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:

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:

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.


Lectures and further reading

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Talk by Sascha Lill

Lecture 5A

Lecture 5B

Lecture 6

Lecture 7

Talk by Markus Nöth

Some historical references


Exercises 1

Exercises 2

How to get there:

Buses: From the city center, take lines 5, 13, 18 or 19 to the stop "Auf der Morgenstelle". Bus plans

Interactive Campus map

From the bus stop to the Mathematics Building


Dinner 1: Wednesday, 6:30 p.m., Restaurant Zum Alten Fritz, Gartenstr. 13, 72074 Tübingen

Dinner 2: Thursday, 6:45 p.m., Restaurant Neckarmüller, Gartenstr. 4, 72074 Tübingen

Accommodation for external participants

Participants are expected to make their own arrangements for accommodation. In the city center of Tübingen, there are many hotels which are well-connected to public transport. The DJH Youth Hostel is a particularly affordable option.

This project has received funding from the European Union's Framework for Research and Innovation "Horizon 2020" (2014-2020) under the Marie Sklodowska-Curie Grant Agreement No. 705295.