It’s a difficult paper to understand fully. However, their main point seems to be that they have an algorithm for computing key properties like superconductivity of non-relativistic v/c << 1 Galilean relativity many-particle systems with spontaneous broken ground state symmetries. Unlike the special relativity case, the number of massless Goldstone bosons need not be equal to the number of broken symmetry Lie algebra generators dim G - dim H when G ---> G/H (coset space of degenerate macro-quantum coherent ground states), H is the residual unbroken symmetry group where G is the initial symmetry group prior to the quantum ground state phase transition.

Also, localizing the global symmetry group is not studied in their paper.

It’s important to realize - and this is my insight not in the paper, that in every case the actual ground states in the coset space is a ROBUST not FRAGILE coherent condensate of VIRTUAL zero frequency Goldstonebosons as distinct from real Goldstone bosons.

Furthermore the condensates are essentially generalized Glauber states for the relevant Lie algebra of the particular many-particle system.

For example, space crystals are Glauber states of zero frequency virtual phonons (1 longitudinal 2 transverse) with finite wave vectors that are the reciprocals of the lattice unit cell base vectors. This is analogous to the electrostatic Coulomb field in the rest frame of a point charge that is a Glauber state of virtual zero frequency photons with a continuum of wave vectors in the Fourier transform of e/r.

On Jun 18, 2012, at 7:34 PM, JACK SARFATTI wrote:

The new theorem expands on Nambu’s ideas to the more general case, Watanabe said, proving that in weird materials, the number of Nambu-Goldstone bosons is actually less than the number of broken symmetries.

“What Nambu showed was true, but only for specialized cases applicable to particle physics,” he said. “Now we have a general explanation for all of physics; no exceptions.”