Willkommen zum Oberseminar Mathematische Physik

We consider a non-relativistic quantum particle surrounded by a detecting surface and ask how to compute, from the particle's initial wave function, the probability distribution of the time and place at which the particle gets detected. In principle, quantum mechanics makes a prediction for this distribution by solving the Schrodinger equation of the particle of interest together with the 10^23 (or more) particles of the detectors, but this is impractical to compute. Is there a simple rule for computing this distribution approximately for idealized detectors? I will argue in favor of a particular proposal of such a rule, the "absorbing boundary rule," which is based on a 1-particle Schrodinger equation with a certain "absorbing" boundary condition on the detecting surface. The mere existence of such a rule may seem surprising in view of the quantum Zeno effect. Time permitting, I may also be able to explain extensions of this rule to the cases of several particles, moving detectors, particles with spin, Dirac particles, curved space-time, and discrete space (a lattice). Some of the results are based on joint work with Abhishek Dhar and Stefan Teufel.

Suppose that particle detectors are placed along a Cauchy surfaces \Sigma in Minkowski space-time. Given an appropriate wave function \psi_\Sigma on \Sigma, one can easily guess the respective "curved Born rule": the probability distribution of the detected configuration on \Sigma has density |\psi_\Sigma|^2 (with |.|^2 suitably understood). However, simply postulating the curved Born rule seems neither appropriate nor necessary, as in principle the usual measurement postulates at equal times in any single Lorentz frame should already determine the results of all conceivable experiments. This situation has been the motivation of a recently finished joint work with Roderich Tumulka (arXiv:1706.07074), about which I will report in this talk. We define an idealized detection process by approximating a curved Cauchy surface by small horizontal (equal-time) pieces and prove that the probability distribution coincides with |\psi_\Sigma|^2. For this result, we make use of two crucial hypotheses on the time evolution: 1. no interaction faster than the speed of light, and 2. no particle creation from the vacuum.