A free particle in infinite space flat space is an idealization we will use here.

Its quantum wave form can be a plane wave

psi(

**r**,t) ~

e^{i}^{(k.r - ft)}
If the particle is real then it is on mass shell

i.e. in special relativity

f^2 = (

ck)^2 + (mc^2/h)^2

m = rest mass from Higgs-mechanism and quantum

chromodynamics - purely local

If the particle is virtual then the there is no constraint between f (energy/h) and wave number (momentum/h) where h is Planck's quantum of action.

example 1

Spatial crystals are

Higgs-Goldstone Bose-Einstein analog -

macro-quantumcoherent Glauber states of VIRTUAL PHONONS of zero frequency f = 0 and finite base wave vectors k in the reciprocal lattice.

A phonon is a normal collective mode of pure vibration of all the atoms/molecules that order into the crystal when the spatial translation subgroup of the

Poincare group is spontaneously broken below the freezing point.