Abstract
Taking the BCS Hamiltonian written in second-quantized form, a modified form of Umezawa's self-consistent field theory method is applied, and a unitarily nonequivalent representation is selected in which the Hamiltonian obviously describes a superconducting system. This result is not at all obvious, since the original Hamiltonian is completely symmetric, and there is no reason a priori for expecting it to describe an asymmetric superconducting configuration. All higher order terms are accounted for, and in doing so, one finds the existence of the energy-gap condition for Cooper pairs. The representation is picked out without using the adiabatic theorem or pair approximation, as is typically done in the self-consistent method for superconductivity.
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Benson, A.K. Self-consistent selection of a superconducting representation for the BCS model. Found Phys 8, 653–666 (1978). https://doi.org/10.1007/BF00717497
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DOI: https://doi.org/10.1007/BF00717497