neo-Copenhagen view, one c an take the “quantum Bayesianism” of Fuc hs and c o-workers (e.g., Fuc hs 2010). (49) Modulo the c aveat about Wigner's friend in footnote 47. G uid o Bac c iag aluppi Guido Bacciagaluppi is Reader in Philosophy at The University of Aberdeen. His field of research is the philosophy of physics, in particular the philosophy of quantum theory. He also works on the history of quantum theory and has published a book on the 1927 Solvay conference (together with A. Valentini). He also has interests in the foundations of probability and in issues of tim e sym m etry and asym m etry. Page 31 of 32

The Everett Interpretation David Wallace The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter examines the Everett interpretation of quantum mec hanic s, or the many-worlds interpretation. It analyz es problems that have been raised for the Everett interpretation, inc luding the problem of providing a preferred basis and the probability problem. The c hapter argues for a straightforward, fully realist interpretation of the bare mathematic al formalism of quantum mec hanic s, whic h, it explain, c an make sense of superposed c ats without c hanging the theory and without c hanging our overall view of sc ienc e. K ey words: Ev erett i n terpretati on , qu an tu m mech an i cs, man y -worl ds i n terpretati on , preferred basi s, probabi l i ty probl em, math emati cal formal i sm 1. Intro ductio n The Everett interpretation of quantum mechanics—better known as the Many-Worlds Theory—has had a rather uneven rec eption. Mainstream philosophers have sc arc ely heard of it, save as sc ienc e fic tion. In the philosophy of physic s it is well known but has historic ally been fairly widely rejec ted.1 Among physic ists (at least, among those c onc erned with the interpretation of quantum mec hanic s in the first plac e), it is taken very seriously indeed, arguably tied for first plac e in popularity with more traditional operationalist views of quantum mec hanic s.2 For this reason, my task in this c hapter is twofold. Primarily I wish to provide a c lear introduc tion to the Everett interpretation in its c ontemporary form; in addition, though, I aim to give some insight into just why it is so popular among physic ists. For that reason, I begin in sec tion 2 by briefly reprising the measurement problem in a way that (I hope) gives some insight into just why Everett's idea, if workable, is so attrac tive. In sec tion 3 I introduc e that idea and state “the Everett interpretation”—whic h, I argue in that sec tion, is really just quantum mec hanic s itself understood in a c onventionally realist fashion. In sec tions 4– 10 I explore the c onsequenc es of the Everett interpretation via c onsiderations of its two traditional diffic ulties: the “preferred basis problem” (sec tions 4– 6) and the “probability problem” (sec tions 8– 10). I c onc lude (sec tions 11– 12) with a brief introduc tion to other issues in the Everett interpretation and with some further reading. Little about the Everett interpretation is unc ontroversial, but I deal with the c ontroversy rather unevenly. The c onc epts of decoherence theory , as I note in sec tions 5– 6, have signific antly c hanged the debate about the preferred-basis problem, but these insights have only entered philosophy of physic s relatively rec ently, and relatively little in the way of c ritic ism of a dec oherenc e-based approac h to Everettian quantum mec hanic s has appeared as yet (rec ent exc eptions are Hawthorne (2010), Maudlin (2010), and Kent (2010)). By c ontrast (perhaps bec ause the salient issues are c loser to mainstream topic s in metaphysic s and philosophy of sc ienc e) the probability problem has been vigorously disc ussed in the last dec ade. As suc h, my disc ussion of the former fairly unc ritic ally lays out what I see as the c orrec t approac h to the definition of the Everett interpretation and to the preferred basis problem. Readers will, I suspec t, be better served by forming their own c ritic isms, and seeking them elsewhere, than by any imperfec t attempt of mine to pre-empt c ritic isms. My disc ussion of the latter is (somewhat) Page 1 of 21

The Everett Interpretation less opinionated and attempts to give an introduc tion to the shape of the debate on probability. I use little tec hnic al mac hinery, but I assume that the reader has at least enc ountered quantum theory and the measurement problem, at about the level of Albert (1992) or Penrose, (1989, c h. 6). 2. The Measurement Pro blem There are philosophic al puz z les, perhaps, in how physic al theories other than quantum mec hanic s represent the world, but it is generally agreed that there is no paradox. States of any suc h theory—be it Newtonian partic le mec hanic s or c lassic al elec trodynamic s or general relativity—are mathematic al objec ts of some kind: perhaps func tions from one spac e to another, perhaps N-tuples of points in a three-dimensional spac e, perhaps single points in a high-dimensional, highly-struc tured spac e. And, insofar as the theory is c orrec t in a given situation, these states represent the physical world, in the sense that different mathematically defined states correspond to different ways the world c an be.3 There is spac e for debate as to the nature of this representation—is it direc tly a relationship between mathematic s and the world, or should it be understood as proc eeding via some linguistic desc ription of the mathematic s?4 — but these details c ause no problems for the straightforward (naive, if you like) view that a theory in physic s is a desc ription, or a representation, of the world. Quantum mec hanic s, it is widely held, c annot be understood this way. To be sure, it has a c lean mathematic al formalism—most c ommonly presented as the evolution of a vec tor in a highly struc tured, high-dimensional c omplex vec tor spac e. To be sure, some of the states in that spac e seem at least struc turally suited to represent ordinary mac rosc opic systems: physic ists, at least, seem relaxed about regarding so-c alled “wave pac ket” states of macroscopic systems as representing situations where those systems have conventional, classically describable c harac teristic s. But c entral to quantum mec hanic s is the superposition princ iple, and it tells us (to borrow a famous example) that if x is a state representing my c at as alive, and ϕ is a state representing my c at as dead, then the “superposition state” (1) is also a legitimate state of the system (where α and β are c omplex numbers satisfying | α|2 +| β|2 = 1)—and what c an it represent? A c at that is alive and dead at the same time? An undead c at, in an indefinite state of aliveness? These don't seem c oherent ways for the world to be; they c ertainly don't seem to be ways we observe the world to be. Nor does the prac tic e of physic s seem to treat suc h states as representing the state of the physic al world. Confronted with a c alc ulation that says that the final state of a system after some proc ess has oc c urred is some superposition like ψ , a theoretic ian instead dec lares that the state of the system after the proc ess c annot be known with c ertainty, but that it has probability | α |2 of being in the mac rosc opic physic al state c orresponding to x, and probability | β| 2 of being in the mac rosc opic physic al state c orresponding to ϕ . (If he is more c autious, he may c laim only that it has probabilities | α | 2 , | β| 2 of being observed to be in those mac rosc opic physic al states.) That is, the theoretic ian treats the mathematic al state of the system less like the states of c lassic al mec hanic s, more like those of c lassic al statistical mec hanic s, whic h represent not the way the world is but a probability distribution over possible ways it might be. But quantum mechanics cannot truly be understood that way either. The most straightforward way to understand why is via quantum interferenc e—the α and β c oeffic ients in ψ c an be real or imaginary or c omplex, positive or negative or neither, and c an reinforc e and c anc el out. Ordinary probability doesn't do that. Put more physic ally: if some partic le is fired at a sc reen c ontaining two slots, and if c onditional on it going through slot 1 it's detec ted by detec tor A half the time and detec tor B half the time, and if c onditional on it going through slot 2 it's likewise detec ted by eac h detec tor half the time, then we shouldn't need to know how likely it is to go through slot 1 to predic t that it will have a 50% c hanc e of being detec ted by A and a 50% c hanc e of being detec ted by B. But a partic le in an appropriately weighted superposition of going through eac h slot c an be 100% likely to be detec ted at A, or 0% likely to be, or anything in between. So it seems that our standard approac h to understanding the c ontent of a sc ientific theory fails in the quantum c ase. That in turn suggests a dilemma: either that standard approac h is wrong or inc omplete, and we need to understand quantum mechanics in a quite different way; or that approach is just fine, but quantum mechanics itself Page 2 of 21

The Everett Interpretation is wrong or inc omplete, and needs to be modified or augmented. Call these strategies “c hange the philosophy” and “c hange the physic s,” respec tively. Famous examples of the change-the-philosophy strategy are the original Copenhagen interpretation, as espoused by Niels Bohr, and its various more-or-less operationalist desc endants. Many physic ists are attrac ted to this strategy: they rec ogniz e the virtues of leaving quantum mec hanic s—a profoundly suc c essful sc ientific theory— unmodified at the mathematic al level. Few philosophers share the attrac tion: mostly they see the philosophic al diffic ulties of the strategy as prohibitive. In partic ular, attempts to promote terms like “observer” or “measurement” to some privileged position in the formulation of a sc ientific theory are widely held to have proved untenable. Famous examples of the c hange-the-physic s strategy are de Broglie and Bohm's pilot-wave hidden variable theory, and Ghirardi, Rimini, and Weber's dynamic al-c ollapse theory. Many philosophers are attrac ted to this strategy: they rec ogniz e the virtue of holding on to our standard pic ture of sc ientific theories as representations of an objec tive reality. Few physic ists share the attrac tion: mostly they see the sc ientific diffic ulties of the strategy as prohibitive. In partic ular, the task of c onstruc ting alternative theories that c an reproduc e the empiric al suc c esses not just of nonrelativistic partic le mec hanic s but of Lorentz -c ovariant quantum field theory has proved extremely c hallenging.5 But for all that both strategies seem to have profound diffic ulties, it seems nonetheless that one or the other is unavoidable. For we have seen (haven't we?) that if neither the physics of quantum mechanics nor the standard philosophic al approac h to a sc ientific theory is to be modified, we do not end up with a theory that makes any sense, far less one that makes c orrec t empiric al predic tions. 3. Everett's Insight It was Hugh Everett's great insight to rec ogniz e that the apparent dilemma is false— that, contra the arguments of sec tion 2, we c an after all interpret the bare quantum formalism in a straightforwardly realist way, without either c hanging our general c onc eption of sc ienc e or modifying quantum mec hanic s. How is this possible? Haven't we just seen that the linearity of quantum mec hanic s c ommits us to mac rosc opic objec ts being in superpositions, in indefinite states? Ac tually, no. We have indeed seen that states like ψ —a superposition of states representing mac rosc opic ally different objec ts—are generic in unitary quantum mec hanic s, but it is ac tually a non sequitur to go from this to the c laim that mac rosc opic objec ts are in indefinite states. An analogy may help here. In elec tromagnetism, a c ertain c onfiguration of the field—say, F1(x, t) (here F is the elec tromagnetic 2-form) might represent a pulse of ultraviolet light z ipping between Earth and the Moon. Another c onfiguration, say F2 (x, t), might represent a different pulse of ultraviolet light z ipping between Venus and Mars. What then of the state of affairs represented by (2) What weird sort of thing is this? Must it not represent a pulse of ultraviolet light that is in a superposition of traveling between Earth and Moon, and of traveling between Mars and Venus? How c an a single pulse of ultraviolet light be in two plac es at onc e? Doesn't the existenc e of superpositions of mac rosc opic ally distinc t light pulses mean that any attempt to give a realist interpretation of c lassic al elec tromagnetism is doomed? Of c ourse, this is nonsense. There is a perfec tly prosaic desc ription of F: it does not desc ribe a single ultraviolet pulse in a weird superposition, it just desc ribes two pulses, in different plac es. And this, in a nutshell, is what the Everett interpretation c laims about mac rosc opic quantum superpositions: they are just states of the world in whic h more than one mac rosc opic ally definite thing is happening at onc e. Mac rosc opic superpositions do not desc ribe indefiniteness, they desc ribe multiplic ity. The standard terminology of quantum mec hanic s c an be unhelpful here. It is often tempting to say of a given macroscopic system—like a cat, say—that its possible states are all the states in some “cat Hilbert space,” H cat. Some states in H cat are “macroscopically definite” (states where the cat is alive or dead, say); most are “mac rosc opic ally indefinite.” From this perspec tive, it is a very small step to the inc oherenc e of unitary quantum mec hanic s: quantum mec hanic s predic ts that c ats often end up in mac rosc opic ally indefinite states; even if it makes sense to imagine a c at in a mac rosc opic ally indefinite state, we have c ertainly never seen one in suc h a Page 3 of 21

The Everett Interpretation state; so quantum mec hanic s (taken literally) makes c laims about the world that are c ontradic ted by observation. From an Everettian perspective this is a badly misguided way of thinking about quantum mechanics. This H cat is presumably (at least in the nonrelativistic approximation) some sort of tensor produc t of the Hilbert spac es of the elec trons and atomic nuc lei that make up the c at. Some states in this box c ertainly look like they c an represent live c ats, or dead c ats. Others look like smallish dogs. Others look like the Mona Lisa. There is an awful lot that c an be made out of the atomic constituents cat of a cat, and all such things can be represented by states in H cat, and so c alling it a “c at Hilbert spac e” is very misleading. But if so, it is equally misleading to describe a macroscopically indefinite state cat of H cat as representing (say) “a c at in a superposed state of being alive and being dead.” It is far more ac c urate to say that suc h a state is a superposition of a live c at and a dead c at. One might still be tempted to objec t: very well, but we don't observe the universe as being in superpositions of c ontaining live c ats and c ontaining dead c ats, any more than we observe c ats as being in superpositions of alive and dead. But it is not at all c lear that we don't observe the universe in suc h superpositions. After all, cats are the sort of perfec tly ordinary objec ts that we seem to see around us all the time—a theory that c laims that they are normally in mac rosc opic ally indefinite states seems to make a nonsense of our everyday lives. But the universe is a very big plac e, as physic s has c ontinually reminded us, and we inhabit only a very small part of it, and it will not do to c laim that it is just “obvious” that it is not in a superposition. This becomes clearer when we consider what actually happens, dynamically, cat to H cat, to its surroundings, and to those observing it, when it is prepared in a superposition of a live-c at and a dead-c at state. In outline, the answer is that the system's surroundings will rapidly bec ome entangled with it, so that we do not just have a superposition of live and dead c at, but a superposition of extended quasi-c lassic al regions—“worlds,” if you like— some of whic h c ontain live c ats and some of whic h c ontain dead c ats. If the c orrec t way to understand suc h superpositions is as some sort of multiplic ity, then our failure to observe that multiplic ity is explained quite simply by the fac t that we live in one of the “worlds” and the other ones don't interac t with ours strongly enough for us to detec t them. This, in short, is the Everett interpretation. It c onsists of two very different parts: a c ontingent physic al postulate, that the state of the Universe is faithfully represented by a unitarily evolving quantum state; and an a priori c laim about that quantum state, that if it is interpreted realistic ally it must be understood as desc ribing a multiplic ity of approximately c lassic al, approximately noninterac ting regions that look very muc h like the “c lassic al world.” And this is all that the Everett interpretation c onsists of. There are no additional physic al postulates introduc ed to desc ribe the division into “worlds,” there is just unitary quantum mec hanic s. For this reason, it makes sense to talk about the Everett interpretation, as it does not to talk about the hidden-variables interpretation or the dynamic al- c ollapse interpretation. The “Everett interpretation of quantum mec hanic s” is just quantum mec hanic s itself, “interpreted” the same way we have always interpreted scientific theories in the past: as modeling the world. Someone might be right or wrong about the Everett interpretation—they might be right or wrong about whether it suc c eeds in explaining the experimental results of quantum mec hanic s, or in desc ribing our world of mac rosc opic ally definite objec ts, or even in making sense—but there c annot be multiple logic ally possible Everett interpretations any more than there are multiple logic ally possible interpretations of molec ular biology or c lassic al elec trodynamic s.6 This in turn makes the study of the Everett interpretation a rather tightly c onstrained ac tivity (a rare and welc ome sight in philosophy!). For it is not possible to solve problems with the Everett interpretation by c hanging the interpretative rules or changing the physics: if there are problems with solving the measurement problem Everett- style, they c an be addressed only by hard study—mathematic al and c onc eptual—of the quantum theory we have. Two main problems of this kind have been identified: 1. the preferred basis problem (whic h might better be c alled the problem of branching)—what ac tually justifies our interpretation of quantum superpositions in terms of multiplic ity? 2. The probability problem—how is the Everett interpretation, which treats the Schrödinger equation as deterministic , to be rec onc iled with the probabilistic nature of quantum theory? Page 4 of 21

The Everett Interpretation My main task in the remainder of this c hapter is to flesh out these problems and the c ontemporary Everettian response to eac h. 4. The Preferred Basis Pro blem If the preferred basis problem is a question (“how c an quantum superpositions be understood as multiplic ities?”), then there is a traditional answer, more or less explic it in muc h c ritic ism of the Everett interpretation (Barrett (1999), Kent (1990), Butterfield (1996)): they c annot. That is: it is no good just stating that a state like (1) desc ribes multiple worlds: the formalism must be explic itly modified to inc orporate them. Adrian Kent put it very c learly in an influential c ritic ism of Everett-type interpretations: one c an perhaps intuitively view the c orresponding c omponents [of the wave func tion] as desc ribing a pair of independent worlds. But this intuitive interpretation goes beyond what the axioms justify: the axioms say nothing about the existence of multiple physical worlds corresponding to wave function components. (Kent, 1990) This position dominated disc ussion of the Everett interpretation in the 1980s and early 1990s: even advoc ates like Deutsc h (1985) ac c epted the c ritic ism and rose to the c hallenge of providing suc h a modific ation. Modific atory strategies c an be divided into two c ategories. Many-exact-worlds theories augment the quantum formalism by adding an ensemble of “worlds” to the state vec tor. The “worlds” are eac h represented by an element in some partic ular c hoic e of “world basis” | ψi(t)〉 at eac h time t: the proportion of worlds in state | ψi(t))〉 at time t is |〈ψ(t)|ψi(t)〉|2, where |ψ(t)〉 is the (unitarily evolving) universal state. Our own world is just one element of this ensemble. Examples of many-exac t-worlds theories are given by the early Deutsc h (1985, 1986), who tried to use the tensor-produc t struc ture of Hilbert spac e to define the world basis,7 and Barbour (1994, 1999) who chooses the position basis. In many-minds theories, by c ontrast, the multiplic ity is to be understood as illusory. A state like (1) really is indefinite, and when an observer looks at the c at and thus enters an entangled state like (3) then the observer too has an indefinite state. However: to eac h physic al observer is assoc iated not one mental state, but an ensemble of them: eac h mental state has a definite experienc e, and the proportion of mental states where the observer sees the c at alive is | α | 2 . Effec tively, this means that in plac e of a global “world-defining basis” (as in the many-exac t-worlds theories) we have a “c onsc iousness basis” for eac h observer.8 When an observer's state is an element of the c onsc iousness basis, all the minds assoc iated with that observer have the same experienc e and so we might as well say that the observer is having that experienc e. But in all realistic situations the observer will be in some superposition of c onsc iousness-basis states, and the ensemble of minds assoc iated with that observer will be having a wide variety of distinc t experienc es. Examples of many-minds theories are Albert and Loewer (1988), Lockwood (1989, 1996), Page (1996), and Donald (1990, 1992, 2002). It can be helpful to see the many-exac t-worlds and many-minds approac hes as embodying two horns of a dilemma: either the many worlds really exist at a fundamental level (in whic h c ase they had better be inc luded in the formalism), or they do not (in whic h c ase they need to be explained away as somehow illusory). Both approac hes have largely fallen from favor. Partly, this is on internal, philosophic al grounds. Many-minds theories, at least, are explic itly c ommitted to a rather unfashionable anti-func tionalism—probably even some kind of dualism—about the philosophy of mind, with the relation between mental and physic al states being postulated to fit the interests of quantum mec hanic s rather than being deduc ed at the level of neurosc ienc e or psyc hology. If it is just a fundamental law that c onsc iousness is assoc iated with some given basis, c learly there is no hope of a func tional explanation of how c onsc iousness emerges from basic physic s (and henc e muc h, perhaps all, of modern AI, c ognitive sc ienc e, and neurosc ienc e is a waste of time9 ). And on c loser inspec tion, many-exac t-worlds theories seem to be c ommitted to something as strong or stronger: if “worlds” are to be the kind of thing we see around us, the kind of thing that ordinary macroscopic objects inhabit, then the relation between those ordinary mac rosc opic objec ts and the world will likewise have to be postulated rather than derived. But more important, both approac hes undermine the basic motivation for the Everett interpretation. For suppose that a wholly satisfac tory many-exac t-worlds or many-minds theory were to be developed, spec ifying an exac t Page 5 of 21

The Everett Interpretation “preferred basis” of worlds or minds. Nothing would then stop us from taking that theory, disc arding all but one of the worlds/minds10 and obtaining an equally empiric ally effec tive theory without any of the ontologic al exc ess that makes Everett-type interpretations so unappealing. Put another way: an Everett-type theory developed along the lines that I have sketc hed would really just be a hidden-variables theory with the additional assumption that a c ontinuum of many noninterac ting sets of hidden variables exists, eac h defining a different c lassic al world. (This point is made with some c larity by Bell (1981b) in his c lassic attac k on the Everett interpretation.) At time of writing, almost no advoc ate of “the many-worlds Interpretation” ac tually advoc ates anything like the many-exac t-worlds approac h11 (Deutsc h, for instanc e, c learly abandoned it some years ago) and many-minds strategies that elevate c onsc iousness to a preferred role c ontinue to find favor mostly in the small group of philosophers of physics strongly committed for independent reasons to a nonfunctionalist philosophy of mind. Advoc ates of the Everett interpretation among physic ists (almost exc lusively) and philosophers (for the most part) have returned to Everett's original c onc eption of the Everett interpretation as a pure interpretation: something that emerges simply from a realist attitude to the unitarily evolving quantum state. How is this possible? The c ruc ial step oc c urred in physic s: it was the development of decoherence theory. 5. The Ro le o f Deco herence A detailed review of decoherence theory lies beyond the scope of this chapter, but in essence, decoherence theory explores the dynamic s of systems that are c oupled to some environment with a high number of degrees of freedom. In the most c ommon models of dec oherenc e, the “system” is something like a massive partic le and the “environment” is an external environment like a gas or a heat bath, but it is equally valid to take the “system” to be the mac rosc opic degrees of freedom of some large system and to take the “environment” to be the residual degrees of freedom of that same system. For instanc e, the large system might be a solid body, in whic h c ase the “system” degrees of freedom would be its c entre-of-mass position and its orientation and its “environment” degrees of freedom would be all the residual degrees of freedom of its c onstituents; or it might be a fluid, in whic h c ase the “system” degrees of freedom might be the fluid density and veloc ity averaged over regions a few microns across. Whatever the system-environment split, “decoherence” refers to the tendency of states of the system to become entangled with states of the environment. Typic ally no system state is entirely immune to suc h entanglement, but c ertain states—normally the wave-pac ket states, whic h have fairly definite positions and momentums— get entangled fairly slowly. Superpositions of such states, on the other hand, get entangled with the environment extremely quic kly, for straightforward physic al reasons: if, say, some stray photon in the environment is on a path that will take it through point q, then its future evolution will be very different ac c ording to whether or not there is a wave-pac ket loc aliz ed at q. So if the system is in a superposition of being loc aliz ed at q and being loc aliz ed somewhere else, pretty soon system-plus-environment will be in a superposition of (system loc aliz ed at q, photon sc attered) and (system loc aliz ed somewhere else, photon not sc attered). Intuitively, we c an think of this as the system being c onstantly measured by the environment, though this “measurement” is just one more unitary quantum-mec hanic al proc ess. Mathematic ally, this looks something like the following. If |q,p〉 represents a wave-pac ket state of our mac rosc opic system with position q and momentum p, then an arbitrary nonentangled state of the system will have state (4) so that if the environment state is initially | env0 〉, the c ombined system-plus-environment state is (5) But very rapidly (very rapidly, that is, c ompared to the typic al timesc ales on whic h the system evolves), this state evolves into something like (6) Page 6 of 21

The Everett Interpretation where 〈env(q,p)| env(q′,p′)〉 ≃ 0 unless q ≃ q′ and p ≃ p′. In this way, the environment rec ords the state of the system, and it does so quic kly, repeatedly, and effec tively irreversibly (more ac c urately, it is reversible only in the sense that other macroscopic-scale processes, like the melting of ice, are reversible). Why does this matter? Bec ause as long as the environment is c onstantly rec ording the state of the system in the wave-pac ket basis, interferenc e experiments c annot be performed on the system: any attempt to c reate a superposition of wave-pac ket states will rapidly be undone by dec oherenc e. The overall quantum system (that is, the system-plus-environment) remains in a superposition, but this has no dynamic al signific anc e (and, in partic ular, c annot be empiric ally detec ted) without c arrying out in-prac tic e-impossible experiments on an indefinitely large region of the universe in the system's vicinity. And this matters, in turn, bec ause it is interferenc e phenomena that allow the different struc tures represented by a quantum state in a superposition to interac t with one another, so as to influenc e eac h other and even to c anc el out with one another. If interferenc e is suppressed with respec t to a given basis, then evolving entangled superpositions of elements of that basis c an be regarded as instantiating multiple independently evolving, independently existing struc tures. As suc h, if mac rosc opic superpositions are dec ohered—as they inevitably will be—then suc h superpositions really c an be taken to represent multiple, dynamic ally isolated, mac rosc opic states of affairs. For this reason, by the mid-1990s dec oherenc e was widely held in the physic s c ommunity to have solved the preferred basis problem, by providing a definition of Everett's worlds. (It was just as widely held to have solved the measurement problem entirely, independent of the Everett interpretation; since decoherence does not actually remove mac rosc opic superpositions, though, it was never c lear how dec oherenc e alone was supposed to help.) Philosophers of physic s were rather more skeptic al (Simon Saunders was a notable exc eption; c f. Saunders Saunders (1993), 1995), essentially bec ause dec oherenc e seems to fall foul of Kent's c ritic ism: however suggestive it might be, it does not seem to suc c eed in defining an “explic it, prec ise rule” (Kent 1990) for what the worlds ac tually are. For dec oherenc e is by its nature an approximate proc ess: the wave-pac ket states that it pic ks out are approximately defined; the division between system and environment cannot be taken as fundamental; interferenc e proc esses may be suppressed far below the limit of experimental detec tion but they never quite vanish. The previous dilemma remains (it seems): either worlds are part of our fundamental ontology (in whic h c ase dec oher-enc e, being merely a dynamic al proc ess within unitary quantum mec hanic s, and an approximate one at that, seems inc apable of defining them), or they do not really exist (in whic h c ase dec oherenc e theory seems beside the point). Outside the philosophy of physic s, though (notably in the philosophy of mind, and in the philosophy of the spec ial sc ienc es more broadly), it has long been rec ogniz ed that this dilemma is mistaken, and that something need not be fundamental to be real. In the last dec ade, this insight was c arried over to the philosophy of physic s. 6. Higher- Order Onto lo gy and the Ro le o f Structure On even c ursory examination, we find that sc ienc e is replete with perfec tly respec table entities that are nowhere to be found in the underlying microphysics. Douglas Hofstader and Daniel Dennett make this point very c learly: Our world is filled with things that are neither mysterious and ghostly nor simply c onstruc ted out of the building bloc ks of physic s. Do you believe in voic es? How about hairc uts? Are there suc h things? What are they? What, in the language of the physic ist, is a hole—not an exotic blac k hole, but just a hole in a piec e of c heese, for instanc e? Is it a physic al thing? What is a symphony? Where in spac e and time does “The Star-Spangled Banner” exist? Is it nothing but some ink trails in the Library of Congress? Destroy that paper and the anthem would still exist. Latin still exists but it is no longer a living language. The language of the c avepeople of Franc e no longer exists at all. The game of bridge is less than a hundred years old. What sort of a thing is it? It is not animal, vegetable, or mineral. These things are not physic al objec ts with mass, or a c hemic al c omposition, but they are not purely abstrac t objec ts either—objec ts like the number pi, whic h is immutable and c annot be loc ated in spac e and Page 7 of 21

The Everett Interpretation time. These things have birthplac es and histories. They c an c hange, and things c an happen to them. They c an move about—muc h the way a spec ies, a disease, or an epidemic c an. We must not suppose that sc ienc e teac hes us that every thing anyone would want to take seriously is identifiable as a c ollec tion of partic les moving about in spac e and time. (Hofstadter and Dennett 1981, 6– 7) The generic philosophy-of-sc ienc e term for entities suc h as these is emergent: they are not direc tly definable in the language of microphysics (try defining a haircut within the Standard Model!) but that does not mean that they are somehow independent of that underlying mic rophysic s. To look in more detail at a partic ularly vivid example, c onsider tigers, whic h are (I take it!) unquestionably real, objec tive physic al objec ts, even though the Standard Model c ontains quarks, elec trons, and the like, but no tigers. Instead, tigers should be understood as patterns, or struc tures, within the states of that mic rophysic al theory. To see how this works in prac tic e, c onsider how we c ould go about studying, say, tiger hunting patterns. In princ iple—and only in princ iple — the most reliable way to make predic tions about these would be in terms of atoms and elec trons, applying molec ular dynamic s direc tly to the swirl of molec ules that make up, say, the Kanha National Park (one of the sadly diminishing plac es where Bengal tigers c an be found). In prac tic e, however (even ignoring the measurement problem itself!), this is c learly insane: no remotely imaginable c omputer would be able to solve the 103 5 or so simultaneous dynamic al equations that would be needed to predic t what the tigers would do. Ac tually, the problem is even worse than this. For in a sense, we do have a c omputer c apable of telling us how the positions and momentums of all the molec ules in the Kanha National Park c hange over time. It is c alled the Kanha National Park. (And it runs in real time!) Even if, per impossibile, we managed to build a c omputer simulation of the Park ac c urate down to the last elec tron, it would tell us no more than what the Park itself tells us. It would provide no explanation of any of its c omplexity. (It would, of c ourse, be a superb vindic ation of our extant mic rophysic s.) If we want to understand the c omplex phenomena of the Park, and not just reproduc e them, a more effec tive strategy c an be found by studying the struc tures observable at the multi-trillion-molec ule level of desc ription of this “swirl of molec ules.” At this level, we will observe robust—though not 100% reliable—regularities, whic h will give us an alternative desc ription of the tiger in a language of c ell membranes, organelles, and internal fluids. The princ iples by whic h these interac t will be deduc ible from the underlying mic rophysic s (in princ iple at least; in prac tic e there are usually many gaps in our understanding), and will involve various assumptions and approximations; henc e very oc c asionally they will be found to fail. Nonetheless, this slight riskiness in our desc ription is overwhelmingly worthwhile given the enormous gain in usefulness of this new desc ription: the language of c ell biology is both explanatorily far more powerful, and prac tic ally far more useful, than the language of physic s for desc ribing tiger behavior. Nonetheless it is still ludic rously hard work to study tigers in this way. To reac h a really prac tic al level of desc ription, we again look for patterns and regularities, this time in the behavior of the c ells that make up individual tigers (and other living c reatures that interac t with them). In doing so we will reac h yet another language, that of z oology and evolutionary adaptationism, whic h desc ribes the system in terms of tigers, deer, grass, c amouflage, and so on. This language is, of c ourse, the norm in studying tiger hunting patterns, and another (in prac tic e very modest) inc rease in the riskiness of our desc ription is happily ac c epted in exc hange for another phenomenal rise in explanatory power and prac tic al utility. The moral of the story is: there are struc tural fac ts about many mic rophysic al systems whic h, although perfec tly real and objec tive (try telling a deer that a nearby tiger is not objec tively real) simply c annot be seen if we persist in analyz ing those systems in purely mic rophysic al terms. Zoology is of c ourse grounded in c ell biology, and c ell biology in molec ular physic s, but the entities of z oology c annot be disc arded in favor of the austere ontology of molec ular physic s alone. Rather, those entities are struc tures instantiated within the molec ular physic s, and the task of almost all sc ienc e is to study struc tures of this kind. Of which kind? (After all,“struc ture” and “pattern” are very broad terms: almost any arrangement of atoms might be regarded as some sort of pattern.) The tiger example suggests the following answer, whic h I have previously (Wallac e, 2003a, 93) c alled “Dennett's c riterion” in rec ognition of the very similar view proposed by Daniel Dennett (1991): Page 8 of 21

The Everett Interpretation Dennett's criterio n: A mac ro-objec t is a pattern, and the existenc e of a pattern as a real thing depends on the usefulness—in partic ular, the explanatory power and predic tive reliability—of theories whic h admit that pattern in their ontology. Nor is this ac c ount restric ted to the relation between physic s and the rest of sc ienc e: rather, it is ubiquitous within physic s itself. Statistic al mec hanic s provides perhaps the most important example of this: the temperature of bulk matter is an emergent property, salient bec ause of its explanatory role in the behavior of that matter. (It is a c ommon error in textbooks to suppose that statistic al-mec hanic al methods are used only bec ause in prac tic e we c annot c alc ulate what eac h atom is doing separately: even if we c ould do so, we would be missing important, objec tive properties of the system in question if we abstained from statistic al-mec hanic al talk.) But it is somewhat unusual bec ause (unlike the c ase of the tiger) the princ iples underlying statistic al-mec hanic al c laims are (relatively!) straightforwardly derivable from the underlying physic s. For an example from physic s that is c loser to the c ases already disc ussed, c onsider the c ase of quasi-partic les in solid-state physic s. As is well known, vibrations in a (quantum-mec hanic al) c rystal, although they c an in princ iple be desc ribed entirely in terms of the individual c rystal atoms and their quantum entanglement with one another, are in prac tic e overwhelmingly simpler to desc ribe in terms of “phonons”—c ollec tive exc itations of the c rystal that behave like “real” partic les in most respec ts. And furthermore, this sort of thing is c ompletely ubiquitous in solid- state physic s, with different sorts of exc itation desc ribed in terms of different sorts of “quasi-partic le”—c rystal vibrations are desc ribed in terms of phonons; waves in the magnetiz ation direc tion of a ferromagnet are desc ribed in terms of magnons, c ollec tive waves in a plasma are desc ribed in terms of plasmons, and so on.12 Are quasi-partic les real? They c an be c reated and annihilated; they c an be sc attered off one another; they c an be detec ted (by, for instanc e, sc attering them off “real” partic les like neutrons); sometimes we c an even measure their time of flight; they play a c ruc ial part in solid-state explanations. We have no more evidenc e than this that “real” partic les exist, and indeed no more grip than this on what makes a partic le “real,” and so it seems absurd to deny that quasi-partic les exist—and yet, they c onsist only of a c ertain pattern within the c onstituents of the solid- state system in question. When exactly are quasi-partic les present? The question has no prec ise answer. It is essential in a quasi-partic le formulation of a solid-state problem that the quasi-partic les dec ay only slowly relative to other relevant timesc ales (suc h as their time of flight) and when this c riterion (and similar ones) is met then quasi-partic les are definitely present. When the dec ay rate is muc h too high, the quasi-partic les dec ay too rapidly to behave in any “partic ulate” way, and the desc ription bec omes useless explanatorily; henc e, we c onc lude that no quasi-partic les are present. It is c learly a mistake to ask exactly when the dec ay time is short enough (2.54 × the interac tion time?) for quasi-partic les not to be present, but the somewhat blurred boundary between states where quasi- partic les exist and states when they don't should not undermine the status of quasi-partic les as real, any more than the absenc e of a prec ise boundary to a mountain undermines the existenc e of mountains. What has all this got to do with dec oherenc e and Everett? Just this: that the branc hes whic h appear in dec oherenc e are prec isely the kind of entities that spec ial sc ienc es in general tell us to take seriously. They are emergent, robust structures in the quantum state, and as suc h, we have (it seems) as muc h reason to take them ontologic ally seriously as we do any other suc h struc ture in sc ienc e—suc h as those struc tures that we identify as c hairs and tables, c ats and dogs and tigers. So—on pain of rejec ting the c oherenc e of the spec ial sc ienc es as a whole—we should ac c ept that unitary quantum mec hanic s is already a many-worlds theory: not a many-exac t- worlds theory in whic h the worlds are part of the basic mathematic al struc ture, but an emergent-worlds theory in whic h the worlds are instantiated as higher-level struc tures within that basic struc ture. In this sense, advoc ac y of the Everett interpretation has c ome full c irc le: the rise and fall of many-exac t-worlds and many-minds theories has returned us to Everett's original insight that unitary quantum mechanics should be understood as, not modified to bec ome, a many-worlds theory. 7. Aspects o f the Pro bability Pro blem Conc erns about probability, and attempts to resolve c onc erns about probability, have been part of the Everett interpretation sinc e its inc eption, and the bulk of philosophic al work on the interpretation c ontinues to foc us on this Page 9 of 21

The Everett Interpretation issue, so that I c an do no more here than provide an introduc tion. I will do so by briefly c onsidering three questions that might be (and indeed have been) raised by c ritic s: 1. How c an probability even make sense in the Everett interpretation, given that it is deterministic and that all possible outc omes oc c ur? 2. What justifies the ac tual form of the quantum probability rule in the Everett interpretation? 3. How c an the Everett interpretation make sense of the sc ientific proc ess by whic h quantum mec hanic s was experimentally tested? Before doing so, however, I make two more general observations. First, if there is a problem of probability in the Everett interpretation, then it is an essentially philosophical problem. There is no mystery about how probabilistic theories are mathematically represented in theoretical physics: they are represented by a space of states, a set of histories in that spac e of states (that is, paths through, or ordered sequenc es of elements drawn from, that spac e), and a probability measure over those histories (that is, a rule assigning a probability to eac h subset of histories, c onsistent with the probability c alc ulus). Given dec oherenc e, quantum mec hanic s provides all three (at the emergent level where branches can be defined) just fine, using the standard modulus-squared amplitude rule to define the probability of eac h branc h; indeed, historic ally muc h of the motivation of the dec oherenc e program was to ensure that the probability c alc ulus was indeed satisfied by the modulus-squared amplitudes of the branc hes. So a physic ist who objec ted to the rather philosophic al tenor of the debates on probability in the Everett interpretation would be missing the point: insofar as he is unc onc erned with philosophical aspec ts of probability, he should have no qualms about Everettian probability at all. Sec ond, it has frequently been the c ase that what appear to be philosophic al problems with probability in Everettian quantum mechanics in fac t turn out to be philosophic al problems with probability simpliciter. Probability poses some very knotty philosophic al issues, whic h often we forget just bec ause we are so used to the c onc ept in prac tic e; sometimes it takes an unfamiliar c ontext to remind us of how problematic it c an be. Note that it is of no use for a c ritic to respond that all the same we have a good prac tic al grasp of probability in the non-Everettian c ontext but that that grasp does not extend to Everettian quantum physic s. For that is exac tly the point at issue: the great majority, if not all, of the objec tive probabilities we enc ounter in sc ienc e and daily life ultimately have a quantum-mec hanic al origin, so if the Everett interpretation is c orrec t, then most of our prac tic al experienc e of probability is with Everett-type probability. 8. Pro bability, Uncertainty, and Po ssibility How c an there be probabilities in the Everett interpretation? (asks the c ritic ): there is nothing for them to be probabilities of! Defenders will reply that the probabilities are probabilities of branc hes (understood via dec oherenc e), but the objec tion is that somehow it is illegitimate to assign probabilities to the branc hes, either bec ause probabilities require unc ertainty and it makes no sense to be unc ertain of whic h outc ome will oc c ur in a theory like Everett's, or because somehow probabilities quantify alternative possibilities and there are no alternative possibilities in the Everett interpretation. The c onc iliatory approac h here would be to argue that these c onc epts do after all find a home in Everettian quantum mec hanic s; that is, to argue that people in an Everettian universe should indeed regard different branc hes as different alternative possibilities, and be unc ertain as to whic h one will ac tually oc c ur. To my knowledge this was first argued for by Saunders (1998), via an ingenious thought experiment related to traditional intuition pumps in the philosophy of personal identity; Saunders' goal was to make it intuitive that someone in an Everettian universe should indeed be uncertain about their future, even if they knew the relevant facts about the future branc hes (though see Greaves (2004) for an attempted rebuttal). Subsequent work (muc h of it building on Saunders') has tried to go beyond intuitive plausibility and give a positive ac c ount of what would ground unc ertainty in the Everett interpretation. I am aware of three broad strategies of this kind. First, and most direc tly, Lev Vaidman points out (Vaidman 2002) that someone who carried out a quantum measurement but did not observe the result would be in a state of genuine (albeit indexical) uncertainty. (There would be multiple copies of the experimenter, some in branches with one result and some in branc hes with another, but eac h would be in subjec tively identic al states.) It is unc lear Page 10 of 21

The Everett Interpretation whether this notion of uncertainty (which does not appear to apply to pre-measurement situations) is sufficient to assuage concerns. An alternative approac h via indexic al unc ertainty—this time also applying to the pre-measurement situation—is to think about branc hes as four-dimensional rather than three-dimensional entities (thus entailing that branc hes overlap in some sense13 prior to whatever quantum event c auses them to diverge. Unc ertainty is then to be understood as unc ertainty as to whic h four-dimensional branc h an observer is part of. For exploration and defense of this position, see Saunders and Wallac e (2008a, 2008b), Saunders (2010), and Wilson (2010a, 2010b); for c ritic ism, see Lewis (2007b) and Tappenden (2008). The third strategy is c losely related to the sec ond, but takes its c ue from semantic s rather than from metaphysic s: namely, c onsider the way in whic h words like “unc ertainty” would func tion in an Everettian universe (possibly given some theory of semantic c ontent along the “c harity” lines advoc ated by Lewis (1974), Davidson (1973), and others) and argue that they would in fac t func tion in suc h a way as to make c laims like “one or other outc ome of the measurement will oc c ur, but not both” ac tually turn out c orrec t. I explore this idea in Wallac e (2005, 2006) and in c hapter 7 of Wallac e (2012); see also Ismael (2003) for a position that c ombines aspec ts of the sec ond and third strategies. Whether such semantical considerations are metaphysically (let alone physically) relevant depends on one's view of metaphysic s; Albert (2010), for instanc e, argues that they are irrelevant. A c onc iliatory approac h of a rather different kind is to c onc ede that probability has no plac e in an Everettian world and to show how one c an do without it; typic ally, this is done by arguing that human ac tivity in general, and sc ienc e in partic ular, would proc eed as if quantum-mec hanic al mod-squared amplitude was probability, even if “really” it was not. Deutsc h (1999) and Greaves (2004) advoc ate positions of this kind; both regard “probability” as something to be understood dec ision-theoretic ally, via an agent's ac tions. If it c an be argued that (rational) agents in an Everettian world would ac t as if eac h branc h has a c ertain probability, then (Deutsc h and Greaves argue) this is sufficient. Of c ourse, there is also a dec idedly nonc onc iliatory response available: just to deny the c laim that genuine probability requires either alternative probabilities or any form of unc ertainty. One seldom hears ac tual arguments for these requirements; typic ally they are just stated as if they were obvious. And perhaps they are intuitively obvious, but it is not c lear that this has any partic ular bearing on anything. Someone who adopts the (hopelessly unmotivated) epistemologic al strategy of regarding intuitive obviousness as a guide to truth in theoretic al physic s will presumably have given up on the Everett interpretation long ago in any c ase. This response is ac tually fairly c lose to Deutsc h's and Greaves's position: if it c an be argued that mod-squared amplitude func tions exac tly like probability but lac ks c ertain standardly required philosophic al features that probability has, it is open to us just to deny that those philosophic al features are required, and to adopt the position that insofar as mod-squared amplitude functions exactly like probability, that's all that's required to establish that it is probability. This is my own view on the problem,14 developed in extenso in Wallac e (2012). 9. The Quantitative Pro blem Grant, if only for the sake of argument, that it is somehow legitimate to attac h probabilities to branc hes. There is a further question: Why should those probabilities be required to equal those given by quantum mechanics? One version of this objec tion—going right bac k to Graham (1973)—is that the quantum probabilities cannot be the right probabilities, bec ause the right probabilities must give eac h branc h equal probability. There is generally no positive argument given for this c laim, beyond some gesture to the effec t that the versions of me on the different branc hes are all “equally me”; still, it has a strong intuitive plausibility. It c an, however, be swiftly dismissed. It is possible to argue that the rule is ac tually inc onsistent when branc hing events at multiple times are c onsidered,15 but more c ruc ially, dec oherenc e just does not lic ense any notion of branc h c ount. It makes sense, in the presenc e of dec oherenc e, to say that the quantum state (or some part of it) branc hes into a part in whic h measurement outc ome X oc c urs and a part in whic h it does not oc c ur, but it makes no sense at all to say how many branches c omprise the part in whic h X oc c urs. Study of a given branc h at a finer level of detail will inevitably show it to c onsist of many sub-branc hes; eventually this will c ease to be the c ase as Page 11 of 21

The Everett Interpretation dec oherenc e c eases to be applic able and interferenc e between branc hes bec omes nonnegligible; but there is no well-defined point at whic h this oc c urs and different levels of toleranc e—as well as small c hanges in other details of how we define “branc h”—lead to wildly differing answers as to how many branc hes there are. Put plainly,“branc h number,” insofar as it is defined at all in a given dec oherenc e formalism, is an artifac t of the details of that formalism. (And it is not by any means defined in all suc h formalisms; many use a c ontinuum framework in whic h the c onc ept makes no sense even inside the formalism. For more details on this and on the general question of branc h c ounting, see c hapter 3 of Wallac e (2012).) So muc h for branc h c ounting. The question remains: What positive justific ation c an be given for identifying mod- squared amplitude with probability? One might answer, as did Simon Saunders in the 1990s (Saunders 1995, 1997, 1998), by rejec ting the idea that any “positive justific ation” is needed: after all,in general we do not argue that the probabilities in a physic al theory are what they are (nor indeed, in general, that the other physic al magnitudes in a theory have the interpretation they have); we just postulate it. It is not immediately c lear why this response is any less justified in the Everett interpretation than in non-Everettian physic s; indeed, arguably it works rather better as a postulate, sinc e it is at least c lear what c ategoric al, previously understood magnitude is to be identified with probability. By c ontrast, in classical physics the only real c andidate seems to be long-run relative frequenc ies or some related c onc ept, and even establishing that those have the formal properties required of probability has proven fraught. The most promising c andidate so far is Lewis's “best systems analysis” (Lewis 1986, 55, 128– 131), whic h c onstruc ts probabilities indirec tly from relative frequenc ies and related c ategoric al data; even if that analysis suc c eeded fully, though, it would deliver no more than quantum physic s (together with dec oherenc e) has already delivered, namely a set of quantities with the right formal properties to be identified with probability but no further justific ation for making suc h an identific ation.16 Papineau (1996, 2010) puts essentially the same point in a more pessimistic way. He identifies two c riteria that a theory of probability must satisfy: a “dec ision-theoretic link” (why do we use probability as a guide to ac tion?) and an “inferential link” (why do we learn about probabilities from observed relative frequenc ies?) and c onc edes that Everettian quantum mechanics has no good explanation of why either is satisfied—but, he continues, neither does any other physic al theory, nor any other extant philosophic al theory of probability. The Everett interpretation (Papineau argues) therefore has no special problem of probability. In fac t, in rec ent years the possibility has arisen that probability may ac tually be in better shape in Everettian quantum mec hanic s than in non-Everettian physic s. Arguments originally given by David Deutsc h (1999) and developed in Wallac e (2003b, 2007) suggest that it may be possible to derive the quantum probability rule from general princ iples of dec ision theory, together with the mathematic al struc ture of quantum mec hanic s shorn of its probabilistic interpretation. A fully formaliz ed version of this argument c an be found in Wallac e (2010) and in c hapters 5 and 6 of Wallac e (2012). In philosophic al terms, what suc h arguments attempt to do is to show that rational agents, c ogniz ant of the fac ts about quantum mec hanic s and c onditional on believing those fac ts to be true, are required to treat mod-squared amplitude operationally as probability. Spec ific ally, they are required to use observed relative frequenc ies as a guide to working out what the unknown mod-squared amplitudes are (Papineau's inferential link), and to use known mod-squared amplitudes as a guide to ac tion (his dec ision-theoretic link).17 Spac e does not permit detailed disc ussion of this approac h to probability, but at essenc e it relies on the symmetries of quantum mec hanic s. There is a long tradition of deriving probability from c onsiderations of symmetry, but in the c lassic al c ase these approac hes ultimately struggle with the fac t that something must break the symmetry, simply to explain why one outc ome oc c urs rather than another. This is, of c ourse, not an issue for Everettian quantum mec hanic s! From this perspec tive, the role of dec ision theory is less c entral in the arguments than it might appear: its main func tion is to justify the applic ability of probabilistic c onc epts to Everettian branc hes at all. (And c onversely, if one were c onc erned purely with the question of what the probabilities of eac h branc h were, and prepared to grant that branc hes did have probabilities and that they satisfy normal sync hronic and diac hronic properties, it is possible to prove the quantum probability rule without any mention of dec ision theory; c f. Wallac e (2012, c h. 4).) If this last approac h to probability works (and fairly obviously, I believe it does), it marks a rather remarkable shift in the debate; probability, far from being something that makes the Everett interpretation unintelligible, becomes Page 12 of 21

The Everett Interpretation something that c an be understood in Everettian quantum mec hanic s in a way whic h does not seem available otherwise. (See Saunders (2010) for further development of this theme.) I feel obliged to note that it is highly c ontroversial whether the approac h does indeed work; for rec ent c ritic ism, see Albert (2010), Pric e (2010), and other artic les in Saunders et al. (2010). 10. Epistemic Puzzles The rise of dec ision-theoretic approac hes to Everettian probability (whether to make sense of probability or to derive the probability rule) has led to a new worry about probability in the Everett interpretation. Suppose for the sake of argument that it really c an be shown, or legitimately postulated, that someone who ac c epts the Everett interpretation as c orrec t should behave, at least for all prac tic al purposes, as if mod-squared amplitude were probability. What has that to do with the question of why we should believe the Everett interpretation in the first plac e? Put another way, how would it lic ense us to interpret the usual evidenc e for quantum mec hanic s as evidenc e for Everettian quantum mec hanic s? This suggests a division of the probability problem into practical and epistemic problems (Greaves 2007a), where the former c onc erns how rational agents should ac t given that Everettian quantum theory is correct, and the latter c onc erns how evidenc e bears on the truth of quantum theory in the first plac e, given that it is to be interpreted à la Everett. Arguably, Deutsc h's dec ision-theoretic program (and my development of it) speaks only to the prac tic al problem; indeed, arguably most of the tradition in thinking about Everettian probability speaks only to the prac tic al problem. The last dec ade has seen the development of a small, but c omplex, literature on this subjec t. In essenc e, there are two strategies that have been developed for solving the epistemic problem. The first is highly philosophic al: if it c an be established that mod-squared amplitude is probability, then (it is c laimed) no more is required of the Everett interpretation than of any other physic al theory as regards showing why probability plugs into our epistemology in the way it does. Strategies of this form rely on a mixture of solutions to the prac tic al problem (c f. sec tion 9), arguments that branching leads to genuine uncertainty about the future and/or genuine probabilities (cf. section 8), and appeal to the no-double-standards princ iple I mentioned in sec tion 7. The strategy is tac it in Saunders (1998); I defended an explic it version in Wallac e (2006) (and, in less developed form, in Wallac e (2002)); Wilson (2010b) defends a similar thesis. The other strategy is signific antly more tec hnic al and formal: namely, c onstruc t a formal dec ision-theoretic framework to model the epistemic situation of agents who are unsure whether the Everett interpretation is c orrec t, and show that in that situation (perhaps c ontingent on a solution to the prac tic al problem), agents regard “ordinary” evidenc e as c onfirmatory of quantum mec hanic s in a standard way, even when quantum mec hanic s is understood ac c ording to the Everett interpretation. This strategy was pioneered by Greaves (2004) and brought to a mature state in Greaves (2007a) and Greaves and Myrvold (2010). The latter two papers, on slightly different starting assumptions (inc luding in both c ases the Bayesian approac h to statistic al inferenc e) take it as given that c onditional on the Everett interpretation being true, mod-squared amplitude func tions as probability in dec ision- making c ontexts, and derive that agents will update their personal probability in quantum mec hanic s via standard update proc edures, whether or not quantum probabilities are to be understood in Everettian terms. As suc h, these arguments take as input a solution to the prac tic al problem (whether postulated or derived via Deutsc h's and/or my arguments) and give as output a solution to the epistemic problem. It is also possible (c f. Wallac e, 2012, c h. 6) to c ombine the two strategies into one theorem, whic h makes standard dec ision-theoretic assumptions and derives solutions to the epistemic and prac tic al problems in a unified fashion. 11. Other To pics While the bulk of contemporary work on the Everett interpretation has been concerned with the preferred-basis and probability problems (and, more generally, has been c onc erned with whether the interpretation is viable, rather than with its philosophic al implic ations if viable), there are a goodly number of other areas of interest within the Everett interpretation (or, as I would prefer to put it: within quantum mec hanic s, onc e it is understood that it should be interpreted Everett-style), and I briefly mention some of these here. Page 13 of 21

The Everett Interpretation • If Everettian quantum mec hanic s is only emergently a theory of branc hing universes, what is its fundamental ontology, insofar as that question has meaning? That is: what kind of physic al entity is represented by the quantum state? Of c ourse, this question c an be asked of any approac h to quantum theory that takes the state as representing something physic ally real, but it takes on a partic ular urgenc y in the Everett interpretation given that the theory is supposed to be pure quantum mechanics, shorn of any additional mathematical structure. For various approac hes to the problem, see Deutsc h and Hayden (2000), Deutsc h (2002), Wallac e and Timpson (2007, 2010), Maudlin (2010) (who argues that there is no coherent understanding of the Everett interpretation's ontology), Hawthorne (2010) (who is at least sympathetic to Maudlin), Allori et al. (2009), and (in the general c ontext of the ontology of the quantum state) Albert (1996) and Lewis (2004b). (I should add one c autionary note: it is very c ommon in the literature to phrase the question as, what is the ontology of the wave-func tion? But rec all that the wave-func tion is only one of a great many ways to represent the quantum state, and one which is much more natural in nonrelativistic physics than in quantum field theory.) • It is generally (and in my view c orrec tly) held that the experimental violation of Bell's inequalities18 shows not just that hidden variable theories must involve superluminal dynamics, but that any empirically adequate theory must involve superluminal dynamic s.19 But the Everett interpretation is generally (and again c orrec tly, in my view) viewed as an exc eption, essentially bec ause it violates a tac it premise of Bell's derivation, that only one outc ome ac tually oc c urs.2 0 There has, however, been rather little exploration of this issue; Bac c iagaluppi (2002) is a notable exc eption.2 1 • There is an ongoing (and somewhat sensationalist) discussion in the literature about so-called “quantum suic ide”: the idea that an agent in an Everettian universe should expec t with c ertainty to survive any proc ess whic h third-party observers regard him as having nonz ero probability of surviving. The idea has been around in the physic s c ommunity for a long time (see, e.g., Tegmark (1998); it was first introduc ed to philosophers by David Lewis, in his only paper on the Everett interpretation (Lewis 2004a) and has been disc ussed further by Lewis (2000) and Papineau (2003). • Everett was originally motivated in part by a desire for an interpretation of quantum mec hanic s that was suitable for c osmology in that it did not assume an external observer. The Everett interpretation has been widely influential in quantum c osmology ever sinc e: for an introduc tion, see Hartle (2010). It is not universally ac knowledged that quantum c osmology does require the Everett interpretation, though; for dissenting views (from widely differing perspectives), see Fuchs and Peres (2000), Smolin (1997: 240–266), and Rovelli (2004: 209–222). • The de Broglie-Bohm “pilot wave” theory (aka Bohmian mec hanic s) has sometimes been c ritic iz ed for being “Everett in denial”: that is, being the Everett interpretation with some additional epiphenomenal struc ture. For examples of this c ritic ism, see Deutsc h (1996) and Brown and Wallac e (2004); for responses, see Lewis (2007a) and Valentini (2010) (see also Brown's (2010) response to Valentini). Allori et al. (2008) c an also be read as a response, insofar as it advoc ates a position on the ontology of a physic al theory far removed from that of sec tion 6 and from whic h the Everett-in-denial objec tion c annot be made. 12. Further Reading Saunders et al. (2010) is an up-to-date and edited c ollec tion of artic les for and against the Everett interpretation, inc luding c ontributions from a large frac tion of the physic ists and philosophers involved in the c ontemporary debate; Saunders's introduction to the book provides an overview of the Everett interpretation complementary to this c hapter. Barrett (1999) is a c omprehensive guide to disc ussions of the Everett interpretation in (mostly) the philosophy of physic s literature, up to the late 1990s. DeWitt and Graham (1973) is a c lassic c ollec tion of original papers. Wallac e (2012) is my own book-length defense of the Everett interpretation; Wallac e (2008) is a review of the measurement problem more generally, focused on the role of decoherence theory. Greaves (2007b) reviews work in the probability problem. Afterwo rd I have left undisc ussed the often-unspoken, often-felt objec tion to the Everett interpretation: that it is simply unbelievable. This is bec ause there is little to disc uss: that a scientific theory is wildly unintuitive is no argument at all against it, as twentieth-c entury physic s proved time and again. David Lewis is memorably reported to have said Page 14 of 21

The Everett Interpretation that he did not know how to refute an inc redulous stare; had he been less c haritable, he might have said explic itly that an inc redulous stare is not an argument, and that if someone says that they are inc apable of believing a given theory—philosophical or scientific—they are but reporting on their psychology. References Abrikosov, A. A., L. P. Gorkov, and I. E. Dz yalohinski (1963). Methods of quantum field theory in statistical physics. New York: Dover. Revised English edition; translated and edited by R. A. Silverman. Albert, D. Z. (1992). Quantum mechanics and experience. Cambridge, MA: Harvard University Press. ———. (1996). Elementary quantum metaphysic s. In J. T. Cushing, A. Fine, and S. Goldstein (Eds.), Bohmian mechanics and quantum theory: An appraisal , 277– 284. Dordrec ht: Kluwer Ac ademic Publishers. ———. (2010). Probability in the Everett pic ture. In S. Saunders, J. Barrett, A. Kent, and D. Wallac e (Eds.), Many worlds? Everett, quantum theory and reality. Oxford: Oxford University Press. Albert, D. Z., and B. Loewer (1988). Interpreting the many worlds interpretation. Synthese 77: 195–213. Allori, V., S. Goldstein, R. Tumulka, and N. Zanghi (2008). On the c ommon struc ture of Bohmian mec hanic s and the Ghirardi–Rimini–Weber theory. British Journal for the Philosophy of Science 59: 353–389. ———. (2009). Many-worlds and Sc hrödinger's first quantum theory. Forthc oming in British Journal for the Philosophy of Science; available online at http://arxiv.o rg/abs/0903.2211. Bac c iagaluppi, G. (2002). Remarks on spac e-time and loc ality in Everett's interpretation. In J. Butterfield and T. Plac ek (Eds.), Non-locality and modality, Vol. 64 of Nato Sc ienc e Series II. Mathematic s, Physic s and Chemistry. 105–122. Dordrecht: Kluwer. Barbour, J. B. (1994). The timelessness of quantum gravity: II. The appearanc e of dynamic s in static c onfigurations. Classical and Quantum Gravity 11: 2875–2897. ———. (1999). The end of time. London: Weidenfeld and Nicholson. Barrett, J. A. (1999). The quantum mechanics of minds and worlds. Oxford: Oxford University Press. Bell, J. S. (1981a). Bertlmann's soc ks and the nature of reality. Journal de Physique 42, C2: 41– 61. Reprinted in Bell (1987), 139–158. ———. (1981b). Quantum mec hanic s for c osmologists. In C. J. Isham, R. Penrose, and D. Sc iama (Eds.), Quantum gravity 2: A second Oxford Symposium. Oxford: Clarendon Press. Reprinted in Bell (1987), 117–138; page referenc es are to that version. ———. (1987). Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press. Brown, H. (2010). Reply to Valentini: “de Broglie-Bohm theory: Many worlds in denial?”. In Many worlds? Everett, quantum theory, and reality. Oxford: Oxford University Press. 510–517. Brown, H. R., and D. Wallac e (2004). Solving the measurement problem: de Broglie-Bohm loses out to Everett. Forthc oming in Foundations of Physics; available online at http://arxiv.o rg/abs/quant- ph/0403094. Butterfield, J. N. (1996). Whither the minds? British Journal for the Philosophy of Science 47: 200–221. Colin, S. (2003). Beables for quantum elec trodynamic s. Available online at http://arxiv.o rg/abs/quant- ph/0310056. Colin, S., and W. Struyve (2007). A Dirac sea pilot-wave model for quantum field theory. Journal of Physics A 40: 7309–7342. Davidson, D. (1973). Radic al interpretation. Dialectica 27: 313– 328. Page 15 of 21

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