Nonlinear Bogolyubov-Valatin transformations: Two modes Original Research Article

Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper, we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for image fermionic modes, which can be implemented by means of unitary image transformations, is isomorphic to image. The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra image] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington’s image-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions image. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonians. Relying on Jordan-Wigner transformations for two-spin-image and single-spin-image systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-image and single-spin-image Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of image and image matrices (of respective sizes 4×4 and 6×6) and their mutual relation.