lambda Phi⁴ Theory From a Particle-Gas Viewpoint

We discuss the physics of the 3+1 dimensional lambda Phi^4 quantum field theory in terms of the statistical mechanics of a gas of particles (`atoms') that interact via a -1/r^3-plus-hard-core potential. The hard-core potential, delta^(3)(r), arises from the bare vertex diagram, while the attractive, long-range -1/r^3 potential is due to exchange of a particle pair via the t,u-channel "fish" diagram. (Higher-order diagrams preserve this form of the interparticle potential.) For sufficiently small atom mass, the lowest-energy state is not the `empty' state with no atoms, but a state with a non-zero density of spontaneously created atoms, Bose-condensed in the zero-momentum mode. This corresponds to the spontaneous-symmetry-breaking phase transition, and the `phonon' excitations of the Bose condensate correspond to Higgs particles. The important point is that the phase transition happens while the atom's physical mass m is still positive: it does not wait until m^2 passes through zero and becomes negative, contrary to the assumption of a second-order transition, on which renormalization-group-improved perturbation theory is based.