Spring School on Multi-Time Wave Functions

Idea of the Spring School

The Spring School is particularly suitable for Master and PhD students in mathematical and theoretical physics. It provides the opportunity to learn more about a new and interesting topic at the intersection of quantum theory and relativity.

Multi-time wave functions were first suggested by Dirac in 1932 but investigated in detail only in recent years. The key idea is to study, for N quantum particles, a wave function psi(x_1,...,x_N) where each x_i is a spacetime point. One arrives at a new and manifestly covariant picture of quantum theory. Contrary to the usual wave function of Schrödinger's equation, psi then depends on many times which opens up several interesting mathematical and physical questions, such as:

• Which evolution equations for psi are appropriate?
• In which cases are these equations consistent?
• How can one implement interaction?
• Can one re-formulate quantum field theory using multi-time wave functions?
• What is the relation of psi with physical experiments? Does |psi|^2 represent a probability density?

In the Spring School, we will develop the topic of multi-time wave functions from scratch, answering the above-mentioned questions. Exercise sessions will complement the lectures. Prerequisites are: background knowledge of calculus, quantum mechanics, and relativity.

The topics of the lectures will include:

• consistency conditions
• inconsistency of potentials
• examples of consistent multi-time evolution with interaction
• connection with quantum field theory and the Tomonaga-Schwinger equation
• multi-time integral equations
• |psi|^2 and detection probability for detectors placed along a spacelike hypersurface

Essential information

Program of the Spring School

Date: April 10, 13:30 - April 12, 16:00, 2019

Location: University of Tübingen

Venue: Campus Auf der Morgenstelle, C-building (Fachbereich Mathematik), room N14 (ground floor).

Registration: please send an email to register!

Deadline: February 28, 2019

Organizers: Matthias Lienert and Roderich Tumulka

Poster: download pdf

Funding: External participants can apply for a contribution towards their traveling and accommodation expenses. Please indicate if you wish to apply in your registration email.

Speakers

• Matthias Lienert, University of Tübingen
• Sören Petrat, Jacobs University Bremen
• Roderich Tumulka, University of Tübingen

Lectures and further reading

Lecture 1

• Overview paper from 2017 (including many but not the most recent developments): M. Lienert, S. Petrat, and R. Tumulka, Multi-time wave functions versus multiple timelike dimensions, https://doi.org/10.1007/s10701-017-0120-5

Lecture 2

• Paper about the consistency condition, the impossibility of interaction potentials and potentials with small ranges: S. Petrat, and R. Tumulka, Multi-Time Schroedinger Equations Cannot Contain Interaction Potentials, arXiv:1308.1065
• Improved result on the impossibility of interaction potentials, also those with spin-spin coupling: D.-A. Deckert and L. Nickel: Consistency of multi-time Dirac equations with general interaction potentials, arXiv:1603.02538

Lecture 3

• Paper about the 1+1 dimensional model with relativistic point interactions M. Lienert: A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions, arXiv:1411.2833

Lecture 4

• S. Petrat, R. Tumulka: Multi-Time Wave Functions for Quantum Field Theory arXiv:1309.0802
• S. Petrat, R. Tumulka: Multi-Time Formulation of Pair Creation arXiv:1401.6093

Talk by Sascha Lill

Lecture 5A

• S. Petrat, R. Tumulka: Multi-Time Wave Functions for Quantum Field Theory arXiv:1309.0802

Lecture 5B

• M. Lienert and L. Nickel: Multi-time formulation of particle creation and annihilation via interior-boundary conditions arXiv:1808.04192

Lecture 6

• M. Lienert and R. Tumulka: Born's Rule on Arbitrary Cauchy Surfaces arXiv:1706.07074

Lecture 7

• Paper introducing the idea of multi-time integral equations: M. Lienert: Direct interaction along light cones at the quantum level, arXiv:1801.00060
• First mathematical results about multi-time integral equations: M. Lienert and R. Tumulka: A new class of Volterra-type integral equations from relativistic quantum physics, arXiv:1803.08792

Talk by Markus Nöth

• Paper about multi-time integral equations for Dirac particles: M. Lienert and M. Nöth: Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions, arXiv:1903.06020

Some historical references

• Paper introducing the concept multi-time wave functions: P. A. M. Dirac: Relativistic Quantum Mechanics, https://doi.org/10.1098/rspa.1932.0094
• A multi-time model for quantum electrodynamics with fixed electron number P. A. M. Dirac, V. A. Fock and B. Podolsky: On Quantum Electrodynamics.In Julian Schwinger, editor,Selected Papers on Quantum Electrodynamics. Dover Publica-tions, 1958.
• Paper about consistency conditions and statistical meaning of multi-time wave functions: F. Bloch. Die physikalische Bedeutung mehrerer Zeiten in der Quantenelektrodynamik.Phys. Zeitsch. d. Sowjetunion, 5:301–315, 1934
• Paper introducing hypersurface dependent wave functions S. Tomonaga: On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields https://doi.org/10.1143/PTP.1.27
• J. Schwinger: Quantum electrodynamics. I. A covariant formulation. https://link.aps.org/doi/10.1103/PhysRev.74.1439

Exercises

Exercises 1

Exercises 2