A. Zee, Quantum Field Theory in a Nutshell · Problem V.2.3

Field Theory Redux

Physics — QFT

Problem

It is interesting to consider quantum field theory at finite density, as may occur in dense astrophysical objects or in heavy ion collisions. (In the previous chapter we studied a system of bosons at finite density and zero temperature.) In statistical mechanics we learned to go from the partition function to the grand partition function Z=tr e−β(H−μN)Z = \text{tr } e^{-\beta(H-\mu N)}, where a chemical potential μ\mu is introduced for every conserved particle number NN. For example, for noninteracting relativistic fermions, the Lagrangian is modified to L=ψˉ(i∂̸−m)ψ+μψˉγ0ψL = \bar{\psi}(i\not{\partial} - m)\psi + \mu\bar{\psi}\gamma^0\psi. Note that finite density, as well as finite temperature, breaks Lorentz invariance. Develop the subject of quantum field theory at finite density as far as you can.

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