On Eigenvectors, Approximations and the Feynman Propagator

Abstract

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product <x(0) vertical bar K-t vertical bar x(1)> of the eigenvector vertical bar x(0)> of the position operator with eigenvalue x(0) and K-t (vertical bar x(1)>), where K-t is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new. (C) 2018 The Authors. Published by Elsevier B.V.