• Mikko Laine
  • Aleksi Vuorinen
Part of the Lecture Notes in Physics book series (LNP, volume 925)


A fermionic (spin-1/2) field is considered at finite temperature. Starting with a fermionic analogue of the harmonic oscillator and proceeding to the case of a field satisfying a Dirac equation, an imaginary-time path integral representation is derived for the partition function. This leads to the concept of Grassmann variables satisfying antiperiodic boundary conditions. The corresponding Matsubara frequencies are introduced, the partition function is evaluated in the low and high-temperature expansions, and the structures of these expansions are compared with those of a scalar field theory.


Fermionic oscillator Grassmann variables Antiperiodic boundary conditions Dirac field Dirac matrices Low and high-temperature expansions for fermions 


  1. 1.
    F.A. Berezin, The Method of Second Quantization (Academic, New York, 1966)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mikko Laine
    • 1
  • Aleksi Vuorinen
    • 2
  1. 1.AEC, Institute for Theoretical PhysicsUniversity of BernBernSwitzerland
  2. 2.Department of PhysicsUniversity of HelsinkiHelsinkiFinland

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