On Eigenvectors 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. We study ways of calculating the Feynman propagator (for the free particle and the harmonic oscillator) as defined by physicists as an inner product $\langle x_{0}\vert K^{t}\vert x_{1}\rangle $ of the eigenvector $\vert x_{0}\rangle $ of the position operator with eigenvalue $x_{0}$ and $K^{t}(\vert x_{1}\rangle )$, where $K^{t}$ is the time evolution operator. We look at scalings needed to get the right result and finally end up showing that the eigenvector method does not work as expected. Complementing this, we show how to calculate the kernel of the time evolution operator in the limit model for the free particle and the harmonic oscillator. We believe that our method of calculating these is new.