# Philosophy of physics

What are good theories of the world?

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## Theories of matter

### Ancient atomism

• Empedocles (c. 494-434 BCE)
• theory of the four elements
• Leucippus (fl. 5th century BCE)
• Democritus (c. 460-370 BCE)
• Epicurus (341-270 BCE)
• Lucretius (c. 99-55 BCE)
• De Rerum Natura translated by Esolen1

Discussion:

### Contemporary views of matter

• Quantum field theory
• Statistical mechanics and condensed matter physics
• TODO: Brief nod to upcoming sections.

## Classical physics

### Mechanics

History:

Lagrangian mechanics:

• TODO: explain
• Complaint about explanations of the Lagrangian:
Howe, A.R. (2020). Why does the Lagrangian equal T-V?
• Relationship to the path-intergral formulation of quantum mechanics.

Pedagogy:

Dimensional analysis:

History:

Pedagogy:

• TODO

### Special relativity

History:

Stein:

And this is the crucial difference, as I see it, between Poincaré’s relation to the special theory of relativity and Einstein’s. Both of them discovered this theory—and did so independently. So far as its mathematical structure is concerned, Poincaré’s grasp of the theory was in some important respects superior to Einstein’s. But Einstein “took the theory seriously” in the sense that he looked to it for NEW INFORMATION about the physical world—that is, in Poincaré’s language, he regarded it as “fertile”: as a source of new “real generalizations”—of empirically testable consequences. And in doing so, Einstein attributed physical significance to the basic notions of the theory itself in a way that Poincaré did not.17

Pedagogy:

• Maudlin18

## Statistical physics

### Introduction

TODO:

• The goal of statistical mechanics.
• How statistical mechanics can be seen as pure mathematics.
• Statistical mechanics and thermodynamics
• Entropy

### Thermodynamics

• Denker, J. (2021). Modern Thermodynamics.
• The 2nd Law of Thermodynamics said simply: Things tend to happen in ways for which there are many ways to happen like that.

### Canonical ensemble

• Canonical ensemble

### Phase translations

• Phase transitions
• Renormalization
• Universality

## Symmetry-first physics

### Noether’s theorems

• Principle of least action, Lagrangians
• Canonical dynamics
• Noether, E. (1918). Invariante variationsprobleme.21
• Goyal, P. (2020). Derivation of classical mechanics in an energetic framework via conservation and relativity.25

### Gauge principle

• Weyl, H. (1929). Elektron und gravitation.26
• ’t Hooft, G. (1994). Under the Spell of the Gauge Principle.27
• Teller, P. (2000). The gauge argument.28
• ’t Hooft, G. (2007). Lie groups in physics.29
• Afriat, A. (2013). Weyl’s gauge argument.30
• Physics from Symmetry31
• Dewar, N. (2019). Sophistication about symmetries.32

Weyl:

It seems to me that this new principle of gauge invariance, which follows not from speculation but from experiment, compellingly indicates that the electromagnetic field is a necessary accompanying phenomenon, not of gravitation, but of the material wave field represented by $$\psi$$. Since gauge invariance includes an arbitrary function $$\lambda$$ it has the character of “general” relativity and can naturally only be understood in that context.33

### Wigner-Stone theorems

• Wigner, E.P. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra.34
• Simon, B. (1976). Quantum dynamics: from automorphism to Hamiltonian.35
• Summers, S.J. (1999). On the Stone-von Neumann uniqueness theorem and its ramifications.36
• Keller, K.J., Papadopoulos, M.A., & Reyes-Lega, A.F. (2007). On the realization of symmetries in quantum mechanics.37
• Wigner-Stone theorems as cornerstones of QM (Ovrut)38
• Wigner’s classification
• Schweber, S.S. (1961). An Introduction to Relativistic Quantum Field Theory.39
• Schroeren, D. (2021). Symmetry fundamentalism in quantum mechanics.40

## Quantum mechanics

### Introduction

• Hilbert spaces. Wigner’s theorem. The Born rule.
• Wave-particle duality misconceptions. Fields are more fundamental than particles.
• Philosophy of QM traditionally focus on NRQM.41
• The measurement problem. Decoherence. The Born rule again.
• Uncertainty principle
• Decoherence brings quantum logic to classical logic?

Feynman and Hibbs on wave-principle duality:

What is remarkable is that this dual use of wave and particle ideas does not lead to contradictions. This is so only if great care is taken as to what kind of statements one is permitted to make about the experimental situation.42

Feynman and Hibbs on the uncertainty principle:

Any determination of the alternative taken by a process capable of following more than one alternative destroys the interference between the alternatives.43

### Hydrogen atom

• Factorizable in spherical coordinates, leading to solutions as a product of spherical harmonics in ($$\theta$$, $$\phi$$) and Laguerre polynomials in $$r$$.
• Fine structure
• Lamb shift
• Hyperfine structure

### Foundations of QM

• Hilbert spaces:

$\hat{H} \: |n\rangle = E_{n} \: |n\rangle$

• Superposition principle:

$|\psi\rangle = \sum_{n} a_{n} \: |n\rangle$

• Born rule

$P(n) = | \langle n | \psi \rangle |^{2} = |a_{n}|^{2}$

• Wigner’s theorem

The generators of the representation of a transformation in a Hilbert space are the operators representing the classical Noether charges that are conserved under that transformation.

$\hat{U}(x^{\mu}) = e^{ -i \, \hat{P}_\mu \, x^\mu }$

• Somehow, QM is about complex numbers:

### Secondary properties of QM

• Wave function:

$\langle x | n \rangle = \psi_{n}(x)$

• Schrödinger equation:

$i \hbar \: \partial_{t} \: |\psi\rangle = \hat{H} \: |\psi\rangle$

• Heisenberg picture:

$i \hbar \: \partial_{t} \: \hat{U}(t) \: |\psi\rangle = \hat{H} \: \hat{U}(t) \: |\psi\rangle$

• Decoherence

$\mathcal{H} = \mathcal{H}_\mathrm{S} \otimes \mathcal{H}_\mathrm{E}$

$|\psi\rangle \otimes |\alpha\rangle \rightarrow |\psi; \alpha\rangle \otimes |\alpha\rangle$

See Dutailly,50 for example, for a demonstration that the Schrödinger equation is derivable from Wigner’s theorem.

## Quantum field theory

### Fields

#### Introduction

• Richard Feynman (1918-1988)
• Julian Schwinger (1918-1994)
• Shin’ichirō Tomonaga (1906-1979)
• Feynman’s Nobel Lecture on QED63
• Weinberg’s folk theorem: QFT is the right way to combine Lorentz invariance, quantum mechanics, and the cluster decomposition principle.64

Baez, Segal, & Zhou:

Quantum field theory is quintessentially the algebra and analysis of infinite-dimensional dynamical systems, as constrained by quantum phenomenology, causality, and symmetry. Although it has a clear-cut central goal, that of the realistic description of particle production and annihilation in terms of the localized interactions of fields in space-time, it is clear from this description that it is a multifaceted subject.65

#### Pedagogy

• Peskin and Schroeder66
• Zee67
• Schwartz68
• David Tong69
• Zeidler, vol 1,70 2,71 and 372

### Symmetry

#### Introduction

• TODO
• Noether’s theorem, again
• Wigner-Stone theorems, again

#### Coleman-Mandula theorem

• Coleman-Mandula theorem73

#### CPT theorem

• Bell, J.S. (1955). Time reversal in field theory.75
• Streater, R. & Wightman, A. (1964). PCT, spin and statistics, and all that.76
• Greaves, H. & Thomas, T. (2012). The CPT Theorem.77

### Spin

#### Spinors

Michael Atiyah:

No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of -1 took centuries, the same might be true of spinors.81

#### Spin-statistics theorem

• Spin-statistics theorem - Pauli

### Path intergrals

• Feynman
• Feynman and Hibbs (1965)90
• Partition functions and generating functionals
• Show this way of deriving the Feynman rules
• Nguyen91

### Renormalization

• Dyson
• Wilson
• ’t Hooft, G. (1971). Renormalizable Lagrangians for massive Yang-Mills fields.92
• Huggett and Weingard93
• Goldenfeld94
• Butterfield95
• Butterfield96
• ’t Hooft, again97
• ’t Hooft, G. (1999). A confrontation with infinity (Nobel lecture).98
• The “renormalization group” isn’t a group; it’s actually a semigroup. The reason that renormalization produces a semigroup is that a block transformation loses information.99
• Borcherds, R.E. & Barnard, A. (2002). Lectures on quantum field theory.100

### Foundations of QFT

#### Introduction

Baez:

Nobody has found a fully rigorous formulation of QED, nor has anyone proved such a thing cannot be found.108

Baez:

In practice, quantum field theory is marvelously good for calculating answers to many physics questions. The answers involve approximations. These approximations seem to work very well: that is, the answers match experiments. Unfortunately we do not fully understand, in a mathematically rigorous way, what these approximations are supposed to be approximating.109

#### Wave-particle duality

• Einstein, A. (1905). On a heuristic point of view about the creation and conversion of light.110
• Wolchover, N. (2020). What is a particle?

Weinberg on wave-particle duality:

In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles.111

Baez, Segal, & Zhou on wave-particle duality:

The treatment of the dynamics of quantum systems turns out to be naturally undertaken in terms of field rather than particle concepts, by virtue of the local character of relativistic interactions. In mathematical terms, the field is diagonalizcd in the functional integration representation, just as the particle numbers are diagonalized in the tensor product representation.113

#### Haag’s theorem

• Haag’s theorem120
• The interaction picture does not exist in interacting relativistic QFT.
• States in the free theory are unitarily inequivalent to those in interacting relativistic QFT.
• Discussion:
• Malament121
• Teller122
• Earman and Fraser’s analysis123
• Klaczynski’s analysis124
• Resolution:
• Bain125
• Duncan126
• Wallace
• QFT requires an ultraviolet regulator (a cutoff, a lattice), and Haag’s theorem does not apply when the regulator is in place.

#### Quantization

• Canonical quantization
• Path integral quantization
• No “2nd quantization”
• Instead of quantizing classical theories, should we be finding the classical limit of quantum theories?
• Geometric quantization

#### Algebraic vs constructive QFT

• AQFT vs LQFT
• Local Quantum Physics129
• Wallace130
• Fraser131
• Buchholz132

Kastler:

Rudolf [Haag] is not satisfied by a notion of local observables relying plainly on space and time. Instead he wishes to base the theory on concepts related to individual processes. This attitude seems to me to move towards a basic “algebra of procedures,” pointing towards a theory of (non-commutative) space-time. I know that, coming from a very different angle, Alain Connes also believes the ultimate algebra of basic physics to be a discrete algebra of elements standing for experimental procedures—following the idea that the spatial notions man acquires in his cradle are less basic than his procedures at [particle] accelerators.133

## Exotics in quantum field theory

### Higher gauge theory

#### Aharanov-Bohm effect

Wikipedia discussion in the magnetic moment article:

A gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.

In electrodynamics, the group is $$U(1)$$, unit complex numbers under multiplication. For infinitesimal paths, the group element is $$1 + i\,A_\mu\,dx^\mu$$ which implies that for finite paths parametrized by $$s$$, the group element is:

$$\prod _{s}\left(1+i\,e\,A_\mu\,\frac{dx^\mu}{ds}\,ds\right) = \exp\left(i\,e\int A\cdot dx\right) \,.$$

The map from paths to group elements is called the Wilson loop or the holonomy, and for a $$U(1)$$ gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:

$$e\oint_{\partial D}A\cdot dx = e\int_{D}(\nabla \times A)\,dS = e\int_{D}B\,dS \,.$$

So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a small solenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.

#### Fiber bundles

• Fiber bundles in physics - nLab
• Fiber bundles embody two central principles of modern physics:
1. the principle of locality
2. the gauge principle.
• Vector bundle
• Frankel, T. (2004). The Geometry of Physics.138

Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.139

Maudlin on fiber bundles:

If we adopt the metaphysics of the fiber bundle to represent chromodynamics, then we must reject the notion that quark color is a universal, or that there are color tropes which can be duplicates, or that quarks are parts of ‘natural sets’ which include all and only the quarks of the same color, for there is no fact about whether any two quarks are the same color or different. Further, we must reject the notion that there is any metaphysically pure relation of comparison between quarks at different points, since the only comparisons available are necessarily dependent on the existence of a continuous path in space-time connecting the points. So it seems that there are no color properties and no metaphysically pure internal relations between quarks.142

But if one asks whether, in this picture, the electromagnetic field is a substance or an instance of a universal or a trope, or some combination of these, none of the options seems very useful. If the electromagnetic field is a connection on a fiber bundle, then one understands what it is by studying fiber bundles directly, not by trying to translate modern mathematics into archaic philosophical terminology.143

### Non-perturbative features

• Extended objects149
• ’t Hooft150
• Sphalerons
• Instanton
• Shifman151
• Q-balls

### Supersymmetry

• $$\mathbb{Z}/2\mathbb{Z}$$
• The supersymmetry algebra is a graded Lie algebra which closes under a combination of commutation and anti-commutation relations.
• Haag-Łopuszański-Sohnius theorem152
• The unique loop-hole in the Coleman-Mandula theorem
• Deligne’s theorem on tensor categories

Urs Schreiber:

not just that local spacetime supersymmetry is one possibility to have sensible particle content under Wigner classification, but that the class of (algebraic) super-groups precisely exhausts the moduli space of possible consistent local spacetime symmetry groups.153

## Interpretations of quantum mechanics

The withdrawal of philosophy into a “professional” shell of its own has had disastrous consequences. The younger generation of physicists, the Feynmans, the Schwingers, etc., may be very bright; they may be more intelligent than their predecessors, than Bohr, Einstein, Schrödinger, Boltzmann, Mach and so on. But they are uncivilized savages, they lack in philosophical depth—and this is the fault of the very same idea of professionalism which you are now defending.

– from a letter in Appendix B of Feyerabend’s Against Method

• TODO: Maudlin161

### Measurement problem

• Maudlin, T. (1995). Three measurement problems.162
• Schrödinger’s cat
• Penrose: $$U$$ and $$R$$ operators
• Dürr, D. & Lazarovici, D. (2020). Understanding Quantum Mechanics: The World According to Modern Quantum Foundations.163

### Copenhagen “interpretation”

• Einstein, A., Podolsky, B. & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete?165
• Bohm, D. & Aharonov, Y. (1957). Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky.166
• Mermin, N.D. (1985). Is the moon there when nobody looks?167
• Caulton, A. (2014). Physical entanglement in permutation-invariant quantum mechanics.168

### Bell’s theorem

• Bell, J.S. (1964). On the Einstein Podolsky Rosen Paradox.169
• Bell, J.S. (1966). On the problem of hidden variables in quantum mechanics.170
• Kochen, S. & Specker, E.P. (1967). The problem of hidden variables in quantum mechanics.171
• Clauser, J., Horne, M., Shimony, A., & Holt, R. (1969). Proposed experiment to test local hidden-variable theories.172
• Epistemological Letters
• Aspect experiments (1982)
• Gisin’s theorem173
• La Nouvelle Cuisine174
• Maudlin, T. (2014). What Bell did.175
• Ahmed, A., & Caulton, A. (2014). Causal decision theory and EPR correlations.176
• Wigner’s friend

### Everettian interpretation

A theory containing many ad hoc constants and restrictions, or many independent hypotheses, in no way impresses us as much as one which is largely free of arbitrariness.190

• Schrödinger himself suggested in 1952 that the different terms of a superposition evolving under the Schrödinger equation are “not alternatives but all really happen simultaneously” (Wikipedia)
• Hugh Everett, III
• Everett, H. (1956). Theory of the Universal Wave Function. Ph.D. thesis.191
• Everett, H. (1957). “Relative state” formulation of quantum mechanics.192
• Wheeler, J.A. (1957). Assessment of Everett’s “relative state” formulation of quantum theory.193
• Everett’s collected works194
• DeWitt, B.S. (1970). Quantum mechanics and reality.195
• DeWitt, B.S. & Graham, N. (1973). The Many-Worlds Interpretation of Quantum Mechanics.196
• Barrett, J.A. (2011). Everett’s pure wave mechanics and the notion of worlds.197
• Barrett, J.A. (2016). Quantum worlds.198

It is therefore improper to attribute any less validity or “reality” to any element of a superposition than any other element, due to this ever present possibility of obtaining interference effects between the elements. All elements of a superposition must be regarded as simultaneously existing.199

• Wallace, D. (2012). The Emergent Multiverse.200
• Carroll, S.M. (2019). Something Deeply Hidden.202
• Everett’s later influence on the theory of decoherence
• Wilhelm, I. (2022). Centering the Everett interpretation.203

A way out of this dilemma [the measurement problem] within quantum mechanical concepts requires one of two possibilities: a modification of the Schrödinger equation that explicitly describes a collapse (also called “spontaneous localization”), or an Everett type interpretation, in which all measurement outcomes are assumed to exist in one formal superposition, but to be perceived separately as a consequence of their dynamical autonomy resulting from decoherence. While this latter suggestion has been called “extravagant” (as it requires myriads of co-existing quasi-classical “worlds”), it is similar in prin- ciple to the conventional (though nontrivial) assumption, made tacitly in all classical descriptions of observation, that consciousness is localized in certain semi-stable and sufficiently complex subsystems (such as human brains or parts thereof) of a much larger external world. Occam’s razor, often applied to the “other worlds,” is a dangerous instrument: philosophers of the past used it to deny the existence of the interior of stars or of the back side of the moon, for example. So it appears worth mentioning at this point that environmental decoherence, derived by tracing out unobserved variables from a universal wave function, readily describes precisely the apparently observed “quantum jumps” or “collapse events” (as will be discussed in great detail in various parts of this book).204

### Collapse interpretations

• Ghirardi-Rimini-Weber theory (GRW)205
• TODO: find ref that GRW is empirical

## The standard model of particle physics

### Mixing

• Cabibbo angle (1963)220
• CP violation
• CKM matrix
• Kaons
• B-mesons

### Higgs mechanism

In 1964, three groups: Robert Brout and Francois Englert;221 Peter Higgs;222 and Gerald Guralnik, Carl R. Hagen, and Tom Kibble,223 independently demonstrated an exception to Goldstone’s theorem, showing that Goldstone bosons do not occur when a spontaneously broken symmetry is local. Instead, the Goldstone mode provides the third polarization of a massive vector field, resulting in massive gauge bosons. The other mode of the original scalar doublet remains as a massive spin-zero particle, the Higgs boson. This is the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism, or Higgs mechanism. In the Standard Model, the Higgs boson also couples to the fermions, generating their bare masses.

On July 4 of 2012, the ATLAS227 and CMS228 experiments both announced discovering a new particle consistent with the long-sought-after Higgs boson, a key to explaining electroweak symmetry breaking in the Standard Model of particle physics.

### Quantum chromodynamics

• QCD: SU(3)
• SU(3) $$\times$$ SU(2) $$\times$$ U(1)
• Asymptotic freedom

### Three generations of fermions

• Discovery of charm: “November revolution” at SLAC and BNL (1974)
• Discovery of tau at SLAC + LBL (1975)
• Discovery of bottom at Fermilab (1977)
• Discovery of three neutrino generations from the $$Z$$ width at LEP (1989)
• Discovery of top at Fermilab (1995)

More:

## Beyond the standard model

• Beyond the standard model (BSM)

### Neutrino masses

• Why SU(3) $$\times$$ SU(2) $$\times$$ U(1)?
• Strong $$CP$$ problem
• Axions
• Matter-antimatter asymmetry
• 3 generations
• Hierarchy problem(s)
• Dark matter and dark energy

### Quantum gravity

• Start of string phenomenology255
• Maldacena, J.M. (1998). The large $$N$$ limit of superconformal field theories and supergravity.256
• Witten, E. (1998). Anti-de Sitter space and holography.257
• Ney, A. (2021). From quantum entanglement to spatiotemporal distance.258

## Gravity and cosmology

### General relativity

• Recall Special relativity.
• Bernhard Riemann (1826-1866)
• Einstein, A. & Grossmann, M. (1913). Outline of a generalized theory of relativity and of a theory of gravitation.259
• Misner, C.W., Thorne, K.S., & Wheeler, J.A. (1973). Gravitation.260
• Carroll, S.M. (2004). Spacetime and Geometry.261
• Arntzenius, F. (2012). Space, Time, and Stuff.262
• Diffeomorphism invariance, background independence

### Newtonian gravity

• History of Newton and calculus.
• Newtonian gravity is the right law to conserve gravitational force flux in three dimensions.
• Derive Newtonian gravity as the low-velocity, low-curavature limit of GR.

### Big bang model

• Alexander Friedmann solves the Einstein field equations for an expanding universe in 1922.
• Edwin Hubble discovered that our galaxy is one of many in 1923.
• Georges Lemaître independently solved the Einstein field equations in 1927.
• Edwin Hubble observationally confirmed the expansion of the universe in 1929.
• Arno Penzias and Robert Wilson discovered the cosmic background radiation (CMB) in 1964.
• See history reviewed by Frè263
• The First Three Minutes264
• Ryden, B. (2003). Introduction to Cosmology.265
• Big Bang Nucleosynthesis (BBN)
• Bahcall, N.A., Ostriker, J.P., Perlmutter, S., & Steinhardt, P.J. (1999). The cosmic triangle: Revealing the state of the universe.266

### Inflation

• Proposed by Alan Guth in 1979.
• Dark energy
• Vaccuum energy
• Inflaton field and slow-roll inflation
• $$\Lambda$$-CDM Cosmological Standard Model

### Alternative theories of gravity

• Einstein-Cartan theory
• Modified Newtonian dynamics (MOND)
• TODO: Sabine Hossenfelder
• Entropic gravity

## Complexity and emergence

Anderson:

The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe. The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. At each level of complexity entirely new properties appear. Psychology is not applied biology, nor is biology applied chemistry. We can now see that the whole becomes not merely more, but very different from the sum of its parts.273

## Annotated bibliography

### Einstein, A., Podolsky, B. & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete?

• Einstein et al. (1935)

• TODO.

### Anderson, P. (1972). More is different.

• Anderson (1972)

• TODO.

• TODO.

### Joos, E., Zeh, H.D., Kiefer, C., Kupsch, J., Stamatescu, I.O. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory.

• Joos, E. et al. (2003).

• TODO.

• TODO.

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1. Lucretius (1995), p. TODO.↩︎

2. Nail (2018).↩︎

3. Yock (2018).↩︎

4. Einstein (1905b).↩︎

5. Perrin (1913).↩︎

6. Patterson (2007).↩︎

7. Russell (1992).↩︎

8. Feynman (1963).↩︎

9. Holm (2011a) and Holm (2011b).↩︎

10. Buckingham (1914).↩︎

11. Kasprzak, Lysik, & Rybaczuk (1990).↩︎

12. Duff, Okun, & Veneziano (2001).↩︎

13. Janyska, Modugno, & Vitolo (2007).↩︎

14. Zapata-Carratala (2021).↩︎

15. Einstein (1905d).↩︎

16. Einstein (1905a).↩︎

17. Stein (2021), p. 69.↩︎

18. Maudlin (2012), p. TODO.↩︎

19. Caulton (2015).↩︎

20. Caulton & Butterfield (2012).↩︎

21. Noether (1918).↩︎

22. Wigner (1954).↩︎

24. Baez (2018).↩︎

25. Goyal (2020).↩︎

26. Weyl (1929).↩︎

27. ’t Hooft (1994).↩︎

28. Teller (2000).↩︎

29. ’t Hooft (2007).↩︎

30. Afriat (2013).↩︎

31. Schwichtenberg (2015).↩︎

32. Dewar (2019).↩︎

33. Weyl (1929), p. TODO.↩︎

34. Wigner (1959).↩︎

35. Simon (1976).↩︎

36. Summers (1999).↩︎

37. Keller, Papadopoulos, & Reyes-Lega (2007).↩︎

38. Reece (2007), p. X.↩︎

39. Schweber (1961), p. TODO.↩︎

40. Schroeren (2021).↩︎

41. Ney & Albert (2013).↩︎

42. Feynman & Hibbs (1965), p. 6.↩︎

43. Feynman & Hibbs (1965), p. 9.↩︎

44. Kelvin (1901).↩︎

45. Bacciagaluppi & Valentini (2009).↩︎

46. von Neumann (1955).↩︎

47. van Hove (1958).↩︎

48. Jordan, Neumann, & Wigner (1934).↩︎

49. Baez (2011).↩︎

50. Dutailly (2014), p. 11–13.↩︎

51. Zurek (2003).↩︎

52. Joos, E. et al. (2003).↩︎

53. Tegmark (1993).↩︎

54. Schlosshauer (2005).↩︎

55. Drossel (2015), p. 51–2.↩︎

56. Friedrich (2016).↩︎

57. C. Cao, Hu, Li, & Schwarz (2019).↩︎

58. Aaronson (2011).↩︎

59. Coecke & Kissinger (2017).↩︎

60. Preskill (2018).↩︎

61. Arute, F. et al. (2019).↩︎

62. Broughton, M. et al. (2020).↩︎

63. Feynman (1965).↩︎

64. Weinberg (1997b), p. 8.↩︎

65. Baez, Segal, & Zhou (1992), p. 1.↩︎

66. Peskin & Schroeder (1995).↩︎

67. Zee (2003).↩︎

68. Schwartz (2014).↩︎

69. Tong (2006).↩︎

70. Zeidler (2007).↩︎

71. Zeidler (2008).↩︎

72. Zeidler (2011).↩︎

73. Coleman & Mandula (1967).↩︎

74. Wigner (1939) and Bargmann & Wigner (1948).↩︎

75. Bell (1955).↩︎

76. Streater & Wightman (1964).↩︎

77. Greaves & Thomas (2012).↩︎

78. Ohanian (1986).↩︎

79. Peskin (1994).↩︎

80. Sebens (2019).↩︎

81. Dutailly (2014), p. 37.↩︎

82. Kontsevich & Segal (2021).↩︎

83. Dyson (1949).↩︎

84. Dyson (1952).↩︎

85. Lehmann, Symanzik, & Zimmermann (1955).↩︎

86. Weinberg (1964a) and Weinberg (1964b).↩︎

87. Martin (2011).↩︎

88. Reece (2007).↩︎

89. Jaeger (2019).↩︎

90. Feynman & Hibbs (1965).↩︎

91. Nguyen (2016).↩︎

92. ’t Hooft (1971).↩︎

93. Huggett & Weingard (1995).↩︎

94. Goldenfeld (1992).↩︎

95. Butterfield (2014).↩︎

96. Butterfield & Bouatta (2015).↩︎

97. ’t Hooft (1994).↩︎

98. ’t Hooft (1999).↩︎

100. Borcherds & Barnard (2002).↩︎

101. Huggett & Weingard (1995).↩︎

102. Weinberg (1997b).↩︎

103. Bain (2013a) and Bain (2013b).↩︎

104. Preskill (2013).↩︎

105. Williams (2019).↩︎

106. Rosaler (2022).↩︎

107. Baez (2016).↩︎

108. Baez (2016), p. 17.↩︎

109. Baez (2016), p. 18.↩︎

110. Einstein (1905c).↩︎

111. Weinberg (1997b), p. 2.↩︎

112. Weinberg (1997a).↩︎

113. Baez et al. (1992), p. 59.↩︎

114. Fraser (2008).↩︎

115. Pessa (2009).↩︎

116. Myrvold (2015).↩︎

117. Lazarovici (2018).↩︎

118. Baker (2009).↩︎

119. Caulton (2018).↩︎

120. Haag (1955).↩︎

121. Malament (1996).↩︎

122. Teller (1997), p. 115.↩︎

123. Earman & Fraser (2006).↩︎

124. Klaczynski (2016).↩︎

125. Bain (2000).↩︎

126. Duncan (2012).↩︎

129. Haag (1992).↩︎

130. Wallace (2011).↩︎

131. Fraser (2011).↩︎

132. Buchholz (1998).↩︎

133. Kastler (2003), p. 6.↩︎

134. Aharonov & Bohm (1959).↩︎

135. Healey (2007), ch. 2-4.↩︎

136. Batterman (2003).↩︎

137. Maudlin (2018).↩︎

138. Frankel (2004).↩︎

139. nLab authors (2021).↩︎

140. Way (2010).↩︎

141. Vákár (2011).↩︎

142. Maudlin (2007), p. 96.↩︎

143. Maudlin (2007), p. 101.↩︎

144. Baez & Muniain (1994).↩︎

145. Baez & Schreiber (2005).↩︎

146. Baez & Huerta (2011).↩︎

147. Schreiber (2020).↩︎

148. Baez & Stay (2009).↩︎

149. ’t Hooft (1978).↩︎

150. ’t Hooft (1994).↩︎

151. Shifman (2012).↩︎

152. Haag, Łopuszański, & Sohnius (1975).↩︎

153. Schreiber (2016).↩︎

154. Dimopoulos & Georgi (1981).↩︎

155. Murayama (2000).↩︎

156. Freedman, Nieuwenhuizen, & Ferrara (1976).↩︎

157. van Nieuwenhuizen (1981).↩︎

158. Frè (2013), ch. 6.↩︎

159. Martin (2016).↩︎

160. Tong (2022).↩︎

161. Maudlin (2019), p. TODO.↩︎

162. Maudlin (1995).↩︎

163. Dürr & Lazarovici (2020).↩︎

164. Becker (2018).↩︎

165. Einstein, Podolsky, & Rosen (1935).↩︎

166. Bohm & Aharonov (1957).↩︎

167. Mermin (1985).↩︎

168. Caulton (2014).↩︎

169. Bell (1964).↩︎

170. Bell (1966).↩︎

171. Kochen & Specker (1967).↩︎

172. Clauser, Horne, Shimony, & Holt (1969).↩︎

173. Gisin (1991), Gisin & Peres (1992), and Gisin (1999).↩︎

174. Bell (2004), pp. 232–248.↩︎

175. Maudlin (2014).↩︎

176. Ahmed & Caulton (2014).↩︎

177. Deutsch (1985).↩︎

178. Bong, K.W. et al. (2020).↩︎

179. Bohm (1952).↩︎

180. Bohm (1953).↩︎

181. Bell (2004).↩︎

182. Dürr, Goldstein, & Zanghì (1995).↩︎

183. Dürr, Goldstein, Tumulka, & Zanghì (2004).↩︎

184. Dürr, Goldstein, Tumulka, & Zanghì (2005).↩︎

185. Dürr, Goldstein, & Zanghì (2013).↩︎

186. Tumulka (2017).↩︎

187. Das & Dürr (2019).↩︎

188. Stopp, Ortiz-Gutiérrez, Lehec, & Schmidt-Kaler (2021).↩︎

189. Ananthaswamy (2021).↩︎

190. Everett (2012), p. 171.↩︎

191. Everett (1956).↩︎

192. Everett (1957).↩︎

193. Wheeler (1957).↩︎

194. Everett (2012).↩︎

195. DeWitt (1970).↩︎

196. DeWitt & Graham (1973).↩︎

197. Barrett (2011).↩︎

198. Barrett (2016).↩︎

199. Everett (2012), p. 150.↩︎

200. Wallace (2012).↩︎

201. Carroll & Singh (2019).↩︎

202. Carroll (2019).↩︎

203. Wilhelm (2022).↩︎

204. Joos, E. et al. (2003), p. 22.↩︎

205. Ghirardi, Rimini, & Weber (1986).↩︎

206. Caves, Fuchs, & Schack (2001).↩︎

207. Fuchs (2002).↩︎

208. Fuchs (2010).↩︎

209. Fuchs & Schack (2013).↩︎

210. Fuchs, Mermin, & Schack (2014).↩︎

211. Fuchs & Stacey (2016).↩︎

212. Harrigan & Spekkens (2010).↩︎

213. Leifer & Spekkens (2013).↩︎

214. Pusey, Barrett, & Rudolph (2012).↩︎

215. Schlosshauer & Fine (2012).↩︎

216. Nigg, D. et al. (2015).↩︎

217. ’t Hooft (2021).↩︎

218. Proietti, M. et al. (2019).↩︎

219. Zyla, P.A. et al. (Particle Data Group) (2021).↩︎

220. Cabibbo (1963).↩︎

221. Englert & Brout (1964).↩︎

222. Higgs (1964).↩︎

223. Guralnik, Hagen, & Kibble (1964).↩︎

224. Georgi (1999), p. 280.↩︎

225. Lyre (2008).↩︎

226. T. Y. Cao (2016).↩︎

227. ATLAS Collaboration (2012).↩︎

228. CMS Collaboration (2012).↩︎

229. Glashow (1961).↩︎

230. Weinberg (1967).↩︎

231. Salam & Ward (1964b).↩︎

232. Salam & Ward (1964a).↩︎

233. Weinberg (1979).↩︎

234. Rubbia (1984).↩︎

235. Chalmers (2017).↩︎

236. LSND Collaboration (1996).↩︎

237. LSND Collaboration (2001).↩︎

238. MiniBooNE Collaboration (2018).↩︎

239. MicroBooNE Collaboration (2021).↩︎

240. Vitagliano, Tamborra, & Raffelt (2020).↩︎

241. TODO: Pierre Auger Collaboration (2007), Pierre Auger Collaboration (2010), Pierre Auger Collaboration (2020a), and Pierre Auger Collaboration (2020b).↩︎

242. Capdevilla, Curtin, Kahn, & Krnjaic (2021).↩︎

243. CDF Collaboration (2022).↩︎

244. Baez & Huerta (2009a).↩︎

245. Baez & Huerta (2010).↩︎

246. Pati & Salam (1974).↩︎

247. Georgi & Glashow (1974).↩︎

248. Slansky (1981).↩︎

249. Georgi (1999).↩︎

250. Baez & Huerta (2009b).↩︎

251. Lisi (2007).↩︎

252. Martin (2016), p. 66.↩︎

253. Dine & Kusenko (2004).↩︎

254. Baggott (2013).↩︎

255. Candelas, Horowitz, Strominger, & Witten (1985).↩︎

256. Maldacena (1998).↩︎

257. Witten (1998).↩︎

258. Ney (2021).↩︎

259. Einstein & Grossmann (1913).↩︎

260. Misner, Thorne, & Wheeler (1973).↩︎

261. Carroll (2004).↩︎

262. Arntzenius (2012).↩︎

263. Frè (2013), ch. 4.↩︎

264. Weinberg (1977).↩︎

265. Ryden (2003).↩︎

266. Bahcall, Ostriker, Perlmutter, & Steinhardt (1999).↩︎

267. Clowe, D. et al. (2006).↩︎

268. Martens (2022).↩︎

269. Debono & Smoot (2016), figure 4.↩︎

270. Penington (2019).↩︎

271. Bedau (1997).↩︎

272. Lisi (2017).↩︎

273. Anderson (1972), p. 393.↩︎

274. Bokulich (2011).↩︎