关于特征向量、近似和Feynman算子
On Eigenvectors, Approximations and the Feynman Propagator
摘要：
试图解释B。在Zilber的量子力学模型论项目中，我们研究了一种从有限维近似建立极限模型的方法。我们的观点是度量模型论的观点，并且我们发展了一种取无界算子超积的方法。我们首先计算自由粒子的费曼传播子，由物理学家定义为位置算子的本征向量$| x_{0}\rangle $与本征值$x_{0}$和$K^{t}(| x_{1}\rangle )$的内积$\langle x_{0}| K^{t}| x_{1}\rangle $，其中$K^{t}$是时间演化算子。然而，由于离散化效应，特征向量法不能像预期的那样工作，并且没有大量的逐例缩放，它给出了错误的值。我们来看看这个现象，然后通过展示如何在极限模型中计算时间演化算子的核（对于自由粒子和谐振子）来补充这个现象。我们认为，我们的计算方法是新的。
英文摘要：
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.
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