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On Model Theory, Zilber and Physics

Åsa Hirvonen, Tapani Hyttinen

Published Jun 19, 2014• 1 citation
OpenAlex

Abstract

We study B. Zilber's method of calculating the Feynman propagator for the free particle and harmonic oscillator.

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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 x0x_{0}x0​ and Kt(∣x1⟩)K^{t}(\vert x_{1}\rangle )Kt(∣x1​⟩), where KtK^{t}Kt 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.
In this paper, the propagators for a particle moving in a time-dependent linear potential and a free particle with linear damping are calculated by the application of the integrals of the motion of a quantum system. The propagator for a charged harmonic oscillator is derived from the Feynman path integrals and the propagator for a damped harmonic oscillator is evaluated by the Schwinger method. The relation between the integrals of the motion, Feynman path integrals, and Schwinger method are also described.