A. Zee, Quantum Field Theory in a Nutshell · Problem V.2.2
Field Theory Redux
Physics — QFT
Problem
It probably does not surprise you that for fermionic fields the periodic boundary condition (6) is replaced by an antiperiodic boundary condition ψ(x,0)=−ψ(x,β) in order to reproduce the results of chapter II.5. Prove this by looking at the simplest fermionic functional integral. [Hint: The clearest exposition of this satisfying fact may be found in appendix A of R. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D12: 2443, 1975.]