Statistical mechanics of the nonrelativistic Zachariasen model

The statistical mechanics of the nonrelativistic Zachariasen model is studied for the purpose of clarifying a mechanism proposed some years ago by Omn\`es to effect a phase transition in hot, dense matter with zero net baryon number from a homogeneous thermodynamic state to one with local inhomogenities in the baryon number density. The model contains $B$ and $B$ particles, which play the role of baryons and antibaryons, respectively, and $A$ particles, which play the role of mesons. The $A$ particles are coupled in the Hamiltonian to $B\overline{B}$ pairs through virtual annihilation and creation. The Helmholtz free energy is calculated in a virial expansion for the cases in which real $B\overline{B}$ pair annihilation and creation is (1) important enough to achieve full thermodynamic equilibrium, and (2) unimportant enough so that over an appropriate time scale the $A$ particles and $B\overline{B}$ pairs are independent of one another. To analyze case (2) a unitary transformation is made on the original Hamiltonian which completely renormalizes the $A$-particle rest energy and transforms the original interaction into an infinite sum of potentials that act within groups containing increasing numbers of particles. We then study case (2) by dropping potentials that involve real creation and annihilation. Within the context of the virial expansion we conclude that no Omn\`es transition occurs for case (1), while such a transition is possible for case (2).