The Philosophical Significance of the Concept of Superposition in Quantum Field Theory Douglas Scotohu Earl Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds School of Philosophy, Religion and History of Science September 2020 The candidate confirms that the work submitted is his/her/their own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. The right of Douglas Scotohu Earl to be identified as Author of this work has been asserted by Douglas Scotohu Earl in accordance with the Copyright, Designs and Patents Act 1988 ii Acknowledgements I would like to thank my supervisors Juha Saatsi and Steven French for their patience, support and guidance. I am grateful to the Arts and Humanities Research Council for funding this work through the Realism and the Quantum project. iii Abstract I analyse the applicability of the concept of superposition within quantum field theory (QFT) in order to consider the physical interpretation of a simple interacting theory. First, I consider the significance of ‘superposition’ in classical physics motivated by the philosophical framework developed by Wilson (2006; 2017), via the analyses of ‘superposition’ developed by Volkmann (1896; 1910) and Simons (1987) in addition to a historical survey of its application with special reference to Fourier techniques. The concept has a patchwork or façade structure, with application subject to ‘prolongation’ and ‘semantic mimicry’. Proper usage of ‘superposition’ is associated with identifying partial states and laws that provide a natural description of complicated phenomena supporting physically salient explanations, inductive inferences and counterfactual reasoning. Secondly, I demonstrate that application of ‘superposition’ in quantum physics is a prolongation of its classical usage involving new rules of application. Thirdly, I analyse the historical origins of QFT and the mature theory to indicate where, and how, proper application of ‘superposition’ is made to free theories, whilst semantic mimicry is involved in the interpretation of interacting theories. Improper and often implicit appeals are made to ‘superposition’ to incorrectly claim physical interpretations of interacting theories. Two major related failures of ‘superposition’ are identified, associated with the initial postulation of putative fields and corresponding states associated with different particle types, and with the nonlinearity of the coupled field equations, so that natural descriptions of interacting states using familiar concepts are unavailable. Renormalization is interpreted as a symptom and pragmatic partial remedy for the failure of ‘superposition’ such that empirically successful calculations are supported using LSZ scattering theory and the Gell-Mann and Low theorem. Finally, I suggest that the interpretation of QFT is best approached within the context of an engineer’s perspective rather than a metaphysician’s, having implications for wider philosophical debates. iv Table of Contents 1. Introduction 1 1.1 Context and concerns of the thesis 1 1.2 A ‘Wilsonian’ approach outlined 3 1.2.1 ‘Theory T syndrome’ 4 1.2.2 Concepts, patches and façades 7 1.2.3 Prolongation, property dragging and semantic mimicry 10 1.2.4 Realism, metaphysics and Wilson’s approach 13 1.2.5 Concepts and metaphysics 15 1.2.6 An example: particle concepts 16 1.3 Thesis overview 17 2. Isolation and Superposition in Mathematical Physics: Overview 21 2.1 Introduction 21 2.2 An overview of the façade structure of ‘superposition’ 23 and its history 2.3 Philosophical analyses of classical ‘superposition’ 28 2.3.1 Paul Volkmann on isolation and superposition 28 2.3.2 Peter Simons on ‘superposition’ 36 2.4 Summary 39 3. The façade structure of ‘superposition’ (1): Classical systems 40 with a finite number of degrees of freedom 3.1 Force vector composition and superposition 40 3.1.1 Conjoined gravitational and electrostatic forces and rocket1 42 3.1.2 Motion north-east, rocket1 and arbitrary vector decomposition 46 3.1.3 Vector decomposition into physically significant components: 49 A body sliding down an inclined plane 3.1.4 Summary of vector decomposition 50 3.2 Modulation of radio signals: conflicting superpositions 52 3.3 Principal axis transformations 55 3.3.1 Overview 55 v 3.3.2 Example: Spring-mass system 57 3.3.3 Hilbert superposition and principal axes 61 3.4 Summary 62 4. The façade structure of superposition (2): Fourier techniques, 64 superposition and semantic mimics 4.1 Overview 64 4.2 Patch 1: Fourier’s original example 68 4.3 Patch 3: The mathematical architecture of Fourier representation 74 4.4 Patch 1 prolonged: Sturm-Liouville theory 79 4.4.1 The vibrating string (1-dimensional wave equation) 83 4.4.2 The vibrating circular membrane (2-dimensional wave equation) 87 4.5 Patch 2 and semantic mimicry 89 4.5.1 Epicyclical astronomy and Fourier analysis 90 4.5.2 Iterative series techniques contrasted with Fourier series 92 4.5.3 Semantic mimicry: Summary 93 4.6 Summary 94 5. The façade structure of superposition (3): Linear and nonlinear systems 95 5.1 Linear systems analysis overview 95 5.2 Linear systems analysis 97 5.2.1 Frequency domain (eigenfunction) decomposition (patch 4a) 97 5.2.2 Time-domain (impulse) decomposition (patch 4c) 100 5.2.3 Integral transforms: Generalizations on patch 4 103 5.2.4 Linear systems summary 104 5.3 Nonlinear systems and semantic mimics 105 5.3.1 The contrast between linear and nonlinear systems 105 5.3.2 Volterra series and semantic mimicry 107 5.4 Summary 108 6. Quantum superposition 110 6.1 Overview 110 6.2 The origins of quantum superposition in NRQM 112 6.3 ‘Superposition’ in ‘orthodox’ or ‘textbook’ NRQM 115 vi 6.3.1 Overview of ‘quantum superposition’ in NRQM 115 6.3.2 Schrödinger picture 116 6.3.3 Energy eigenfunction decomposition 118 6.3.4 Example: The hydrogen atom 119 6.3.5 Other observables 123 6.4 Time evolution and the Heisenberg picture 124 6.5 Dirac / interaction picture 126 6.6 The quantized simple harmonic oscillator 126 6.7 Summary 130 7. Superposition and the early foundations of QFT 131 7.1 Overview 131 7.2 The origins of QFT and particle descriptions via Fourier techniques 131 7.3 ‘Second quantization’ 133 7.4 Yukawa: Interaction as virtual quanta exchange 136 7.4.1 Interaction as exchange 136 7.4.2 Virtual processes / quanta 140 7.5 Divergences, self-energy and the roots of renormalization 142 7.6 Perturbation theory, nonlinearity and Heitler’s worry 143 7.7 Summary and anticipation in relation to ‘superposition’ 145 8. Superposition, field quanta and particles in free QFTs 147 8.1 Orientation and overview 147 8.2 Initial application of the ‘Volkmann device’ 150 8.3 Fock space construction via the Klein-Gordon equation for a 151 neutral scalar field 8.3.1 Overview: From second quantization to the Fock construction 151 8.3.2 Single particle state construction 153 8.3.3 Construction of realistic particles 160 8.3.4 Provisional construction of multi-quanta / particle 161 eigenstates of 4-momentum 8.3.5 Imposing the CCRs via the permutation operator and the 163 Fock space construction 8.4 The Fock construction for the charged scalar field 168 vii 8.5 Quantum systems with different particle types: The initial 172 use of the ‘Volkmann device’ revisited 8.6 The ‘group theoretic’ characterization of particles 175 8.7 Summary and anticipation of interacting QFTs 178 9. Introducing interactions, Dyson’s expansion, Feynman diagrams 180 and virtual particles 9.1 Introduction and overview 180 9.2 Introducing an interaction 182 9.3 Interactions in Scalar Yukawa Theory 185 9.3.1 Introduction of an interaction as a perturbation 185 9.3.2 Consequences of the introduction of an interaction 187 as a perturbation: Overview 9.4 The interaction picture, Dyson’s series and asymptotic states 190 9.4.1 The interaction picture 190 9.4.2 Solving for !"#(t, t'): Dyson’s series as a sum of correction 192 terms rather than a ‘superposition’ 9.4.3 Choice of a coordinate representation for |*+(,)⟩ 193 9.4.4 Asymptotic states 194 9.5 The S-matrix and Dyson’s series 199 9.6 The Feynman propagator and ‘virtual particles’ 202 9.6.1 Two-point functions: Introduction 202 9.6.2 The free Feynman propagator 203 9.6.3 Virtual particles 206 9.7 Feynman diagrams and interactions 208 9.7.1 Feynman diagrams in the context of Dyson’s series 208 9.7.2 Renormalization 210 9.7.3 The failure of Feynman diagrams to describe interacting states 212 9.8 Conclusion 216 10. Conceptual analysis of perturbative and non-perturbative QFT 217 in relation to the applicability of ‘superposition’ 10.1 Introduction and overview 217 viii 10.2 The absence of a particle description of interacting states 219 10.2.1 The nonlinearity of the coupled field equations and the 219 decomposition of ./ 10.2.2 Comments on Haag’s theorem 224 10.2.3 Other attempts to establish a particle patch 227 10.2.4 The nonlinear superposition principle 229 10.2.5 Summary 230 10.3 Non-perturbative QFT 231 10.3.1 The ‘state axioms’ 231 10.3.2 The ‘field axioms’ 235 10.3.3 The ‘particle-field duality axioms’ 236 10.3.4 Limitations of the non-perturbative approach 238 10.4 A natural description of general interacting states 240 10.5 Failure of the Volkmann device and its consequences 242 10.6 Summary 250 11. Unstable particles, scattering theory and bound states: