Location and Causality in Quantum Theory A dissertation submitted to the University of Manchester for the degree of Master of Science by Research in the Faculty of Engineering and Physical Sciences 2013 Robert A Dickinson School of Physics and Astronomy Contents 1 Introduction 8 1.1 Causality . .8 1.2 Relativistic causality . .9 1.3 The structure of this document . 10 2 Correlations 12 2.1 Operational descriptions . 12 2.2 Restrictions on non-local correlations in a classical theory . 12 2.3 Parameter spaces of general correlations . 14 2.3.1 Imaginable parameter spaces for one binary observer . 15 2.3.2 One-way signalling between two observers . 15 2.3.3 Imaginable parameter spaces for two spacelike-separated binary observers . 17 2.3.4 The parameter space in nature for two spacelike-separated binary observers . 21 2.3.5 The parameter space in quantum theory for two spacelike- separated binary observers . 22 2.3.6 Parameter spaces for three or more spacelike-separated binary observers . 24 3 Hilbert Space 26 3.1 The Postulates and their implications . 26 3.2 Operational stochastic quantum theory . 28 3.2.1 Internal unknowns regarding the initial state . 29 3.2.2 External unknowns regarding the initial state . 30 3.2.3 Internal unknowns regarding the measurement . 30 3.2.4 External unknowns regarding the measurement . 31 3.2.5 The generalised update rule . 31 3.3 Causality in quantum mechanics . 33 3.3.1 The no-signalling condition . 33 3.3.2 Special cases . 35 3.4 From discrete to continuous sets of outcomes . 36 3.4.1 Partitions of discrete sets . 36 3.4.2 Partitions of continuous sets . 37 3.4.3 Unbounded operators . 39 3.4.4 Continuous observables . 40 2 4 The relationship between Hilbert space and spaces of gener- alised coordinates 43 4.1 Relationship to classical mechanics . 43 4.2 Non-classical degrees of freedom . 48 4.3 The single particle . 51 5 Propagators 53 5.1 Canonical quantum mechanics and causality . 53 5.2 The propagator in canonical quantum mechanics . 56 5.2.1 If the Hamiltonian is a quadratic function of momentum 58 5.2.2 free-particle propagators . 59 5.3 Propagators involving non-classical degrees of freedom . 61 5.4 Propagators as Green's functions . 62 6 Relativistic free particles and their propagators 65 6.1 The relativistic scalar particle propagator . 65 6.1.1 Positive and negative energy propagators . 65 6.1.2 The path integral for a relativistic particle . 68 6.1.3 Propagation backwards in time? . 71 6.2 The square root Hamiltonian . 74 6.3 Other relativistic Hamiltonians . 77 6.4 The Feshbach{Villars Hamiltonian . 78 6.4.1 Introduction . 78 6.4.2 Notation . 78 6.4.3 The two momentum representations . 79 6.4.4 The structure of the space of states . 82 6.4.5 The two position operators and their eigenstates . 83 6.4.6 A theory with two position spaces . 89 6.4.7 A small adjustment to restore Lorentz invariance . 91 6.5 The Dirac Hamiltonian . 94 6.6 The breakdown of relativistic quantum particle mechanics . 97 7 Quantum Field Theory 99 7.1 Covariant field theory . 100 7.1.1 Classical fields . 100 7.1.2 Quantum scalar fields and observables . 101 7.1.3 Non-scalar fields and observables . 103 7.2 A field theory with a classical source . 106 7.3 Location-specific information in Hilbert space . 113 3 7.4 Particles in quantum field theory . 116 7.4.1 The Fock representation . 116 7.4.2 The correspondence between quantum field theory and quantum mechanics . 119 7.4.3 Particle creation . 123 7.5 The return of relativistic causality violations . 124 7.5.1 Signalling using more than two measurements . 124 7.5.2 The limits of canonical quantum theory? . 127 8 Conclusions 129 A Appendix 134 A.1 Causal relation between observables in quantum mechanics . 134 A.2 The square root of the Klein{Gordon Hamiltonian as a convo- lution operation . 136 A.3 Note on the structure of the Feshbach{Villars space of states and the particle interpretation . 140 A.4 Interaction of a Feshbach{Villars particle with an electromag- netic field . 142 A.4.1 Minimal coupling in the canonical position space . 142 A.4.2 The non-relativistic limit . 143 A.5 Observables in a spinor field . 144 Word Count: 35151 4 Abstract An operational development of quantum theory is presented, from general correlations through single-particle theories to ele- mentary quantum field theory, with a focus throughout on causal relations. The objective is to establish the extent to which quan- tum theories are consistent with the principle of relativistic causal- ity (that no two events separated by a distance greater than their separation in time multiplied by the speed of light may have a causal influence on each other) and to examine the assumptions that this analysis requires. It is necessary to pay particular atten- tion to the notions of spatial location and measurement. It is found that in a relativistically causal theory, any measure- ment made in a finite spatial region must have the capacity for particle creation. A number of derivations are presented, including some rela- tivistic single-particle propagators and a Hamiltonian based on the square root of the Klein Gordon equation. 5 Declaration No portion of the work referred to in this dissertation has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. Copyright Statement The author of this dissertation (including any appendices and/or schedules to this dissertation) owns any copyright in it (the \Copyright") and he has given The University of Manchester the right to use such Copyright for any administrative, promotional, educational and/or teaching purposes. Copies of this dissertation, either in full or in extracts, may be made only in accordance with the regulations of the John Rylands University Library of Manchester. Details of these regulations may be obtained from the Librarian. This page must form part of any such copies made. The ownership of any patents, designs, trade marks and any and all other intellectual property rights except for the Copyright (the \Intellectual Prop- erty Rights") and any reproductions of copyright works, for example graphs and tables (\Reproductions"), which may be described in this dissertation, may not be owned by the author and may be owned by third parties. Such Intellectual Property Rights and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property Rights and/or Reproductions. Further information on the conditions under which disclosure, publication and exploitation of this dissertation, the Copyright and any Intellectual Prop- erty Rights and/or Reproductions described in it may take place is available from the Head of School of Physics and Astronomy. 6 Acknowledgements I owe a great deal of thanks to Jeff Forshaw, for his patience and encourage- ment, for the time and insight he has put into supervising this project, and most of all for posing a problem that fascinated and confused me so much I had to go back and look at quantum theory afresh. Thanks also to the Manchester HEP group for providing a positive and stimulating atmosphere, with excellent lectures, seminars and discussions. Special thanks to Tamora, my family and friends for all their help and support, and for accepting that I would be doing something strange for a little while. 7 1 Introduction 1.1 Causality The notion that some phenomena arise in dependence upon others is basic to any attempt to understand the world we find ourselves in. The relationship between the dependent events (`effects’) and the sets of events that they de- pend upon (`causes') is known as causality. The purpose of this work is to investigate this relationship in the context of quantum theory. Perhaps the most striking quality of causality is that we intuitively asso- ciate its direction { from cause to effect { with the direction of time. Causes appear to precede effects. Quantum theory is not known for its respect of what we might intuitively think, but in this case we will find our intuitions to be correct. Before I introduce the content and scope of this work, let us take a moment to narrow down what we mean by cause and effect. The philosophy of causal relations has deep roots and many subtleties [1, 2], but from the outset I wish to adopt a simplification that I suggest preserves the essential characteristics of causality without exposing us to crippling am- biguities. Consider the following causal relation: stepping outside in the rain causes you to get wet. Two characteristics contribute the majority of what is meaningful about this relation: • the effect (getting wet) is something we could detect and respond to in some way, and • the causes (stepping outside, rain) include some that are at least partly responsive to something we could do.1 Following this example, our simplification will be to focus only on effects that can be responded to, either by a device that can detect the event and is programmed to respond in a certain way or by a human being who can observe the effect and choose to act on what he or she finds, and to causes that are responsive to such a device or to something a human being could choose to do. There need be no limit on how quickly a response can take place. 1For example, if we chose to stay indoors, one of the causes would cease in response to our choice.