On Eigenvectors, Approximations and the Feynman Propagator

TL;DR

This paper develops a metric model theory approach to quantum mechanics, focusing on ultraproducts of unbounded operators and providing a novel method to compute the Feynman propagator for free particles and harmonic oscillators.

Contribution

It introduces a new method for calculating the Feynman propagator using ultraproducts of unbounded operators within metric model theory.

Findings

Calculated the Feynman propagator for free particles and harmonic oscillators.

Identified limitations of eigenvector methods due to discretising effects.

Proposed a new approach to compute the kernel of the time evolution operator.

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 $\langle x_{0}| K^{t}| x_{1}\rangle $ of the eigenvector $| x_{0}\rangle $ of the position operator with eigenvalue $x_{0}$ and $K^{t}(| x_{1}\rangle )$, where $K^{t}$ is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and without heavy case-by-case scaling, it gives the wrong value. We look at this phenomenon, and then complement this by showing how to instead 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.