philosophy of physics

Index Fields, c lassic al mec hanic s, 44 50-50 volume mixture, the tyranny of sc ales, 268 Fine-graining, indistinguishability, 346 Finiteness c ondition, radiation theory, 124 First-order phase transition, 147 Fisher, Mic hael, 166, 170, 208 Fixed points matter, infinities and renormaliz ation, 178 renormaliz ation group theory, 177 Flatness problem, early universe c osmology, 635 Flexible beam, c lassic al mec hanic s, 44 Flexible bodies, c lassic al mec hanic s, 44 Flows and flow diagrams matter, infinities and renormaliz ation, 180–81 one-dimensional Ising model, 180 two-dimensional Ising model, 181 Fluctuations boiling, 151 c ritic al opalesc enc e, 151– 52 liquid gas, 150n6, 151 Fluid mec hanic s. See Hydrodynamic s, philosophy of Fluid motion, hydrodynamic s, 13– 14, 22 Fluids mean field theory, 165 veloc ity, hydrodynamic s, 13 Forc ed c losed, rigid body mec hanic s, 70n27 Forc e of rolling fric tion, axiomatic presentation, 54, 54 Forc es, rigid body mec hanic s, 73– 74 Foundational point of view, axiomatic presentation, 48– 49 Four-forc e, relativity, 554 Fourier spac e, renormaliz ation group theory, 172–74 Fourier transformations advanc ed Green's func tions, 111 dispersion theory, 133, 135 effective field theory (EFT), 239 radiation theory, 126–27 retarded Green's func tions, 113, 114 Fraser, D., 251 Free body diagrams, rigid body mec hanic s, 73, 74 Freez e out of partic les, Standard Model, 614 Freez ing to a sc ale level, 55 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 610, 612–13, 614n16, 617, 623 early universe c osmology, 634– 36 global struc ture, c osmology, 631– 33 Friedman, Mic hael, 548 Page 27 of 112

Index Frisc h, Mathias c ausation in c lassic al mec hanic s, 108nn7 and 9, 109n10 dispersion relations, 132 dispersion theory, 135–37 point-partic le elec trodynamic s, 130– 31 Froude, William boundary layers, 24–26 surfac e waves, 16 turbulence, 21 Fruitless definition, symmetry and equivalenc e, 322 Future Cauc hy horiz on, relativistic spac etime, 592 Future-directed curve, relativistic spacetime, 589 Future distinguishability c ondition, spac etime properties, 596 Future domain of dependenc e, relativistic spac etime, 592 Future endpoint, relativistic spac etime, 592 Future incomplete, spacetime properties, 598 Galaxies and c lusters of galaxies, length sc ale, 613 Galilean boosts, symmetry and equivalenc e, 327, 331 Galilean c ovarianc e spacetime, substantivalist and relationalist approaches to, 523 Galilean idealiz ation, 210n7 Galilean invarianc e, 527– 31 dynamic al symmetries, spac etime and, 527– 29 dynamic al symmetry group, 529 Euc lidean c oordinate systems, 528, 529 Galilei group, 529 kinematic ally privileged systems, 527 kinematic shift argument, 529–30 Leibniz group of transformations, 528 Leibniz ian relationalism, 527–28, 530 Newton group, 528 Princ iple of Suffic ient Reason (PSR), 530n14 relative partic le c onfigurations, 527 spac etime symmetry groups, 528 Galilean spac etime have-it-all relationalism, 570 Hole Argument (Einstein), 576n104 spacetime substantivalism, 531 Galilei group enric hed relationalism, c lassic al mec hanic s, 549– 50 physic al equivalenc e, 330 spac etime, substantivalist and relationalist approac hes to, 529, 533 symmetries, spacetime substantivalism, 534 Galileo, Churc h's c ondemnation of, 524n3 Games and gambling, the tyranny of sc ales, 275– 76, 277 “ Page 28 of 112

Index Gauge argument,” symmetry, 291, 298–300 Lagrangian, 298–99 kinetic component, 300n17 Gauge hierarc hy problem electroweak theory, 404 Gauge theories, symmetry, 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 Gauss-Codac c i c onstraint equations, 629n47 Gaussian bell c urves, quantum mec hanic s, 432 Gaussian distribution, the tyranny of sc ales, 276, 279 Gaussian wave pac kets, quantum mec hanic s, 432– 34 Gauss's Law, effec tive field theory (EFT), 231– 32 Gell-Mann/Low formulation, effec tive field theory (EFT), 408– 9 Gell-Mann, M. gauge theories, 302 “Eightfold Way,” 302n22 omega minus hadron, predic tion of, 306 predic tion, totalitarian princ iple, 309n39 symmetry, predic tive reasoning, 308– 9 Gell-Mann “strangeness,” 302 Generaliz ed c oordinates, rigid body mec hanic s, 71 Generally c ovariant equations, symmetries, 533 General relativity (GR), 325, 537–39, 541, 543–44 dark matter and dark energy, 619 have-it-all relationalism, 568–69 Hole Argument (Einstein), 573–75, 578 Mac hian relationalism, 563– 64 unific ation in physic s, 383 Georgi, H., 238–39, 411n25 Geroc h, Robert, 613 Gibbs, J. Willard. See also Boltz mann-Gibbs formulation Elementary Principles in Statistical Mechanics, 352–53 on indistinguishability, 341–52. See also Indistinguishability phase transitions infinite idealiz ation, 207 Page 29 of 112

Index statistic al mec hanic al treatment, 195 thermodynamic treatment, 191 statistic al mec hanic s, 146 thermodynamics, 142, 145 Gibbs paradox, indistinguishability, 378 Ginz burg, Vitaly, 167. See also Landau-Ginz burg-Wilson free energy Glashow model, elec troweak theory, 397 Glashow, S. See Glashow model; Glashow-Salam-Weinberg (GSW) model Glashow-Salam-Weinberg (GSW) model, 303 Gleason, A. M., 422, 425, 449 Global c ontinuous symmetry, 295– 97 Global hyperbolic spac etime, 10, 597, 629 Global isotropy, reduc tion of, 611 Global spac etime struc ture, 10, 587– 606 Big Bang singularity, 600 boundary conditions, 600 c ompac tly generated Cauc hy horiz on, 602 hole-freeness, 603 Klein-Gordon fields, 601 manifold M, 588–90 c ompac t manifold, 588 Hausdorff, 588 metric, 588–90 Lorentz ian metric , 588 Minkowski spac etime singularities, 601, 601 time travel, 603 Misner spacetime, 603 nakedly singular spac etime, 601 reasonable properties, 598–603 Big Bang singularity, 600 boundary conditions, 600 c ompac tly generated Cauc hy horiz on, 602 hole-freeness, 603 Klein-Gordon fields, 601 Minkowski spac etime, 601, 603 Misner spacetime, 603 nakedly singular spac etime, 601 singularities, 598–601 time travel, 601–3 trapped surfac e, 600 relativistic spac etime, 587– 93 c ausal c urve, 591 c ausal future, 590– 91 c hronology violating region of spac etime, 591 Page 30 of 112

Index c losed c ausal c urve, 591 c onformal fac tor, 592 c ylindric al Minkowski spac etime, 591 dependence, 590–93 future Cauc hy horiz on, 592 future-directed curve, 589 future domain of dependenc e, 592 future endpoint, 592 influence, 590–93 isometric spac etimes, 589– 90 loc ally isometric spac etime, 590 manifold M, 588–90 maximal spac etimes, 590 metric, 588–90 Minkowski spac etime, 591, 591– 93, 592 Möbius strip, 589 null vec tors, 589, 589 past Cauc hy horiz on, 592 past-directed curve, 589 past domain of dependence, 592 past endpoint, 592 spacelike surface, 592 spac elike vec tors, 589, 589 temporarily orientable spacetime, 589 timelike future, 590–91 timelike vec tors, 589, 589 singularities, 598–601 spacetime properties, 593–98 Cauc hy surfac e, 597 c ausal c ontinuity c ondition, 597 c ausal simplic ity c ondition, 597 c ausal struc ture, 596 c hronology c ondition, 596 c onstraint solutions, 595 c onvex normal, 593 distinguishability c onditions, 596 dominant energy c onditions, 595 Einstein's equation, 595 Einstein tensor, 594 energy-momentum tensor, 594 future distinguishability c ondition, 596 future incomplete, 598 global hyporbolic ity, 597 global properties, 596–98 loc al properties, 593– 95 Page 31 of 112

Index Minkowski spac etime, 597 null geodesic inc ompleteness, 598 past distinguishability, 596 past inc omplete, 598 Ric c i tensor, 594 singularities, 596 spacelike geodesic incompleteness, 598 stable c ausality c ondition, 596– 97 strong c ausality c ondition, 596 strong energy c onditions, 595 timelike geodesic ally inc omplete, 598 time travel, 601–3 trapped surfac e, 600 Global struc ture, c osmology, 628– 33 Big Bang model, 631 Cauc hy surfac e, 629 “c ausality c onditions,” 629 “c haotic c osmology” program, 631– 32 Copernic an princ iple, 631– 32 c osmic bac kground radiation (CBR), 632 “c osmologic al princ iple,” 630 Ehlers-Geren-Sachs (EGS) theorem, 630–32 Einstein Field Equations (EFEs), 628–29 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 631–33 Gauss-Codac c i c onstraint equations, 629n47 global hyperbolic spac etime, 10, 629 homogeneity, 630 loc al property of spac etime, 629n48 Minkowski spacetime, 628n46, 629 observationally indistinguishable (OI) spac etime, 629– 30 spacetime geometry, 628 Sunyaey-Zel'dovich effect, 632 “Gluing” role, symmetry, 301 Glymour, Clark, 496, 499n9 Goldstone boson, symmetry, 304 GR. See General relativity (GR) Gravitational wave, relativity, 556 Green, George, 269 Green, Melville, 168 Green's func tions, point-mass mec hanic s, 67 Greenwood, Donald, 80–82 Greenwood's proofs, rigid body mec hanic s, 80– 82 Group orbits, Mac hian relationalism, 559 Group theory, symmetry, 289 Guggenheim, E. A. Page 32 of 112

Index c orresponding states, princ iple of, 162 mean field theory, 165 Guns, Germs and Steel (Diamond), 141 Guth, Alan, 634, 636 Haag, Rudolph, 492 Hac king, Ian, 13 Haec c eitism, indistinguishability, 356– 60 Hagen, Gotthilf, 20 Hamel, Georg, 48 Hamiltonian equations, symmetry, 289 Hamiltonian func tion bloc k transforms and sc aling, 171 free energy, 173 Ising model, 153 matter, infinities and renormaliz ation, 146 flows and flow diagrams, 180, 181 mean field theory, 157 naming of func tion, 146n3 phase transitions, 190 statistic al mec hanic al treatment, 193 renormaliz ation group theory, 196–97 and statistic al mec hanic s, 146 the tyranny of sc ales, 276 universality c lasses, 178 Hamiltonians, effec tive field theory (EFT), 410 Hamiltonian symmetries outlook, 334 and physic al equivalenc e, 331– 32 symmetry and equivalence, 326–28 Hamiltonian systems, “uniqueness results,” 493–94 Hamiltonian theory, unitary inequivalenc e, 507 Hamilton's “Princ iple of the Least Ac tion,” 295 Hamilton, William Rowan. See Hamiltonian equations; Hamiltonian func tion Hausdorff, 588 Have-it-all relationalism Best Systems prescription, 566 dynamic al approac h to relativity, 569– 74 dynamical laws, 565 Galilean spac etime, 570 general relativity (GR), 568–69 homogeneous matter, 573–74 Humean approac h, 566 Leibniz ian relationalism, 567–68 Lorentz c ontrac t, 570n92, 572 Mill-Ramsey-Lewis's Best Systems prescription, 566 Page 33 of 112

Index Minkowski geometries, 569n91, 570–71 regularity approac h, 564– 69 rod, c onstituents of, 570 rotation disks argument, 566 Hawking radiation exterior, 518 Hawking, Stephen, 599, 613 Heat c apac ity as measured, mean field theory, 166 Heat transfer, indistinguishability, 344–45 Heisenberg model of ferromagnetism, 163 Heisenberg's “c ut,” quantum mec hanic s, 440– 42 Copenhagen interpretation, 441n30 Heisenberg, Werner, 20 gauge theories, 301 quantum mec hanic s c lassic al regime, 430– 31 partic le trac ks, 439 Helmholtz equation, radiation theory, 123, 125–27, 129–30 Helmholtz , Hermann boundary layers, 23 and explanatory progress, 28–29 instabilities, 20 thermodynamic c alc ulations, 146 turbulence, 21 vortex motion, 18–19 Helmholtz -Kelvin instability, 19 Hermann, Grete, 422 Heterogeneous spec ializ ations, 30 Hierarc hy problem, effec tive field theory (EFT), 228n6 Higgs boson electroweak theory, 394–95, 403–6 symmetry, 291 Higgs field, elec troweak theory, 398– 401 Higgs mec hanism, symmetry, 303– 6 Higgs partic le, unific ation in physic s, 381– 82 Higgs term, effec tive field theory (EFT), 228n6 High-energy partic le ac c eleration, funding for, 2 Higher-order ontology, Everett interpretation, 470–74 “Higher or lower” pairs, rigid body mec hanic s, 70n28 Hilbert, David, 104 axiomatic enc apsulations, 56 c hoic e of length sc ale, 50 decompositional programs, 53, 53 degeneration, 53 dispersion theory, 134, 136 homogeniz ation, 53 Page 34 of 112

Index indistinguishability, 362, 363 ontology, 369 list of problems, 47–48, 81–82 sixth problem, 52–53 symmetry, 289 Hilbert spac e Everett interpretation, 464, 467n8 indistinguishability, 359 quantum mechanics, 420–21, 425–26 environment, 434 ideal spin measurements, 443 measurement, 417, 450n38 unitary equivalenc e, physic al equivalenc e and, 9, 489– 90, 503n12, 510, 518 c ompeting c riteria of equivalenc e, 513 “uniqueness results,” 494–95 unitary inequivalenc e, as example of, 503 Hofstader, Douglas, 471 Hole Argument (Einstein), 300n17, 523, 574–79 and dynamic al symmetries, 576 Euc lidean symmetries, 577 and Galilean spac etime, 576n104 general relativity (GR), 573–75, 578 hole diffeomorphism, 575 individualistic fac ts, 577n108 and kinematic shift argument, 576 Leibniz and Clarke c orrespondenc e, 577n106 Mac hian 3-spac e approac h, 578 and Maxwell group, 576n104, 578 Newton-Cartan theory, 576–77 pseudo-Reimannian metric field, 574n100, 578 sophisticated substantivalism, 575 struc tural realist interpretation of spac etime, 577 Hole-freeness, global spacetime structure, 603 Homogeneity global struc ture, 630 have-it-all relationalism, 573–74 Homogeneous spec ializ ations, 30 Homogeniz ation axiomatic presentation, 51, 53, 55 the tyranny of sc ales, 256, 280– 83 averaging and homogeniz ation, differenc es, 277–78 limit, 281 Hooke's law c ontinuum mec hanic s, 99 steel beams, 258 Page 35 of 112

Index Horiz on problem, early universe c osmology, 634–35, 635 “Hot” versus “c old,” dark matter and dark energy, 618n24 Hubble, E., 613 Hubble radius, 617n21 Huggett, Nick effective field theory (EFT), 234 enric hed relationalism, c lassic al mec hanic s, 548 have-it-all relationalism, 564–69, 571–72, 573n97 indistinguishability, haec c eitism, 356 regularity approac h to relationalism, 564– 69, 571– 72 relationalism, 545 relational spac etime, 544 Hume, David, 56, 107 have-it-all relationalism, 566 Hydrodynamic s, philosophy of, 12– 42 approximation methods, 31 artic ulation, 13 Bernoulli's Law, 15–16, 27 boundary layers, 13, 23–27 airplane wings, 26 disc ontinuity surfac e, 24 eddy resistanc e, 24, 25 and modules, 37 ships, 24 skin resistance, 24 wave resistanc e, 24 c alc ulation, 13 c omputational templates, 32– 33n41 explanatory progress, 27–30 c omponents of explanation, 28– 29 heterogeneous spec ializ ations, 30 homogeneous spec ializ ations, 30 pragmatic definition of, 29– 30 sourc es of, 27– 28 fluid motion, 13–14, 22 fluid veloc ity, 13 history, 13–27 Bernoulli's Law, 15–16, 27 boundary layers, 13, 23–27 fluid motion, 13–14, 22 fluid veloc ity, 13 instabilities, 19–21 kinetic theory of gases, 22 “losses of head,” 16 nonviscous fluid, 13 Page 36 of 112

Index pipe flow, 20 plane parallel flow, 20 plane Poiseuille flow, 21 surfac e waves, 16– 17 turbulence, 21–23 vortex motion, 17–19 instabilities, 19–21 interpretive sc hemes, 31 kinetic theory of gases, 22 “losses of head,” 16 modules, 33–38 c orrespondenc e princ iple, 35– 36 defining, 33 idealiz ing modules, 33 and models, 36–38 reduc ing modules, 34, 36n46 shared module, 36 spec ializ ing modules, 33 struc ture, 35 nonviscous fluid, 13 overview, 3 physic al theories, 31– 32 pipe flow, 20 plane parallel flow, 20 plane Poiseuille flow, 21 surfac e waves, 16– 17 symbolic universe, 31 theoretic al laws, 31 theory artic ulation, 13 turbulence, 21–23 the tyranny of sc ales, 279 vortex motion, 17–19 wing theory (Prandtl), 13, 30 Ideal spin measurements, quantum mec hanic s, 443– 44 Identity c onditions, indistinguishability, 372– 76 Imprec ision, Ising model, 155n10 Improper mixtures, quantum mec hanic s, 427 indistinguishability from proper mixtures, 429 Inc oming waves, radiation theory, 127 Indexic al unc ertainty, Everett interpretation, 476 Indistinguishability, 340–80 “ac c ount of equality,” 372 asymmetry, 365n33 c lassic al indistinguishability, argument against, 354– 56 c lassic al partic le indistinguishability, 355– 56 Page 37 of 112

Index c oarse-graining equilibrium entropy, 348 statistic al mec hanic s, 346 c onventions, system of, 246 Ehrenfest-Trkal-van Kampen approac h, 350– 51, 366 eliminativism, 376–78 and Gibbs paradox, 378 Pauli exc lusion princ iple, 377 “preferred basis problem,” 377 quantum fields, 377–78 entropy, 343–45 Boltz mann definition of, 346–47 equilibrium entropy, 348–49 c oarse-graining, 348 fermions, 364–65 Feynman diagrams, 367, 368 fine-graining, 346 Gibbs paradox, 7, 341–52 Ehrenfest-Trkal-van Kampen approac h, 350– 51, 366 and eliminativism, 378 equilibrium entropy, 348–49 Feynman diagrams, 367, 368 haecceitism, 356 N! puz z le, 349–52 ontology, 366–69 permutable coins, 366–67, 368 quantum, indistinguishability and, 341–43 statistic al mec hanic s, 346– 47 thermodynamics, 343–46 uniform symmetry, 352–53 haecceitism, 356–60 Hilbert spac e, 359, 362, 363 identity c onditions, 372– 76 Leibniz 's princ iple, 373–74 Maxwell-Boltz mann statistic s, 371n42 N! puz z le, 349–52 ontology, 365–78 “ac c ount of equality,” 372 eliminativism, 376–78 Gibbs paradox, 366–69 identity c onditions, 372– 76 Leibniz 's princ iple, 373–74 Maxwell-Boltz mann statistic s, 371n42 permutability of objec ts, 369– 70 philosophic al logic , 369– 72 Page 38 of 112

Index quantities invariant, 365 quantum partic les, 371– 72 spin, 375 total symmetry, 369 overview, 6–7 parastatistics, 364n32 Pauli exc lusion princ iple, 377 permutability of objec ts, 369– 70 permutable coins, 366–67, 368 phase spac e, reduc ed, 361– 62, 362 philosophic al logic , 369– 72 “preferred basis problem,” 377 quantities invariant, 365 quantum fields, 377–78 quantum, indistinguishability and, 341–43 “light quanta,” 342 microstate, 342n3 and “permutable,” 342n4 quantum partic les, 371– 72 quantum statistic s, explanation of, 360– 64 Hilbert spac e, 362, 363 phase spac e dimension, 361– 62 subspace dimension, 362–63 volume measures, 362–63 weighting, 361, 362 spin, 375 statistic al mec hanic s, 346– 47 c oarse-graining, 346 entropy, 346–47 fine-graining, 346 subspace dimension, 362–63 thermodynamics, 343–46 c onventions, system of, 246 entropy, 343–45 heat transfer, 344–45 total symmetry, 369 as uniform symmetry, 352–53 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 volume measures, 362–63 weighting, 361, 362 Individualistic fac ts, Hole Argument (Einstein), 577n108 Page 39 of 112

Index Inertia c ontinuum mec hanic s, 97n44 Descartes, René, 524 point-mass mec hanic s, 65 spac etime explanation of, 541– 44 Infinite idealiz ation, 204–17 c rossover theory, 220– 21 and emergenc e of phase transitions, 202 explanatory irreduc ibility, 210– 14 irreduc ibility, 221 phase transitions, 202 renormaliz ation group, 216–23 c ritic al behavior of partic ular systems, 218 c rossover theory, 220– 21 irreduc ibility, 221 universality, 217–18 Infinite limits, the tyranny of sc ales, 261– 62 Infinitesimal c ubes, c ontinuum mec hanic s, 96 Infinitesimal trac tion vec tors, 85 Infinite spin c hain, as example of unitary inequivalenc e, 501– 5, 508 Inflation early universe cosmology, 636n64, 637 multiverse, 644 Influenc e, relativistic spac etime, 590– 93 Instabilities, hydrodynamic s, 19– 21 Instantaneous relative distanc es, 540n35 Instantaneous states, symmetry and equivalenc e, 326n31 Intensive properties, phase transitions, 201 Interpretive sc hemes, hydrodynamic s, 31 Intertheoretic relation, effective field theory (EFT), 245–47 Irreduc ibility infinite idealiz ation, 221 phase transitions, 199, 203 explanatory irreduc ibility, 210– 14 infinite idealiz ation, 221 Irrelevant scalings, 177–78 Irreversibility, 142 Ising, Ernst, 145 Ising model, 152–56 Boltz mann-Gibbs formulation, 154 defined, 152–53 extended singularity theorem, 152, 154–55 ferromagnets, 156 and Hamiltonian func tion, 153 imprec ision, 155n10 Page 40 of 112

Index one-dimensional Ising model, 180 Ornstein-Zernike infinity, 156 partic le spin, 154 phase diagram, 156 sc aling, 171 singularity in phase transition, 155 two-dimensional Ising model, 153, 181 Isometric spac etimes, relativistic spac etime, 589– 90 Isospin electroweak theory, 396n14 symmetry, 301 Jac obi's princ iple, Mac hian relationalism, 558– 59 Jeans, J., 289n3 Jet bundle, symmetry and equivalenc e, 324 Jordan-Wigner theorem, 9, 490–91, 501 Kadanoff, Leo P. the tyranny of sc ales, 266 unific ation in physic s, 408– 9 Kant, Immanuel c ausation, 107 c lassic al mec hanic s, 89n39 indistinguishability, haec c eitism, 356 and sc ientific c osmology, 625– 26 symmetry, inc ongruent c ounterparts, 293n9 Kármán, Theodore von, 22, 26 Kellers, L. F., 165, 168 Kelvin, Lord. See Thomson, William (Lord Kelvin) Kepler, Johannes, 310 Mysterium Cosmographicum, 287–88 Kepler problem, symmetry and equivalenc e differential equations, symmetries of, 325 and physic al equivalenc e, 332 Kinematic ally possible models (KPMs) diffeomorphism group (DPM), 533–34 Mac hian relationalism, 558 relativity, enric hed relationalism, 555 spacetime substantivalism, 531–33 Kinematic ally privileged systems, 527 Kinematic s rigid body mechanics, 79 unitary equivalenc e, kinematic pair, 495– 96 unitary equivalenc e, physic al equivalenc e and, 492 Kinematic shift argument enric hed relationalism, 546– 47 Hole Argument (Einstein), 576 Page 41 of 112

Index spac etime, 523, 544 substantivalist and relationalist approac hes to, 529– 30 Kinetic theory of gases, 22 King's College (London) sc hool, 166– 67 Kirc hhoff, G., 23– 24 Klein, Felix, 289 Klein-Gordon fields, 601 Kolmogorov, Andrei, 167 Korteweg-de Vries vec tor, 325, 333 Koyré, Alexander, 524 KPMs. See Kinematic ally possible models (KPMs) Kramers, Hendrik, 163 Kramers-Kronig relations, 135–36 Kretsc hmann, E., 537 Kuhn, Thomas c ritic isms of, 30n38 dark matter and dark energy, 619 matter, theories of, 150 and “normal” phases of sc ienc e, 12 phase transitions, 168 Labyrinth of the c ontinuum, 90, 101 Lagrange, Joseph Louis instabilities, 20 rigid body mec hanic s, 78– 80, 82 surfac e waves, 16 vortex motion, 17 Lagrangian density, effec tive field theory (EFT), 226, 235, 251 formalism, effec tive field theory (EFT), 247 initial and Lagrangian for superfluid Helium, effec tive field theory (EFT), 246 invarianc e, elec troweak theory, 395, 397– 99 methodology symmetry, 295–96, 299 symmetry and equivalenc e and physic al equivalenc e, 331 symmetry kinetic component, 300n17 variational symmetries, 326–27 unitary equivalenc e, physic al equivalenc e and, 491– 92 Lamb, Willis, 307n35 Lanc z os, Cornelius, 82 Landau-Ginz burg-Wilson free energy, 173–74 Landau, Lev different phase transition problems, 160–61 mean field theory, 158 experimental facts, 164–65 Page 42 of 112

Index as model, 164 summary of, 161–62 phase transitions, 144, 152 order parameter, 192 universality, 161–62 Laplac e transform tec hnique, radiation theory, 125 Large Hadron Collider (LHC), 305, 381–82, 384 electroweak theory, 405–6 unific ation in physic s, future output, 406 “Large Number Hypothesis,” 638–39 Lattic e defec ts, c ontinuum mec hanic s, 98 Laws of motion. See Newton, Isaac Lebesgue measure z ero, 118 Lebowitz , J. L., 195, 198 Leeuwen, Hans van, 151n7 Leibniz , Gottfried Wilhelm axiomatic presentation, 48, 52 Clarke, correspondence, 577n106 c ontinuum mec hanic s labyrinth of the c ontinuum, 90 loaded beam, 91n40 indistinguishability haecceitism, 356 Leibniz 's princ iple, 373–74 Leibniz group of transformations, 528 spacetime, substantivalist and relationalist approaches to, 333 Leibniz ian relationalism have-it-all relationalism, 567–68 Mac hian relationalism, 558 spac etime, substantivalist and relationalist approac hes to, 527– 28, 530, 544– 45, 548, 567– 68 symmetries, spacetime substantivalism, 535 Lennard-Jones potential, 60 Lensing effec t, dark matter and dark energy, 621 Lenz -Runge vec tor, symmetry and equivalenc e, 325 Lenz , Wilhelm, 145, 153 Levanyuk, A. P., 167 Le Verrier, Urbain, 618, 619, 623 Lewis, David K., 482, 483, 566 LHC. See Large Hadron Collider (LHC) Lie-Bäcklund transformations, 324–26 Lie group, elec troweak theory, 395 Light-bending, dark matter and dark energy, 620–21 “Light quanta,” 342 Linear operators, quantum mec hanic s, 417n4 Lipkin, H., 307 Page 43 of 112

Index Liquid gas fluctuations, 150n6, 151 infinite idealiz ation, phase transitions, 215 Liu, C., 198 Loaded beam, c ontinuum mec hanic s, 91n40 Loc ally isometric spac etime, relativistic spac etime, 590 Loc al properties, spac etime, 593– 95, 629n48 Local symmetry, 323–25 London, Fritz , 297 Loop quantum gravity (LQG), elec troweak theory, 405 Lorentz c ontrac t, have-it-all relationalism, 570n92, 572 Lorentz -Dirac equation c ausation in c lassic al mec hanic s, 132 description, 109 point-partic le elec trodynamic s, 132 Lorentz ian metric , 588 Lorentz invariant gauge theories, 299 Maxwell's equation, form of, 538 symmetry, 292 Lorentz -Poinc aré relativity, 292 Lorentz , quantum mec hanic s, 433 Lorentz transformations, relativistic spac etimes, 536 “Losses of head,” hydrodynamic s, 16 Low-energy superfluid helium-4 film, 230–32 Mac h, Ernst, 55, 67, 541 Mac hian relationalism, 557– 64 best matching, 559, 559–60 BSW ac tion, 562– 64 diffeomorphism group (DPM), 557–58, 561 Euc lidean c oordinate systems, 557 general relativity (GR), 563–64 group orbits, 559 Jac obi's princ iple, 558– 59 kinematic ally possible models (KPMs), 558 and Leibniz ian relationalism, 558 Reimannian 3-metrics, 561–62 shape spac e, 558 similarity groups, 559 superspace, 562 z ero field strength, 557n75 Mac hian 3-spac e approac h, 578 Mach's Principle: From Newton's Bucket to Quantum Gravity (Barbour), 557 Mac h's princ iple, Standard Model, 610 Mac romec hanic s, quantum mec hanic s, 430 Page 44 of 112

Index Macroscopic scale behaviors, 256 Magnitude F, point-mass mec hanic s, 65, 65 Magnitude of parameter, 144 Malament, David, 629n50 Manifold M global spac etime struc ture, 588– 90 c ompac t manifold, 588 Hausdorff, 588 relativistic spac etime, 588– 90 Manohar, A., 234, 246 Many-exac t worlds theories, 467– 68 Many-minds theories, 467–68 Many spins, mean field theory, 157 Many Worlds Theory. See Everett interpretation Marginal sc alings, 177– 78 Mars's motion, 626–27 Mass-dependent schemes, effective field theory (EFT), 236–37 Mass-independent schemes, effective field theory (EFT), 237–39 Mass point lattic e, c lassic al mec hanic s, 44 Matc hed asymptotic s, point-mass mec hanic s, 69 Material derivatives, c ontinuum mec hanic s, 87 “Material partic les,” c ontinuum mec hanic s, 270n18 Materials, disc overy and invention of, 141– 43 Mathematic al c omplexity, c lassic al mec hanic s, 45 Matter, infinities and renormaliz ation, 141–88 Boltz mann-Gibbs formulation, 180 broken symmetries, 144–45 Callen-Symanz ik equation, 180 c anonic al ensemble, 147 use of term, 147n4 c ontinuous phase transition, 147 c orrelation func tion c alc ulations, 168– 69 c orresponding states, princ iple of, 162 c ritic al fixed points, 179, 180 c ritic al point, 147 c rystalline materials, 143– 44 different phases of matter, 143–51 broken symmetries, 144–45 c anonic al ensemble, 147 c ontinuous phase transition, 147 c ritic al point, 147 c rystalline materials, 143– 44 dynamics, 145–47 equilibrium, 145–47 first mean field theory, 147–49 Page 45 of 112

Index first-order phase transition, 147 Hamiltonian func tion, 146 magnitude of parameter, 144 Maxwell-Boltz mann distribution, 147 Maxwell's improvement, 149–51 mean field theory, 147–49 order parameters, 144–45 orientation of order parameter, 144 phase spac e, 146 phase transitions, 147 and statistic al mec hanic s, 146 water, c artoon PVT diagram for, 148 different scalings, 177–78 dynamic al systems theory, 180 dynamics, 145–47 ensemble generates averages, 180 ensemble generates ensemble, 180 equilibrium, 145–47 extended singularities, renormaliz ation group, 183 first mean field theory, 147–49 first-order phase transition, 147 fixed points, 178 flows and flow diagrams, 180–81 fluctuations, 151 boiling, 151 c ritic al opalesc enc e, 151– 52 liquid gas, 150n6, 151 Hamiltonian func tion, 146 irrelevant sc alings, 177– 78 irreversibility, 142 Ising model, 152–56 Boltz mann-Gibbs formulation, 154 defined, 152–53 extended singularity theorem, 152, 154–55 and Hamiltonian func tion, 153 imprec ision, 155n10 Ornstein-Zernike infinity, 156 partic le spin, 154 phase diagram, 156 singularity in phase transition, 155 two-dimensional Ising model, 153 Landau's generaliz ation, 160–61 magnitude of parameter, 144 marginal sc alings, 177– 78 materials, disc overy and invention of, 141– 43 Page 46 of 112

Index Maxwell-Boltz mann distribution, 147 Maxwell's improvement, 149–51 mean field theory, 147–49, 156–60 c orrelation length, 162 experimental facts, 164–65 first mean field theory, 147–49 fluids, behavior of, 165 and Hamiltonian func tion, 157 heat capacity as measured, 166 many spins, 157 mean field results, 157–58 one spin, 157 order parameter jump, 162 power laws, representing c ritic al behavior by, 159– 60 renormaliz ation c alc ulations, 159–60 scale transformation, 159–60 sc aling, 161 spatial structures, 167 symmetry, 160n12, 161 theoretic al fac ts, 165– 67 turbulence, 167 universality, 161–62 mean field theory, summary of, 161–62 1937, events of, 160–64 c orresponding states, princ iple of, 162 Landau's generaliz ation, 160–61 mean field theory, summary of, 161–62 Netherlands meeting, 162–64 universality, 161–62 order parameters, 144–45 orientation of order parameter, 144 phases of matter different phases, 143–51, 160–61 fluctuations, 151 thermodynamics, 145–46 thermodynamic s phases, 142 phase spac e, 146 phase transitions, 141–42, 147, 168 word “phase,” use of, 141n1 relevant scalings, 177–78 renormaliz ation group as not a group, 181–82 sc alings, different, 170– 72 short-distance expansion, 182 snowflake, 143, 143–44 splash, 143 Page 47 of 112

Index statistical equilibrium, 142 statistic al mec hanic s, 142, 146 statistic al physic s, theory of, 141 sub-universe, 182 thermodynamic phases, 142 universality, 161–62 c lasses, 178– 79 U.S. National Bureau of Standards c onferenc e, 168 Virasoro algebra, 182 water, c artoon PVT diagram for, 148 Widom scaling, 168–70 Wilson on renormaliz ation group theory, 172–77 c alc ulational method, 174– 75 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 weak c oupling fixed points, 177 Maudlin, Tim, 545n49 Maximal spac etimes, relativistic spac etime, 590 Maxwell-Boltz mann distribution, 147 Maxwell-Boltz mann statistic s, indistinguishability, 371n42 Maxwell equations, quantum mec hanic s, 446 Maxwell group enric hed relationalism, 550– 52 Hole Argument (Einstein), 576n104, 578 Maxwell, James Clerk different phases of matter boiling, 150 improvement, 149–51 “A Dynamic al Theory of the Elec tromagnetic Field,” 390 Green's functions, 113 modules, 34 “On Physic al Lines of Forc e,” 387– 93 phase transitions, 147 statistic al physic s, theory of, 141 Treatise on Electricity and Magnesium, 391 turbulence, 22 vortex motion, 17 Maxwell's elec trodynamic s, 385– 93 Ampère law, 389 c hain reac tion, 387– 88 Page 48 of 112

Index c urrents, induc tion of, 386n5 d'Alembert's principle, 389–90 displacement, 391n9 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 equation, relativistic spac etimes, 538 Maxwell's improvement, 149–51 Maxwell's theory elec troweak theory, 395, 399 symmetry and equivalence, 325n21 unific ation in physic s, 7, 383 McMullin, Ernan, 636 Mean field c alc ulations, the tyranny of sc ales, 266 Mean field theory, 147–49, 156–60 c orrelation length, 162 experimental facts, 164–65 first mean field theory, 147–49 fluids, behavior of, 165 and Hamiltonian func tion, 157 heat capacity as measured, 166 and long-range forc es, 167n14 many spins, 157–58 mean field results, 157–58 one spin, 157 order parameter jump, 162 power laws, representing c ritic al behavior by, 159– 60 renormaliz ation c alc ulations, 159–60 scale transformation, 159–60 sc aling, 161 spatial structures, 167 summary of, 161–62 symmetry, 160n12, 161 theoretic al fac ts, 165– 67 turbulence, 167 universality, 161–62 Meaningfully c ombined, rigid body mec hanic s, 77 Melnyk, Andrew, 198 Mercury's perihelion motion, 619 Metaphysical Foundations of Natural Science (Kant), 89n39 “Method of extensive abstrac tion,” 96n43 Metric , global spac etime struc ture, 588– 90 Lorentz ian metric , 588 Page 49 of 112

Index Mic romec hanic s, quantum mec hanic s, 430 Milgrom, M., 624 Mill-Ramsey-Lewis's Best Systems prescription, 566 Minkowski distanc es, 553– 56 Minkowski geometries, have-it-all relationalism, 569n91, 570–71 Minkowski metric struc ture, relativistic spac etimes, 536 Minkowski spac etime effective field theory (EFT), 231 global spacetime structure, 628n46, 629 singularities, 601 time travel, 603 relativistic spac etime, 538, 591, 591– 93, 592 spacetime properties, 597, 597 substantivalist-relationalist debate, 522n1 unific ation in physic s, 393 unitary equivalenc e, physic al equivalenc e and, 518 Misner spac etime “c haotic c osmology” program, 631– 32 global spacetime structure, 603 Mobility spac e, rigid body mec hanic s, 79 Möbius strip, relativistic spac etime, 589 Modific ation of Newtonian dynamic s (MOND), 623– 24 Mod-squared amplitude, Everett interpretation, 479n17, 480 Modules, hydrodynamic s, 33– 36 c orrespondenc e princ iple, 35– 36 defining, 33 idealiz ing modules, 33 and models, 36–38 reduc ing modules, 34, 36n46 shared module, 36 spec ializ ing modules, 33 struc ture, 35 Monte Carlo method. infinite idealiz ation, 211–12 “More is Different” (Anderson), 2 Motion as c hange of plac e, 524 Motte, Andrew, 58 Multiple realiz ation, phase transitions, 198n1 Multiplets electroweak theory, 396 symmetry, 307–9 Multiverse, 639, 643–47 De Sitter universe, 644 Everett interpretation, 644n86 inflation, 644 and string theory, 644 Page 50 of 112

Index Munitz . Milton K., 624, 626–27 Mysterium Cosmographicum (Kepler), 287–88 Nagel, Ernest, 1, 198, 260 Naimark dilation, quantum mec hanic s, 448 Nakedly singular spac etime, global spac etime struc ture, 601, 601 “Natural c oordinates,” point-mass mec hanic s, 63 Naturalness, electroweak theory, 404n19 Navier-Cauchy equation, 6 Navier, Claude Louis instabilities, 19–20 internal fluid forc es, 14 point-mass model, 67 Navier-Stokes equation, 256 boundary layers, 25–27 described, 14 and explanatory progress, 27–28 instabilities, 19–20 low-density gas spec ializ ation, 34–35n44 modules, 34–35 turbulence, 22 the tyranny of scales, 269–70, 275, 278–79 Ne'eman, Y., 306, 308–9 Neo-Newtonian spacetime, 531–33 Neptune disc overy of, 618 predic tion of, 310 Netherlands meeting, 162–64 Neutrino, postulation of, 310 Newton-Cartan theory enric hed relationalism, 551, 553 Hole Argument (Einstein), 576–77 Newton group, 528 Newtonian behavior, quantum mec hanic s, 439 Newtonian boosts, symmetry and equivalenc e, 327n33 Newtonian gravity dark matter and dark energy, 619–20 uniqueness of universe, 627, 632 Newtonian mec hanic s, unific ation in physic s, 406 Newtonian sc heme, symmetry, 295 Newtonian theory, symmetry and equivalenc e, 322n12, 324, 325, 327– 28n36 particles, 324, 325, 330 physic al equivalenc e, 328 Newton, Isaac billiard c ollisions, 68, 69 boundary layers, 23 Page 51 of 112

Index De Grav, 526 law of gravitation, 60, 98 laws of motion, 43, 58–59, 311 Second Law, 77–78, 433 third law, 63–64, 66, 81, 552 “natural coordinates,” 63 planets, treatment of, 55 Principia, 523 Sc holium, 523, 525n4, 526 rotation, 48 spacetime, substantivalist and relationalist approaches to diametric al symmetries, 527– 29 dynamic al symmetries, spac etime and, 527– 31 Galilean invarianc e, 527– 31 have-it-all relationalism, 564–74 Hole Argument (Einstein), 574–79 inertia, spac etime explanation of, 541– 44 kinematic shift argument, 529–30 Mac hian relationalism, 557– 64 neo-Newtonian spacetime, 531–33 Newton's buc ket, 523– 27 rationality, failure of, 539–40 relationalism, varieties of, 544–74 Newton's buc ket, 523– 27 Noether, Emmy, 289, 291 electroweak theory, 396–97 “gauge argument,” 291 obituary (by Einstein), 297 symmetry c ontinuous symmetry, 295– 97 global c ontinuous symmetry, 295– 97 local symmetry, 324–25 symmetry and equivalenc e differential equations, symmetries of, 324–25, 328 Hamiltonian symmetries, 328 No-go theorem for safe bit c ommitment protoc ols, 430 No-hidden variables theorem, 422 Non-Abelian c ase, elec troweak theory, 396, 401 Noninertial motion, 542 Non-interac ting partic les, Everett interpretation, 463n5 Non-relativistic QCD, effec tive field theory (EFT), 229n7 Nonsimultaneous events, enric hed relationalism, 549 Nontrivial spin properties, quantum mec hanic s, 428 Nonvisc ous fluid, hydrodynamic s, 13 Normaliz ation, quantum mec hanic s, 422n10 Page 52 of 112

Index Normal sc ienc e, hydrodynamic s, 30n38 Norton, John D. anthropic reasoning, 642–43 have-it-all relationalism, 572–73 Hole Argument (Einstein), 523, 575 No-signaling theorem, quantum mec hanic s, 425 N! puz z le, indistinguishability, 349–52 NRQCD. See Non-relativistic QCD Null geodesic incompleteness, spacetime properties, 598 Null vec tors, relativistic spac etime, 589, 589 Objec tive-probability role, Everett interpretation, 479n17 Observationally indistinguishable (OI) spac etime, 629– 30 Oc kham's raz or, 523, 539 ODEs. See Ordinary differential equations (ODEs) “Old” c osmologic al c onstant problem, dark matter and dark energy, 621– 22 Oldenburg group, infinite idealiz ation, 209 Olver, P., 322n10 Omega minus hadron, symmetry, 307 One-dimensional Ising model, 180 One spin, mean field theory, 157 Onnes, Heike Kamerlingh, 165 “On Physic al Lines of Forc e” (Maxwell), 387– 93 Onsager, Lars, 165–67, 166, 195 Ontology and ontologic al issues c lassic al mec hanic s, 44n2 ontologic ally mixed c irc umstanc es, 44n2 effective field theory (EFT), 239–43 antifoundationalism, 240 antireductionism, 240 approximations, EFTs and, 243 c utoff, realistic interpretations of, 241– 43 decoupling, 240–41 quantum field theory (QFT), 240–43 quasi-autonomous domains, 240–41 Wilsonian approac h, 240 Everett interpretation, higher-order ontology, 470–74 indistinguishability, 365–78, 374 “ac c ount of equality,” 372 eliminativism, 376–78 Gibbs paradox, 366–69 identity c onditions, 372– 76 Leibniz 's princ iple, 373–74 Maxwell-Boltz mann statistic s, 371n42 permutability of objec ts, 369– 70 philosophic al logic , 369– 72 Page 53 of 112

Index quantities invariant, 365 quantum partic les, 371– 72 spin, 375 total symmetry, 369 infinite idealiz ation, phase transitions, 214–17 retarded Green's func tions, c ontours for damped osc illator, 115 symmetry, 306–7 Open channel flow, 28 Order parameters matter, infinities and renormaliz ation, 144–45 mean field theory, order parameter jump, 162 orientation of order parameter, 144 Ordinary differential equations (ODEs) c ontinuous variables, lift from, 55n14 and foundational princ iples, 45n4 PDE's distinguished, 45 and Schrödinger equation, 46 and spin, 45n5 tasks governed by, 49 Ordinary QM unitary equivalenc e, physic al equivalenc e and, 495, 496, 500– 502, 504, 510– 13 unitary inequivalenc e, as example of, 508 Ornstein, Leonard, 152, 158 Ornstein-Zernike infinity, Ising model, 156 Ostriker, P., 620n31 Our Knowledge of the “External World (Russell), 96n43 Outgoing waves, radiation theory, 127 “Overlapping domains” argument, early universe c osmology, 633 Pairs of solutions, symmetry and equivalenc e, 333n55 Parallelly transported vector, spacetime properties, 593 Parallel tradition, Everett interpretation, 478n16 Paramagnetism, 192 Parastatistic s, indistinguishability, 364n32 Parity, symmetry, 293n9 Partial differential equations (PDEs) and foundational princ iples, 45n4 ODE's distinguished, 45 Partic le spin, Ising model, 154 Passivity and c ausality, dispersion theory, 137 Past Cauc hy horiz on, relativistic spac etime, 592 Past-direc ted c urve, relativistic spac etime, 589 Past distinguishability, spac etime properties, 596 Past domain of dependenc e, relativistic spac etime, 592 Past endpoint, relativistic spac etime, 592 Past inc omplete, spac etime properties, 598 Page 54 of 112

Index PAS, unitary equivalenc e, 504, 508, 514 Patashinskii, Alexander, 168–69 Patriz i, Franc esc o, 526 Pauli relations, unitary equivalenc e, 494 Pauli spins, as example of unitary inequivalenc e, 494, 501– 4 Pauli, Wolfgang, 310, 377 Pearson, Karl, 271 Peebles, Phillip James Edward, 620n31 Penrose, Roger, 601, 613, 626, 637 Permutability of objec ts, indistinguishability, 369– 70 Permutable c oins, indistinguishability, 366– 67, 368 PEV, unitary equivalence, 496–500, 496n6, 504, 508, 514 Phase diagram, Ising model, 156 Phase space matter, infinities and renormaliz ation, 146 quantum statistic s, explanation of, 361– 62 reduc ed phase spac e, 361– 62, 362 symmetry and equivalence, 327 Phase transformation, elec troweak theory, 396n13 Phase transitions, 189–223 additivity, 201n2 boiling, 147 c onc eptual novelty, 199– 200 defined, 189 emergence of, 197–204 ensemble equivalent, 207 experimental studies, 147 explanatory irreduc ibility, 210– 14 extensivity, 201n2 infinite idealiz ation, 204–17 back-bending, 206–8, 207 Bose-Einstein, 206, 208 c aloric c urve, bac k-bending of, 207 c omplex inverse temperature, 209 c onc eptual novelty, 204– 10 distribution of z eros, 208–10 and emergenc e of phase transitions, 202 explanatory irreduc ibility, 210– 14 Galilean idealiz ation, 210n7 liquid-gas system, 215 Monte Carlo method, 211–12 Oldenburg group, 209 ontologic al irreduc ibility, 214– 17 renormaliz ation group, 216–23 smooth phase transitions, 206 Page 55 of 112

Index Yang-Lee theorem, 208 intensive properties, 201 irreduc ibility, 199, 203 explanatory irreduc ibility, 210– 14 infinite idealiz ation, 221 materials, disc overy and invention of, 141– 42 multiple realiz ation, 198n1 Nagel's theory, 198 ontologic al irreduc ibility, 214– 17 paramagnetism, 192 and reduction, 198–201 as reduc tionism in the c ore sense of, 198 renormaliz ation group, infinite idealiz ation in, 216–23 c ritic al behavior of partic ular systems, 218 c rossover theory, 220– 21 irreduc ibility, 221 universality, 217–18 renormaliz ation group theory, 195–97 c oarse-graining, 197 and Hamiltonian func tion, 196– 97 spin, 193–94 statistic al mec hanic al treatment, 193– 95 and Hamiltonian func tion, 193 spin, 193–94 Yang-Lee theorem, 195 and thermodynamic properties, 201 thermodynamic treatment, 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 U.S. National Bureau of Standards c onferenc e, 168 word “phase,” use of, 141n1 Phenomenologic al arrow of time, quantum mec hanic s, 433n22 Phenomenological theories, generally, 2 Phillips, Rob, 272–73, 284–85 Physic al equivalenc e symmetry and equivalence, 328–33 differential equation, 329–30 D2, 329–30 Galilei group, 330 Hamiltonian symmetries, 331–32 Page 56 of 112

Index Kepler problem, 332 Lagrangian treatment, 331 and Newtonian theory, 328 pairs of solutions, 333n55 spacetime symmetries, 331–32n49, 332 and unitary equivalenc e. See Unitary equivalenc e, physic al equivalenc e and Physic al motivation, retarded Green's func tions, 113 Physic al spac e, renormaliz ation group theory, 172–74 Pic kering, Andy, 302n24 Pilot-wave theories, quantum mec hanic s, 440, 482 “Pinned c onstraint,” rigid body mec hanic s, 71 Pipe flow, hydrodynamic s, 20 Planc k, M. on indistinguishability, 7, 340 N! puz z le, 350 quantum, indistinguishability and, 341–43 quantum statistic s, explanation of, 360 unitary equivalenc e, 494 Planc k sc ale dark matter and dark energy, 621 electroweak theory, 403 Plane parallel flow, hydrodynamic s, 20 Plane Poiseuille flow, hydrodynamic s, 21 Plüc ker, Julius, 49 Poinc aré relativity, 292, 423 Poincaré sphere, 423 effective field theory (EFT), 238 quantum mec hanic s, 423 rationality, failure of, 539 Point-mass mechanics, 44–46, 57–69 billiard c ollisions, 68– 69, 69 Cartesian locations, 63 Cassini spac e probe to Saturn, 61, 61 c oeffic ient of restitution, 69 c onstitutive modeling c onditions, 66, 82 inertial reac tion, 65 isolated point mass, 59 Lennard-Jones potential, 60 magnitude F, 65, 65 matc hed asymptotic s, 69 methodologies of avoidanc e, 62 “natural coordinates,” 63 purely elastic c ollision, 69n26 representative center, 59 rigid body mechanics, 71 Page 57 of 112

Index rotating rigid objec ts, 58 spec ial forc e laws, 63 steel ball pendulums, 64–65, 64–66 Point-mass swarm, axiomatic presentation, 51 Point-partic le elec trodynamic s, 129– 32 Points, axiomatic presentation, 49– 50 Poisson bracket unitary equivalenc e, physic al equivalenc e and, 493– 94 unitary inequivalenc e, as example of, 505– 6 Poisson, Siméon Denis point-mass mec hanic s, 67– 68 rigid body mechanics, 83 surfac e waves, 17 “two c onstant,” 67 Pokrovsky, Valery, 168–69 Polariz ation of a system, 502 Polc hinski, J., 227 Ponc elet, Jean Vic tor, 22, 23 Possibility and Everett interpretation, 475–77 Post fac to strategy, the tyranny of sc ales, 263 Post, H., 357 POV measure, quantum mec hanic s, 8, 448– 49, 454 Power laws, representing c ritic al behavior by, 159– 60 Prandtl, Ludwig boundary-layer theory, 13, 25–27, 37, 53n12 explanatory progress, 28–29 instabilities, 20 turbulence, 22 wing theory, 13, 30 Predic tion and predic tability effective field theory (EFT), 232–35 symmetry from multiplet sc heme, 307– 9 Neptune, predic tion of, 310 of omega minus hadron, 306 spin-3/2 baryon decuplet, 307–8, 309 “Preferred basis problem” eliminativism, 377 Everett interpretation, 466–68 and many-exac t worlds theories, 467– 68 and many-minds theories, 467–68 overview, 461 Pressures, rigid body mec hanic s, 74 A Primer on Determinism (Earman), 1 Principia (Newton), 523 Page 58 of 112

Index Sc holium, 523, 525n4, 526 Principle of Equivalence (Einstein), 537 Princ iple of Limiting Amplitude, 128 “Princ iple of medioc rity” (PM), 639– 42 Princ iple of Suffic ient Reason (PSR), 530n14 Princ iple of virtual work, rigid body mec hanic s, 79 Privileging, retarded Green's func tions, 113– 20 Probabilities Everett interpretation, 466, 474–77 objec tive-probability role, 479n17 parallel tradition, 478n16 philosophic al aspec ts of probability, 474– 75 and possibility, 475–77 probability simplic iter, 475 and uncertainty, 475–77 quantum mec hanic s, 446 rigid body mechanics, 76 Probability simplic iter, 475 Problem of the Physic al Infinitesimal, 86, 90, 93 Projec tion postulate, quantum mec hanic s, 417 Proper mixtures, quantum mec hanic s, 423– 28 indistinguishability from proper mixtures, 429 Pseudo-Reimannian metric field, Hole Argument (Einstein), 574n100, 578 Punc tiform point of view, rigid body mec hanic s, 79, 80 Purely elastic c ollision, point-mass mec hanic s, 69n26 Puz z le solving, hydrodynamic s, 30n38 PV measures, quantum mec hanic s, 446n34 Quantitative problem, Everett interpretation, 477–79 Quantiz ing, unitary equivalenc e, 492–94 Quantum c hromodynamic s (QCD) electroweak theory, 394, 402–3 symmetry, 302 Quantum elec trodynamic s (QED) effective field theory (EFT), 411 unific ation in physic s, 396– 97 Quantum fields, eliminativism, 377–78 Quantum field theory (QFT), 240–43 dark matter and dark energy, 621–22 effec tive field theory (EFT), 240– 43, 251 unification in physics, 382, 408–10, 412–13 uniqueness of universe, 626 unitary equivalenc e, physic al equivalenc e and, 491, 518 Quantum, indistinguishability and, 341–43 “light quanta,” 342 microstate, 342n3 Page 59 of 112

Index and “permutable,” 342n4 Quantum locality, 1 Quantum mec hanic s bit c ommitment problem, 429– 30 Bloc h sphere, 423 Born rule, 8, 417, 419, 431 c onsistenc y, 442 general phenomenology of measurements, 446, 448 measurement, 451–52, 454 proper mixtures, 424 unsharp spin measurements, 445 Broglie-Bohm theory, 431, 440, 482 Brownian partic le, 437 c at example (Sc hrödinger), 435, 440 c lassic al dynamic al behavior, 438 c lassic al regime, 8, 416– 59 Born rule, 8, 417, 419, 431 Brownian partic le, 437 c at example (Sc hrödinger), 435, 440 c lassic al dynamic al behavior, 438 c oherent states, 432– 34 Ehrenfest's theorem, 432–34 environment, entanglement with, 434–37 Gaussian bell c urves, 432 Gaussian wave pac kets, 432– 34 Heisenberg's “cut,” 440–42 macromechanics, 430 micromechanics, 430 overview, 8 phenomenologic al arrow of time, 433n22 pilot-wave theories, 440 uniqueness of, 439n28 c oherent states, 432– 34 c ollapse postulate, 8, 417, 418, 424, 454 density operators, 425 Copenhagen interpretation, 420 decoherence, 437–41 c ontinuous models of, 438– 39 Everett interpretation. See Everett interpretation Newtonian behavior, 439 density operators, 420–23 bit c ommitment problem, 429– 30 c ollapse postulate, 425 entangled states, 427 Hilbert spac e, 425– 26 Page 60 of 112

Index Hilbert-space vectors, 420–21 improper mixtures, 427 no-go theorem for safe bit c ommitment protoc ols, 430 no-hidden variables theorem, 422 nontrivial spin properties, 428 normaliz ation, 422n10 no-signaling theorem, 425–26 proper mixtures, 423–28 reduced states, 424 simplex, 423–24 spin-1/2 systems, 422–23, 425, 443 Dirac-von Neumann interpretation, 419 disc retiz ed position measurements, 443 Ehrenfest's theorem, 432–34 entangled states, 427 environment, entanglement with, 434–37 Everett interpretation. See Everett interpretation Gaussian bell c urves, 432 Gaussian wave pac kets, 432– 34 generally, 1, 45 Gleason's theorem, 449 Heisenberg's “cut,” 440 Copenhagen interpretation, 441n30 Hilbert space, 425–26, 450n38 environment, 434 ideal spin measurements, 443 Hilbert-space vectors, 419n7, 420–21 ideal spin measurements, 443–44 improper mixtures, 427 indistinguishability from proper mixtures, 429 linear operators, 417n4 macromechanics, 430 Many Worlds Theory. See Everett interpretation Maxwell equations, 446 measurement, 8, 416–59 Born rule, 451–52, 454 Broglie-Bohm theory, 482 c ollapse postulate, 417, 418, 454 disc retiz ed position measurements, 443 eigenvectors, 418 Everett interpretation. See Everett interpretation Gleason's theorem, 449 Hilbert space, 417, 450n38 ideal spin measurements, 443–44 linear operators, 417n4 Page 61 of 112

Index Maxwell equations, 446 Naimark dilation, 448 phenomenology of, 417–19, 446–49 POV measure, 8, 448–49, 454 probabilities, 446 problem, 451–55 projection postulate, 417 PV measures, 446n34 Sc hrödinger equation, 420, 451– 52 self-adjoint operators, 417–18 statistic al algorithm of quantum mec hanic s, 417 Stern-Gerlach magnetic field, 418, 419n6, 442–44, 446, 453 subspace, 417–18 transformation, 446 “unsharp” spin measurements, 444–446 up and down spin states, 418–19 micromechanics, 430 minimal interpretation, 419–20 Naimark dilation, 448 no-go theorem for safe bit c ommitment protoc ols, 430 no-hidden variables theorem, 422 nontrivial spin properties, 428 normaliz ation, 422n10 no-signaling theorem, 425 phenomenologic al arrow of time, 433n22 pilot-wave theories, 440, 482 Poincaré sphere, 423 POV measure, 8, 448–49, 454 probabilities, 446 projection postulate, 417 proper mixtures, 423–28 indistinguishability from proper mixtures, 429 PV measures, 446n34 radioac tive dec ay, 437 reduced states, 420–30 resolution of the identity, 446 Sc hrödinger equation, 420, 451– 52 ideal spin measurements, 444 Schrödinger evolution, 418 self-adjoint operators, 417–18 simplex, 423–24 spin-1/2 systems, 422–23, 425, 443 standard interpretation, 419–20 statistic al algorithm of quantum mec hanic s, 417 Stern-Gerlach magnetic field, 418, 419n6, 442–44, 446, 453 Page 62 of 112

Index subspace, 417–18 transformation, 446 uniqueness of, 439n28 “unsharp” spin measurements, 444–446 up and down spin states, 418–19 Quantum partic les, indistinguishability, 371– 72 Quantum statistic al mec hanic s (QSM), 491 Quantum statistic s explanation of Hilbert spac e, 362, 363 phase spac e dimension, 361– 62 subspace dimension, 362–63 volume measures, 362–63 weighting, 361, 362 indistinguishability and, 360–64 Quasi-autonomous domains, effec tive field theory (EFT), 240– 41, 243 and emergenc e, 244 Wilsonian approac h, 241 Quasi partic les, Everett interpretation, 473– 74 Quine, W. van, 340–41 Radiation theory, 123–29 boundary conditions, 123–24 description, 109 finiteness c ondition, 124 Fourier transformation tec hnique, 126– 27 Helmholtz equation, 123, 125–27, 129–30 inc oming waves, 127 Laplac e transform tec hnique, 125 outgoing waves, 127 Princ iple of Limiting Amplitude, 128 reasons for, 123–29 reduc ed wave equation, 125. See also Helmholtz equation Sommerfeld radiation c ondition. See Sommerfeld radiation c ondition time dependent boundary condition, 124 time-harmonic waves, 127–29 waveguide, 124 Radioac tive dec ay, 437 Radioac tive sc attering, 52 Ramsey, Jeffrey, 13, 30 Random variables, the tyranny of sc ales, 278 Rankine, William John Mac quorn, 24– 25 “Rari-c onstanc y” theorists, 270 Raychaudhuri equation, 611 Rayleigh, Lord boundary layers, 23–24 Page 63 of 112

Index instabilities, 20 surfac e waves, 17 Reasonable properties, global spac etime struc ture. See Global spac etime struc ture “Re-combination,” Standard Model, 615 Redirec tion of thrust, rigid body mec hanic s, 71 Reduc ed phase spac e, 361– 62, 362 Reduc ed states, quantum mec hanic s, 420– 30 Reduc ed wave equation, radiation theory, 125. See also Helmholtz equation Reduc tion phase transitions, 198–201 the tyranny of sc ales, 260 Reductionism effective field theory (EFT), 411n25 phase transitions, 198 Reduc tive unity, unific ation in physic s, 385– 93 Regularity approach, 564–69 Reic henbac h, H., 358– 59, 637 Reimannian 3-metrics, 561–62 Relationalism Barbour's Mac hian relationalism, 557– 64 dynamic al approac h to relativity, 569– 74 enric hed relationalism, 545– 57 c lassic al mec hanic s, 545– 53 relativity, 553–57 have-it-all relationalism Best Systems prescription, 566 dynamic al approac h to relativity, 569– 74 dynamical laws, 565 Galilean spac etime, 570 general relativity (GR), 568–69 homogeneous matter, 573–74 Humean approac h, 566 Leibniz ian relationalism, 567–68 Lorentz c ontrac t, 570n92, 572 Mill-Ramsey-Lewis's Best Systems prescription, 566 Minkowski geometries, 569n91, 570–71 regularity approac h, 564– 69 rod, c onstituents of, 570 rotation disks argument, 566 Mac hian relationalism, 557– 64 best matching, 559, 559–60 BSW ac tion, 562– 64 diffeomorphism group (DPM), 557–58, 561 Euc lidean c oordinate systems, 557 general relativity (GR), 563–64 Page 64 of 112

Index group orbits, 559 Jac obi's princ iple, 558– 59 kinematic ally possible models (KPMs), 558 and Leibniz ian relationalism, 558 Reimannian 3-metrics, 561–62 shape spac e, 558 similarity groups, 559 superspace, 562 z ero field strength, 557n75 regularity approac h, 564– 69 Relationalist approac hes to spac etime. See Spac etime, substantivalist and relationalist approaches to Relationalist, reasons for being, 539–40 Relative partic le c onfigurations spacetime, substantivalist and relationalist approaches to, 527 Relativistic c osmology, 625– 26 Relativistic quantum field theories (RQFTs) emergence, EFTs, 250–51 overview, 224–25 Relativistic spac etime, 587– 93 Cartesian coordinates, 536 c ausal c urve, 591 c ausal future, 590– 91 c hronology violating region of spac etime, 591 c loc ks, 536 c losed c ausal c urve, 591 c onformal fac tor, 592 c ylindric al Minkowski spac etime, 591 dependence, 590–93 Einstein Field Equations (EFEs), 537 Euc lidean spac e, 536 future Cauc hy horiz on, 592 future-directed curve, 589 future domain of dependenc e, 592 future endpoint, 592 influence, 590–93 isometric spac etimes, 589– 90 loc ally isometric spac etime, 590 Lorentz -invariant form of Maxwell's equation, 538 Lorentz transformations, 536 manifold M, 588–90 maximal spac etimes, 590 Maxwell's equation, 538 metric, 588–90 Minkowski metric struc ture, 536 Page 65 of 112

Index Minkowski spac etime, 538, 591, 591– 93, 592 Möbius strip, 589 null vec tors, 589, 589 past Cauc hy horiz on, 592 past-directed curve, 589 past domain of dependence, 592 past endpoint, 592 spacelike surface, 592 spac elike vec tors, 589, 589 strong equivalenc e princ iple, 538 temporarily orientable spacetime, 589 timelike future, 590–91 timelike vec tors, 589, 589 “twin paradox” sc enario, 536 Relativity, enric hed relationalism, 553– 57 four-forc e, 554 gravitational wave, 556 kinematic ally possible models (KPMs), 555 Minkowski distanc es, 553– 56 Relevant sc alings matter, infinities and renormaliz ation, 177–78 Renormaliz ation c alc ulations, mean field theory, 159–60 Renormaliz ation group (RG) e-expansion, 176–77 effective field theory (EFT), 232–35 nonrenormaliz able EFTs, 233 elementary particles, 176 fixed-points, 177 Fourier space, 172–74 Landau-Ginz burg-Wilson free energy, 173–74 matter, infinities and renormaliz ation, 181–82 overview, 5–6 phase transitions, 195–97 c oarse-graining, 197 and Hamiltonian func tion, 196– 97 phase transitions, infinite idealiz ation, 216–23 c ritic al behavior of partic ular systems, 218 c rossover theory, 220– 21 irreduc ibility, 221 universality, 217–18 physical space, 172–74 running c oupling c onstants, 175– 76 the tyranny of scales, 264–66, 269, 275, 280 unific ation in physic s, 382, 385, 407, 410 weak c oupling fixed points, 177 Page 66 of 112

Index Wilson, Kenneth on, 172–77 c alc ulational method, 174– 75 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 weak c oupling fixed points, 177 Renormaliz ation sc hemes, effec tive field theory (EFT), 235–39 c ontinuum EFTs, 237– 39 Green's function, 235 Lagrangian density, 235 mass-dependent schemes, 236–37 mass-independent schemes, 237–39 Wilsonian approach, 236–37 Renormaliz ation, unific ation in physic s, 406–13 Representative c enter, point-mass mec hanic s, 59 Representative points, rigid body mec hanic s, 81, 93 Representative volume element (REV), 264, 267–68, 280 Resolution of the identity, quantum mec hanic s, 446 Response planes, c ontinuum mec hanic s, 95 Retarded Green's functions Abraham-Lorentz equation, 119 bounded spatial domain, wave equation in, 121–23 Cauc hy problem, 120 c ausal direc tionality, 116 “c ausality” as initial value problem, 116– 20 c ontours for damped osc illator, 115 damping, added, 113–16, 135–36 Dirac delta func tion, 113, 119 “final c ondition,” 119 “final value problem,” 119 Fourier transformations, 113, 114 initial value problem, 116–20 Lebesgue measure z ero, 118 overview, 109 physic al motivation, 113 privileging, 113–20 Residue Theorem, 114 spatial propagation, 120–21 undamped harmonic osc illator, 110 wave equation, 120–23 Reynolds, Osborne Page 67 of 112

Index boundary layers, 26 instabilities, 20 turbulenc e, 21, 22 RG. See Renormaliz ation group (RG) Ric c i tensor, spac etime properties, 594 Riemann-Hugoniot approac h to shoc k waves, 53n12 Riemann tensor, 539n34 Rigid body mec hanic s, 44, 67, 70– 83 ac tion-at-a-distanc e forc es, 73 arc length along the slot, 71– 72, 72 bead sliding on rigid wire, 71 bulls-eyes, 75, 75–76 c lassic al distillations of quantum proc esses, 74– 75 c onstraint relationships, 70 c ontac t forc es, 73 d'Alembert's principle, 79 dead load, 73 dimensional inharmonious quantities, 76 dynamic loading, 73 Eulerian c uts, 73, 74 forc ed c losed, 70n27 forces, 73–74 free body diagrams, 73, 74 generaliz ed c oordinates, 71 Greenwood's proofs, 80–82 “higher or lower” pairs, 70n28 independently variable, 72 kinematic s of mec hanisms, 79 Lagrange's princ iple, 78– 80, 82 meaningfully c ombined, 77 mobility spac e, 79 “pinned constraint,” 71 point-mass perspective, 71 pressures, 74 princ iple of virtual work, 79 probability differences, 76 punc tiform point of view, 79, 80 redirec tion of thrust, 71 representative points, 81, 93 sewing mac hine mec hanism, 71 static load, 73 stress, 74 theory of measure, 76 torque r, 77, 78 trac tion forc es, 73 Page 68 of 112

Index turning moment, 77, 78 virtual displac ement, 80, 81 virtual variations, 79n33 virtual-work reasoning, 72, 72 Rigid c rystalline forms, axiomatic presentation, 51 Robin and Marian analogy, 75, 75–76 Rod, c onstituents of, 570 Rolling on a rigid trac k, axiomatic presentation, 54– 55 Rosenfeld, L., 356 Rotating rigid objec ts, point-mass mec hanic s, 58 Rotation c urves, dark matter and dark energy, 620n30 Rotation disks argument, have-it-all relationalism, 566 Rotations axiomatic presentation, 48 symmetry, 288 RQFTs. See Relativistic quantum field theories (RQFTs) Running c oupling c onstants, renormaliz ation group theory, 175–76 Russell, Bertrand on causation, 4, 107–8, 137–38 dispersion relations, 137 indistinguishability, 369 ontology, 372 Our Knowledge of the “External World, 96n43 Russell, Sc ott, 17 Saint-Venant, Barré, Adhémar de, 22, 23 Saunders, Simon, 478, 480 Sc alar field theory, 227n3 Sc ale siz es, relationships between, 51–52 Sc ale transformation, mean field theory, 159– 60 Sc aling, 170– 72 block transforms, 170–72 mean field theory, 161 Schrödinger equation c lassic al mec hanic s, 46 electroweak theory, 395 ordinary differential equations (ODEs), 46 quantum mec hanic s, 438– 39 ideal spin measurements, 444 measurement, 420, 451–52 unitary equivalenc e, physic al equivalenc e and, 494 Sc hrödinger, Erwin, 46 electroweak theory, 395 indistinguishability N! puz z le, 349–50 quantum, indistinguishability and, 343 Page 69 of 112

Index as uniform symmetry, 354 and ordinary differential equations (ODEs), 46 quantum mec hanic s c at example, 435, 440 c lassic al regime, 430– 31 failure of, possible, 437 measurement, 420 wave function, 432n19 wave functions, 433–34 unitary equivalenc e, 494 Schrödinger evolution, 418 Sc hweber, S., 243, 244, 249– 41, 407, 411 Sc hwinger, J., 397, 491, 492 Segal, Irving E., 516–17 Self-adjoint operators, quantum mec hanic s, 417– 18 Semantic s, Everett interpretation, 476 Sewing mac hine mec hanism, 71 Shape spac e, Mac hian relationalism, 558 Shearing pattern, 85 Ships, boundary layers, 24 Shoc k waves, Riemann-Hugoniot approac h to, 53n12 Short-distance expansion, 182 Short-range forc es, 97, 97, 98 Similarity groups, Mac hian relationalism, 559 Simplex, quantum mec hanic s, 423– 24 Singularities global spac etime struc ture, 598– 601 phase transition, Ising model, 155 spacetime properties, 596 Siz e of group, symmetry and equivalenc e, 322n10 Skin resistanc e, 24 Sklarations, 548–49 Sklar, Lawrenc e, 548 Smolin, Lee, 625–26 Smoluc howski, Marian, 151 Smooth phase transitions, 206 Snowflake, 143, 143–44 symmetry, 288 Sommerfeld radiation c ondition applic ation of, 123 description, 109 time harmonic waves, 127, 129 Sophistic ated substantivalism, Hole Argument (Einstein), 575 Spac elike geodesic inc ompleteness, spac etime properties, 598 Spac elike surfac e, relativistic spac etime, 592 Page 70 of 112

Index Spac elike vec tors, relativistic spac etime, 589, 589 Spac etime geometry, global struc ture, 628 Spac etime properties, global spac etime struc ture, 593– 98 Cauc hy surfac e, 597 c ausal c ontinuity c ondition, 597 c ausal simplic ity c ondition, 597 c ausal struc ture, 596 c hronology c ondition, 596 c onstraint solutions, 595 c onvex normal, 593 distinguishability c onditions, 596 dominant energy c onditions, 595 Einstein's equation, 595 Einstein tensor, 594 energy-momentum tensor, 594 future distinguishability c ondition, 596 future incomplete, 598 global hyporbolic ity, 597 global properties, 596–98 loc al properties, 593– 95 Minkowski spac etime, 597, 597 null geodesic inc ompleteness, 598 parallelly transported vec tor, 593 past distinguishability, 596 past inc omplete, 598 Ric c i tensor, 594 singularities, 596 spacelike geodesic incompleteness, 598 stable c ausality c ondition, 596– 97 strong c ausality c ondition, 596 strong energy c onditions, 595 timelike geodesic ally inc omplete, 598 Spac etime, substantivalist and relationalist approac hes to, 9– 10, 522– 86 Cartesian coordinates, 536 Cartesian motion, 524–26 c irc ular motion, 525 c loc ks, 536 De Sitter universe, 541n37 diffeomorphism group (DPM), 533–34, 547, 557–58, 561 dynamic al symmetry group, 529 Einstein Field Equations (EFEs), 537, 543 Euc lidean c oordinate system, 529 Euc lidean c oordinate systems, 528 Euc lidean spac e, 536 Galilean c ovarianc e, 523 Page 71 of 112

Index Galilean invarianc e, 527– 31 dynamic al symmetries, spac etime and, 527– 29 dynamic al symmetry group, 529 Euc lidean c oordinate system, 529 Euc lidean c oordinate systems, 528 Galilei group, 529 kinematic ally privileged systems, 527 kinematic shift argument, 529–30 Leibniz group of transformations, 528 Leibniz ian relationalism, 527–28 Leibniz ian relationalist, 530 Newton group, 528 Princ iple of Suffic ient Reason (PSR), 530n14 relative partic le c onfigurations, 527 spac etime symmetry groups, 528 Galilean spac etime, 531– 33 Galilei group, 529, 533 and general relativity, 537–39, 541 gravitational field, 539n34 Hole Argument (Einstein), 300n17, 523, 574–79 and dynamic al symmetries, 576 Euc lidean symmetries, 577 and Galilean spac etime, 576n104 general relativity (GR), 573–75, 578 hole diffeomorphism, 575 individualistic fac ts, 577n108 and kinematic shift argument, 576 Leibniz and Clarke c orrespondenc e, 577n106 Mac hian 3-spac e approac h, 578 and Maxwell group, 576n104, 578 Newton-Cartan theory, 576–77 pseudo-Reimannian metric field, 574n100, 578 sophisticated substantivalism, 575 struc tural realist interpretation of spac etime, 577 inertia, spac etime explanation of, 541– 44 instantaneous relative distanc es, 540n35 kinematic ally possible models (KPMs), 531– 33 kinematic ally privileged systems, 527 kinematic shift argument, 523, 529–30, 544, 546–47 Leibniz group of transformations, 333, 528 Leibniz ian relationalism, 527–28, 544–45, 548, 567–68, 572 Leibniz ian relationalist, 530 Lorentz -invariant form of Maxwell's equation, 538 Lorentz transformations, 536 Maxwell's equation, 538 Page 72 of 112

Index Minkowski metric struc ture, 536 Minkowski's spac e time, 538 motion as c hange of plac e, 524 neo-Newtonian spacetime, 531–33 Newton group, 528 Newton's bucket, 523–27 Oc kham's raz or, 523, 539 Princ iple of Suffic ient Reason (PSR), 530n14 rationality, failure of, 539–40 relationalism, varieties of, 544–74 Barbour's Mac hian relationalism, 557– 64 enric hed relationalism, 545– 57 have-it-all relationalism, 564–74 Leibniz ian relationalism, 527–28, 544–45, 548 relationalist, reasons for being, 539–44 inertia, spac etime explanation of, 541– 44 rationality, failure of, 539–40 relative partic le c onfigurations, 527 relativistic spac etimes, 536– 39 Cartesian c oordinates, 536 c loc ks, 536 Einstein Field Equations (EFEs), 537 Euclidean space, 536 Lorentz -invariant form of Maxwell's equation, 538 Lorentz transformations, 536 Maxwell's equation, 538 Minkowski metric struc ture, 536 Minkowski's space time, 538 strong equivalenc e princ iple, 538 “twin paradox” sc enario, 536 Riemann tensor, 539n34 spac etime substantivalism diffeomorphism group (DPM), 533–34 Galilean spac etime, 531– 33 kinematic ally possible models (KPMs), 531– 32, 558 neo-Newtonian spacetime, 531–33 symmetries, 533–36 transtemporal struc ture, 531 spac etime symmetry groups, 528 strong equivalenc e princ iple, 538 symmetries coordinate independent transformations, 533 Galilei group, 534 generally c ovariant equations, 533 Leibniz group of transformations, 533 Page 73 of 112

Index Leibniz relationalist, 535 trajectories of force-free bodies, spacetime as, 542n41 transtemporal struc ture, 531 “twin paradox” sc enario, 536 Spacetime symmetries, 292, 321–22 and physical equivalence, 331–32n49, 332 struc ture of, 319 Spacetime symmetry groups, 528 Spatial propagation advanc ed Green's func tions, 120– 21 retarded Green's functions, 120–21 Spatial struc tures, mean field theory, 167 Spec ial forc e laws c ontinuum mec hanic s, 99 point-mass mec hanic s, 63 Spin indistinguishability, 375 and ordinary differential equations (ODEs), 45n5 phase transitions, 193–94 “unsharp” spin measurements, 444–446 Spin-1/2 systems, 422–23, 425, 443 Spin-3/2 baryon decuplet, 307–8, 309 Splash, 143 SSB insight, 304 Stable c ausality c ondition, 596– 97 Stage-setting, symmetry and equivalenc e, 320– 21 “Standard candle” c osmology, philosophy of, 609n1 dark matter and dark energy, 618 Standard Model, 10–11 ACDM model, 617 barriers, 615 Big Bang, 609, 615 black-body spectrum of radiation, 615 Boltz mann equation, 614n16 c old dark matter, 617 c osmic bac kground radiation (CBR), 614– 16 c osmologic al princ iple, 610 early universe c osmology, 633– 36 effective field theory (EFT), 228 Einstein Field Equations (EFEs), 610–11, 613 electroweak theory, 394, 402–6 Everett interpretation, 471 expanding universe models, 609–14 freez e out of partic les, 614 Page 74 of 112

Index Friedman-Lemaitre-Robertson-Walker (FLRW) models, 610, 612–13, 614n16, 617 galaxies and c lusters of galaxies, length sc ale, 613 global isotropy, reduc tion of, 611 Mac h's princ iple, 610 overview of, 609–17 parameters of, determining, 632 Raychaudhuri equation, 611 “re-combination,” 615 structure formation, 616–17 thermal history, 614–16 unific ation in physic s, 381, 383 Static load, rigid body mec hanic s, 73 Statistic al algorithm of quantum mec hanic s, measurement, 417 Statistical equilibrium, 142 Statistic al mec hanic al treatment, phase transitions, 193– 95 and Hamiltonian func tion, 193 spin, 193–94 Yang-Lee theorem, 195 Statistic al mec hanic s, 142, 146 effective field theory (EFT), 249–50 indistinguishability, 346–47 c oarse-graining, 346 entropy, 346–47 fine-graining, 346 Statistic al physic s, theory of, 141 Statistical Thermodynamics (Sc hrödinger), 349– 50 Steel ball pendulums, point-mass mec hanic s, 64– 65, 64– 66 Steel beams, the tyranny of sc ales, 258– 59, 264, 278, 279 Steel, Gaussian and, 279 Stein, H., 524, 526n5 Stern-Gerlac h magnetic field, quantum mec hanic s, 418, 419n6, 442– 44, 446, 453 Stern, Otto, 350 Stieltjes-Lesbeque integration, axiomatic presentation, 53n12 Stoker, J. J., 127–29 Stokes, George Gabriel boundary layers, 23 fluid veloc ity, 14 Navier-Stokes equation boundary layers, 25–27 instabilities, 19–20 low-density gas spec ializ ation, 34–35n44 modules, 34–35 point-mass mec hanic s, 67 turbulence, 22 point-mass mec hanic s, 67 Page 75 of 112

Index surfac e waves, 17 turbulence, 22 the tyranny of sc ales, 271 vortex motion, 17 Stone-von Neumann theorem. See also “Uniqueness” results unitary equivalenc e, physic al equivalenc e and, 490– 91, 501, 505– 6 Strain tensors, 85, 95, 95 Stress-energy tensors c ontinuum mec hanic s, 84 dark matter and dark energy, 621n35 Stress, rigid body mec hanic s, 74 Stress tensors, 85, 95, 95, 96 String theory electroweak theory, 405 and multiverse, 644 unific ation in physic s, 383n3, 406 Strong c ausality c ondition, spac etime properties, 596 Strong energy conditions, spacetime properties, 595 Strong equivalenc e princ iple, relativistic spac etimes, 538 Struc tural realist interpretation of spac etime, 577 Struc ture formation, Standard Model, 616–17 role of, 470–74 Strutt, John William. See Rayleigh, Lord SU(2) and SU(3) color groups, 394, 396–97 Subspace dimension, indistinguishability, 362–63 quantum mec hanic s, measurement, 417– 18 Substantivalist approac hes to spac etime. See Spac etime, substantivalist and relationalist approaches to Substantivalist-relationalist debate. See Spac etime, substantivalist and relationalist approac hes to Sub-universe, 182 Sun's gravitational field, 626–27 Sunyaey-Zel'dovich effect, 632 Superc onduc tivity, unific ation in physic s, 384 Superfluid, EFT of, 245–46, 246 Superfluidity, unific ation in physic s, 384 “Superposition state,” Everett interpretation, 462 Superspac e, Mac hian relationalism, 562 Supersymmetry (SUSY), 383, 405 Surfac e forc es, c ontinuum mec hanic s, 87 Surfac e waves, hydrodynamic s, 16– 17 Sykes, Martin, 166 Symbolic universe, 31 Page 76 of 112