The natural generalization of the quantum-mechanical N-particle wave function to relativistic space-time is a function of N space-time points, and thus of N time variables. This book, based on a collection of lectures given at a spring school in Tübingen in 2019, provides an accessible and concise introduction to the recent development of the theory of multi-time wave functions, their use in quantum field theory, their relation to detection probabilities, and the mathematical question of consistency of their time evolution equations. The book is intended for advanced students and researchers with an interest in relativity and quantum physics.
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1 Introduction and Overview |
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1 | (16) |
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1.1 Relativistic Space-Time |
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1 | (1) |
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2 | (3) |
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1.3 What is a Multi-time Wave Function? |
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5 | (5) |
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1.4 Multi-time Schrodinger Equations |
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10 | (4) |
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14 | (3) |
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2 Consistency Conditions and Interaction Potentials |
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17 | (14) |
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2.1 Rigorous Formulations of Consistency Conditions |
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17 | (6) |
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2.2 Interaction Potentials |
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23 | (3) |
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2.3 Short Range Interactions |
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26 | (1) |
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27 | (4) |
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3 Relativistic Point Interactions in 1+1 Dimensions |
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31 | (10) |
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31 | (1) |
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3.2 Definition of the Model |
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31 | (2) |
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33 | (1) |
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3.4 Boundary Conditions from Local Probability Conservation |
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34 | (3) |
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37 | (1) |
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38 | (1) |
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38 | (1) |
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39 | (2) |
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4 Multi-time Quantum Field Theory |
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41 | (14) |
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41 | (2) |
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4.2 Emission-Absorption Model |
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43 | (2) |
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4.3 Multi-time Emission-Absorption Model |
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45 | (2) |
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47 | (1) |
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48 | (2) |
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4.6 Tomonaga-Schwinger Approach |
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50 | (2) |
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52 | (3) |
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5 Interior-Boundary Conditions for Multi-time Wave Functions |
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55 | (6) |
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55 | (1) |
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55 | (1) |
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5.3 Derivation of the IBC from Local Probability Conservation |
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56 | (1) |
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5.4 Relation to Creation and Annihilation Operators |
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57 | (1) |
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5.5 Sketch of How to Construct the Solution of the Model |
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58 | (1) |
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5.6 Conclusion and Outlook |
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59 | (1) |
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60 | (1) |
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6 Born's Rule for Arbitrary Cauchy Surfaces |
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61 | (10) |
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61 | (2) |
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63 | (3) |
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6.3 Hypersurface Evolution |
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66 | (2) |
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68 | (1) |
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68 | (1) |
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69 | (2) |
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7 Multi-time Integral Equations |
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71 | (10) |
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71 | (1) |
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7.2 Derivation of a Suitable Integral Equation |
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71 | (2) |
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7.3 How to Understand the Time Evolution |
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73 | (1) |
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7.4 Non-relativistic Limit |
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74 | (1) |
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7.5 Action-at-a-Distance Form of the Multi-time Equations |
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74 | (2) |
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7.6 Mathematical Analysis of the Integral Equation |
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76 | (1) |
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77 | (1) |
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7.8 On the Cauchy Problem |
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78 | (1) |
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78 | (1) |
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79 | (1) |
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79 | (2) |
| Conclusion |
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81 | (2) |
| References |
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83 | (4) |
| Index |
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87 | |
Born in 1988 in Würzburg, Germany, Matthias Lienert studied physics in Göttingen and theoretical physics in Cambridge, UK. He obtained his PhD in Mathematics at Ludwig Maximilians University in Munich in 2015. Since then he has been a postdoctoral researcher in mathematical physics, in particular relativistic quantum theory, at Rutgers University, USA, and the University of Tübingen, Germany.
Sören Petrat was born in 1985 in Castrop-Rauxel (Germany). After obtaining a PhD in mathematics from Ludwig Maximilians University Munich, he worked as a postdoctoral researcher at IST Austria, IAS Princeton, and Princeton University. Since 2017 he is Professor at Jacobs University Bremen. His field of research is mathematical physics, specifically many-body quantum mechanics and relativistic quantum physics.
Roderich Tumulka was born in 1972 in Frankfurt am Main (Germany), earned a Ph.D. in mathematics from Ludwig Maximilians University in Munich, taught at Rutgers University (USA) in 20072016, and has since taught at Eberhard Karls University in Tübingen (Germany). His field of research is mathematical physics, particularly the foundations of quantum mechanics, quantum field theory, and quantum statistical mechanics.
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