7  Philosophy of physics

Published

October 10, 2025

What are good theories of the world?

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

7.1 Theories of matter

7.1.1 Ancient atomism

  • Kanada (c. 700-100 BCE)
  • 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 Esolen 1

1 Lucretius (1995), p. TODO.

Discussion:

2 Nail (2018).

7.1.2 Modern atomism

3 Yock (2018).

4 Einstein (1905b).

5 Perrin (1913).

Discussion:

  • Russell, B. (1927). The Analysis of Matter. 6
  • Weyl, H. (1927). Philosophy of Mathematics and Natural Science. 7
  • Patterson, G. (2007). Jean Perrin and the triumph of the atomic doctrine. 8

6 Russell (1992).

7 Weyl (2009).

8 Patterson (2007).

7.1.3 Contemporary views of matter

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

See also:

7.2 Classical physics

7.2.1 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:

9 Feynman (1963).

10 Holm (2011a) and Holm (2011b).

Dimensional analysis:

11 Buckingham (1914).

12 Kasprzak, Lysik, & Rybaczuk (1990).

13 Duff, Okun, & Veneziano (2001).

14 Janyska, Modugno, & Vitolo (2007).

15 Zapata-Carratala (2021).

See also:

7.2.2 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

16 Heaviside (1893), pp. vi–vii.

Pedagogy:

  • TODO

7.2.3 Special relativity

History:

17 Einstein (1905d).

18 Einstein (1905a).

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

19 Stein (2021), p. 69.

Pedagogy:

20 Carnap (1966).

21 Maudlin (2012), p. TODO.

See also:

7.3 Statistical physics

7.3.1 Introduction

TODO:

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

7.3.2 History

7.3.3 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.

7.3.4 Canonical ensemble

  • Canonical ensemble

7.3.5 Phase translations

22 J. Wu (2021).

See also:

7.4 Symmetry-first physics

7.4.1 Curie’s principle

23 Caulton (2015).

24 Caulton & Butterfield (2012).

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

See also:

7.4.2 Relativity

See also:

7.4.3 Noether’s theorems

  • Principle of least action, Lagrangians
  • Canonical dynamics
  • Noether, E. (1918). Invariante variationsprobleme. 26
    • TODO: Noether’s first and second theorem.
  • Wigner, E.P. (1954). Conservation laws in classical and quantum physics. 27
  • Brading, K.A. (2002). Which symmetry? Noether, Weyl, and conservation of electric charge. 28
  • Baez, J.C. (2018). Getting to the bottom of Noether’s theorem. Talk given at The Philosophy and Physics of Noether’s Theorems. 29
  • Goyal, P. (2020). Derivation of classical mechanics in an energetic framework via conservation and relativity. 30

26 Noether (1918).

27 Wigner (1954).

28 Brading (2002).

29 Baez (2018).

30 Goyal (2020).

7.4.4 Gauge principle

  • Weyl, H. (1918). Raum, Zeit, Materie. 31
  • Weyl, H. (1929). Elektron und gravitation. 32
  • Pauli, W. (1941). Relativistic field theories of elementary particles. 33
  • Yang C.N. & Mills R.L. (1954). Conservation of isotopic spin and isotopic gauge invariance. 34
  • ’t Hooft, G. (1994). Under the Spell of the Gauge Principle. 35
  • Yang, C.N. (1996). Symmetry and physics. 36
  • O’Raifeartaigh, L. (1997). The Dawning of Gauge Theory. 37
  • Teller, P. (2000). The gauge argument. 38
  • ’t Hooft, G. (2007). Lie groups in physics. 39
  • Greaves, H. & Wallace, D. (2011). Empirical consequences of symmetries. 40
  • Afriat, A. (2013). Weyl’s gauge argument. 41
  • Schwichtenberg, J. (2015). Physics from Symmetry. 42
  • Dewar, N. (2019). Sophistication about symmetries. 43

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).

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

44 Weyl (1929), p. TODO.

7.4.5 Wigner-Stone theorems

45 Wigner (1959).

46 Kadison (1965).

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

47 Reece (2007), p. 27.

Discussion:

  • Schweber, S.S. (1961). An Introduction to Relativistic Quantum Field Theory. 48
  • Simon, B. (1976). Quantum dynamics: From automorphism to Hamiltonian. 49
  • Summers, S.J. (1999). On the Stone-von Neumann uniqueness theorem and its ramifications. 50
  • Keller, K.J., Papadopoulos, M.A., & Reyes-Lega, A.F. (2007). On the realization of symmetries in quantum mechanics. 51
  • Schroeren, D. (2021). Symmetry fundamentalism in quantum mechanics. 52
  • Wigner’s theorem - nLab

48 Schweber (1961), p. TODO.

49 B. Simon (1976).

50 Summers (1999).

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

52 Schroeren (2021).

See also:

7.5 Quantum mechanics

7.5.1 Introduction

  • Hilbert spaces; Wigner’s theorem; Born rule
  • Wave-particle duality misconceptions. Fields are more fundamental than particles.
  • Philosophy of QM traditionally focus on NRQM. 53
  • The measurement problem. Decoherence. The Born rule again.
  • Uncertainty principle
    • Relationship to the Gabor limit
  • Decoherence brings quantum logic to classical logic?

53 Ney & Albert (2013).

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

54 Feynman & Hibbs (1965), p. 6.

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

55 Feynman & Hibbs (1965), p. 9.

7.5.2 History

56 Kelvin (1901).

57 Bacciagaluppi & Valentini (2009).

Figure 7.1: 1927 Solvay Conference on Quantum Mechanics (source: Wikimedia).

58 von Neumann (1955).

59 van Hove (1958).

7.5.3 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

7.5.4 Foundations of QM

7.5.4.1 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:

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

7.5.4.2 Operators

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

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

7.5.4.3 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.

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

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

7.5.4.4 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.

7.5.4.5 Discussion

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).

See also:

7.5.5 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:

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

Schrödinger vs Heisenberg pictures is like Heraclitus vs Parmenides.

  • Heisenberg uncertainty principle - Derivable from the Gabor limit in time-frequency analysis

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

70 Dutailly (2014), p. 11–13.

7.5.6 Decoherence

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 Leggett (2002).

81 Schlosshauer (2005).

82 Drossel (2015), p. 51–2.

83 Wallace (2018).

84 Zurek (2022).

85 Nagele, Janssen, & Kleban (2023).

See also:

7.5.7 Quantum chemistry

86 Cohen (2015).

87 Friedrich (2016).

88 C. Cao, Hu, Li, & Schwarz (2019).

89 Seifert (2024).

7.5.8 Quantum computing

90 Coecke & Kissinger (2017).

91 Preskill (2018).

92 Arute, F. et al. (2019).

93 Broughton, M. et al. (2020).

94 Preskill (2021).

7.6 Quantum field theory

7.6.1 Fields

7.6.1.1 Introduction

  • Field concept/definition: Laplace, Faraday
  • Richard Feynman (1918-1988)
  • Julian Schwinger (1918-1994)
  • Shin’ichirō Tomonaga (1906-1979)
  • Feynman’s Nobel Lecture on QED 95
  • Weinberg’s folk theorem: QFT is the right way to combine Lorentz invariance, quantum mechanics, and the cluster decomposition principle. 96
  • Euler-Lagrange equations of motion
    • The Dirac equation does not describe a relativistic wavefunction (hence the obsolete “Dirac sea” interpretation). 97

95 Feynman (1965).

96 Weinberg (1997b), p. 8.

97 Auyang (1995), p. 50.

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. 98

98 Baez, Segal, & Zhou (1992), p. 1.

7.6.1.2 Pedagogy

99 Peskin & Schroeder (1995).

100 Weinberg (1995).

101 Zee (2003).

102 Schwartz (2014).

103 Tong (2006).

104 Zeidler (2007).

105 Zeidler (2008).

106 Zeidler (2011).

107 T. Y. Cao (1999).

108 ’t Hooft (2005).

109 Mulders (2011).

7.6.2 Symmetry

7.6.2.1 Introduction

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

See also:

7.6.2.2 Coleman-Mandula theorem

  • Coleman-Mandula theorem 110

110 Coleman & Mandula (1967).

See also:

7.6.2.3 Wigner’s classification

111 Wigner (1939) and Bargmann & Wigner (1948).

112 Gross (1996).

7.6.2.4 CPT theorem

  • Bell, J.S. (1955). Time reversal in field theory. 113
  • Streater, R. & Wightman, A. (1964). PCT, spin and statistics, and all that. 114
  • Greaves, H. & Thomas, T. (2012). The CPT theorem. 115

113 Bell (1955).

114 Streater & Wightman (1964).

115 Greaves & Thomas (2012).

7.6.3 Spin

7.6.3.1 Introduction

  • Stern-Gerlach experiment (1922)
  • Ohanian 116
  • Peskin 117
  • Sebens 118

116 Ohanian (1986).

117 Peskin (1994).

118 Sebens (2019).

7.6.3.2 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. 119

119 Dutailly (2014), p. 37.

7.6.3.3 Spin-statistics theorem

  • Spin-statistics theorem - Pauli

7.6.4 Scattering

120 Schlingemann (1998).

121 Kontsevich & Segal (2021).

122 Gell-Mann & Low (1951).

123 Molinari (2006).

124 Dyson (1949).

125 Dyson (1952).

126 Lehmann, Symanzik, & Zimmermann (1955).

127 Buchholz & Dybalski (2005).

128 Weinberg (1964b) and Weinberg (1964a).

129 S. P. Martin (2011).

130 S. P. Martin & Wells (2023).

131 Reece (2007).

132 Jaeger (2019).

7.6.5 Path intergrals

133 Feynman & Hibbs (1965).

134 Nguyen (2016).

135 Gill (2017).

7.6.6 Renormalization

136 Dirac (1963).

137 ’t Hooft (1971).

138 Wilson (1974).

139 Wilson (1979).

140 Goldenfeld (1992).

141 ’t Hooft (1994).

142 ’t Hooft (1999).

143 Borcherds & Barnard (2002).

144 Kadanoff (2013), p. 50.

145 Butterfield (2014).

146 Butterfield & Bouatta (2015).

147 J. D. Fraser (2021).

148 Phillips (2023).

7.6.7 Effective field theory

149 Huggett & Weingard (1995).

150 Weinberg (1997b).

151 T. Y. Cao (2003).

152 Bain (2013a) and Bain (2013b).

153 Preskill (2013).

154 Glick (2016).

155 Williams (2019).

156 Ruetsche (2018).

157 J. D. Fraser (2018).

158 Halvorson (2019).

159 Rosaler (2022).

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. 160

160 J. D. Fraser (2018), p. 10.

7.6.8 Foundations of QFT

7.6.8.1 Introduction

  • Weinberg
  • Reeh-Schlieder theorem
    • Taj Mahal principle
  • Struggles with the continuum 161
  • Auyang, S.Y. (1995). How Is Quantum Field Theory Possible? 162

161 Baez (2016).

162 Auyang (1995).

Baez:

Nobody has found a fully rigorous formulation of QED, nor has anyone proved such a thing cannot be found. 163

163 Baez (2016), p. 17.

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. 164

164 Baez (2016), p. 18.

7.6.8.2 Wave-particle duality

165 Einstein (1905c).

166 Schrödinger (1953).

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. 167

167 Auyang (1995), p. 80.

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. 168

168 Weinberg (1997b), p. 2.

169 Weinberg (1997a).

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. 170

170 Baez et al. (1992), p. 59.

171 Albert (1992).

172 D. Fraser (2008).

173 Baker (2009).

174 Pessa (2009).

175 Duncan (2012), p. 163–4.

176 Lazarovici (2018).

7.6.8.3 Quantization

  • Canonical quantization
  • Path integral quantization
  • No “2nd quantization”
    • Redhead, M. (1982). Quantum field theory for philosophers. 177
    • Redhead, M. (1988). A philosopher looks at quantum field theory. 178
  • Geometric quantization
  • Instead of quantizing classical theories, should we be finding the classical limit of quantum theories?

177 Redhead (1982).

178 Redhead (1988).

179 Todorov (2012).

180 Schreiber (2016a).

7.6.8.4 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: 181

181 Myrvold (2015), p. 15.

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

182 Newton & Wigner (1949).

183 Fleming (2000).

184 Myrvold (2015).

185 Moretti (2023).

7.6.8.5 Haag’s theorem

186 Haag (1955).

187 Malament (1996).

188 Teller (1997), p. 115.

189 Earman & Fraser (2006).

190 Klaczynski (2016).

191 Ruetsche (2002).

192 Bain (2000).

193 Duncan (2012), p. 359.

194 Seidewitz (2017).

7.6.8.6 Bootstrap theory

  • Chew, G. (1961). S-Matrix Theory of Strong Interactions. 195
  • Chew, G. (1964). Nuclear democracy and bootstrapping dynamics. 196

195 Chew (1961).

196 Chew (1964).

7.6.8.7 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. 197

197 nLab authors (2021a).

First rigorous perturbation theory in QFT according to Schreiber:

198 Epstein & Glaser (1973).

7.6.8.8 Algebraic vs constructive QFT

  • AQFT vs LQFT
  • Haag-Ruelle scattering theory
  • Haag, R. (1992). Local Quantum Physics: Fields, Particles, Algebras. 199
  • Buchholz, D. (1998). Current trends in axiomatic quantum field theory. 200
  • Wallace, D. (2011). Taking particle physics seriously: A critique of the algebraic approach to quantum field theory. 201
  • Fraser, D. (2011). How to take particle physics seriously: A further defence of axiomatic quantum field theory. 202

199 Haag (1992).

200 Buchholz (1998).

201 Wallace (2011).

202 D. Fraser (2011).

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. 203

203 Kastler (2003), p. 6.

7.7 Exotics in quantum field theory

7.7.1 Gauge theory

7.7.1.1 Aharonov-Bohm effect

204 Aharonov & Bohm (1959).

205 Maudlin (1998).

206 Healey (2007), ch. 2-4.

207 Batterman (2003).

208 Wallace (2014).

209 Maudlin (2018).

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.

7.7.1.2 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.

  • Fiber bundle
  • Fiber bundles in physics - nLab
    • Fiber bundles embody two central principles of modern physics:
      1. the principle of locality
      2. the gauge principle.
    • In Yang-Mills theories, the gauge fields are not just local differential 1-forms, \(A_{\mu}\), but are globally really connections on principle bundles, and this is all-important once one passes to non-perturbative Yang-Mills theory, hence to the full story, instead of its infinitesimal or local approximation.
  • Wu, T.T. & Yang, C.N. (1975). Concept of nonintegrable phase factors and global formulation of gauge fields. 210

210 T. T. Wu & Yang (1975).

Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality. 211

211 nLab authors (2021b).

Figure 7.2: An illustration of the fiber bundle formulation of interacting quantum fields. 212

212 Auyang (1995), p. 220.

213 Auyang (1995).

214 Auyang (2000).

215 Frankel (2004).

216 Way (2010).

217 Vákár (2011).

218 Marsh (2016).

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. 219

219 Maudlin (2007), p. 96.

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. 220

220 Maudlin (2007), p. 101.

7.7.1.3 Higher gauge theory

  • Higher gauge field - nLab
    • An ordinary gauge field is a field which is locally represented by a differential 1-form, the gauge potential, and whose field strength is locally a differential 2-form.
  • Baez, J.C. & Muniain, J.P. (1994). Gauge Fields, Knots and Gravity. 221
  • Baez, J.C. & Schreiber, U. (2005). Higher gauge theory. 222
  • Baez, J.C. & Huerta, J. (2011). An invitation to higher gauge theory. 223
  • Relationship to branes and string theory

221 Baez & Muniain (1994).

222 Baez & Schreiber (2005).

223 Baez & Huerta (2011).

7.7.1.4 Topological QFT

224 Witten (1989).

225 Freed (2001).

226 Baez & Stay (2009).

227 Schreiber (2020).

See also:

7.7.2 Non-perturbative features

228 ’t Hooft (1978).

229 ’t Hooft (1994).

230 Shifman (2012).

231 Manton (2019).

232 Percacci (2024).

7.7.3 Supersymmetry

233 Haag, Łopuszański, & Sohnius (1975).

234 Deligne (2002).

235 Ostrik (2004).

236 Schreiber (2016b).

237 Lepine (2016).

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. 238

238 Schreiber (2016b).

nLab:

By Deligne’s theorem on tensor categories it is precisely the context of supersymmetry in which tensor categories over the complex numbers exhibit full Tannaka duality.

239 S. P. Martin (2016).

240 Tong (2022).

241 Bertolini (2022).

242 Dimopoulos & Georgi (1981).

243 Murayama (2000).

244 Arkani-Hamed, Cachazo, & Kaplan (2008).

245 Wall (1964).

246 Deligne (1999).

247 Freed & Moore (2012).

248 Geiko & Moore (2020).

249 Baez (2020).

250 Freedman, Nieuwenhuizen, & Ferrara (1976).

251 van Nieuwenhuizen (1981).

252 Frè (2013), ch. 6.

253 Connes (1985).

See also:

7.8 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: Maudlin 254

254 Maudlin (2019), p. TODO.

7.8.1 Measurement problem

255 Maudlin (1995).

256 d’Espagnat (1999).

257 Dürr & Lazarovici (2020).

258 Mermin (2022).

7.8.2 Copenhagen “interpretation”

  • Niels Bohr (1885-1962)
  • Complementarity
  • Reichenbach, H. (1944). Philosophic Foundations of Quantum Mechanics. 259
  • Barad, K. (2007). Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning. 260
    • Some strange defenses of Bohr
  • Becker, A. (2018). What is Real? 261

259 Reichenbach (1944).

260 Barad (2007).

261 Becker (2018).

Figure 7.3: Interpretations of quantum mechanics (philosophy-in-figures.tumblr.com).

Criticisms:

262 Bunge (1955a).

263 Bunge (1955b).

7.8.3 Von Neumann’s no hidden variables “proof”

  • von Neumann, J. (1932). The Mathematical Foundations of Quantum Mechanics.
  • Hermann, G. (1933). Determinism and quantum mechanics.
  • Hermann, G. (1935). The Natural-Philosophical Foundations of Quantum Mechanics.

7.8.4 EPR paradox

  • Einstein, A., Podolsky, B. & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? 264
  • Schrödinger, E. (1935). Discussion of probability relations between separated systems. 265
    • coined entanglement
  • Schrödinger, E. (1936). Probability relations between separated systems. 266
  • Bohm, D. & Aharonov, Y. (1957). Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky. 267
  • Mermin, N.D. (1985). Is the moon there when nobody looks? 268
  • Caulton, A. (2014). Physical entanglement in permutation-invariant quantum mechanics. 269
  • Ismael, J. & Schaffer, J. (2020). Quantum holism: Nonseparability as common ground. 270

264 Einstein, Podolsky, & Rosen (1935).

265 Schrödinger (1935).

266 Schrödinger (1936).

267 Bohm & Aharonov (1957).

268 Mermin (1985).

269 Caulton (2014).

270 Ismael & Schaffer (2020).

7.8.5 Von Neumann-Wigner interpretation

271 Wigner (1961).

272 Deutsch (1985).

273 Bong, K.W. et al. (2020).

274 Stacey (2014).

Criticisms:

  • Similar crticisms to Copenhagen: collapse is arbitrary; dualism of observables and observer.
  • TODO: Discuss how Everett and his Ph.D. adviser, Wigner, disagreed about how to interpret the Wigner’s friend thought experiment.

7.8.6 Bell’s theorem

275 Bell (1964).

276 Bell (1966).

277 Kochen & Specker (1967).

278 Clauser, Horne, Shimony, & Holt (1969).

279 d’Espagnat (1979).

280 Shimony (1984).

281 Bell (2004b), pp. 232–248.

282 Greenberger, Horne, & Zeilinger (1989).

283 Greenberger, Horne, Shimony, & Zeilinger (1990).

284 Mermin (1990).

285 Gisin (1991), Gisin & Peres (1992), and Gisin (1999).

286 Conway & Kochen (2006).

287 Maudlin (2014).

288 Ahmed & Caulton (2014).

7.8.7 Bohmian mechanics

  • de Broglie-Bohm theory
  • version of quantum theory discovered by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952.
  • Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of ‘hidden’ variables, I and II. 289
  • Bohm, D. (1953). Proof that probability density approaches \(|\psi|^2\) in causal interpretation of quantum theory. 290
  • Schönberg, M. (1954). On the hydrodynamical model of the quantum mechanics. 291
  • Bohm, D. & Hiley, B.J. (1993). The Undivided Universe. 292

289 Bohm (1952).

290 Bohm (1953).

291 Schönberg (1954).

292 Bohm & Hiley (1993).

Discussion:

293 Raman & Forman (1969).

294 Bell (2004b).

295 Dürr, Goldstein, & Zanghì (1995).

296 Allori, Dürr, Goldstein, & Zanghì (2002).

297 Brown & Hiley (2004).

298 Dürr, Goldstein, & Zanghì (2013).

299 Tumulka (2017).

300 Del Santo & Krizek (2023).

Attempts at QFT:

301 Bell (1984).

302 Dürr, Goldstein, Tumulka, & Zanghì (2004).

303 Dürr, Goldstein, Tumulka, & Zanghì (2005).

304 Dürr, D. et al. (2014).

305 Nikolić (2022).

Attempts at empirical proposals:

306 Das & Dürr (2019).

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

308 Ananthaswamy (2021).

Primitive ontology:

309 Esfeld, Lazarovici, Lam, & Hubert (2017).

310 Reichert & Lazarovici (2022).

Virtues:

  • No collapse; universal unitary Schrödinger evolution
  • Consistent histories of particle trajectories.

Criticisms:

  • Postulates a new theoretical apparatus, the guiding equation.
  • Committed to particle onotology.
  • Focuses on NRQM; Not yet demonstrated any relativistic extensions; No working QFT.
  • Nonlocality baked right into the guiding equation.
  • TODO: Ignores decoherence? Or if it doesn’t how naturally is it used?
  • Ignores renormalization and EFT?
  • TODO: Criticisms of primitive ontology as being apriori metaphysics.
  • Caulton, A. (2018). A persistent particle ontology for quantum field theory. 311
  • Deutsch: “Bohmian mechanics is Everett’s many worlds in denial.”

311 Caulton (2018).

7.8.8 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. 312

312 Everett (2012), p. 171.

  • 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. 313
    • Everett, H. (1957). “Relative state” formulation of quantum mechanics. 314
    • Wheeler, J.A. (1957). Assessment of Everett’s “relative state” formulation of quantum theory. 315
    • Everett’s collected works 316
    • Shikhovtsev, E. (2003). Biographical sketch of Hugh Everett, III.
  • DeWitt, B.S. (1970). Quantum mechanics and reality. 317
  • DeWitt, B.S. & Graham, N. (1973). The Many-Worlds Interpretation of Quantum Mechanics. 318
  • Gell-Mann, M. & Hartle, J.B. (1989). Quantum mechanics in the light of quantum cosmology. 319
  • Barrett, J.A. (2011). Everett’s pure wave mechanics and the notion of worlds. 320
  • Barrett, J.A. (2016). Quantum worlds. 321

313 Everett (1956).

314 Everett (1957).

315 Wheeler (1957).

316 Everett (2012).

317 DeWitt (1970).

318 DeWitt & Graham (1973).

319 Gell-Mann & Hartle (1989).

320 Barrett (2011).

321 Barrett (2016).

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. 322

322 Everett (2012), p. 150.

  • Everett’s later influence on the theory of decoherence
  • Wallace, D. (2012). The Emergent Multiverse. 323

323 Wallace (2012).

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). 324

324 Joos, E. et al. (2003), p. 22.

325 Barrett (2019).

326 Carroll & Singh (2019).

327 Carroll (2019).

328 Saunders (2021).

329 Wilhelm (2022).

330 Wallace (2022).

331 Gisin & Del Santo (2023).

Videos:

Virtues:

  • Minimal; No additions to quantum theory
  • No collapse; universal unitary Schrödinger evolution
  • Relativistic QFT works naturally.
  • Decoherence does a lot of work at explaining the appearance of collapse into classical states.
  • Classical concepts are derived; not dual; no complementarity.
  • Applies QM to the system + aparatus + observer.
  • First interpretation to allow for quantum cosmology, the quantum mechanics of the universe as a closed system.

Criticisms:

332 Boughn (2018).

333 Frauchiger & Renner (2018).

334 Bub (2019).

335 Barbado & Del Santo (2023).

See also:

7.8.9 Collapse interpretations

336 Ghirardi, Rimini, & Weber (1986).

337 Ghirardi, Pearle, & Rimini (1990).

338 Bassi (2005).

339 Putnam (1975).

340 Putnam (2005).

341 Wuthrich (2014).

342 Allori (2022).

Virtues:

  • Tries to explain collapse stochastically.

Criticisms:

343 Tegmark (1993).

7.8.10 Epistemic interpretations

344 Caves, Fuchs, & Schack (2001).

345 Fuchs (2002).

346 Fuchs (2010).

347 Fuchs & Schack (2013).

348 Fuchs, Mermin, & Schack (2014).

349 Fuchs & Stacey (2016).

350 Harrigan & Spekkens (2010).

351 Leifer & Spekkens (2013).

Criticisms:

  • Anti-Copernican; anthropomorphism
  • PBR theorem
  • Ignores decoherence?

7.8.11 PBR theorem

352 Pusey, Barrett, & Rudolph (2012).

353 Schlosshauer & Fine (2012).

354 Wallace (2013).

355 Nigg, D. et al. (2015).

Videos:

7.8.12 Other interpretations

356 Cramer (1986).

357 Maudlin (1996).

358 Palmer (2009).

359 Palmer (2016).

360 ’t Hooft (2014).

361 Martin-Dussaud, Rovelli, & Zalamea (2018).

362 ’t Hooft (2021).

363 Hossenfelder & Palmer (2020).

364 Adlam, Hance, Hossenfelder, & Palmer (2023).

365 Hossenfelder (2023).

366 Del Santo & Schwarzhans (2022).

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

367 Brans (1988).

368 Palmer (1995).

369 Degorre, Laplante, & Roland (2005).

370 Hall (2010).

371 Ciepielewski (2020).

372 Donadi & Hossenfelder (2022).

7.8.13 Bad takes

373 Proietti, M. et al. (2019).

374 Nikolić (2007).

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.

7.9 The standard model of particle physics

7.9.1 History of particle physics

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

7.9.2 Mixing

  • Cabibbo angle (1963) 376
  • CP violation
  • CKM matrix
  • Kaons
  • B-mesons

376 Cabibbo (1963).

7.9.3 Higgs mechanism

In 1964, three groups: Robert Brout and Francois Englert 377; Peter Higgs 378; and Gerald Guralnik, Carl R. Hagen, and Tom Kibble 379, 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.

377 Englert & Brout (1964).

378 Higgs (1964).

379 Guralnik, Hagen, & Kibble (1964).

380 Georgi (1999), p. 280.

381 Lyre (2008).

382 ATLAS Collaboration (2012).

383 CMS Collaboration (2012).

On July 4 of 2012, the ATLAS 382 and CMS 383 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.

  • Cao, T.Y. (2016). The Englert-Brout-Higgs mechanism: An unfinished project. 384
  • ’t Hooft, G. (2022). A triumph for theory.

384 T. Y. Cao (2016).

7.9.4 A model of leptons

385 Glashow (1961).

386 Weinberg (1967).

387 Salam & Ward (1964b).

388 Salam & Ward (1964a).

389 Weinberg (1979).

390 Rubbia (1984).

391 Chalmers (2017).

7.9.5 Quantum chromodynamics

392 Anzivino, Vaibhav, & Zaccone (2024).

7.9.6 Three generations of fermions

Figure 7.4: The fields in the standard model of particle physics (source: Symmetry Magazine).
  • 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)
Figure 7.5: 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:

7.9.7 Experimental methods

393 Hamamatsu (2007).

7.10 Beyond the standard model

  • Beyond the standard model (BSM)

7.10.1 Neutrino masses

394 LSND Collaboration (1996).

395 LSND Collaboration (2001).

396 MiniBooNE Collaboration (2018).

397 MicroBooNE Collaboration (2021).

398 Vitagliano, Tamborra, & Raffelt (2020).

7.10.2 Ad hoc structures

  • 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

See also:

7.10.3 Experimental anomalies

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

400 Capdevilla, Curtin, Kahn, & Krnjaic (2021).

401 Aimè (2022).

402 CDF Collaboration (2022).

7.10.4 Grand unification

403 Baez & Huerta (2009a).

404 Baez & Huerta (2010).

405 Pati & Salam (1974).

406 Georgi & Glashow (1974).

407 Georgi, Quinn, & Weinberg (1974).

408 Slansky (1981).

409 Georgi (1999).

410 Baez & Huerta (2009b).

411 Lisi (2007).

412 Chester, Marrani, & Rios (2023).

Figure 7.6: 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). 413

413 S. P. Martin (2016), p. 66.

Figure 7.7: 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:

7.10.5 Baryogenesis

414 Dine & Kusenko (2004).

7.10.6 Future colliders and criticisms

415 Baggott (2013).

7.10.7 Quantum gravity

416 Candelas, Horowitz, Strominger, & Witten (1985).

417 Maldacena (1998).

418 Witten (1998).

419 Polyakov (2008).

420 Gopakumar (2011).

421 Gopakumar & Mazenc (2022).

422 Penrose (1971).

423 Ney (2021).

7.11 Gravity and cosmology

7.11.1 General relativity

424 Einstein & Grossmann (1913).

425 Misner, Thorne, & Wheeler (1973).

426 Carroll (2004).

427 Arntzenius (2012).

428 Norton (1993).

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

7.11.2 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.
  • Toth, V.T. (2016). The Newtonian limit in general relativity.

7.11.3 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 Frè 429
  • The First Three Minutes 430
  • Ryden, B. (2003). Introduction to Cosmology. 431
  • 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. 432

429 Frè (2013), ch. 4.

430 Weinberg (1977).

431 Ryden (2003).

432 Bahcall, Ostriker, Perlmutter, & Steinhardt (1999).

7.11.4 Spacetime

433 Romero (2015).

7.11.5 Blackholes

434 Penington (2019).

7.11.6 Gravitational waves

7.11.7 Dark matter

435 Clowe, D. et al. (2006).

436 Bahcall (2015).

437 Arbey & Mahmoudi (2021).

438 Martens (2022).

7.11.8 Inflation

439 J. Martin (2012).

440 Guth (1981).

441 Albrecht & Steinhardt (1982).

442 Linde (1982).

443 Baumann (2009).

444 Steinhardt (1983).

445 Linde (1983).

446 Guth (2007).

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. 447

447 Guth (1997).

Figure 7.8: How the \(\Lambda\)-CDM concordance model of cosmology was developed. 448

448 Debono & Smoot (2016), figure 4.

7.11.9 Alternative theories of gravity

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

7.12 Fine-tuning

7.13 Complexity and emergence

449 H. A. Simon (1962).

450 Anderson (1972).

451 Gell-Mann (1988).

452 Bedau (1997).

453 Bunge (2001), p. 72.

454 Lisi (2017).

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. 455

455 Anderson (1972), p. 393.

See also:

7.14 Bracketing human experience

456 Bokulich (2011).

Figure 7.9: Sean Carroll on the entailment of everyday life by physics.

Videos of talks:

See also:

7.15 My thoughts

7.16 Annotated bibliography

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

  • Einstein et al. (1935)

7.16.1.1 My thoughts

  • TODO.

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

7.16.2.1 My thoughts

  • TODO.

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

7.16.3.1 My thoughts

  • TODO.

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

  • Joos, E. et al. (2003).

7.16.4.1 My thoughts

  • TODO.

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

7.16.5.1 My thoughts

  • TODO.