  We discuss the phase transition in 3+1 dimensional lambda Phi^4 theory from a
very physical perspective. The particles of the symmetric phase (`phions')
interact via a hard-core repulsion and an induced, long-range -1/r^3
attraction. If the phion mass is sufficiently small, the lowest-energy state is
not the `empty' state with no phions, but is a state with a non-zero density of
phions Bose-Einstein condensed in the zero-momentum mode. The condensate
corresponds to the spontaneous-symmetry-breaking vacuum with <Phi> neq 0 and
its excitations ("phonons" in atomic-physics language) correspond to Higgs
particles. The phase transition happens when the phion's physical mass m is
still positive; it does not wait until m^2 passes through zero and becomes
negative. However, at and near the phase transition, m is much, much less than
the Higgs mass M_h. This interesting physics coexists with `triviality;' all
scattering amplitudes vanish in the continuum limit, but the vacuum condensate
becomes infinitely dense. The ratio m/M_h, which goes to zero in the continuum
limit, can be viewed as a measure of non-locality in the regularized theory. An
intricate hierarchy of length scales naturally arises. We speculate about the
possible implications of these ideas for gravity and inflation.
