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The intersection of condensed matter and high-energy physics, for which the path integral method was originally invented, is based on the fundamental formula \( \Omega =-T\ln Z=-T\ln \displaystyle \sum _{n}e^{-\beta E_n} \) where Z is the partition function - statistical sum, \( Z=\sum _n\langle {n}\vert \exp (-\beta H)\vert {n}\rangle =\text{ Tr }\,\exp (-\beta H) \) and \(E_n\) are the eigenvalues of the Hamiltonian H. It is well known that the grand canonical thermodynamic potential \(\Omega (T,V,\mu )\) contains all the information on the equilibrium system confined in the volume V with the chemical potential \(\mu \) [1, 2].

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Path Integrals for Bosons

  • 章节

  • 发表日期：

    2026年

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  • Abdulla Rakhimov,
  • Shukhrat Mardonov

摘要

The intersection of condensed matter and high-energy physics, for which the path integral method was originally invented, is based on the fundamental formula \( \Omega =-T\ln Z=-T\ln \displaystyle \sum _{n}e^{-\beta E_n} \) where Z is the partition function - statistical sum, \( Z=\sum _n\langle {n}\vert \exp (-\beta H)\vert {n}\rangle =\text{ Tr }\,\exp (-\beta H) \) and \(E_n\) are the eigenvalues of the Hamiltonian H. It is well known that the grand canonical thermodynamic potential \(\Omega (T,V,\mu )\) contains all the information on the equilibrium system confined in the volume V with the chemical potential \(\mu \) [1, 2].