# Questions tagged [path-integral]

The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics.

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### How to compute this path integral?

Let $\mathbb{R}^2$ be phase space with coordinates $(p,q)$ and let $\epsilon>0\,.$ Then given any path $\gamma:[0,1]\to \mathbb{R}^2$ and any large enough $N>0\,,$ we can approximate $\gamma$ by ...
273 views

### Finding examples of functions which are infinite or undefined with current extensions of the expected value?

Preliminaries Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
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### How to make estimation of probabilities of atypically large fluctuations in random matrix theory rigorous?

Suppose we try to rigorously answer questions like: Given a random $n\times n$ matrix from GUE (Gaussian Unitary Ensemble), what is the probability $P^{\text{GUE}}_n$ that it is positive semidefinite?...
287 views

### Double integral in a polygon domain

I want to compute a integral of a polynomial $f(x, y)$ over a polygon domain $D$ of $n$ sides. $$I(f) = \int_{D} f(x, \ y) \ dx \ dy$$ The vertex of this polygon are \vec{p}_{i} = (x_i, \ y_i) \ \ ...
241 views

### Reference request: path integral approach to Gaussian processes

Are there any good, rigorous and preferably modern books or papers on path integral approach to Gaussian processes? I am interested in both introductory level and deeper monographs on the subject. I ...
3k views

### Looking for some interesting complex integration contours

I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I ...
1k views

### Condensed/liquid vector spaces and path integrals

[Edited to take into account comments.] Background One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
750 views

### Peter Freyd on path Integral?

In the issue Electronic Notes in Theoretical Computer Science Volume 29, 1999, Page 79 there is a very intriguing abstract by Peter Freyd. Path Integrals, Bayesian Vision, and Is Gaussian Quadrature ...
313 views

### Ratios of Gaussian integrals with a positive semidefinite matrix

Cross-post from MSE https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix Generally speaking, I’m wondering what the usual identities for ...
172 views

### References for computing $n$-point correlations in Chern-Simons theory

I am interested in learning how to compute $n$-point correlation functions in Chern-Simons theory, thought of as a TQFT (similar to Witten's work linking that theory to knot theory). I am mostly ...
439 views

### Path integral as quantum mechanics on the tangent bundle

Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a ...
Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups). However, it is not entirely clear the formulation of Yang-Mills theory with non-...