Condensed matter problem with Feynman integral

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SUMMARY

The discussion centers on deriving the final expression for a particle in a potential coupled to a bath of harmonic oscillators using the Feynman integral approach. Key points include the assumption of periodicity for the variable q on the interval [0,T-1] and the application of Fourier series expansion for q. The transformation of the integral in dt' to an integral in τ via the substitution t' -> -iτ is also highlighted. The Matsubara formalism, as discussed in Fetter and Valecka, is suggested as a reference for further understanding.

PREREQUISITES
  • Understanding of Feynman integrals
  • Knowledge of harmonic oscillators in quantum mechanics
  • Familiarity with Fourier series expansion
  • Basic grasp of the Matsubara formalism
NEXT STEPS
  • Study the application of Feynman integrals in condensed matter physics
  • Explore the derivation of Fourier series expansions in quantum mechanics
  • Review the Matsubara formalism in Fetter and Valecka's texts
  • Investigate the implications of periodic boundary conditions in quantum systems
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Physicists, particularly those specializing in condensed matter physics, quantum mechanics students, and researchers working on Feynman integrals and harmonic oscillator systems.

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Im working on deriving the final expression on the attached picture. The problem is that of a particle in a potential coupled to a "bath" of harmonic oscillators but I'm not sure how you arrive at the final expression. First of all, why are you allowed to assume that q is periodic on the interval [0,T-1]? And secondly how should I use this? Should I plug in the Fourier series expansion of q? And in that case what should I do with the integral in dt' (which I suppose has been transformed to an integral in τ by t'->-iτ)
 

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Did you have a look on how the Matsubara formalism works, e.g. in Fetter/ Valecka?
 

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