"I don't play accurately - any one can play accurately - but I play with wonderful expression."

- Oscar Wilde, The Importance of Being Earnest, Act 1

Wilzcek's time crystals fit into this scheme as well. I will try to put Wilzcek's lecture from yesterday in Boston on youtube soon. He actually says some things relevant to the above.

There is an analogy here to Feynman's treatment of EM gauge invariance eliminating the timelike photon polarization getting photon propagator in k,f space with (in rest frame of electron)

1/k^2 as the NEAR FIELD instantaneous VIRTUAL PHOTON Coulomb part (longitudinal polarization) obviously for electrostatics

these virtual photons have f = 0 and all possible k's with a distribution ~ 1/k^2 (Fourier transform)

1/(f^2 - k^2 ) for FAR FIELD propagating transverse polarization modes as REAL PHOTONS

OK well what is a space crystal?

T3 spontaneously breaks to a discrete crystal group

the photon is the quantum of the locally gauged U1 group - without spontaneous breaking of its symmetry that does happen in a superconductor where the photon gets rest mass. (Anderson-Higgs mechanism)

the phonon is the collective emergent quantum when the 3D translation group spontaneously breaks to a discrete crystal group.

the acoustic longitudinal phonon is then the zero mass Goldstone phase mode

f = csk for speed of sound in material

there are also massive transverse phonons that are the Higgs amplitude quanta

The crystal lattice itself is a Glauber coherent state of VIRTUAL PHONONS

f = 0 with k = 1/a

a = lattice spacing

note there are UMKLAPP processes from APPARENT VIOLATION OF MOMENTUM CONSERVATION - they have to be there in time crystals as well

Umklapp scattering Umklapp scattering (also U-Process or Umklapp process) is an anharmonic phonon-phonon (or electron-phonon) scattering process creating a third phonon with a k-vector outside the first Brillouin zone. (references)

Umklapp process [′u?m‚kläp ‚prä·s?s]

(solid-state physics)

The interaction of three or more waves in a solid, such as lattice waves or electron waves, in which the sum of the wave vectors is not equal to zero but, rather, is equal to a vector in the reciprocal lattice. Also known as flip-over process.

Umklapp process

A concept in the theory of transport properties of solids which has to do with the interaction of three or more waves in the solid, such as lattice waves or electron waves. In a continuum, such interactions occur only among waves described by wave vectors k 1, k 2, and so on, such that the interference condition, given by Eq. (1), is satisfied.

(1)

The sign of k depends on whether the wave absorbs or emits energy. Since ℏ k is the momentum of a quantum (or particle) described by the wave, Eq. (1) corresponds to conservation of momentum. In a crystal lattice further interactions

(2)

occur, satisfying Eq. (2), where b is any integral combination of the three inverse lattice vectors b i, defined by a · b j = 2&pgr;δij, the a 's being the periodicity vectors. The group of processes described in Eq. (2) are the Umklapp processes or flip-over processes, so called because the total momentum of the initial particles or quanta is reversed. See Crystal

The Glauber coherent state is the eigenstate of the non-Hermitian field quantum destruction operator and its time evolution is nonlinear and non-unitary and also local in space-time in an important way that the Fock states are not.

The first order density matrix corresponding to a Glauber macro-quantum coherent state has a macroscopic eigenvalue leading to the local in space order parameter - giant single particle Psi(x) (with a large number of bosons in same single-particle cell h^3 of phase space. (ODLRO of Penrose-Onsager).

Similarly in the Wilczek TIME CRYSTAL

we have a Glauber coherent state of virtual phonons of finite frequency f and zero wave vector k from the spontaneous symmetry breaking of T1 in the ground state of a many particle substrate.

what are the TEMPORAL UMKLAPP apparent ENERGY NON-CONSERVATION scatterings?

f1 + f2 + f3 =/= 0

f1 + f2 + f3 = b

where b is in the 1D reciprocal time lattice?

But in any case, a spontaneous symmetry breaking selects ONE of continuum of energy degenerate ground states that are not unitarily equivalent to each other so we never have the Schrodinger Cat Paradox. This explains why we see ONE WORLD macroscopically because the CLASSICAL WORLD is a set of FROZON OFF-MASS SHELL BOSON CONDENSATES (essentially Glauber coherent states of emergent boson quasi-particles and collective modes of underlying real fermions and bosons.