Particle-Hole Inversion in the Fermi Surface: A Symmetry Operation for Collective Excitations

The operation of transposing particles (or holes) on one side of a Fermi surface into holes (or particles) on the other side is shown to be a symmetry operation of the Bardeen, Cooper, and Schrieffer reduced Hamiltonian, and an approximate symmetry operation of the full two-particle scalar interaction Hamiltonian, insofar as particle-conserving collective excitations are concerned. Calling this operation $f$ inversion, and the conserved quantity associated with it $f$ parity, the zero-rest-energy collective excitations in a Fermi gas with weak attractive interactions prove to have $f\mathrm{parity}=-1\mathrm{}$.