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One of the simplest and most common physical objects is your average crystal, a collection of atoms arranged in an orderly, repeating three-dimensional pattern. Salt, snowflakes, and the quartz in your watch are all crystals. Earlier this year, the Nobel laureate and MIT physicist Frank Wilczek prop...

Jack Sarfatti

There is a very intuitive though not simple way to understand the space-time crystal.
1) spontaneous broken symmetry in complex many particle systems. These are quantum phase transitions like when our observable universe is created in the moment of inflation out of the pre-existing unstable false vacuum in which all particles have zero rest mass because the Higgs field had not yet formed. The appearance of the Higgs field is the effect spontaneous broken symmetry in which the post-inflation quantum vacuum of our expanding universe. The quantum vacuum has less symmetry than do the field equations for some of the matter fields.
2) Quantum field theory shows that matter exists in two very different forms - real and virtual. Matter in virtual form lives inside the quantum vacuum briefly popping into and out of existence. We see this indirectly in small shifts of spectral lines of atoms (Lamb shift) and in the Casimir zero point force between two neutral plates. Virtual particles do not transport energy outside the "near field" and they cannot directly cause a counter to click only real particles can do that. The LHC just showed us a real Higgs boson kicked out of the vacuum by the tremendous focused energy of the machine. It's like chipping a small piece of ice out of a huge glacier that is the VIRTUAL Higgs-Goldstone spontaneous broken symmetry field inside the vacuum. There are two kinds of spontaneous broken symmetry particles. The Goldstone particle has zero rest mass like the photon particle of light. The Higgs particle has a finite rest mass now seen at about 125 Gev in the LHC. There may be several Higgs and Goldstone particles. The Higgs and Goldstone particles come in conjugate pairs like the amplitude and phase of a coherent laser beam wave. In fact the Higgs-Goldstone vacuum field is mathematically somewhat similar to a laser beam field with some important differences of course. The mathematics of these general "coherent states" was worked out in the early 1960's by Nobel Laureate Harvard physics professor Roy Glauber. Basically we have a large number of particles all in the same single-particle quantum state although that actual number is uncertain and in the simplest case has a Poisson distribution.This happens not only in lasers but in superconductors and as we see below even in Frank Wilczek's space-time crystal. The difference is that the Higgs vacuum field that itself gives rest masses to the leptons and quarks is made up of huge numbers of VIRTUAL Higgs-Goldstone conjugate particle pairs that form a set of complex numbers z in the polar representation for those of you who know some high school math where z = Rexp(itheta). R is the amplitude and theta is the phase. The massive Higgs particle in real form are quantized vibrations in the amplitude R like you AM radio. The massless Goldstone particles in real form are quantized vibrations in the phase theta of the coherent vacuum field like your FM radio roughly.
3) A space crystal is a periodic lattice of atoms that forms in a quantum phase transition in which the continuous translational symmetry of the higher temperature gas or liquid is spontaneously broken down to a much smaller discrete crystal group. The phonon is a massless Goldstone particle. The analogous Higgs particle would be a phonon sound wave with an energy gap at infinite wavelength. However, a single phonon is a collective normal mode of all the real atoms that form the crystal lattice. Now real phonons that propagate sound energy have a frequency that is the speed of sound divided by the wavelength. However, virtual phonons do not obey that relationship at all. Indeed, the crystal lattice itself is a Glauber coherent state of a huge uncertain number of VIRTUAL PHONONS all in the same single-phonon quantum state. These particular virtual phonons have zero frequency with finite wavelengths along the three directions of space that are determined by the particular discrete space-crystal group that is not spontaneously broken. A very similar thing happens for electromagnetic photons in the ordinary electrostatic Coulomb field e/r potential energy per unit test charge q in the rest frame of a point charge e where r is the distance between e and q. The longitudinal electrostatic field is a coherent Glauber state of a huge uncertain number of virtual photons of zero frequency with a whole continuum of wavelengths along the three dimensions of space.
4) We now have a unified conceptual framework. The space-time crystal is simply a Glauber coherent state of again virtual phonons but this time with a finite frequency and the same set of discrete wavelengths as in the space-crystal.