# Chemical Potential and interactions

I'm interested in an model with interactions between different kind's of particles. Each particle species has it's own chemical potential. I want to treat the system in the Matsubara formalism. Here, to my current understanding, the chemical potential enters as an additional term in the Hamiltonian to the propagators.
Calculating now quantities involving more kinds of species e.g. certain loop diagrams the relative energy between the species is shifted by the chemical potentials.

Let me give here an examples for my problem:

Consider a cavity filled with atoms. The photons inside the cavity can have a chemical potential, since the cavity mirror reflect the photons. The interaction between the photons and the atoms are the usual absorption and emission processes, which are resonant when the photon energy and the level spacing of the atoms match.

The photon energy is given by the mirror spacing and the atomic spectrum is fixed.

When now using the chemical potential for the photons as described above it seems that the energy of the photon is shifted, w.r.t. to the atom and therefore another resonance condition is present.

This seems to be unphysical in my view, since the chemical potential should just set the particle numbers for the different species and therefore enter over distribution functions.

The practicle problem in this formalism is that the distribution functions, like Fermi or Bose distribution, enter over the evaluation of Matsubara frequencie sums. Here there are just a nice trick to converte the sum to a contour integral and have not much to do with the chemical potential.

I would be glade if somebody could clarify my misconceptions!

## 2 Answers

If you write the action in path integral formalism then this action will have quadratic part for every flavor of particle you have $$\sum_\sigma\int dx \bar{\psi}_\sigma(iD_0+1/2m_\sigma D_j^2 +\mu_\sigma)\psi_\sigma+\text{higher order interaction terms}$$ where $$\sigma$$ denotes the flavor of your particles.

when you make matrubasa sums, yout will have such terms disregarding interactions $$h(\omega_n)=\frac{1}{-i\omega_n-E+\mu}$$ and when you want to evaluate the summation over $$n$$ by using residue theorem you have to be careful with the convergence. If you evaluate the sums properly being careful to convergence you will see that $$\mu$$ plays an important role.

The root of the problem is that the photon number is not conserved even with reflecting mirrors. In other words the photon number operator does not commute with the Hamiltonian. What is conserved is the number of photon plus excited atoms. For this an chemical potential can be introduced. This also preserves the resonance conditions.