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arXiv · 1506.00888

Local-time representation of path integrals

Petr Jizba·Vaclav Zatloukal
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Abstract

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high and low temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle are also discussed.

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Petr Jizba, Vaclav Zatloukal. 2015-06-02. Local-time representation of path integrals. https://doi.org/10.1103/physreve.92.062137

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