philosophy of physics

Index Symmetry, 287– 317. See also Symmetry and equivalenc e anti-particles, 294n10 CERN's Large Hadron Collider, 305 c harge c onjugation, 292– 94 c lassific ation of, 292– 95 c harge c onjugation, 292– 94 CPT theorem (Christenson, Cronin, Fitc h, and Turlay), 294– 95 discrete symmetries, 292–94 Lorentz invariant, 292 parity, 292–94 spacetime symmetries, 292 time-reversal, 292–94 c ontinuous symmetry, 295– 97 gauge theories, 299 CPT theorem (Christenson, Cronin, Fitc h, and Turlay), 294– 95 “Curie Princ iple,” 290 discrete symmetries, 292–94 dynamic al ac c ount, 302n22 “Eightfold Way,” 302n22 Einstein's theories of spec ial and general relativity, 290 electroweak theory, 395–97 elec troweak unific ation, 303– 6 equal areas law, 288n2 Euler-Lagrange equations, 296 Fermilab's Tevatron, 305 “gauge argument,” 291, 298–300 Lagrangian, 288–99 Lagrangian, kinetic c omponent, 300n17 gauge theories, 300–303 c ontinuous symmetry, 299 dynamic al ac c ount, 302n22 “Eightfold Way,” 302n22 Gell-Mann “strangeness,” 302 Glashow-Salam-Weinberg (GSW) model, 303 “gluing” role, 301 isospin idea, 301 Lorentz invariant, 299 “power of the gauge,” 299 quantum c hromodynamic s (QCD), 302 weak neutral c urrents, 302n24 Gell-Mann “strangeness,” 302 Glashow-Salam-Weinberg (GSW) model, 303 global c ontinuous symmetry, 295– 97 “gluing” role, 301 Goldstone boson, 304 Page 77 of 112

Index group theory, 289 Hamiltonian equations, 289 Hamilton's “Princ iple of the Least Ac tion,” 295 Higgs boson, 291 Higgs mec hanism, 303– 6 Higgs partic le, 6, 381– 82 isospin idea, 301 Lagrangian, 295–96, 299 kinetic component, 300n17 variational symmetries, 326–27 Lorentz invariant, 292 mean field theory, 160n12, 161 multiplet sc heme, predic tion from, 307– 9 Newtonian scheme, 295 Noether's theorem, 289, 291, 295–97 omega minus hadron, 307 ontologic al issues, 306– 7 overview, 6 parity, 292–94 permutation symmetry, 6–7 prediction from multiplet sc heme, 307– 9 Neptune, predic tion of, 310 of omega minus hadron, 306 spin-3/2 baryon decuplet, 307–8, 309 quantum c hromodynamic s (QCD), 302 reversal of a trend, 310–12 rotations, 288 snowflake, 288 spacetime substantivalism, 533–36 c oordinate independent transformations, 533 Galilei group, 533, 534 generally covariant equations, 533 Leibniz group of transformations, 533 Leibniz relationalisms, 535 spacetime symmetries, 292 square invariant, 288 SSB insight, 304 STR, formulation of, 311–12 symmetry group of the square, 289 time-reversal, 292–94 totalitarian princ iple, 309n39 total symmetry, indistinguishability, 369 unification, 303–6 “variational problem,” 297 Page 78 of 112

Index weak neutral c urrents, 302n24 Symmetry and equivalence, 318–39 Cartan distribution, 324 c lassic al symmetries, 323– 24, 334– 37 differential equations, symmetries of, 323–26 Cartan distribution, 324 c lassic al symmetries, 323– 24, 334– 37 generaliz ed symmetries, 323–26 Kepler problem, 325 Korteweg-de Vries vector, 325 Lenz -Runge vec tor, 325 Lie-Bäcklund transformations, 324–26 local symmetries, 324–25 Maxwell's theory, 325n21 nonloc al symmetries, 323, 325 and physic al equivalenc e, 329– 30 disaster, rec ipe for, 321– 22 divergence symmetries, 327nn33 and 34 D1 and D2, 319–20 D2, physical equivalence, 329–30 fecundity and generality, trade-off between, 335n13 Fruitless definition, 322 Galilean boosts, 327, 331 Galilei group and physic al equivalenc e, 330 generaliz ation, 321–22 generaliz ed symmetries, 323–26 Hamiltonian symmetries, 326–27 outlook, 334 and physic al equivalenc e, 331– 32 instantaneous states, 326n31 jet bundle, 324 Kepler problem differential equations, symmetries of, 325 and physic al equivalenc e, 332 Korteweg-de Vries vec tor, 325, 333 Lagrangian treatment and physic al equivalenc e, 331 Lenz -Runge vec tor, 325 Lie-Bäcklund transformations, 324–26 local symmetries, 324–25 Maxwell's theory, differential equations, 325n21 Newtonian boosts, 327n33 Newtonian theory, 322n12, 324, 327–28n36 particles, 324, 325, 330 physic al equivalenc e, 328 Noether's theorem, 324–25, 328 Page 79 of 112

Index nonloc al symmetries, 323, 325 outlook, 333–34 overview, 6 pairs of solutions and physic al equivalenc e, 333n55 phase spac e, 327 and physic al equivalenc e, 328– 33 differential equation, 329–30 D2, 329–30 Galilei group, 330 Hamiltonian symmetries, 331–32 Kepler problem, 332 Lagrangian treatment, 331 and Newtonian theory, 328 pairs of solutions, 333n55 spacetime symmetries, 331–32n49, 332 siz e of group, 322n10 spacetime symmetries, 321–22 and physical equivalence, 331–32n49, 332 struc ture of, 319 stage-setting, 320–21 symmetries abstractly speaking, 321 symplectic form, 327n35 variational symmetries, 326–27 Symmetry group of the square, 289 Symplectic form, 327n35 Synthetic unity, unific ation in physic s, 393– 401 Systematic error, dark matter and dark energy, 620 Tait, Peter, 59, 66, 105 Taylor, Geoffrey, 22 Temporarily orientable spac etime, relativistic spac etime, 589 Tensor fields, c ontinuum mec hanic s, 98 Theoretic al laws, hydrodynamic s, 31 “Theoretic al unific ation,” early universe c osmology, 633 Theory fac ades, axiomatic presentation, 48, 57 Theory of Everything (TOE), 381, 383, 405 “Theory of initial c onditions,” 637 Thermal history, Standard Model, 614–16 Thermodynamic phases, 142 Thermodynamic properties, phase transitions, 201 Thermodynamics, 145–46 effective field theory (EFT), 249–50 indistinguishability, 343–46 c onventions, system of, 246 entropy, 343–45 heat transfer, 344–45 Page 80 of 112

Index limits, 259–69 phases, 142 reduc tion of, 260 Thermodynamic treatment, phase transitions, 191– 93 c ontinuous phase transitions, 191– 92 c ritic al exponent, 193 ferromagnetic transitions, 192 first-order phase transitions, 191 Helmholtz free energy, 192 order-disorder transitions, 192 order parameter, 192–93 Thomson, J. J., 30 Thomson, William (Lord Kelvin) explanatory progress, 28 Helmholtz -Kelvin instability, 19 instabilities, 20 surfac e waves, 17 Treatise on Natural Philosophy, 59, 66 unified theory, 387 vortex motion, 17 Tidal forc es, 539n34 Time dependent boundary c ondition, radiation theory, 124 Time-harmonic waves, radiation theory, 127– 29 Timelike future, relativistic spac etime, 590– 91 Timelike geodesic ally inc omplete, spac etime properties, 598 Timelike vec tors, relativistic spac etime, 589, 589 Time-reversal c ausation, 136n37 symmetry, 292–94 Time travel, 601–3 Todhunter, Isaac , 271 Tollmien, Walter, 20 Toothpaste, c ontinuum mec hanic s, 89, 89 Top-down approach effective field theory (EFT) naturalness, hypothesis of, 228 overview, 225–29 quantum c hromodynamic s (QCD), 228– 29 symmetry considerations, 228 Wilsonian approach, 228–29 “rari-c onstanc y” theorists, 270 the tyranny of sc ales, 257 Torque r, 77, 78 Totalitarian princ iple, symmetry, 309n39 Trac tion forc es, rigid body mec hanic s, 73 Page 81 of 112

Index Trac tion vec tors, 93 c ontinuum mec hanic s, 83, 83– 84 Trajectories of force-free bodies, spacetime as, 542n41 Transformation, quantum mec hanic s, 446 Transtemporal structure, spacetime substantivalism, 531 Trapped surfac e, global spac etime struc ture, 600 Treatise on Electricity and Magnesium (Maxwell), 391 Treatise on Natural Philosophy (Tait and Thomson), 59, 66, 105 Truesdell, Clifford, 93 Turbulence hydrodynamic s, 21– 23 mean field theory, 167 Turning moment, rigid body mec hanic s, 77, 78 “ Twin paradox” sc enario, relativistic spac etimes, 536 Two-dimensional Ising model, 153, 181 Tyndall, John, 20 Type Ia supernovae, 609, 618n25 The tyranny of sc ales, 255– 86 ab initio strategy, 263–64 averaging and homogeniz ation, differenc es, 277–78 “between” sc ale struc tures, 256, 284 bottom-up approach, 257 bubbles within bubbles, 267, 284 Cauchy's equation, 270–75 c ontinuum mec hanic s, “material partic les,” 270n18 c ontinuum model equations, 256– 57 c ontroversy, 269– 73 Cauchy's equation, 270–75 Navier-Stokes equations, 269–70 empiric al investigation of means, 276 Euler's recipes, 273–75 and Cauchy, Augustin, 273–75 c ontinuum, 273– 74, 278 discrete, 273–74 and Navier-Stokes equations, 275 and Young's modulus, 275 ferromagnet model, 265–66, 278 spontaneous magnetiz ation, 265 50-50 volume mixture, 268 games and gambling, 275–76, 277 Gaussian distribution, 276, 279 Hamiltonian func tion, 276 homogeniz ation, 256, 280–83 averaging and homogeniz ation, differenc es, 277–78 limit, 281 Page 82 of 112

Index hydrodynamic theory, 279 infinite limits, 261–62 “material partic les,” 270n18 mean field c alc ulations, 266 Navier-Stokes equations, 269–70, 275 Navier-Stokes theory, 278–79 overview, 5–6 post fac to strategy, 263 random variables, 277 reduction, 260 renormaliz ation group (RG), 264–66, 269, 275, 280 representative volume element (REV), 264, 267–68, 280, 284 resolution to, 275–83 averaging and homogeniz ation, differenc es, 277–78 “between” scale structures, 284 empiric al investigation of means, 276 Euler's c ontinuum rec ipe, 278 games and gambling, 275–76, 277 Gaussian distribution, 276, 279 Hamiltonian func tion, 276 homogeniz ation, 280–83 hydrodynamic theory, 279 Navier-Stokes theory, 278–79 random variables, 277 renormaliz ation group (RG), 280 representative volume element (REV), 280, 284 steel, Gaussian and, 279 steel beams, 258–59, 264, 278, 279 steel, Gaussian and, 279 thermodynamic limits, 259– 69 top-down approach, 257 and Young's modulus, 272, 275 Uhlenbec k, George, 162, 163 Unc ertainty, Everett interpretation and, 475– 77 Undamped harmonic osc illator advanc ed Green's func tions, 110 retarded Green's functions, 110 Unific ation in physic s, 381– 415 c ondensed matter physic s, 383– 84 effec tive field theory (EFT), 384, 406– 13 Gell-Mann/Low formulation, 408–9 Hamiltonians, 410 quantum elec trodynamic s (QED), 411 reduc tionism, problems of, 411n25 Wilson-Kadanoff model, 412 Page 83 of 112

Index electroweak theory, 393–401 Big Bang, 303, 403 “big dessert” assumption, 404 Cabibbo-Kobayashi-Maskawa framework, 402 CP symmetry, 402–3 gauge hierarchy problem, 404 Glashow model, 397 Higgs boson, 394–95, 403–6 Higgs field, 398–401 isospin, 396n14 Lagrangian, 395 Lagrangian invarianc e, 397– 99 Large Hadron Collider (LHC), 405–6 Lie group, 395 loop quantum gravity (LQG), 405 from mathematic s to physic s, 397– 401 and Maxwell's theory, 395, 399 multiplets, 396 naturalness, 404n19 Noether's theorem, 396–97 non-Abelian c ase, 396, 401 phase transformation, 396n13 Planc k sc ale, 403 problems with, 403–6 QCD vac uum, 394 quantum c hromodynamic s (QCD), 402– 3 Schrödinger equation, 395 Standard Model, 394, 402–6 string theory, 405 SU(2) and SU(3) color groups, 394, 396–97 supersymmetry (SUSY), 405 symmetry as tool for unific ation, 395– 97 Theory of Everything (TOE), 405 Yukawa couplings, 402 Yukawa interac tion, 402n17 General Relativity, 383 Higgs partic le, 6, 381– 82 Large Hadron Collider (LHC), 381–82, 384 future output, 406 Maxwell's elec trodynamic s, 385– 93 Ampère law, 389 c hain reac tion, 387– 88 c urrents, induc tion of, 386n5 d'Alembert's principle, 389–90 displacement, 391n9 Page 84 of 112

Index displacement current, 386–89 electrodynamics, 386–87 and Faraday's ac c ount of elec tromagnetism, 385, 387 and fic tional models, 387– 93 Lagrangian mechanics, 386–87, 390 Maxwell's theory, 383 Newtonian mec hanic s, 406 overview, 7–8 physic al problem, analysis of, 384n3 possibility of reduc tion, 384 quantum elec trodynamic s (QED), 396– 97 quantum field theory (QFT), 382, 408–10, 412–13 reductive unity, 385–93 renormaliz ation, 406–13 renormaliz ation groups (RGs), 382, 385, 407, 410 Standard Model, 381, 383 string theory, 406 superconductivity, 384 superfluidity, 384 supersymmetry (SUSY), 383 symmetry, 303–6 synthetic unity, 393–401 Theory of Everything (TOE), 381, 383 universal behavior, 384 universality c lass, 412 “Unified theory.” See Unific ation in physic s Uniform symmetry, indistinguishability as, 352–65 c lassic al indistinguishability, argument against, 354– 56 fermions, 364–65 Gibbs' solution, 352–53 haecceitism, 356–60 quantum statistic s, explanation of, 360– 64 Uniqueness of universe, 624–27 “c eteris paribus” laws, 626n44 and “Laws of Physic s,” 625 Mars's motion, 626–27 and quantum field theory (QFT), 626 and relativistic c osmology, 625– 26 Sun's gravitational field, 626–27 Uniqueness results, unitary equivalenc e, 491– 501 ac tion princ iple, 491 analyz ing physic al equivalenc e, 496–501 c anonic al antic ommutation relations (CARS), 494, 497 Hamiltonian system, 493–94 Hilbert space, 494–95 Page 85 of 112

Index kinematic pair, 495– 96 kinematic s, 492 Lagrangian, 492 ordinary QM, 495, 496, 500–502, 504 Pauli relations, 494 PEV, 496–500, 496n6, 504, 508 physic al equivalenc e, unitary equivalenc e as, 494– 95 Poisson bracket, 493–94 preliminaries, 491–92 quantiz ing, 492–94 Schrödinger equation, 494 weak operator topology, 495 Unitarily inequivalent representations, 491 Unitary equivalenc e, physic al equivalenc e and, 9, 489– 521 ac tion princ iple, 491 “Algebraic Imperialism,” 513 analyz ing physic al equivalenc e, 496–501 blac k hole evaporation, 518 Bohm-Aharonov effec t, 507 c anonic al antic ommutation relations (CARS), 489, 490, 494, 497, 501, 502, 504 c losing uniform topology, 510n23 c ompeting c riteria of equivalenc e, 513– 14 GNS representation, 511–12 Hadamard condition, 517–18 Hamiltonian system, 493–94 Hawking radiation exterior, 518 Hilbert space, 494–95, 503n12, 510, 518 c ompeting c riteria of equivalenc e, 513 representations, 9, 489–90 Jordan-Wigner theorem, 9, 490–91, 501 kinematic pair, 495– 96 kinematic s, 492 Lagrangian, 492 Minkowski spac etime, 518 ordinary QM, 495, 496, 500–502, 504, 510–16 PAS, 504, 508, 514 Pauli relations, 494 PEV, 496–500, 496n6, 504, 508, 514 physic al equivalenc e, unitary equivalenc e as, 494– 96 Poisson bracket, 493–94 preliminaries, 491–92 princ iples, 514– 19 Hadamard condition, 517–18 Hawking radiation exterior, 518 Minkowski spac etime, 518 Page 86 of 112

Index ordinary QM, 514–16 quantum field theory (QFT), 518 quantiz ing, 492–94 quantum field theory (QFT), 491, 518 quantum statistic al mec hanic s (QSM), 491 reasons for unitary equivalenc e, 513– 14 Schrödinger equation, 494 Stone-von Neumann theorem, 490–91, 501, 505–6 technical interlude, 509–13 uniqueness results, 491–501 ac tion princ iple, 491 analyz ing physic al equivalenc e, 496–501 c anonic al antic ommutation relations (CARS), 494, 497 Hamiltonian system, 493–94 Hilbert space, 494–95 kinematic pair, 495– 96 kinematic s, 492 Lagrangian, 492 ordinary QM, 495, 496, 500–502, 504 Pauli relations, 494 PEV, 496–500, 496n6, 504, 508 physic al equivalenc e, unitary equivalenc e as, 494– 96 Poisson bracket, 493–94 preliminaries, 491–92 quantiz ing, 492–94 Schrödinger equation, 494 weak operator topology, 495 unitary inequivalenc e, examples of, 501– 9 bead on a c irc le, 505– 7 Bohm-Aharonov effec t, 507 Circ ular Canonic al Commutation Relations (CCCRs), 506 ferromagnet, 508, 509 Hamiltonian theory, 507 Hilbert spac e, 503 infinite spin c hain, 501– 5, 508 ordinary QM, 508 Pauli spins, 494, 501–4 Poisson brac ket, 505– 6 polariz ation of a system, 502 Yang-Mills theory, 507 weak operator topology, 495 Unitary inequivalenc e bead on a c irc le as example of unitary inequivalenc e, 505– 7 examples of. See Unitary equivalenc e, physic al equivalenc e and Universal behavior, unific ation in physic s, 384 Page 87 of 112

Index Universality, 161–62 phase transitions, 217–18 Universality c lasses matter, infinities and renormaliz ation, 178–79 unific ation in physic s, 412 Universe, uniqueness of. See Uniqueness of universe “Unsharp” spin measurements, quantum mec hanic s, 444– 446 Up and down spin states, quantum mec hanic s, 418– 19 U.S. National Bureau of Standards c onferenc e, 168 Vaidman, Lev, 476 van der Waals, Johannes c orresponding states, princ iple of, 162 first mean field theory, 147–49, 149 memorial meeting, Netherlands meeting (1937), 162–64 phase transitions, 142, 147 van Fraassen, Bas C., 565 van Kampen, N. Gibbs paradox, 368 haecceitism, 356 N! puz z le, 350–51 thermodynamic s, 345 as uniform symmetry, 354–55 “Variational problem,” symmetry, 297 Variational symmetries, 326–27 Veblen, O., 289n3 Versc haffelt, J. E., 164 Vibrating strings, c ontinuum mec hanic s, 92, 92, 101 Virasoro algebra, 182 Virtual displac ement, rigid body mec hanic s, 80, 81 Virtual variations, rigid body mec hanic s, 79n33 Virtual-work reasoning, rigid body mec hanic s, 72, 72 Visc osity of fluid, axiomatic presentation, 54 Volume measures, indistinguishability, 362–63 von Neumann, John. See also Dirac -von Neumann interpretation; Stone-von Neumann theorem+ disc retiz ed position measurements, 443 and Heisenberg's “cut,” 441, 442n31 no-hidden variables theorem, 422 quantum mec hanic s measurement, 452, 454 Voronel, Alexander, 165 Vortex motion Helmholtz -Kelvin instability, 19 hydrodynamic s, 17– 19 vortex filaments, defined, 18 Page 88 of 112

Index Vortic ity, 17 Water, c artoon PVT diagram for, 148 Water Waves (Stoker), 127–28 Wave equation advanc ed Green's func tions, 120– 23 retarded Green's functions, 120–23 Waveguide, radiation theory, 124 Wave movements, c ontinuum mec hanic s, 88, 88 Wave packets, Everett interpretation, 461 Wave resistance, 24 Weak c oupling fixed points, 177 Weak neutral c urrents, symmetry, 302n24 Weak operator topology, unitary equivalenc e, 495 Weeks, John, 163 Weighting, indistinguishability, 361, 362 Weinberg, Steven anthropic reasoning, 638–40, 642, 643 unification in physics, 383, 398, 400–401 Weiss, Pierre, 157 Weyl, Hermann, 289, 297, 626 Whitehead, A. N., 96n43 Widom, Benjamin, 168–70, 170, 173 Widom scaling, 168–70 Wien's law, 342 Wigner, Eugene, 6, 291 c lassifying, 292 discussion, 310–12 group theoretic approach, 306n33 representations, forms of, 307n34 “superprinciples,” 291 Wilc ox, C., 128– 29 Wilsonian approac h, effec tive field theory (EFT), 252 c utoff, realistic interpretations of, 242 mass-independent schemes, 237–39 ontologic al implic ations, 240 quasi-autonomous domains, 241 and renormaliz ation group (RG) tec hniques, 232 renormaliz ation sc hemes, 236–37 top-down approach, 226, 228–29 Wilson-Kadanoff model, 412 Wilson, Kenneth, 175. See also Landau-Ginz burg-Wilson free energy mean field theory, 152 replacement theory, 164 on renormaliz ation group theory, 172–77 c alc ulational method, 174– 75 Page 89 of 112

Index e-expansion, 176–77 elementary particles, 176 fixed-points, 177 Fourier space, 172–74 Landau-Ginz burg-Wilson free energy, 173–74 physical space, 172–74 running c oupling c onstants, 175– 76 Wing theory, 13, 30 airplane wings, 26 Xia, Zhihong, 62, 68 Yang, C. N., 166 Yang-Lee theorem, 195, 208 Yang-Mills theory “gauge argument, 299, 301, 302n22 unific ation in physic s, 397 unitary inequivalenc e, as example of, 507 Young's modulus, 272, 275 Yukawa c ouplings, elec troweak theory, 402 Yukawa interac tion, elec troweak theory, 402n17 Zemanian, A., 137 Zernike, Frederik, 152, 158 Zero field strength, Mac hian relationalism, 557n75 “Zodiacal masses,” 619 Notes: (1) A “standard c andle” is an objec t whose intrinsic luminosity c an be determined; the observed apparent magnitude then provides an ac c urate measurement of the distanc e to the objec t. (1) Stric tly speaking, the c ontroversy has c onc erned two c andidate entities. Prior to Minkowski's reformulation of Einstein's spec ial theory of relativity in four-dimensional form, the debate was about the existenc e of spac e. Sinc e then, the debate has been about the existenc e of spac etime. For the sake of brevity, I will often only mention spac etime, leaving the “and/or spac e” implic it. (1) The word “phase” is interesting. Ac c ording to the Oxford Dictionary of Word Histories (and the Oxford English Dictionary) it entered English language in the nineteenth c entury to desc ribes the phases of the moon. The Oxford English Dictionary lists a very early use in J. Willard Gibbs's writings about thermodynamic s as the “phases of matter.” Apparently Gibbs then extended the meaning to get “extension in phase” that then got further extended into the modern usages “phase transition” and “phase spac e.” (1) As an example, c onsider multiple realiz ation, often presented as a failure of reduc tion. However, it is only a failure if we believe that a lower-level explanation of the higher-level law must be unified (i.e., the explanation must be the same for every instanc e of the higher-level Page 90 of 112

Index law). If we are willing to allow for disunified explanation, then we may indeed have a genuine lower-level explanation of the higher-level law, preserving the c ore sense of reduc tion. (2) Stric tly speaking, additivity and extensivity are different properties; see Touc hette (2002). Sinc e they overlap for many real systems, they are c ommonly run together; however, it is a mistake to do so in general, for some quantities sc ale with partic le number N (and henc e are extensive), yet are not additive. (2) In textbooks, ontologically mixed circumstances (a point mass sliding upon a rigid plane) often appear. Usually these need to be viewed as degenerations of dimensionally c onsistent sc hemes (i.e., a ball sliding on a plane or a free mass floating above a lattic e of strongly attracting masses). (2) See Earman (2008) for a disc ussion of this c ondition. (2) This is also known as the “equal areas” law: a segment c onnec ting the Sun and a planet on an elliptic al orbit sweeps out equal areas in equal time intervals. (3) If a mathematic al treatment happens to make two point masses c oinc ide, that oc c urrenc e is generally viewed as a blowup (= breakdown of the formalism) rather than a true c ontac t. It is often possible to push one's treatment through suc h blowups through appeal to sundry c onservation laws and the rationale for these popular proc edures will be sc rutiniz ed in sec tion 3. (3) Consider, again, sc alar field theory From note 2, the dimension of a sc alar fields is given by D/2 − 1; hence, in general, an operator Oi constructed from Mϕ's and N derivatives will have dimension δi = M(D/2 − 1) + N. For D ≥ 3, there are only a finite number of ways in whic h Si 〈 D and δi = D. (3) In 1633, on hearing of the Churc h's c ondemnation of Galileo for c laiming that the Earth moved, Desc artes suppressed an early statement of his physic s, whic h did not c ontain his later relational c laims about the nature of motion. It is frequently (and plausibly) c onjec tured that Desc artes's offic ial views on motion were devised to avoid Churc h c ensure. However, the prec ise manner in whic h Desc artes's definitions sec ure the Earth's lac k of true motion suggest that he was genuinely c ommitted to a relational c onc eption of motion. What does the work in sec uring the Earth's rest is not that, in Desc artes's c osmology, there is no relative motion with respec t to immediately c ontiguous bodies (Desc artes explic itly says there is suc h motion; ibid., III: 28); it is that Cartesian true motion is motion with respec t to those c ontiguous bodies that are regarded as at rest. (3) This func tion is named after William Rowan Hamilton who desc ribed how to formulate c lassic al mec hanic s using this Hamiltonian func tion. (3) A mic rostate as just defined c an be spec ified as a string of Ns symbols “p” and Cs − 1 symbols “| ” (thus, for example, Ns = 3, Cs = 4, the string p| | pp| c orresponds to one partic le in the first c ell, none in the sec ond, two in the third, and none in the fourth). The number of distinc t strings is (Ns + Cs − 1)! divided by (Cs − 1)!Ns !, bec ause permutations of the symbol “|” among themselves or the symbol “p” among themselves give the same string. (This Page 91 of 112

Index derivation of (1) was given by Ehrenfest in 1912.) (3) Dyson (1964, 129) reports a c onversation between O. Veblen and J. Jeans in 1910 about the reformation of the mathematic s c urric ulum at Princ eton. Jeans was of the opinion that “we may as well c ut out group theory. This is a subjec t whic h will never be of any use in physic s.” (3) Perhaps the most c ited problem with string theory is that it has a huge number of equally possible solutions, c alled string vac u, that may be suffic iently diverse to explain almost any phenomena one might observe at lower energies. If so, it would have little or no predic tive power for low-energy partic le physic s experiments. Other c ritic isms inc lude the fac t that it is bac kground dependent, requiring a spec ific starting point. This is inc ompatible with general relativity, whic h is bac kground independent. The problems assoc iated with loop quantum gravity also involve c omputational diffic ulties in making predic tions direc tly from the theory and the fac t that its desc ription of spac etime at the Planc k sc ale has a c ontinuum limit that is not c ompatible with general relativity. Obviously, there are many more detailed issues here that I have not mentioned. For more disc ussion, see Dine (2007) on string theory and supersymmetry and Rovelli (2007) on quantum gravity. See Smolin (2001) for a popular ac c ount of the latter. (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) Modern investigations have shown that true ODEs and PDEs are usually the resultants of foundational princ iples that require more sophistic ated mathematic al c onstruc tions for their proper expression (integro-differential equations; variational princ iples, weak solutions, etc .). We shall briefly survey some of the reasons for these c omplic ations when we disc uss c ontinua in sec tion 4 (although suc h c onc erns c an even affec t point-mass mec hanic s as well). For the most part, the simple rule “ODEs = point masses or rigid bodies; PDEs = c ontinua” remains a valuable guide to basic mathematic al c harac ter. (4) I take “indistinguishable” and “permutable” to mean the same. But others take “indistinguishable” to have a broader meaning, so I will give up that word and use “permutable” instead. (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. (4) The word “c anonic al” seems to be a somewhat old-fashioned usage for something set to a given order or rule. The Oxford English Dictionary trac es it bac k to Chauc er. Page 92 of 112

Index (4) The paragraph desc ribing the buc ket experiment c ompletes Newton's arguments for his ac c ount of true motion in terms of absolute spac e but it is not the end the Sc holium. After a brief paragraph that explic itly c onc ludes: “Henc e relative quantities are not the quantities themselves, whose names they bear, but are only sensible measures of them,” there follows a long, final paragraph desc ribing a thought experiment involving two globes attac hed by a c ord in a universe in whic h no other observable objec ts exist. The purpose of this thought experiment is not to further argue for absolute spac e by, e.g., desc ribing a situation in whic h there is absolute motion (revealed by a tension in the c ord) and yet no relative motion whatsoever. Instead, Newton's purpose is to demonstrate how true motion c an (partially) be empiric ally determined, despite the imperc eptibility of the spac e with respec t to whic h it is defined: the tension in the c ord is a measure of the rate of rotation and, by measuring how this tension c hanges as different forc es are applied to opposite fac es of the globes, one c an also determine the axis and sense of the rotation. (5) Often internal variables suc h as spin are tolerated in these ODEs, even though they lac k c lear c ounterparts within true c lassic al tradition. (5) In the c ase of dynamic al-c ollapse theories, Tumulka (2006) has produc ed a relativistic ally c ovariant theory for non-interacting partic les, but to my knowledge there is no dynamic al- c ollapse theory empiric ally equivalent to any relativistic theory with interac tions. There has been rather more progress in the c ase of hidden variable theories (perhaps unsurprisingly, as these supplement but do not modify the already-known unitary dynamic s); for three different rec ent approac hes, see Dürr et al. (2004, 2005) (hidden variables are partic le positions), Struyve and Westman (2006) (hidden variables are bosonic field strengths), and Colin (2003) and Colin and Struyve (2007) (hidden variables are loc al fermion numbers). As far as I know, no suc h approac h has yet been demonstrated to be empiric ally equivalent to the Standard Model to the satisfac tion of the wider physic s c ommunity. (5) Maxwell (1965, 1: 564). The experimental fac ts c onc erned the induc tion of c urrents by inc reases or dec reases in neighboring c urrents, the distribution of magnetic intensity ac c ording to variations of a magnetic potential and the induc tion of statistic al elec tric ity through dielec tric s. (5) As has been emphasiz ed by Stein (1967, 269–271); the argument is also singled out by Barbour (1989, 616–617). (6) For interesting disc ussion of the status of determinism in c lassic al physic s, see Earman (1986, 2007), Norton (2006, 2007a), and Wilson (1989). For disc ussion of restric tions on the veloc ity of propagation, see Earman (1987) and Maudlin (2002). Obviously, in a relativistic c ontext, if there is c ausation over spac elike separation, then there is bac kward c ausation in some frames of referenc e unless one “reinterprets” the direc tion of c ausation in suc h frames. I will limit myself, however, to c ases of bac kward c ausation that are within or on the light c one. (6) Perhaps in some sense there are multiple interpretations of c lassic al elec tromagnetism: perhaps realists c ould agree that the elec tromagnetic field is physic ally real but might disagree about its nature. Some might think that it was a property of spac etime points; others might regard it as an entity in its own right. I am deeply skeptic al as to whether this really expresses Page 93 of 112

Index a distinc tion, but in any c ase, I take it this is not the problem that we have in mind when we talk about the measurement problem. (6) While these aspec ts of the Standard Model suggest it c an be viewed as a natural EFT, other aspec ts famously prec lude this view. In partic ular, terms representing massive sc alar partic les like the Higgs boson are not protec ted by any symmetry and thus should not appear in an EFT. That they do, and that the order of the Higgs term is proportional to the elec troweak c utoff, generates the “hierarchy problem” for the Standard Model. (6) In fac t, the liquid– gas c ase is one of the most subtle of the phase transitions sinc e the symmetry between the two phases, gas and liquid, is only an approximate one. In magnets and most other c ases the symmetry is essentially exac t, before is it broken by the phase transition. (6) What if my theory differs from yours only in a trivial sc ale transformation? That is, we don't satisfy PEV, but there are bijections iobs: iobs :  Q → Q′ and istate : S → S ′ such that (say) istate(ω)(iobs(A)) = 2 × ω(A). Wouldn't it be mad to take this failure to satisfy PEV to disqualify our theories from physic al equivalenc e?! I am not sure it would be. Notic e that at least one of the theories entertains only states that fail to be normaliz ed. And notic e as well that we c an restore unitary equivalence by attributing the theorists the same observable algebra but different c onventions for c oordinating self-adjoint elements of that algebra and measurement proc edures. (Thanks to Dave Baker and Bryan Roberts, who independently raised this point.) (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 c orresponds 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. (7) Another example is Non-Relativistic QCD (NRQCD), whic h is an EFT of quark/gluon bound systems for whic h the relative veloc ity is small. The low-energy fields are obtained by splitting the gluon field into four modes and identify three of these as light variables. Rothsein (2003, 61) desc ribes this proc ess of identific ation as an “art form” as opposed to a systematic procedure. (7) Mathias Frisc h prefers to frame the princ iple as “the c ause does not c ome after the effec t.” This way of stating the princ iple seems to have as its sole motivation the desire to maintain that nothing is amiss with equations like Newton's Sec ond Law, f =ma, even though the forc e does Page 94 of 112

Index c ause the ac c eleration but does not c ome before it, sinc e they are simultaneous. However, bec ause I will not disc uss alleged c ases of simultaneous c ausation, I will stic k with the more standard wording, sinc e nothing will depend upon the differenc e. (7) There are some potential c onnec tions between “explanatory irreduc ibility” and notions in the literature on idealiz ation. In partic ular, depending upon how one understands Galilean idealiz ation, it is possible that a c onc eptual novelty is explanatorily irreduc ible just in c ase it is not a “harmless” Galilean idealiz ation. Coined by Mc Mullin, a Galilean idealiz ation in a sc ientific model is a deliberate distortion of the target system that simplifies, unifies or generally makes more useful or applic able the model. Cruc ially, a Galilean idealiz ation is also one that allows for c ontrolled “de-idealiz ation.” In other words, it allows for adding realism to the model (at the expense of simplic ity or usefulness, to be sure) so that one c an see that the distortions are justified by c onvenienc e and are not ad hoc . Idealiz ations like this are sometimes dubbed “c ontrollable” idealiz ations and are widely viewed as harmless. What to make of suc h non- Galilean idealiz ations is an ongoing projec t in philosophy of sc ienc e. One prominent idea—see, e.g., Cartwright (1983) or Strevens (2009)—is that the model may faithfully represent the signific ant c ausal relationships involved in the real system. The departure from reality need not then ac c ompany a c orresponding lac k of faith in the deliveranc es of the model. It is possible that we could understand the standard explanation of phase transitions as a distortion that nonetheless suc c essfully represents the c ausal relationships of the system. Perhaps the thermodynamic limit is legitimatiz ed by the fac t that surfac e effec ts are not a differenc e-maker (in the sense of Strevens) in the systems of interest. We will leave this line of thought to others to develop. (7) As pointed out to me by Hans van Leeuwen, the opalesc enc e is very c onsiderably enhanc ed by the diffic ulty of bringing the near-c ritic al system to equilibrium. The out-of- equilibrium system tends to have anomalously large droplets analogous to those produc ed by boiling. These droplets then produce the observed turbidity. (8) Given that an “observer” is represented in the quantum theory by some Hilbert spac e many of whose states are not c onsc ious at all, and that c onversely almost any suffic iently large agglomeration of matter c an be formed into a human being, it would be more ac c urate to say that we have a c onsc iousness basis for all systems, but one with many elements that c orrespond to no c onsc ious experienc e at all. (9) Frisc h has appealed to nearly all of these items from physic s in support of the importanc e of c ausality c onsiderations within physic s (Frisc h 2005, 2009a). (9) Clifton and Halvorson (2001) follow Glymour's analysis of physic al equivalenc e, whic h supposes physic al theories to be interpreted as axiomatic systems. For Glymour, suc h theories are physic ally equivalent only if intertranslatable in suc h a way that axioms get translated as axioms and theorems get translated as theorems. Roughly speaking, Clifton and Halvorson assimilate the generators of an observable algebra to axioms and its other elements to theorems, thereby motivating (7) as the axiom-to-axiom demand and PEV as the theorem-to- theorem demand. I take my rec onstruc tion to agree in spirit with theirs. (9) Coughlan and Dodd (1991, 44– 49) provide further tec hnic al details. Parity violation Page 95 of 112

Index (demonstrated experimentally by a team led by Wu, in 1957) has reignited the interest in the disc ussions on the struc ture of physic al spac e and the nature of c hiral objec ts, going bac k to Kant's attempts to ac c ount for the differenc e between the “inc ongruent c ounterparts” by appeal to their relation to absolute spac e (“inc ongruent c ounterparts” are objec ts that are mirror images of eac h other but are not superposable through rigid motion, e.g., a right and left glove). See Nerlic h (1994) for an introduc tion and Hoefer (2000), Huggett (2003), and Pooley (2003) for recent discussions. (9) The methods used in “A Dynamic al Theory” were extended and more fully developed in the Treatise on Electricity and Magnetism (TEM), where the goal was to examine the c onsequenc es of the assumption that elec tric c urrents were simply moving systems whose motion was c ommunic ated to eac h of the parts by c ertain forc es, the 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) One c an rec all here Feynman's famous proposal to understand antipartic les as partic les moving bac kward in time, or, in other words, that the time-reversal operation applied to a partic le state would turn it into the c orresponding antipartic le state (Feynman 1985). For details and c ritic ism, see Arntz enius and Greaves (2009). This disc ussion is related to an earlier debate between Albert and Malament with regard to c lassic al elec tromagnetism. Albert (2000) argued that c lassic al elec tromagnetism is not time-reversal invariant, while Malament (2004) defended the standard view, ac c ording to whic h the theory does possess this feature. (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). (10) Frisc h, someone who thinks that c ausality does play a substantive role in theoriz ing, points to this passage among others (Frisc h 2009a). (10) Imprec ision c an often be used to distinguish between the mathematic ian and the physic ist. The former tries to be prec ise; the latter sometimes uses vague statements that c an then be extended to c over more c ases. However, in prec isely defined situations, for example the situation defined by the Ising model, the extended similarity “theorem” is ac tually a theorem (Isakov 1984). (10) Don't mathematic ians sometimes offer up c harac teriz ations of symmetries along these lines? Yes—but only when speaking loosely and heuristic ally. Thus in the introduc tion to Olver's influential textbook on symmetries of differential equations, we are told that: “Roughly speaking, a symmetry group of a system of differential equations is a group whic h transforms solutions of the system to other solutions” Olver (1993, xviii)—see also, e.g., Bluman and Anc o (2002, 2) and Klainerman (2008, 457). But on the next page we are told that onc e “one has determined the symmetry group of a system of differential equations, a number of applic ations Page 96 of 112

Index bec ome available. To start with, one c an direc tly use the defining property of suc h a group and c onstruc t new solutions to the system from known ones.” Of c ourse, one c annot do this if one's notion of a symmetry is given by the Fruitless Definition—one needs to be working with one of the more spec ializ ed notions that are the foc us of Olver's book, some of whic h are desc ribed below. And likewise for the other applic ations on Olver's list. (12) I do not have the spac e to survey suc h modern studies here, whic h attempt to, for example, rec over the tenets of rigid body mec hanic s from c ontinuum princ iples by allowing c ertain material parameters to bec ome infinitely stiff (thus “degeneration”). Generally the results are quite c omplex, with c orrec tive modeling fac tors emerging in the manner of Prantdl's boundary layer equations. Sometimes efforts are made to weld our different foundational approac hes into unity through employing tools like Stieltjes-Lesbeque integration. More generally, a “homogeniz ation” rec ipe smears out the detailed proc esses oc c urring ac ross a wide region ΔW in an “averaging” kind of way, whereas “degeneration” instead c onc entrates the proc esses within ΔW onto a spatially singular support like a surfac e (the Riemann-Hugoniot approac h to shoc k waves provides a c lassic exemplar). (12) Simon expresses a c ommitment to the tradition as he launc hes into an exposition of those aspec ts of func tional analysis he c onsiders most c entral to physic s: “Throughout, all our Hilbert spac es will be separable unless otherwise indic ated. Many of the results extend to non- separable spac es, but we c annot be bothered with suc h obsc urities” (1972, 18). Although there are some ways in whic h the mathematic s of separable Hilbert spac es are “nic er” (for instanc e, some operator topologies are first-c ountable), I am not aware of a c anonic al explanation of the tradition. (12) The symmetry of the phase transition is reflec ted in the nature of the order parameter, whether it be a simple number (the c ase disc ussed here), a c omplex number (superc onduc tivity and superfluidity), a vec tor (magnetism), or something else. (12) Consider, by way of illustration, the Newtonian theory of three gravitating point partic les of distinct masses. Here a point in the space of kinematically possible fields, K , essentially assigns eac h of the partic les a worldline in spac etime (without worrying about whether these worldlines jointly satisfy the Newtonian laws of motion). The space of solutions, S , is the 18- dimensional submanifold of K consisting of points corresponding to particle motions obeying Newton's laws. So one expects that any diffeomorphism from S itself can be extended to a suitably nice map from K to itself. But for any solutions u1 ,  u2 ∈  S , we can find a diffeomorphism from S to itself that maps u1 to u2, so we again find that arbitrary pairs of solutions are related by symmetries. This seems unac c eptable, sinc e we ordinarily think of this theory as having a relatively small symmetry group (c onsisting just of spac etime symmetries). (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 . (13) I owe this turn of phrase to Jos Uffink. (14) Stric tly speaking, a lift to c ontinuous variables from an ODE-style treatment involving a large number of disc rete variables at the ΔL level should not be c alled a “reduc ed variable” Page 97 of 112

Index treatment, as we ac tually increased the number of degrees of freedom under the lift (normally, a true “reduc ed variable” treatment will supply a ΔL* level manifold lying near to some submanifold c ontained within the ΔL phase spac e). However, the desc riptive advantages of a lift to c ontinuous variables often resembles those supplied within a true “reduc ed variable” treatment, so in the sequel I will often c onsider both forms of lift under a c ommon heading. (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. (14) In his c orrespondenc e with Clarke (Alexander 1956), Leibniz is sometimes read as offering kinematic -shift arguments somewhat different to the one just sketc hed. The idea is that kinematic ally shifted possible worlds would violate the Princ iple of Suffic ient Reason (PSR) and the Princ iple of the Identity of Indisc ernibles (PII). Sinc e these princ iples are a priori true, ac c ording to Leibniz , there c an be no suc h plurality of possibilities. A “Leibniz ian” argument from the PSR would ask us to c onsider what reasons God c ould have had for c reating the ac tual universe rather than one of its kinematic ally shifted c ousins. An argument from the PII would c laim that, sinc e kinematic ally shifted worlds are observationally indistinguishable, they direc tly violate the PII. Neither argument is c onvinc ing (nor is either faithful to Leibniz ; see Pooley, unpublished). The sense of indisc ernibility relevant to kinematic shifts is not that whic h has been the foc us of c ontemporary disc ussion of the PII. This takes two entities to be indisc ernible just if they share all their (qualitative) properties. In general, two kinematic ally shifted worlds do differ qualitatively; given how the qualitative/nonqualitative distinc tion is standardly understood, a body's absolute speed is a qualitative property, and differenc es in absolute veloc ity are (typic ally) qualitative differenc es. Suc h qualitative differenc es are empiric ally inac c essible but, theoretic ally, they c ould ground a reason for an all-seeing God's preferenc e for one possibility over another. A PSR dilemma for God is c reated if we c onsider kinematic ally shifted worlds that differ, not in terms of the magnitude of their objec ts' absolute veloc ities, but only over their direc tions. These are worlds that are qualitatively indistinguishable. Disc ussion of how the substantivalist should treat these is postponed until sec tion 7. (14) There are exc eptions. Mean field theory works quite well whenever the forc es are suffic iently long-ranged so that many different partic les will interac t direc tly with any given partic le. By this c riterion mean field theory works well for the usual superc onduc ting materials studied up through the 1980s(7, 8), exc ept extremely c lose to the c ritic al point. However, mean field theory does not work for the newer “high-temperature superc onduc tors,” a c lass disc overed in 1986 by Georg Bednorz and Alexander Müller(9). (16) The departures from equilibrium are desc ribed using the Boltz mann equation. The Boltz mann equation formulated in an FLRW spac etime inc ludes an expansion term. As long as Page 98 of 112

Index the c ollision term (for some c ollec tion of interac ting partic les) dominates over the expansion term then the interac tions are suffic ient to maintain equilibrium, but as the universe c ools, the c ollision term bec omes subdominant to the expansion term, and the partic les dec ouple from the plasma and fall out of equilibrium. To find the number density at the end of this freez ing out proc ess, one typic ally has to solve a differential equation (or a c oupled set of differential equations for multiple partic le spec ies) derived from the Boltz mann equation. (16) For reasons of spac e I omit detailed disc ussion of the parallel tradition in Everettian quantum mec hanic s of identifying probability via long-run relative frequenc y (notably by Everett himself (1957) and by Farhi, Goldstone, and Gutmann(1989). I disc uss this program in detail in c hapter 4 of Wallac e(2012); my c onc lusion is that it works about as well, or as badly, as equivalent c lassic al attempts, though there is no direc t Everettian analogue to the best- systems approach. (17) Note that it is the “kinetic ” c omponent −1/4FμνFμν of the Lagrangian for the full theory (also featuring an “interac ting” c omponent), whic h, as Martin nic ely puts it, “imbues the field with its own existenc e, ac c ounting for the presenc e of non-z ero elec tromagnetic fields, for the propagation of free photons” (2003, 43). See Quigg (1983, 45– 48) for the tec hnic al details. But, tec hnic alities aside, one of the main c omplaints against this standard story has been that this generation talk is misleading, as the gauge field is put in by hand. For disc ussion, see Brown (1999). A number of further issues arise, having to do with the (in)determinist c harac ter of a gauge theory. A sourc e of c onc ern is the identific ation of those quantities that are ac tually “physic al,” as opposed to mere artifac ts of desc ription. The disc ussions in the literature foc us on Einstein's “hole argument” (Earman and Norton 1987; Butterfield 1989; Belot 1996, esp. c hs. 5, 6, 7; 1998; Saunders 2002; etc .; for a rec ent introduc tion, see Norton 2008). Equally pressing is the question about the right ontologic al interpretation that should be given to those quantities that are not gauge-invariant, the so-c alled (by Redhead 2003) “surplus struc ture.” (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 dominance of matter over antimatter in the Universe. (17) A more prec ise way of stating both is that the program attempts to show that agents are rationally required to ac t as if mod-squared amplitude played the objec tive-probability role in David Lewis's Principal Principle; c f. Lewis (1980). (18) Note that in c ontinuum mec hanic s, generally, a material point or “material partic le” is not an atom or molec ule of the system; rather it is an imaginary region that is large enough to c ontain many atomic subsc ales (whether or not they really exist) and small enough relative to the sc ale of field variables c harac teriz ing the impressed forc es. Of c ourse, as noted, Navier's derivation did make reference to atoms. (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 struc tured, as will happen in Page 99 of 112

Index spontaneous c ollapse theories in the c ase 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). (21) To get a feeling for what this means, c onsider the sort of gauge transformations that normally arise in presentations of Maxwell's theory: if we take the vec tor potential A(x) as our field, then the theory is invariant under infinitesimal transformations of the form A ↦ A + εdΛ where Λ(x) is a real-valued func tion on spac etime. Now suppose that A is a map then when fed a kinematic ally possible A returns a real-valued func tion Λ [A] on spac etime. If for eac h spac etime point x and eac h A, the value of Λ [A] at x depends only on x and on A(x), then the infinitesimal transformation A ↦ A + εd Λ[A] c orresponds to a c lassic al symmetry of Maxwell's theory; if the value of Λ [A] (x) depends also on a finite number of derivatives of A at x, then this map is a generaliz ed symmetry of Maxwell's theory See the disc ussion of generaliz ed gauge symmetries in Pohjanpelto (1995) and in Torre (1995). For a thoroughly worked-out example involving only finitely many degrees of freedom, see Cantwell (2002, §14.4.1). (21) More prec isely, the different perturbation modes have the same density c ontrast when their wavelength equals the Hubble radius, H−1. (22) Cosmologists use “c onc ordanc e model” to refer to the Standard Model of c osmology with the spec ified c ontributions of different types of matter. The c ase in favor of a model with roughly these c ontributions to the overall energy density was made well before the disc overy of c osmic ac c eleration (see, e.g., Ostriker and Steinhardt (1995); Krauss and Turner (1999)). Coles and Ellis (1997) give a useful summary of the opposing arguments (in favor of a model without a dark energy c omponent) as of 1997, and see Frieman, Turner, and Huterer (2008) for a more recent review. (22) To c larify: Gell-Mann's so-c alled “Eightfold Way” SU(3)-based theory mentioned at the beginning of this paragraph is not QCD as developed later on. The degrees of freedom of the Eightfold Way are not the degrees of freedom of the SU(3)-based QCD—though the group is the same, SU(3). This later theory postulates three different types of strong-forc e c harge (the red, green, and blue quarks). The former SU(3) spac e (where only global invarianc e required) is a different entity than the SU(3) spac e of strong c harge, whic h is under the c onstraint of local (“gauge”) invarianc e. Within the former theory, we only c ategoriz e nonfundamental c ollec tions of quarks (for more, see sec tion 5). It is the latter theory whic h is the c urrently ac c epted dynamical ac c ount of the strong nuc lear forc e. Yang and Mills (see the previous paragraph) attempted to make a dynamic al theory out of the SU(2) isospin spac e, but we c an now see that this is c learly wrong-headed, sinc e protons and neutrons are not fundamental partic les. (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 Page 100 of 112

Index 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 existence 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. (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) ‘Closing in the uniform topology’ means adding to the algebra the limit points of all uniformly c onvergent sequenc es of elements that have made their way into the algebra by other means. (24) “Hot” vs. “c old” refers to the thermal veloc ities of relic partic les for different types of dark matter. Hot dark matter dec ouples while still “relativistic ,” in the sense that the momentum is muc h greater than the rest mass, and relic s at late times would still have large quasi-thermal veloc ities. Cold dark matter is “non-relativistic ” when it dec ouples, meaning that the momentum is negligible c ompared to the rest mass, and relic s have effec tively z ero thermal veloc ities. (24) Part of this story provided soc ial-c onstruc tivists with a c ase to uphold their position. As we will see below (next footnote), the experimental demonstration of the so-c alled “weak neutral c urrents” would have c orroborated the unified model. Analyz ing this episode, Andy Pic kering (1998, 136) writes: “There I argue that the ac c eptability of the weak neutral c urrent (and henc e of the assoc iated interpretative prac tic es) was determined by the opportunities its existenc e offered for future experimental and theoretic al prac tic e in partic le physic s. Quite simply, partic le physic ists ac c epted the existenc e of the neutral c urrent bec ause they c ould see how to ply their trade more profitably in a world in whic h the neutral c urrent was real. The key idea here is that of a symbiotic relationship between experimenters and theorists, the two distinc t professional groupings within partic le physic s.” (25) Type Ia supernovae do not have the same intrinsic luminosity, but the shape of the light c urve (the luminosity as a func tion of time after the initial explosion) is c orrelated with intrinsic luminosity. See Kirshner (2009) for an overview of the use of supernovae in c osmology. (25) This subsec tion is mostly based on my entry for the Stanford Encyclopedia of Philosophy (Bacciagaluppi 2003). (26) If no kinetic energy is lost to heat (a so-c alled “purely elastic c ollision”), then we possess enough “c onservation laws” (energy and linear momentum) to guide two c olliding point masses uniquely through a c ollision (as every elementary c ollege text demonstrates). But these princ iples alone are not adequate to three-way c ollisions, energetic losses, or to more oblique modes of scattering. (26) See Stone (2000, 204) for the c ondensed matter c ontext. The bare parameters are the parameters of the theory before resc aling is performed to restore the c utoff bac k to its initial Page 101 of 112

Index value after one iteration of the RG transformations. The renormaliz ed parameters are the rescaled parameters. (27) Constraint relationships are sometimes maintained through fac tors external to the devic e (suc h as the pressures of an ambient fluid or the gravitational attrac tion that binds a c am to its follower), in whic h c ase the devic e is said to be force closed. Desc artes, for example, essentially dissec ted the universe into c omponent mec hanisms, but they were usually held together through forc e c losure rather than internal pinning. (28) Suc h c ontac ts are further c lassified as “higher or lower pairs” ac c ording the c ontac ting geometry they implement. (28) See Sotiriou and Faraoni (2010) for a review of one approac h to modifying GR, namely by adding higher-order c urvature invariants to the Einstein-Hilbert ac tion. These so-c alled “ f (R) theories” (the Ric c i sc alar R appearing in the ac tion is replac ed by a func tion f (R)) have been explored extensively within the last five years, but it has proven to be diffic ult to satisfy a number of seemingly reasonable c on straints. Uz an (2010) gives a brief overview of other ways of modifying GR in light of the observed ac c eleration. (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 decomposition 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). (30) This behavior is usually desc ribed using the rotation c urve, a plot of orbital veloc ity as a func tion of the distanc e from the galac tic c enter. The “expec ted” behavior (dropping as r−1/2 after an initial maximum) follows from Newtonian gravity with the assumption that all the mass is c onc entrated in the c entral region, like the luminous matter. The disc repanc y c annot be evaded by adding dark matter with the same distribution as the luminous matter; in order to produc e the observed rotation c urves, the dark matter has to be distributed as a halo around the galaxy. (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). Page 102 of 112

Index (31) Under suc h approac hes, one works with a spac e of instantaneous states (whic h will be infinite-dimensional in the field-theoretic c ase), equips this spac e with a real-valued func tion, L (the Lagrangian), and employs a variational princ iple to find those c urves in the spac e of states that correspond to dynamically possible histories of the system. (31) In a seminal paper, Ostriker and Peebles (1973) argued in favor of a dark matter halo based on an N body simulation, extending earlier results regarding the stability of rotating systems in Newtonian gravity to galaxies. These earlier results established a c riterion for the stability of rotating systems: if the rotational energy in the system is above a c ritic al value, c ompared to the kinetic energy in random motions, then the system is unstable. The instability arises, roughly speaking, bec ause the formation of an elongated bar shape leads to a larger moment of inertia and a lower rotational energy. Considering the luminous matter alone, spiral galaxies appear to satisfy this c riterion for instability; Ostriker and Peebles (1973) argued that the addition of a large, spheroidal dark matter halo would stabiliz e the luminous matter. (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 partic ularly interested in the c ase of the Heisenberg mic rosc ope, where the elec tron 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 decoherence does not kick in until later. (32) In this c ontext, “Euler's First Law” is often viewed as simply “Newton's Sec ond Law” in applic ation to rigid bodies. Credit for regarding the “ F = ma” sc heme as a framework upon whic h “rec ipes” for differential equations for both forms of mec hanic s c an be built is historic ally due to Euler, not Newton. As we shall see, the analogous rec ipe for c ontinua relies upon a formula traditionally c alled “Cauc hy's Law,” whic h many writers regard as yet “another version of F = ma” (although it ac tually employs the tric ky notion of stress that Cauc hy originated). The similarities of these three “rec ipe” formulas support the strong “family resemblanc e” c harac ter of “c lassic al mec hanic s.” Terminologic al issues bec ome more c onfusing within the c ontext of c ontinua, in whic h analogs of Euler's two laws are also applied to the sub-bodies in the interior of c ontainer blobs. In suc h c ontexts, these analogs are often dubbed the “balanc e princ iples” for momentum and angular momentum. In the c ontext of rigid bodies, onc e spec ific values for moments of inertia et al. have been c omputed with respec t to suc h entities, these values remain the same, allowing the import of Euler's princ iples to be expressed as equations of ODE type. Within flexible bodies, in c ontrast suc h values fluc tuate as they flex and so PDEs are required to c apture the requisite relationships. (32) This is to rule out parastatistic s—representations of the permutation group that are not one-dimensional (see, e.g., Greiner and Müller 1994). This would be desirable (sinc e parastatistic s have not been observed, exc ept in 2-dimensions, where spec ial c onsiderations apply), but I doubt that it has really been explained. (33) The situation is a little more c omplic ated, as antisymmetry in the spin partof the overall Page 103 of 112

Index state forc es symmetry in the spatial part—whic h c an lead to spatial bunc hing (this is the origin of the homopolar bond in quantum c hemistry). (33) Although I have quoted Lagrange's princ iple in its standard textbook form, it c onc eals a subtle ambiguity; spec ific ally as to whether the “r” c ited is a true position c oordinate or rather represents something “generaliz ed” like an angle. If the latter (whic h is usually what is needed), then the c orresponding “mass” terms “m” must be read as moments of inertia, etc . Presumably, we require some instruc tion in how these “generaliz ed inertial terms” are to be found. Suc h unnotic ed shifts are often sites of signific ant “lifts” (and sometimes outright errors, whic h are c ommon in this branc h of mec hanic s). The restric tion to “virtual variations” is nec essary bec ause the mechanical advantages of most mec hanisms c ontinuously adjust as they move through their c yc les. This means that inputted forc es F1, F2 , F3 on our c rane will not be able to balanc e quite the same output forc e F4 when the mac hine stands in a different c onfiguration. But the “instantaneous work” performed by the input forc es will always equal the “instantaneous work” expended at the outputs, whic h is the key idea that we need to c apture in our “virtual work” formula for static situations. (33) On sc ale transformations, see Olver (1993, 255). A standard remedy is to introduc e the notion of a divergence symmetry, a transformation that leaves the Lagrangian invariant up to a total divergenc e; many interesting symmetries are divergenc e symmetries but not variational symmetries, inc luding boosts of Newtonian systems and the c onformal symmetries of the wave equation; see Olver (1993, 278– 281). Sc aling symmetries are more subtle. Sc ale transformations are symmetries of general relativity, but are neither variational nor divergenc e symmetries; see Anderson and Torre (1996, § 2.B). Resc aling of spac e and time is a symmetry of the wave equation that is neither a variational nor a divergenc e symmetry, although there is a related sc ale transformation that ac ts on the dependent variables, as well as the independent variables, whic h is a divergenc e symmetry (but not a variational symmetry); see Olver (1993, Examples 2.43, 4.15, and 4.36). (33) The literature on the (Wignerian) group theoretic approac h to the c onstitution of physic al objec ts has been growing in the last dec ade, when a variety of approac hes have been attempted. See Castellani (1998) and espec ially the work on ontic struc tural realism by Frenc h (1998), Frenc h and Ladyman (2010), and Ladyman (2009), esp. sec t. 4 and the bibliography therein. (34) But: c ertain types of variational (or divergenc e) symmetries of theories whose initial value problems are ill-posed are assoc iated with so-c alled trivial c onservation laws; see Olver (1993, 342– 346) on Noether's sec ond theorem. And: there exist tec hniques for assoc iating c onservation laws with symmetries that do not rely on Noether's theorem; see, e.g., Bluman (2005). (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 indices labeling the results) a corresponding projection ∑i∈I Pi. One will talk similarly of Page 104 of 112

Index POV measures when the requirement that the elements of the resolution of the identity be projec tions is relaxed. (34) Some authors favor talk of “tidal forc es” or state that there is a real “gravitational field” just where the Riemann tensor is nonz ero (e.g., Synge, 1960, ix). As far as I c an see, this is simply a misleading way of talking about spac etime c urvature and (typic ally) nothing of c onc eptual substanc e is intended by it. For a disc ussion of some of the pros and c ons of identifying various geometric al struc tures with the “gravitational field,” see Lehmkuhl (2008, 91– 98). Lehmkuhl regards the metric gab as the best c andidate. My own view is that c onsideration of the Newtonian limit (e.g., Misner et al., 1973, 445– 446) favors a c andidate not on his list, viz ., deviation of the metric from flatness: hab, where gab = ηab + hab. That this split is not prec isely defined and does not c orrespond to anything fundamental in c lassic al GR undersc ores the point that, in GR, talk of the “gravitational field” is at best unhelpful and at worst c onfused. The distinc tion between bac kground geometry and the graviton modes of the quantum field propagating against that geometry is fundamental to perturbative string theory, but this is a feature that one might hope will not survive in a more fundamental “bac kground- independent” formulation. (34) In partic ular, physic ists assoc iate these labels (e.g. -1/2 and +1/2 in the isospin c ase) with the values of the invariant properties (isospin) c harac teriz ing physic al systems (in this c ase, the doublet neutron-proton). Wigner (1959) derives a formula that enc odes the general form of the representations. For a more modern approac h, see Joshi (1982, 131). (35) Energy c onditions plac e restric tions on the stress-energy tensor appearing in EFE. They are useful in proving theorems for a range of different types of matter with some c ommon properties, suc h as “having positive energy density” or “having energy-momentum flow on or within the light c one.” In this c ase the strong energy c ondition is violated; for the c ase of an ideal fluid disc ussed above, the strong energy c ondition holds iff ρ +3p ≥ 0. Cf, for example, c hapter 9 of Wald (1984) for definitions of other energy c onditions. (35) Instantaneous relative distanc es and their first derivatives are the natural Leibniz ian relational data. As reviewed in sec tion 6.2, Barbour's preferred framework for understanding c lassic al mec hanic s also dispenses with a primitive temporal metric and an absolute length sc ale. With respec t to these more frugal initial data, five, not three, additional numbers are needed. See Barbour (2011, §2.2). (35) The diversity and the large number of partic les had always bothered the high-energy physic ists. Willis Lamb voic ed this uneasiness in his Nobel speec h, in whic h he reminded the public of a popular saying in the partic le physic s c ommunity: anyone who disc overs a new partic le ought be punished by a $10,000 fine (instead of being awarded a Nobel Priz e!) (35) ω is a symplectic form—a c losed, nondegenerate two-form. ω and H determine a vec tor field XH on J : XH is the vector field that when contracted with ω yields the one-form dH. Integrating this vec tor field gives the c urves mentioned in the text. Note that there is a canonical recipe for constructing J , H, H, and ω given a Lagrangian treatment of the theory. (36) Note that the symplectic space (J, ω) has a vast family of symmetries. Suppose that we J Page 105 of 112

Index are interested in a Newtonian theory of finitely many particles. Then J is finite-dimensional, but the family of smooth permutations of J that preserve ω is infinite-dimensional—it is only when we restric t attention to transformations that also preserve H that we end up with something like what we want. Something similar is of c ourse true in ordinary quantum mec hanic s: while the family of unitary transformations of a Hilbert spac e will be very large, the family of suc h transformations that preserve a given Hamiltonian will be quite small—and only the latter is a good c andidate for the symmetry group of a theory. The situation is more perplexing in the c ase of fanc ier quantum theories. On the one hand, an arbitrary C∗ -algebra automorphism is pretty c learly the analogue of an arbitrary symplec tic or unitary transformation and so is not a good c andidate to be a symmetry of a theory: indeed, in many c ases of interest any two states are related by suc h an automorphism; see Kishimoto, Oz awa, and Sakai (2003). On the other hand, in some c ontexts it is not possible to identify symmetries of a theory with those C∗ -algebra automorphisms that preserve the Hamiltonian bec ause there is no Hamiltonian operator available at the C∗ -algebra level; on this point, see Ruetsc he (2011, §12.3). (37) Time-reversal invarianc e does not hold in fundamental physic s. However, the failure of time-reversal invari-anc e in the dec ay of neutral K mesons is not thought to be responsible for the sort of damping that makes Jac kson's model viable nor for thermodynamic al behavior more generally. (37) De Sitter first pointed out to Einstein that, in addition to spec ific ation of Tab, one needs to spec ify boundary c onditions at infinity in order to determine gab. This prompted Einstein to searc h for spatially c ompac t solutions to the EFEs and to introduc e the c osmologic al c onstant to allow for a static , spatially c losed universe. This in turn led de Sitter to the disc overy of the de Sitter universe: a spatially c ompac t vac uum solution to the modified EFEs. See Janssen (2008, § 5) for a summary of this episode and for further referenc es. (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. (38) Kuhn 1962, 24. Hilary Putnam similarly c ritic iz ed another of Kuhn's c harac teriz ations of normal sc ienc e: “The term ‘puz z le solving’ is unfortunately trivializ ing; searc hing for explanations of phenomena and for ways to harness nature is too important a part of human life to be demeaned” (Putnam 1974, 261). (39) The so-c alled “totalitarian princ iple” (attributed to Gell-Mann), ac c ording to whic h “what is not forbidden must oc c ur” is of notoriety in the partic le physic s c ommunity. It is unc lear, however, whether this dic tum (reminisc ent of the anc ient Princ iple of Plenitude—stating, roughly, that given an infinite time, all genuine possibilities ac tualiz e) played an important role in this episode. Its c onverse—when it seems that an event c an happen but it does not, look for Page 106 of 112

Index a c onservation law that prec ludes it—is also a well-known heuristic tool. (39) One c an witness some of this struggle in Kant's Metaphysical Foundations of Natural Science (Cambridge: Cambridge University Press, 2004) where he is plainly aware that some sourc e of sheer is needed to make sense of c onventional “solidity,” but c annot find a way to inc orporate suc h a quantity into his desc riptive framework. (40) I have patterned my first Bernoulli-Euler “element” after a diagram that Leibniz provides for a loaded beam. Cf. Clifford Truesdell, The Rational Mechanics of Flexible or Elastic Bodies 1638– 1788, (editor's introduc tion to Euler, Opera Omnia II, vol. 12) (Lausanne: 1954). (41) A more detailed disc ussion is given in Darrigol 2008. Interpretive sc hemes supplemented with the requirement of c omputability are similar to Humphreys's “c omputational templates.” Ac c ording to Humphreys 2004, it is at the level of c omputational templates that questions about theoretic al representation, empiric al fitness, realism, and so on must be disc ussed; knowledge “in princ iple” must be subordinated to knowledge “in prac tic e,” whic h involves the available tec hnologies of measurement and c alc ulation. (41) The idea that spac etime geodesic s are defined as the trajec tories of forc e-free bodies is defended by DiSalle (1995, 327), whom Brown quotes approvingly. Elsewhere Brown, ostensibly to make a point against the substantivalist explanation of inertia, stresses that the princ iple that the trajec tories of forc e-free bodies are geodesic s in fac t has limited validity in GR (Brown 2005, 141, see also 161– 168). What this observation in fac t undermines is a relationalist approac h to spac etime geometry that tries to define geodesic s in terms of “basic physic al laws” (DiSalle 1995, 325). More rec ently, DiSalle makes c lear that he differs from the logic al positivists in not regarding the c oordination of geodesic s with free-fall trajec tories as a matter of arbitrary stipulation. Instead it is said to be “a kind of disc overy, at onc e physic al and mathematic al, that … the only objec tively distinguishable state of motion c orresponds to the only geometric ally distinc tive path in a generally c ovariant geometry” (DiSalle, 2006, 131– 132). Nothing in the substantivalist's metaphysic s is inc onsistent with this position; it is less c lear what other metaphysic al views are c ompatible with it. DiSalle does not share the substantivalist's and relationalist's preoc c upation with ontologic al questions but nor does he offer reasons to see such questions as illegitimate. (42) This c onstruc tion was overlooked by Dieks and Lubberdink (2011) in their c ritic isms of the c onc ept of c lassic al indistinguishable partic les. They go further, rejec ting indistinguishability even in the quantum c ase (they c onsider that partic les only emerge in quantum mec hanic s in the limit where Maxwell-Boltz mann statistic s hold sway-where individuating predic ates in our sense can be defined. (42) Philosophers new to the pec uliar world of c ontinuum physic s parlanc e should prepare themselves for phraseology suc h as “dimensionless point c ube” (J. D. Reddy, An Introduction to Continuum Mechanics (Cambridge: Cambridge University Press, 2008), 126—an exc ellent book, by the way). (43) A. N. Whitehead did some foundational work in mec hanic s at the turn of the twentieth c entury and his “method of extensive abstrac tion” was later populariz ed by Bertrand Russell Page 107 of 112

Index in Our Knowledge of the External World (London: Routledge, 2009). I am not sure how Whitehead understood his c onstruc tion (whic h shrinks in on points through dec reasing volumes), but Russell plainly regarded the tec hnique entirely as a logic al proc edure for “defining away points.” Russell's misunderstanding of the underlying physic al problematic c ontinues to reverberate within the halls of analytic philosophy. For a survey, see Mark Wilson, “Beware of the Blob,” in Dean Zimmerman, ed., Oxford Studies in Metaphysics (Oxford: Oxford University Press, 2008). A subtle point : when we c ombine our stress and strain information, should our resultant vec tors situate themselves on the referenc e or the response planes? This matter bec omes important in nonlinear elastic ity and requires the c areful delineation of different stress tensors (“Piola-Kirc hhoff” versus “Cauc hy”) that one finds in modern textbooks. (44) This mistake also underlies muc h of the disc ussion of “c eteris paribus” laws, and here I draw on the line of argument due to Smith (2002); Earman and Roberts (1999). (44) ρg, it will be rec alled, c aptures the summed body forc es ac ting upon q. In following this standard representation, we are tac itly ignoring the third law demands that persuaded us to distinguish V(q) from V*(qn ) earlier (the mathematic s of c ontinua is rough enough without fussing about that!). It is important to realiz e that the ac c elerative term behaves mathematic ally very muc h like g and is often c alled an “inertial forc e” as a result (some of the third law ambiguities surveyed earlier trac e to this drift in the signific anc e of “forc e”). And an important symmetry with respec t to c onstitutive equations is relevant as well: materials (usually) respond to an applied sc hedule of ac c elerations by exac tly the same rules as they reac t to a c omparable array of genuine forc es (this requirement is c alled “material frame indifferenc e” or “objec tivity”). (44) More exac tly, the low-density gas spec ializ ation of the Navier-Stokes theory is an approximating module of the kinetic theory of gases, of whic h the mec hanic s of a set of interac ting molec ules is a reduc ing module. (46) Brown's thesis that inertia rec eives a dynamic al explanation only in GR has rec ently been defended by Sus (2011). Sus emphasiz es that in GR the metric is a genuinely dynamic al entity and that one c an derive ∇a Tab = 0 from the very equations that govern the metric 's behavior. In c ontrast, SR, as standardly c onc eived, involves fixed inertial struc ture whose properties are postulated by fiat. However, this differenc e between the theoriesis c ompatible with the theories agreeing on the fundamental reasons why forc e-free bodies are related to inertial struc ture in just the way they are. (46) In Minkowski spac etime, this set is the past lobe of the light c one at p, inc luding interior points and the point p itself. A point p causally precedes q (p 〈 q), if there is a future-direc ted c urve from p to q with tangent vec tors that are timelike or null at every point. The sets J ± (p) are defined in terms of this relation: J−(p) = q : q 〈 p , J+(p) = q : p 〈 q , the causal past and future of the point p, and the definition generaliz es immediately to spac etime regions. (46) In c onformity with the physic ists’ usage, Morrison and Morgan also c all “models” what I c all a “reduc ing module.” For instanc e, Maxwell's mec hanic al model of 1862 for the Page 108 of 112

Index elec tromagnetic field is a model in this sense. This kind of model widely differs from ad hoc models for limited classes of phenomena. (47) The Gauss-Codac c i c onstraint equations do impose some restric tions on spac elike separated regions, although these would not make it possible to determine the state of one region from the other; see Ellis and Sc iama (1972). (48) A local property of a spac etime is one that is shared by loc ally isometric spac etimes, whereas global properties are not. (Two spac etimes are loc ally isometric iff for any point p in the first spac etime, there is an open neighborhood of the point suc h that it c an be mapped to an isometric open neighborhood of the sec ond spac etime (and vic e versa).) (49) Related versions of relationalism, ac c ording to whic h absolute veloc ity (or even absolute position) is inter preted as a primitive, monadic property of partic les, have been disc ussed by Horwic h (1978, 403) and Friedman (1983, 235) (see also Teller, 1987). In addition to being less natural than the form of Newtonian relationalism identified by Maudlin, they are vulnerable (like Newtonian relationalism) to the kinematic shift argument. The absolute position version is also vulnerable to the static shift argument mentioned in sec tion 7. (49) Just as one might move from variational symmetries to divergenc e symmetries (see fn. 33 above), one might c onsider transformations of a system's phase spac e that leave invariant the set of Hamiltonian trajec tories without worrying about whether they also leave the Hamiltonian itself invariant. Suitably interpreted, this should manage to c apture Galilean boosts in Newtonian mec hanic s and the sc aling symmetry of the Kepler problem; see Abraham and Marsden (1985, 446 f.) and Princ e and Eliez er (1981, §5). Of c ourse, it also inc ludes the various undesirable c harac ters that already c ount as Hamiltonian symmetries (see below). (50) Malament (1977) reviews several different definitions of observational indistinguishability and gives a series of c onstruc tions of OI spac etimes lac king spec ific global properties. Note that Malament defines OI in terms of the chronological rather than causal sets, whic h inc lude the interior of the light c one but not the c one itself (The definition follows the one given in footnote 46, dropping the phrase “or null.”) Manc hak (2009) proves that Malament's tec hnique for c onstruc ting suc h spac etimes fails only in the exc eptional c ase noted in the text. Cf Norton (2011), who argues that the induc tive generaliz ations from J− (p) to other regions of spac etime lac k c lear justific ation. (55) In the examples just c onsidered, it is pretty c lear that one does not want to c ount every pair of solutions related by a generaliz ed symmetry as being physic ally equivalent. Does one ever want to c ount solutions as physic ally equivalent that are related by a generaliz ed symmetry that is not a c lassic al symmetry? Yes—for instanc e, when the solutions in question are also related by a respec table c lassic al symmetry. Consider, e.g., the generaliz ed gauge transformations desc ribed in fn. 21 above—if two solutions are related by suc h a symmetry, then they are also related by an ordinary gauge transformation. Are there pairs of solutions related by a generaliz ed symmetry (but not by any c lassic al symmetry) that one would want to c onsider physic ally equivalent? That appears to be a more diffic ult question. Part of the diffic ulty lies in the fac t that what one has in prac tic e are the Page 109 of 112

Index infinitesimal generators of generaliz ed symmetries: it is in general a nontrivial task to find the c orresponding group ac tions; see, e.g., Olver (1993, 297ff.). Further, even in c ases where the c orresponding groups of transformations c an be determined, their physic al interpretation c an be obsc ure; see, e.g., Olver (1984, 136f.). (64) Inflation solves the horiz on problem bec ause the horiz on distanc e inc reases exponentially during inflation; for a suffic iently long period of inflation, all the points on the surfac e of last sc attering will have overlapping past light c ones. The inflationary phase also reverses the dynamical feature of the FLRW models responsible for the flatness problem. Bec ause γ = 0 (in the equation in f n. 61) for most models of inflation, inflationary expansion drives Ω toward 1, enlarging the range of c hoic es Ω (tp) c ompatible with observations. (75) For related reasons, Earman defines Machian spacetime to be spac etime with simultaneity struc ture and Euc lidean metric al struc ture on its simultaneity surfac es but with no temporal metric (Earman 1989, 27–30). (86) The Everett interpretation of quantum mec hanic s attributes a branc hing struc ture to the universal wave func tion of the universe, and the individual branc hes c an be regarded as something akin to poc ket universes (see Wallac e, this volume, for a disc ussion of the Everett interpretation). However, unlike the other ac c ounts the laws of physic s do not vary in the different branc hes. There is a c lear distinc tion between the two c ases, although rec ently there has been interest in exploring c onnec tions between these two lines of thought. (91) This is not to say that every explanatory question one might ask about the phenomenon of length c ontrac tion requires an appeal to dynamic al laws; in some c ontexts it is enough to c ite the relevant geometric al fac ts in order to provide an explanation. This is a point explic itly emphasiz ed in Brown and Pooley (2006, 78–79, 82), where paradigm explanatory uses of Minkowski diagrams (e.g., to highlight that observers in relative motion c onsider different c ross-sec tions of a rod's world tube when judging its length) are said to c onstitute “perfec tly acceptable explanations (perhaps the only acceptable explanations) of the explananda in question.” Our emphasis of this fac t seems to have been overlooked by some authors (Skow 2006, Frisch 2011). (92) As it was put in Brown and Pooley (2006, 82): “it is suffic ient for these bodies to undergo Lorentz c ontrac tion that the laws (whatever they are) that govern the behavior of their mic rophysic al c onstituents are Lorentz c ovariant. It is the fact that the laws are Lorentz covariant …that explains why the bodies Lorentz c ontrac t. To appeal to any further details of the laws that govern the c ohesion of these bodies would be a mistake.” Janssen's (2009) c arefully argued c ase that phenomena rec ogniz ed to be kinematic al (in his sense) should not be explained in terms of the details of their dynamic s is therefore one that we had antecedently conceded. The explanation of the phenomena in terms of symmetries nonetheless deserves the label “dynamic al” (though not, as ac knowledged in Brown and Pooley (2006, 83), “c onstruc tive”) bec ause the explanantia are (in the first instanc e) the dynamical symmetries of the laws governing the material systems manifesting the phenomena. (97) More radic al options c ould also be pursued. Starting with the idea that there are no primitive fac ts about the c ontiguity or otherwise of distinc t material events, one might n Page 110 of 112

Index nonetheless map them into a single c opy of ℝn. The c oinc idenc e of events (whic h events are to be mapped to the same element in ℝn) is then to be thought of as determined in the same manner as the spac etime interval, that is, determined by those c oordinatiz ations that yield total desc riptions of all events that satisfy some simple set of equations. Perhaps one c ould even view the value of n (that is, the dimensionality of spac etime itself) as determined in this way too. As with generaliz ations of Huggett's proposal (see footnote 88), the more one views as grounded via some kind of Best System presc ription, the more unc onstrained the problem bec omes; it c eases to be plausible that the c omplexity of the postulated supervenienc e base will be suffic ient to underwrite the target quantities and the laws they obey. (100) Rec all (sec tion 4) that the pseudo-Riemannian metric tensor gab enc odes all of the geometrical properties of spacetime, itself represented by the four-dimensional manifold M. Stric tly speaking, the stress– energy tensor Tab does not direc tly represent the fundamental matter c ontent of the model. This will be represented by other fields, in terms of whic h Tab is defined. (104) Note one parallel between the Hole Argument and the argument against Galilean spac etime that exploits the Maxwell group. The fac t that the Maxwell group involves a parameter that is an arbitrary func tion of time means that the Galilean substantivalist interpretation of the models of a Maxwellian invariant theory involves regarding the theory as indeterministic (c f Stein 1977, Saunders, 2003a). The fac t that the indeterminism involves qualitative differenc es (ac c ording to the Galilean substantivalist) arguably makes the argument more effec tive against Galilean substantivalism than the Hole Argument is against GR. (106) An argument like this was made by Leibniz in his c orrespondenc e with Clarke (Alexander, 1956). That Leibniz makes a prec isely parallel argument, exploiting permutation invarianc e, against the existenc e of atoms, should give those sympathetic to the static shift argument pause for thought. Consistenc y should lead one either to embrac e or rejec t both c onc lusions. (108) I am attrac ted to the view that sees individualistic fac ts as grounded in general fac ts (Pooley, unpublished). However, as Dasgupta (whose terminology I adopt) has rec ently stressed (Dasgupta, 2011, 131– 134), this requires that one's understanding of general fac ts does not presuppose individualistic fac ts. Sinc e the stan dard understanding of general fac ts arguably does take individualistic fac ts for granted, the spac etime struc turalist/sophistic ated substantivalist must show that they are not illic itly making the same presupposi tion. (Dasgupta's own view is that something quite radic al is needed (2011, 147– 152).) The rec ent literature on “weak disc ernibility” (see, e.g., Saunders, 2003b) has made muc h of the fac t that numeric al diversity fac ts c an supervene on fac ts statable without the identity predic ate even when traditional forms of the Princ iple of the Identity of Indisc ernibles are violated. Note, however, that merely showing that one set of fac ts supervene on another set of fac ts is not suffic ient to show that the former are grounded in the latter (or even that it is possible to think of them as so grounded). Page 111 of 112