Table Of ContentSPRINGER BRIEFS IN PHYSICS
Matthias Lienert
Sören Petrat
Roderich Tumulka
Multi-time
Wave
Functions
An
Introduction
123
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Matthias Lienert S ren Petrat
(cid:129) (cid:129)
Roderich Tumulka
Multi-time Wave Functions
An Introduction
123
Matthias Lienert SörenPetrat
Fachbereich Mathematik Mathematical Sciences
Eberhard-Karls-UniversitätTübingen JacobsUniversity Bremen
Tübingen,Baden-Württemberg, Germany Bremen, Germany
Roderich Tumulka
Fachbereich Mathematik
Eberhard-Karls-UniversitätTübingen
Tübingen,Baden-Württemberg, Germany
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs inPhysics
ISBN978-3-030-60690-9 ISBN978-3-030-60691-6 (eBook)
https://doi.org/10.1007/978-3-030-60691-6
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Preface
A multi-time wave function is a quantum-mechanical wave function wðx1;...;xNÞ
which, for N particles, depends on N space-time variables xi ¼ðti;xiÞ2R4, and
thus on N time variables. It provides a natural generalization of wave functions to
relativistic space-time, in fact, a covariant position-representation of the quantum
state. As we elucidate, the applicability of this simple and elegant concept is not
limited to quantum mechanics but also comprises quantum field theory, where the
configuration is allowed to consist of any (variable) number N of particles. Since
simultaneous PDEs for several time variables can easily be inconsistent, the
requirementofconsistencyrestrictsthepossibletimeevolutionlawsformulti-time
wave functions and leads us to certain types of laws that can be expressed in
strikinglysimpleequations—aswouldseemverysuitableforfundamentalphysical
laws.Sotheconceptofmulti-timewavefunctionsturnsouttoberatherdeep,anda
powerful tool in the study of relativistic quantum dynamics (and some other
applications as well).
Theconceptwasconsideredalreadyintheearlydaysofquantummechanics,in
fact, since 1929 by illustrious physicists such as Eddington [15], Gaunt [16], Mott
[39], and, in particular, Dirac [8]. Surprisingly, the multi-time formalism was not
studied comprehensively for many years. Much work on these wave functions, on
the mathematically non-trivial consistency question of their evolution equations,
andonwaysofimplementinginteractionforthemisratherrecent.Itisparticularly
this recent work that we aim at presenting here in the first book devoted to
multi-time wave functions.
This book has originated from lectures at a spring school on multi-time wave
functions that took place in Tübingen, Germany, on April 10–12, 2019. The lec-
tures are intended for master-level students, doctoral students, and advanced
undergraduates;inaddition, they may also behelpfultoresearchers lookingfor an
overview of the topic. As prerequisites, the lectures assume familiarity with basic
notions of quantum mechanics and relativity. While in some places some supple-
mentary material has been added, the chapters in this book retain the concise style
of the lectures. We have included a number of exercises that were given to the
v
vi Preface
participants at the spring school, and which may aid the reader’s self-study. Each
chapterinthisbooktreatsoneofthelecturesinthespringschool,andthechapters
were mainly written by the corresponding presenters, i.e., Chaps. 1 and 6 by
Roderich Tumulka, Chaps. 2 and 4 by Sören Petrat, and Chaps. 3, 5, and 7 by
Matthias Lienert.
We hope that this little volume will open a door for many a reader to an
intriguing topic that is an active field of current research.
Tübingen, Germany Matthias Lienert
August 2020 Sören Petrat
Roderich Tumulka
Contents
1 Introduction and Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Relativistic Space-Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dirac Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 What is a Multi-time Wave Function? . . . . . . . . . . . . . . . . . . . . . 5
1.4 Multi-time Schrödinger Equations . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Consistency Conditions and Interaction Potentials . . . . . . . . . . . . . . 17
2.1 Rigorous Formulations of Consistency Conditions . . . . . . . . . . . . 17
2.2 Interaction Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Short Range Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Relativistic Point Interactions in 1+1 Dimensions . . . . . . . . . . . . . . . 31
3.1 Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Definition of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 General Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Boundary Conditions from Local Probability Conservation . . . . . . 34
3.5 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Lorentz Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Upshot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Multi-time Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Emission–Absorption Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Multi-time Emission–Absorption Model. . . . . . . . . . . . . . . . . . . . 45
4.4 Pair Creation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Tomonaga-Schwinger Approach . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
vii
viii Contents
5 Interior-Boundary Conditions for Multi-time Wave Functions. . . . . 55
5.1 What is an IBC?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Derivation of the IBC from Local Probability Conservation . . . . . 56
5.4 Relation to Creation and Annihilation Operators. . . . . . . . . . . . . . 57
5.5 Sketch of How to Construct the Solution of the Model. . . . . . . . . 58
5.6 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Born’s Rule for Arbitrary Cauchy Surfaces . . . . . . . . . . . . . . . . . . . 61
6.1 The Curved Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 Hypersurface Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Interaction Locality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5 Propagation Locality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Multi-time Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2 Derivation of a Suitable Integral Equation . . . . . . . . . . . . . . . . . . 71
7.3 How to Understand the Time Evolution. . . . . . . . . . . . . . . . . . . . 73
7.4 Non-relativistic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.5 Action-at-a-Distance Form of the Multi-time Equations. . . . . . . . . 74
7.6 Mathematical Analysis of the Integral Equation . . . . . . . . . . . . . . 76
7.7 Idea of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.8 On the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.10 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Conclusion.... .... .... .... ..... .... .... .... .... .... ..... .... 81
References.... .... .... .... ..... .... .... .... .... .... ..... .... 83
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 87
Chapter 1
Introduction and Overview
Inthischapter,weintroducetheconceptofamulti-timewavefunctionandrelated
basicnotionsandexamples.
1.1 RelativisticSpace-Time
Webeginbyintroducingsomenotation.LetM =R4beMinkowskispace-timewith
pointsx = xμ =(x0,x)=(ct,x).Henceforthwesetc=1.Themetricis
⎛ ⎞
1
gμν =ημν =⎜⎜⎝ −1 −1 ⎟⎟⎠, (1.1)
−1
see Fig. 1.1. A 4-vector xμ is called causal if it is timelike or lightlike (i.e., non-
spacelike, or xμx ≥0). We follow the convention that the set future(x) contains
μ
alsothefuturelightconeofx andx itself.Foraset A⊆M,onewrites
(cid:8)
future(A)= future(x), (1.2)
x∈A
likewiseforthepast.
TheLorentzgroupL = O(1,3)isthesetofallLorentztransformationsincluding
rotationsinspace,whichformtherotationgroupSO(3).TherestrictedLorentzgroup
orproperLorentzgroupL+ ⊂L containsthosetransformationsthatdonotreverse
eithertimeorspace.ThesymmetrygroupofMinkowskispace-timeisthePoincaré
group
(cid:9) (cid:10)
P = x (cid:6)→a+(cid:2)x : a∈M,(cid:2)∈L , (1.3)
whichincludesspace-timetranslations;correspondingly,onedefinesP+.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2020 1
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