A Stochastic Approach to the Definition of the Path Integral Measure
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918282604314624 |
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| author | Obolenskiy, Timur |
| author_facet | Obolenskiy, Timur |
| contents | We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed endpoints with a referential non-degenerate classical trajectory, formulating a framework around a quadratic Lagrangian. Through fibration, we reduce the infinite-dimensional space under consideration to an $L^2$-isometric flux spaces in which we consider a stochastic process associated to a Gaussian measure. The Path Integral is subsequently defined as an expectation value with respect to the Gaussian measure, allowing us to rigorously formulate the former as a functional integral. We prove mathematical correspondence between the Stochastic Path Integral and the Euclidean Path Integral theory formulated rigorously under the Feynman-Kac theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Stochastic Approach to the Definition of the Path Integral Measure Obolenskiy, Timur Probability Mathematical Physics We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed endpoints with a referential non-degenerate classical trajectory, formulating a framework around a quadratic Lagrangian. Through fibration, we reduce the infinite-dimensional space under consideration to an $L^2$-isometric flux spaces in which we consider a stochastic process associated to a Gaussian measure. The Path Integral is subsequently defined as an expectation value with respect to the Gaussian measure, allowing us to rigorously formulate the former as a functional integral. We prove mathematical correspondence between the Stochastic Path Integral and the Euclidean Path Integral theory formulated rigorously under the Feynman-Kac theorem. |
| title | A Stochastic Approach to the Definition of the Path Integral Measure |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2511.15772 |