Local-time representation of path integrals

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Petr Jizba, Václav Zatloukal

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.

References

Nov 15, 1987·Physical Review A: General Physics·D Sokolovski, L M Baskin
Jun 22, 1987·Physical Review Letters·C M BenderL M Simmons
Mar 4, 2014·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Petr Jizba, Václav Zatloukal

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Citations

Mar 17, 2017·Physical Review. E·Petr JizbaVáclav Zatloukal
Jun 18, 2017·Physical Review. E·Václav Zatloukal

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