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In this work we present two correspondences between the massless Gross-Neveumodel with one or two coupling constants in 1+1 dimensions and nonrelativisticfield theories in 3+1 dimensions. It is shown that on a mean-field level themassless Gross-Neveu model can be mapped onto BCS theory provided thattranslational invariance of the condensate is assumed. The generalized masslessGross-Neveu model with two coupling constants is mapped onto a quasione-dimensional extended Hubbard model used in the description of spin-Peierlssystems. It is shown that the particle hole symmetry of the Hubbard modelimplies self-consistency of the condensate. The dimensional reduction allows anidentification of the phase diagrams of the models.

Dimensional reduction in quantum field theories at finite temperature and density