  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.
