Bogoliubov transformations and fermion condensates in lattice field theories

Abstract

We apply generalized Bogoliubov transformations to the transfer matrix of relativistic field theories regularized on a lattice. We derive the conditions these transformations must satisfy to factorize the transfer matrix into two terms which propagate fermions and antifermions separately, and we solve the relative equations under some conditions. We relate these equations to the saddle point approximation of a recent bosonization method and to the Foldy-Wouthuysen transformations which separate positive from negative energy states in the Dirac Hamiltonian.

Diquarks in the nilpotency expansion of QCD and their role at finite chemical potential

“…so that the open spatial plaquettes, which determine the chromomagnetic fields, are constrained to be gauge equivalent, indeed 27) and therefore their trace is time invariant and they give one and the same contribution to the gauge-field action at all times.…”

“…The trace of the fermion transfer matrix, after the first transformation, was represented in [27,28] as a Berezin integral, with the result…”

“…But needless to say, unlike the latter these terms do not violate any symmetry). Hence at the saddle point, the effect of the Bogoliubov transformations (4.2) is analogous to that of the Foldy-Wouthuysen transformations which separate positive from negative energy states in the Dirac Hamiltonian [27]. The time evolution of the quasiparticle Hamiltonians is slightly different:…”

“…The evaluation of the matrix element of the transfer matrix between coherent states was already performed in [27] with the result…”

“…For this purpose we perform a first Bogoliubov transformation, which is equivalent to a Foldy-Wouthuysen transformation [27]. Bare fermions are replaced by quasiparticles in presence of a background field.…”