Nonlinear Bogolyubov-Valatin transformations: Two modes

Abstract

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 n = 2 fermionic modes which can be implemented by means of unitary SU (2 n = 4) transformations is isomorphic to SO(6; R)/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 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-1 2 and single-spin-3 2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1 2 and single-spin-3 2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU (4) and SO(6; R) matrices (of respective sizes 4 × 4 and 6 × 6) and their mutual relation. † As a consequence of eq. (17) and the anti-Hermiticity of the operatorsĉ k , the operatorsĉ k [i.e., the basis elements of the paravector space associated with the Clifford algebra C(0, 5)] obey the relation (k, l = −1, . . . , 4)(304) 35 Below we have not indicated that the wedge product relates to a different vector space than in eq. (301), i.e.:

Local Molecular Field Theory for Nonequilibrium Systems

“…Finally, in the approach taken here, it will be possible to consider transformations that mix the two Hilbert spaces, which will be further explored. As special cases of transformed basis sets, one could or more general nonlinear canonical transformations [97]. Some general results on canonical transformations are reviewed in appendix C.1.…”

Approximate thermofield dynamics of interacting fermions

“…Finally, in the approach taken here, it will be possible to consider transformations that mix the two Hilbert spaces, which will be further explored. As special cases of transformed basis sets, one could consider Bogoliubov transformations [5][6], local canonical transformations [33][34], or more general nonlinear canonical transformations [35]. Some general results on canonical transformations are reviewed in appendix A.…”

Sketches from the life of hypercomplex numbers

“…Thus, for n=6, the partial solutions of the Clifford equation can be represented with the help of six antisymmetric matrixes. It should be noted that the specified formalism is closely connected with the Bogolyubov-Valatin transformations [30]. It can be shown that for n=8, partial solutions of the Clifford equation can be represented with the help of 8 matrixes: seven antisymmetric matrixes and an one symmetric matrix [5], [3, p. 77, Section 14.4(eng), p. 211(88), Section 14.4 (rus)], [2, p. 88, Section 14.4].…”