Philosophy of physics

What are good theories of the world?

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Contents

  1. Theories of matter
    1. Ancient atomism
    2. Modern atomism
    3. Contemporary views of matter
  2. Classical physics
    1. Mechanics
    2. Electrodynamics
    3. Special relativity
  3. Statistical physics
    1. Introduction
    2. History
    3. Thermodynamics
    4. Canonical ensemble
    5. Phase translations
  4. Symmetry-first physics
    1. Curie’s principle
    2. Relativity
    3. Noether’s theorems
    4. Gauge principle
    5. Wigner-Stone theorems
  5. Quantum mechanics
    1. Introduction
    2. History
    3. Hydrogen atom
    4. Foundations of QM
    5. Secondary properties of QM
    6. Decoherence
    7. Quantum chemistry
    8. Quantum computing
  6. Quantum field theory
    1. Fields
    2. Symmetry
    3. Spin
    4. Scattering
    5. Path intergrals
    6. Renormalization
    7. Effective field theory
    8. Foundations of QFT
  7. Exotics in quantum field theory
    1. Gauge theory
    2. Non-perturbative features
    3. Supersymmetry
  8. Interpretations of quantum mechanics
    1. Measurement problem
    2. Copenhagen “interpretation”
    3. Von Neumann’s no hidden variables “proof”
    4. EPR paradox
    5. Von Neumann-Wigner interpretation
    6. Bell’s theorem
    7. Bohmian mechanics
    8. Everettian interpretation
    9. Collapse interpretations
    10. Epistemic interpretations
    11. PBR theorem
    12. Other interpretations
    13. Bad takes
  9. The standard model of particle physics
    1. History of particle physics
    2. Mixing
    3. Higgs mechanism
    4. A model of leptons
    5. Quantum chromodynamics
    6. Three generations of fermions
    7. Experimental methods
  10. Beyond the standard model
    1. Neutrino masses
    2. Ad hoc structures
    3. Experimental anomalies
    4. Grand unification
    5. Baryogenesis
    6. Future colliders and criticisms
    7. Quantum gravity
  11. Gravity and cosmology
    1. General relativity
    2. Newtonian gravity
    3. Big bang model
    4. Spacetime
    5. Blackholes
    6. Gravitational waves
    7. Dark matter
    8. Inflation
    9. Alternative theories of gravity
  12. Fine-tuning
  13. Complexity and emergence
  14. Bracketing human experience
  15. My thoughts
  16. Annotated bibliography
    1. Einstein, A., Podolsky, B. & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete?
    2. Anderson, P. (1972). More is different.
    3. Redhead, M. (1988). A philosopher looks at quantum field theory.
    4. Joos, E., Zeh, H.D., Kiefer, C., Kupsch, J., Stamatescu, I.O. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory.
    5. Pusey, M.F., Barrett, J., & Rudolph, T. (2012). On the reality of the quantum state.
    6. More articles to do
  17. Links and encyclopedia articles
    1. SEP
    2. IEP
    3. Scholarpedia
    4. Wikipedia
    5. Others
    6. Videos
  18. References

Theories of matter

Ancient atomism

Discussion:

Modern atomism

Discussion:

Contemporary views of matter

See also:

Classical physics

Mechanics

History:

Lagrangian mechanics:

Pedagogy:

Dimensional analysis:

See also:

Electrodynamics

History:

Heaviside:

What is Maxwell’s theory? or, What should we agree to understand by Maxwell’s theory?

The first approximation to the answer is to say, There is Maxwell’s book as he wrote it; there is his text, and there are his equations: together they make his theory. But when we come to examine it closely, we find that this answer is unsatisfactory. To begin with, it is sufficient to refer to papers by physicists, written say during the twelve years following the first publication of Maxwell’s treatise, to see that there may be much difference of opinion as to what his theory is. It may be, and has been, differently interpreted by different men, which is a sign that it is not set forth in a perfectly clear and unmistakeable form. There are many obscurities and some inconsistencies. Speaking for myself, it was only by changing its form of presentation that I was able to see it clearly, and so as to avoid the inconsistencies. Now there is no finality in a growing science. It is, therefore, impossible to adhere strictly to Maxwell’s theory as he gave it to the world, if only on account of its inconvenient form. But it is clearly not admissible to make arbitrary changes in it and still call it his. He might have repudiated them utterly. But if we have good reason to believe that the theory as stated in his treatise does require modification to make it self-consistent, and to believe that he would have admitted the necessity of the change when pointed out to him, then I think the resulting modified theory may well be called Maxwell’s.16

Pedagogy:

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.19

Pedagogy:

See also:

Statistical physics

Introduction

TODO:

History

Thermodynamics

Canonical ensemble

Phase translations

See also:

Symmetry-first physics

Curie’s principle

See also:

Relativity

See also:

Noether’s theorems

Gauge principle

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.44

Wigner-Stone theorems

Ovrut’s version of Wigner’s theorem:

The generators of the representation of a transformation in the Hilbert space are the operators representing the classical Noether’s charges that are conserved under that transformation.47

Discussion:

See also:

Quantum mechanics

Introduction

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.54

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.55

History

Figure 1: 1927 Solvay Conference on Quantum Mechanics (source: Wikimedia).

Hydrogen atom

Foundations of QM

Hilbert spaces

States being represented as vectors in a Hilbert space implies the superposition principle:

\[ |\psi\rangle = \sum_{n} a_{n} \: |n\rangle \]

the definition of a complex inner product:

\[ \langle\psi_1|\psi_2\rangle = \int dx \: \langle\psi_1|x\rangle \, \langle{}x|\psi_2\rangle \]

and a norm:

\[ \langle\psi|\psi\rangle \geq 0 \]

Operators

Observables are represented as self-adjoint operators with the “eigenvector-eigenvalue link.”

\[ \hat{H} \: |n\rangle = E_{n} \: |n\rangle \]

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.

\[ \hat{U}(x^{\mu}) = \exp( -i \, x^\mu \, \hat{P}_\mu ) \]

\[ \hat{U}(\theta^{\mu\nu}) = \exp( \frac{-i}{2} \, \theta^{\mu\nu} \, \hat{M}_{\mu\nu} ) \]

Born rule

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

TODO: Note that Everettian QM would argue the Born rule is secondary and derivable.

Discussion

See also:

Secondary properties of QM

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

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

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

Schrödinger vs Heisenberg pictures is like Heraclitus vs Parmenides.

\[ \mathcal{H} = \mathcal{H}_\mathrm{S} \otimes \mathcal{H}_\mathrm{E} \]

\[ |\psi\rangle \otimes |\alpha\rangle \rightarrow |\psi; \alpha\rangle \otimes |\alpha\rangle \]

See Dutailly,70 for example, for a demonstration that the Schrödinger equation is derivable from Wigner’s theorem.

Decoherence

See also:

Quantum chemistry

Quantum computing

Quantum field theory

Fields

Introduction

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.97

Pedagogy

Symmetry

Introduction

See also:

Coleman-Mandula theorem

See also:

Wigner’s classification

CPT theorem

Spin

Introduction

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.118

Spin-statistics theorem

Scattering

Path intergrals

Renormalization

Effective field theory

J.D. Fraser:

in demonstrating that these large scale properties of a QFT model are insensitive to what is going on at very high energies, the renormalization group is also telling us that these features are largely independent of the details of unknown physics at currently inaccessible energy scales. We thus have reason to be confident that these features of current QFTs will be retained through future theory change, in one way or another, whatever physics beyond the standard model has in store for us.159

Foundations of QFT

Introduction

Baez:

Nobody has found a fully rigorous formulation of QED, nor has anyone proved such a thing cannot be found.162

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.163

Wave-particle duality

The so-called wave-particle paradox is similar to the paradox that arises from attributing life or death to the quantum cat. A system is often said to be a particle if a position eigenvalue is observed and a wave if a momentum eigenvalue is observed, hence it is said to present a paradox. The paradox is the result of fallacious attribution. Both eigenvalues are classical quantities, and neither can be attributed to the system as its property. The property of the quantum system is its wavefunction in the position representation and momentum amplitude in the momentum representation. Either one of them offers a complete description, and the descriptions can be shown to be equivalent by rigorous transformations. Paradox arises only when interpreters insist on considering exclusively classical quantities that we can observe. The insistency closes the mind to quantum properties.166

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.167

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 diagonalized in the functional integration representation, just as the particle numbers are diagonalized in the tensor product representation.169

Quantization

Newton-Wigner localization

TODO: Clean up

Single-particle states:

\[ | \vec{x}, t \rangle = \hat{\psi}(\vec{x}, t) | 0 \rangle \]

and single-particle wavefunction derived from QFT:180

\[ \psi(x) = \langle x | \Psi \rangle = \langle 0 | \hat{\psi}(x) | \Psi \rangle \]

Haag’s theorem

Bootstrap theory

Causal perturbation theory

nLab (Schreiber):

Causal perturbation theory may be regarded as providing a well-defined consistent generalization from quantum mechanics to quantum field theory on Lorentzian spacetimes of the construction of the \(S\)-matrix via the Dyson formula (“time-ordered products”) in the interaction picture.196

First rigorous perturbation theory in QFT according to Schreiber:

Algebraic vs constructive QFT

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.202

Exotics in quantum field theory

Gauge theory

Aharonov-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

A fiber bundle is locally a product space, \(X \times G\), of a base space, \(X\), and a fiber space, \(G\). In general, fiber bundles may have non-trivial global structure (e.g. a Mobius strip), but commonly in the ordinary use in physics, trivial bundles are used where the fiber bundle is just \(X \times G\) globally. The base space is spacetime, and the fiber is the space of possible gauge transformations.

Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.210

Figure 2: An illustration of the fiber bundle formulation of interacting quantum fields.

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.218

and

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.219

Higher gauge theory

Topological QFT

See also:

Non-perturbative features

Supersymmetry

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.237

See also:

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

Measurement problem

Copenhagen “interpretation”

Figure 3: Interpretations of quantum mechanics (philosophy-in-figures.tumblr.com).

Criticisms:

Von Neumann’s no hidden variables “proof”

EPR paradox

Von Neumann-Wigner interpretation

Criticisms:

Bell’s theorem

Bohmian mechanics

Discussion:

Attempts at QFT:

Attempts at empirical proposals:

Primitive ontology:

Virtues:

Criticisms:

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.308

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.318

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 principle 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).320

Videos:

Virtues:

Criticisms:

See also:

Collapse interpretations

Virtues:

Criticisms:

Epistemic interpretations

Criticisms:

PBR theorem

Videos:

Other interpretations

From Sabine Hossenfelder, some examples for models that violate measurement independence are here:

Bad takes

Press release for The Nobel Prize in Physics 2022:

This means that quantum mechanics cannot be replaced by a theory that uses hidden variables.

which is wrong! The violation of Bell’s inequality means that QM cannot be explained by fully local hidden variables. Bohmian mechanics exists as a counter example that hidden variables can explain QM, but require a non-local guiding equation.

The standard model of particle physics

History of particle physics

Mixing

Higgs mechanism

In 1964, three groups: Robert Brout and Francois Englert;372 Peter Higgs;373 and Gerald Guralnik, Carl R. Hagen, and Tom Kibble,374 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 ATLAS377 and CMS378 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.

A model of leptons

Quantum chromodynamics

Three generations of fermions

Figure 4: The fields in the standard model of particle physics (source: Symmetry Magazine).
Figure 5: The total action of the physics of the standard model together with general relativity as presented by Sean Carroll on his blog. In this all encompassing equation, fermions are the quanta of the \psi fields and bosons are the quanta of the g, A, and \Phi fields.

More:

Experimental methods

Beyond the standard model

Neutrino masses

Ad hoc structures

See also:

Experimental anomalies

Grand unification

Figure 6: Two-loop renormalization group evolution of the inverse gauge couplings, \alpha^{-1}, in the Standard Model (dashed lines) and the MSSM (solid lines). In the MSSM case, the sparticle masses are treated as a common threshold varied between 750 GeV (blue) and 2.5 TeV (red).
Figure 7: Ryan’s sketch of how a grand unified theory may come together. Noted in green are parts of the theory that are well verified experimentally. Noted in red are parts yet unseen. This was a slide from my thesis defense, partially to help motivate why I would look for a Z^{\prime} particle (source: Ryan’s thesis defense (2013).).

See also:

Baryogenesis

Future colliders and criticisms

Quantum gravity

Gravity and cosmology

General relativity

\[ R_{\mu\nu} - \frac{1}{2} R \: g_{\mu\nu} + \Lambda \: g_{\mu\nu} = \frac{8 \pi G}{c^4} \: T_{\mu\nu} \label{eq:einstein_field_equations} \]

Newtonian gravity

Big bang model

Spacetime

Blackholes

Gravitational waves

Dark matter

Inflation

Guth:

The peculiar properties of the false vacuum stem from its pressure, which is large and negative… Mechanically such a negative pressure corresponds to a suction, which does not sound like something that would drive the Universe into a period of rapid expansion. The mechanical effects of pressure, however, depend on pressure differences, so they are unimportant if the pressure is reasonably uniform. According to general relativity, however, there is a gravitational effect that is very important under these circumstances. Pressures, like energy densities, create gravitational fields, and in particular a positive pressure creates an attractive gravitational field. The negative pressure of the false vacuum, therefore, creates a repulsive gravitational field, which is the driving force behind inflation.441

Figure 8: How the \Lambda-CDM concordance model of cosmology was developed.

Alternative theories of gravity

Fine-tuning

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.449

See also:

Bracketing human experience

Figure 9: Sean Carroll on the entailment of everyday life by physics.

Videos of talks:

See also:

My thoughts

Annotated bibliography

Click to show annotated bibliography

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

  • Einstein et al. (1935)

My thoughts

  • TODO.

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

  • Anderson (1972)

My thoughts

  • TODO.

Redhead, M. (1988). A philosopher looks at quantum field theory.

  • Redhead (1988)

My thoughts

  • 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).

My thoughts

  • TODO.

Pusey, M.F., Barrett, J., & Rudolph, T. (2012). On the reality of the quantum state.

My thoughts

  • TODO.
Click to show links

SEP

IEP

Scholarpedia

Wikipedia

Others

Videos

References

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  1. Lucretius (1995), p. TODO.↩︎

  2. Nail (2018).↩︎

  3. Yock (2018).↩︎

  4. Einstein (1905b).↩︎

  5. Perrin (1913).↩︎

  6. Russell (1992).↩︎

  7. Weyl (2009).↩︎

  8. Patterson (2007).↩︎

  9. Feynman (1963).↩︎

  10. Holm (2011a) and Holm (2011b).↩︎

  11. Buckingham (1914).↩︎

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

  13. Duff, Okun, & Veneziano (2001).↩︎

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

  15. Zapata-Carratala (2021).↩︎

  16. Heaviside (1893), pp. vi–vii.↩︎

  17. Einstein (1905d).↩︎

  18. Einstein (1905a).↩︎

  19. Stein (2021), p. 69.↩︎

  20. Carnap (1966).↩︎

  21. Maudlin (2012), p. TODO.↩︎

  22. J. Wu (2021).↩︎

  23. Caulton (2015).↩︎

  24. Caulton & Butterfield (2012).↩︎

  25. de Queiroz, Lachieze-Rey, & Simon (2014).↩︎

  26. Noether (1918).↩︎

  27. Wigner (1954).↩︎

  28. Brading (2002).↩︎

  29. Baez (2018).↩︎

  30. Goyal (2020).↩︎

  31. Weyl (1918).↩︎

  32. Weyl (1929).↩︎

  33. Pauli (1941).↩︎

  34. Yang & Mills (1954).↩︎

  35. ’t Hooft (1994).↩︎

  36. Yang (1996).↩︎

  37. O’Raifeartaigh (1997).↩︎

  38. Teller (2000).↩︎

  39. ’t Hooft (2007).↩︎

  40. Greaves & Wallace (2011).↩︎

  41. Afriat (2013).↩︎

  42. Schwichtenberg (2015).↩︎

  43. Dewar (2019).↩︎

  44. Weyl (1929), p. TODO.↩︎

  45. Wigner (1959).↩︎

  46. Kadison (1965).↩︎

  47. Reece (2007), p. 27.↩︎

  48. Schweber (1961), p. TODO.↩︎

  49. B. Simon (1976).↩︎

  50. Summers (1999).↩︎

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

  52. Schroeren (2021).↩︎

  53. Ney & Albert (2013).↩︎

  54. Feynman & Hibbs (1965), p. 6.↩︎

  55. Feynman & Hibbs (1965), p. 9.↩︎

  56. Kelvin (1901).↩︎

  57. Bacciagaluppi & Valentini (2009).↩︎

  58. von Neumann (1955).↩︎

  59. van Hove (1958).↩︎

  60. Moretti (2015).↩︎

  61. Wallace (2016).↩︎

  62. Carcassi, Maccone, & Aidala (2020).↩︎

  63. Wallace (2020).↩︎

  64. Rédei (1996).↩︎

  65. de la Madrid (2005).↩︎

  66. Carcassi, Calderon, & Aidala (2023).↩︎

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

  68. Baez (2011).↩︎

  69. Morales & Zilber (2014).↩︎

  70. Dutailly (2014), p. 11–13.↩︎

  71. Schrödinger (1952a).↩︎

  72. Schrödinger (1952b).↩︎

  73. Born (1953).↩︎

  74. Joos & Zeh (1985).↩︎

  75. Bell (2004a).↩︎

  76. Zurek (1991).↩︎

  77. Zurek (2001).↩︎

  78. Zurek (2003).↩︎

  79. Joos, E. et al. (2003).↩︎

  80. Schlosshauer (2005).↩︎

  81. Drossel (2015), p. 51–2.↩︎

  82. Wallace (2018).↩︎

  83. Zurek (2022).↩︎

  84. Nagele, Janssen, & Kleban (2023).↩︎

  85. Cohen (2015).↩︎

  86. Friedrich (2016).↩︎

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

  88. Seifert (2024).↩︎

  89. Coecke & Kissinger (2017).↩︎

  90. Preskill (2018).↩︎

  91. Arute, F. et al. (2019).↩︎

  92. Broughton, M. et al. (2020).↩︎

  93. Preskill (2021).↩︎

  94. Feynman (1965).↩︎

  95. Weinberg (1997b), p. 8.↩︎

  96. Auyang (1995), p. 50.↩︎

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

  98. Peskin & Schroeder (1995).↩︎

  99. Weinberg (1995).↩︎

  100. Zee (2003).↩︎

  101. Schwartz (2014).↩︎

  102. Tong (2006).↩︎

  103. Zeidler (2007).↩︎

  104. Zeidler (2008).↩︎

  105. Zeidler (2011).↩︎

  106. T. Y. Cao (1999).↩︎

  107. ’t Hooft (2005).↩︎

  108. Mulders (2011).↩︎

  109. Coleman & Mandula (1967).↩︎

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

  111. Gross (1996).↩︎

  112. Bell (1955).↩︎

  113. Streater & Wightman (1964).↩︎

  114. Greaves & Thomas (2012).↩︎

  115. Ohanian (1986).↩︎

  116. Peskin (1994).↩︎

  117. Sebens (2019).↩︎

  118. Dutailly (2014), p. 37.↩︎

  119. Schlingemann (1998).↩︎

  120. Kontsevich & Segal (2021).↩︎

  121. Gell-Mann & Low (1951).↩︎

  122. Molinari (2006).↩︎

  123. Dyson (1949).↩︎

  124. Dyson (1952).↩︎

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

  126. Buchholz & Dybalski (2005).↩︎

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

  128. S. P. Martin (2011).↩︎

  129. S. P. Martin & Wells (2023).↩︎

  130. Reece (2007).↩︎

  131. Jaeger (2019).↩︎

  132. Feynman & Hibbs (1965).↩︎

  133. Nguyen (2016).↩︎

  134. Gill (2017).↩︎

  135. Dirac (1963).↩︎

  136. ’t Hooft (1971).↩︎

  137. Wilson (1974).↩︎

  138. Wilson (1979).↩︎

  139. Goldenfeld (1992).↩︎

  140. ’t Hooft (1994).↩︎

  141. ’t Hooft (1999).↩︎

  142. Borcherds & Barnard (2002).↩︎

  143. Kadanoff (2013), p. 50.↩︎

  144. Butterfield (2014).↩︎

  145. Butterfield & Bouatta (2015).↩︎

  146. J. D. Fraser (2021).↩︎

  147. Phillips (2023).↩︎

  148. Huggett & Weingard (1995).↩︎

  149. Weinberg (1997b).↩︎

  150. T. Y. Cao (2003).↩︎

  151. Bain (2013a) and Bain (2013b).↩︎

  152. Preskill (2013).↩︎

  153. Glick (2016).↩︎

  154. Williams (2019).↩︎

  155. Ruetsche (2018).↩︎

  156. J. D. Fraser (2018).↩︎

  157. Halvorson (2019).↩︎

  158. Rosaler (2022).↩︎

  159. J. D. Fraser (2018), p. 10.↩︎

  160. Baez (2016).↩︎

  161. Auyang (1995).↩︎

  162. Baez (2016), p. 17.↩︎

  163. Baez (2016), p. 18.↩︎

  164. Einstein (1905c).↩︎

  165. Schrödinger (1953).↩︎

  166. Auyang (1995), p. 80.↩︎

  167. Weinberg (1997b), p. 2.↩︎

  168. Weinberg (1997a).↩︎

  169. Baez et al. (1992), p. 59.↩︎

  170. Albert (1992).↩︎

  171. D. Fraser (2008).↩︎

  172. Baker (2009).↩︎

  173. Pessa (2009).↩︎

  174. Duncan (2012), p. 163–4.↩︎

  175. Lazarovici (2018).↩︎

  176. Redhead (1982).↩︎

  177. Redhead (1988).↩︎

  178. Todorov (2012).↩︎

  179. Schreiber (2016a).↩︎

  180. Myrvold (2015), p. 15.↩︎

  181. Newton & Wigner (1949).↩︎

  182. Fleming (2000).↩︎

  183. Myrvold (2015).↩︎

  184. Moretti (2023).↩︎

  185. Haag (1955).↩︎

  186. Malament (1996).↩︎

  187. Teller (1997), p. 115.↩︎

  188. Earman & Fraser (2006).↩︎

  189. Klaczynski (2016).↩︎

  190. Ruetsche (2002).↩︎

  191. Bain (2000).↩︎

  192. Duncan (2012), p. 359.↩︎

  193. Seidewitz (2017).↩︎

  194. Chew (1961).↩︎

  195. Chew (1964).↩︎

  196. nLab authors (2021a).↩︎

  197. Epstein & Glaser (1973).↩︎

  198. Haag (1992).↩︎

  199. Buchholz (1998).↩︎

  200. Wallace (2011).↩︎

  201. D. Fraser (2011).↩︎

  202. Kastler (2003), p. 6.↩︎

  203. Aharonov & Bohm (1959).↩︎

  204. Maudlin (1998).↩︎

  205. Healey (2007), ch. 2-4.↩︎

  206. Batterman (2003).↩︎

  207. Wallace (2014).↩︎

  208. Maudlin (2018).↩︎

  209. T. T. Wu & Yang (1975).↩︎

  210. nLab authors (2021b).↩︎

  211. Auyang (1995), p. 220.↩︎

  212. Auyang (1995).↩︎

  213. Auyang (2000).↩︎

  214. Frankel (2004).↩︎

  215. Way (2010).↩︎

  216. Vákár (2011).↩︎

  217. Marsh (2016).↩︎

  218. Maudlin (2007), p. 96.↩︎

  219. Maudlin (2007), p. 101.↩︎

  220. Baez & Muniain (1994).↩︎

  221. Baez & Schreiber (2005).↩︎

  222. Baez & Huerta (2011).↩︎

  223. Witten (1989).↩︎

  224. Freed (2001).↩︎

  225. Baez & Stay (2009).↩︎

  226. Schreiber (2020).↩︎

  227. ’t Hooft (1978).↩︎

  228. ’t Hooft (1994).↩︎

  229. Shifman (2012).↩︎

  230. Manton (2019).↩︎

  231. Percacci (2024).↩︎

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

  233. Deligne (2002).↩︎

  234. Ostrik (2004).↩︎

  235. Schreiber (2016b).↩︎

  236. Lepine (2016).↩︎

  237. Schreiber (2016b).↩︎

  238. S. P. Martin (2016).↩︎

  239. Tong (2022).↩︎

  240. Bertolini (2022).↩︎

  241. Dimopoulos & Georgi (1981).↩︎

  242. Murayama (2000).↩︎

  243. Arkani-Hamed, Cachazo, & Kaplan (2008).↩︎

  244. Wall (1964).↩︎

  245. Deligne (1999).↩︎

  246. Freed & Moore (2012).↩︎

  247. Geiko & Moore (2020).↩︎

  248. Baez (2020).↩︎

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

  250. van Nieuwenhuizen (1981).↩︎

  251. Frè (2013), ch. 6.↩︎

  252. Connes (1985).↩︎

  253. Maudlin (2019), p. TODO.↩︎

  254. Maudlin (1995).↩︎

  255. Dürr & Lazarovici (2020).↩︎

  256. Mermin (2022).↩︎

  257. Barad (2007).↩︎

  258. Becker (2018).↩︎

  259. Bunge (1955a).↩︎

  260. Bunge (1955b).↩︎

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

  262. Schrödinger (1935).↩︎

  263. Schrödinger (1936).↩︎

  264. Bohm & Aharonov (1957).↩︎

  265. Mermin (1985).↩︎

  266. Caulton (2014).↩︎

  267. Ismael & Schaffer (2020).↩︎

  268. Wigner (1961).↩︎

  269. Deutsch (1985).↩︎

  270. Bong, K.W. et al. (2020).↩︎

  271. Stacey (2014).↩︎

  272. Bell (1964).↩︎

  273. Bell (1966).↩︎

  274. Kochen & Specker (1967).↩︎

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

  276. d’Espagnat (1979).↩︎

  277. Shimony (1984).↩︎

  278. Bell (2004b), pp. 232–248.↩︎

  279. Greenberger, Horne, & Zeilinger (1989).↩︎

  280. Greenberger, Horne, Shimony, & Zeilinger (1990).↩︎

  281. Mermin (1990).↩︎

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

  283. Conway & Kochen (2006).↩︎

  284. Maudlin (2014).↩︎

  285. Ahmed & Caulton (2014).↩︎

  286. Bohm (1952).↩︎

  287. Bohm (1953).↩︎

  288. Schönberg (1954).↩︎

  289. Raman & Forman (1969).↩︎

  290. Bell (2004b).↩︎

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

  292. Allori, Dürr, Goldstein, & Zanghì (2002).↩︎

  293. Brown & Hiley (2004).↩︎

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

  295. Tumulka (2017).↩︎

  296. Del Santo & Krizek (2023).↩︎

  297. Bell (1984).↩︎

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

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

  300. Dürr, D. et al. (2014).↩︎

  301. Nikolić (2022).↩︎

  302. Das & Dürr (2019).↩︎

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

  304. Ananthaswamy (2021).↩︎

  305. Esfeld, Lazarovici, Lam, & Hubert (2017).↩︎

  306. Reichert & Lazarovici (2022).↩︎

  307. Caulton (2018).↩︎

  308. Everett (2012), p. 171.↩︎

  309. Everett (1956).↩︎

  310. Everett (1957).↩︎

  311. Wheeler (1957).↩︎

  312. Everett (2012).↩︎

  313. DeWitt (1970).↩︎

  314. DeWitt & Graham (1973).↩︎

  315. Gell-Mann & Hartle (1989).↩︎

  316. Barrett (2011).↩︎

  317. Barrett (2016).↩︎

  318. Everett (2012), p. 150.↩︎

  319. Wallace (2012).↩︎

  320. Joos, E. et al. (2003), p. 22.↩︎

  321. Carroll & Singh (2019).↩︎

  322. Carroll (2019).↩︎

  323. Saunders (2021).↩︎

  324. Wilhelm (2022).↩︎

  325. Wallace (2022).↩︎

  326. Gisin & Del Santo (2023).↩︎

  327. Boughn (2018).↩︎

  328. Frauchiger & Renner (2018).↩︎

  329. Bub (2019).↩︎

  330. Barbado & Del Santo (2023).↩︎

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

  332. Ghirardi, Pearle, & Rimini (1990).↩︎

  333. Bassi (2005).↩︎

  334. Putnam (1975).↩︎

  335. Putnam (2005).↩︎

  336. Wuthrich (2014).↩︎

  337. Allori (2022).↩︎

  338. Tegmark (1993).↩︎

  339. Caves, Fuchs, & Schack (2001).↩︎

  340. Fuchs (2002).↩︎

  341. Fuchs (2010).↩︎

  342. Fuchs & Schack (2013).↩︎

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

  344. Fuchs & Stacey (2016).↩︎

  345. Harrigan & Spekkens (2010).↩︎

  346. Leifer & Spekkens (2013).↩︎

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

  348. Schlosshauer & Fine (2012).↩︎

  349. Wallace (2013).↩︎

  350. Nigg, D. et al. (2015).↩︎

  351. Cramer (1986).↩︎

  352. Maudlin (1996).↩︎

  353. Palmer (2009).↩︎

  354. Palmer (2016).↩︎

  355. ’t Hooft (2014).↩︎

  356. Martin-Dussaud, Rovelli, & Zalamea (2018).↩︎

  357. ’t Hooft (2021).↩︎

  358. Hossenfelder & Palmer (2020).↩︎

  359. Adlam, Hance, Hossenfelder, & Palmer (2023).↩︎

  360. Hossenfelder (2023).↩︎

  361. Del Santo & Schwarzhans (2022).↩︎

  362. Brans (1988).↩︎

  363. Palmer (1995).↩︎

  364. Degorre, Laplante, & Roland (2005).↩︎

  365. Hall (2010).↩︎

  366. Ciepielewski (2020).↩︎

  367. Donadi & Hossenfelder (2022).↩︎

  368. Proietti, M. et al. (2019).↩︎

  369. Nikolić (2007).↩︎

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

  371. Cabibbo (1963).↩︎

  372. Englert & Brout (1964).↩︎

  373. Higgs (1964).↩︎

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

  375. Georgi (1999), p. 280.↩︎

  376. Lyre (2008).↩︎

  377. ATLAS Collaboration (2012).↩︎

  378. CMS Collaboration (2012).↩︎

  379. T. Y. Cao (2016).↩︎

  380. Glashow (1961).↩︎

  381. Weinberg (1967).↩︎

  382. Salam & Ward (1964b).↩︎

  383. Salam & Ward (1964a).↩︎

  384. Weinberg (1979).↩︎

  385. Rubbia (1984).↩︎

  386. Chalmers (2017).↩︎

  387. Anzivino, Vaibhav, & Zaccone (2024).↩︎

  388. Hamamatsu (2007).↩︎

  389. LSND Collaboration (1996).↩︎

  390. LSND Collaboration (2001).↩︎

  391. MiniBooNE Collaboration (2018).↩︎

  392. MicroBooNE Collaboration (2021).↩︎

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

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

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

  396. Aimè (2022).↩︎

  397. CDF Collaboration (2022).↩︎

  398. Baez & Huerta (2009a).↩︎

  399. Baez & Huerta (2010).↩︎

  400. Pati & Salam (1974).↩︎

  401. Georgi & Glashow (1974).↩︎

  402. Georgi, Quinn, & Weinberg (1974).↩︎

  403. Slansky (1981).↩︎

  404. Georgi (1999).↩︎

  405. Baez & Huerta (2009b).↩︎

  406. Lisi (2007).↩︎

  407. Chester, Marrani, & Rios (2023).↩︎

  408. S. P. Martin (2016), p. 66.↩︎

  409. Dine & Kusenko (2004).↩︎

  410. Baggott (2013).↩︎

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

  412. Maldacena (1998).↩︎

  413. Witten (1998).↩︎

  414. Gopakumar (2011).↩︎

  415. Gopakumar & Mazenc (2022).↩︎

  416. Penrose (1971).↩︎

  417. Ney (2021).↩︎

  418. Einstein & Grossmann (1913).↩︎

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

  420. Carroll (2004).↩︎

  421. Arntzenius (2012).↩︎

  422. Norton (1993).↩︎

  423. Frè (2013), ch. 4.↩︎

  424. Weinberg (1977).↩︎

  425. Ryden (2003).↩︎

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

  427. Romero (2015).↩︎

  428. Penington (2019).↩︎

  429. Clowe, D. et al. (2006).↩︎

  430. Bahcall (2015).↩︎

  431. Arbey & Mahmoudi (2021).↩︎

  432. Martens (2022).↩︎

  433. J. Martin (2012).↩︎

  434. Guth (1981).↩︎

  435. Albrecht & Steinhardt (1982).↩︎

  436. Linde (1982).↩︎

  437. Baumann (2009).↩︎

  438. Steinhardt (1983).↩︎

  439. Linde (1983).↩︎

  440. Guth (2007).↩︎

  441. Guth (1997).↩︎

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

  443. H. A. Simon (1962).↩︎

  444. Anderson (1972).↩︎

  445. Gell-Mann (1988).↩︎

  446. Bedau (1997).↩︎

  447. Bunge (2001), p. 72.↩︎

  448. Lisi (2017).↩︎

  449. Anderson (1972), p. 393.↩︎

  450. Bokulich (2011).↩︎