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Nonlinear Bogolyubov-Valatin transformations: two modes. (English) Zbl 1252.81092

Summary: Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for \(n=2\) fermionic modes, which can be implemented by means of unitary \(SU(2^n = 4)\) transformations, is isomorphic to \(SO(6;\mathbb R)/\mathbb Z_{2}\). 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 \(C(0,4)\)) 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 \(E\)-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions \(\mathbb H\). From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner transformations for two-spin-\(\frac {1}{2}\) and single-spin-\(\frac {3}{2}\) systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-\(\frac {1}{2}\) and single-spin-\(\frac {3}{2}\) Hamiltonians. Finally, from the point of view of calculations, we pay due attention to explicit parametrizations of \(SU(4)\) and \(SO(6;\mathbb R\)) matrices (of respective sizes \(4\times 4\) and \(6\times 6\)) and their mutual relation.

MSC:

81T05 Axiomatic quantum field theory; operator algebras
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81T60 Supersymmetric field theories in quantum mechanics
15A66 Clifford algebras, spinors
15B33 Matrices over special rings (quaternions, finite fields, etc.)
81R15 Operator algebra methods applied to problems in quantum theory
81R25 Spinor and twistor methods applied to problems in quantum theory

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