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Petr Jizba1,2, Václav Zatloukal3

  • 1ITP, Freie Universität in Berlin, Arnimallee 14, D-14195 Berlin, Germany; 2. FNSPE, Czech Technical University in Prague, Bˇrehov´a 7, 115 19 Praha 1, Czech Republic.

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We developed a new local-time path integral method for quantum systems. This approach offers a powerful alternative to existing formulas, especially at extreme temperatures.

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Mathematical physics

Background:

  • The Feynman path integral is a fundamental tool in quantum mechanics.
  • Calculating matrix elements of the Bloch density matrix is crucial for understanding thermal properties of quantum systems.
  • Existing methods like the Feynman-Kac formula have limitations, particularly at high and low temperatures.

Purpose of the Study:

  • To derive a novel local-time path-integral representation for time-independent quantum systems.
  • To rephrase matrix elements of the Bloch density matrix using local-time profiles.
  • To provide a powerful alternative to the Feynman-Kac formula, especially in extreme temperature regimes.

Main Methods:

  • Derivation of a local-time path-integral representation for one-dimensional systems.
  • Expressing Bloch density matrix elements as path integrals over local-time profiles.
  • Generalization to include arbitrary functionals of local time.

Main Results:

  • A new path-integral formulation based on local time is established.
  • The local-time profiles quantify the time spent by sample paths near specific points.
  • The method is shown to be a powerful alternative to the Feynman-Kac formula, particularly for analyzing low-temperature asymptotic behavior.

Conclusions:

  • The derived local-time path-integral representation offers a significant advancement in quantum mechanical calculations.
  • This method provides new insights into the behavior of the Bloch density matrix, especially at low temperatures.
  • Connections to Sturm-Liouville theory and Rayleigh-Ritz variational principle are discussed, highlighting the broad applicability of the approach.