K. Scharnhorst, J.-W. van Holten: Nonlinear Bogolyubov-Valatin transformations: Two modes. Annals of Physics (New York) 326:11(2011)2868-2933 (DOI: 10.1016/j.aop.2011.05.001) [NIKHEF preprint NIKHEF/2010-005, arXiv:1002.2737]. [INSPIRE record]

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(2ⁿ = 4) transformations is isomorphic to SO(6;R)/Z₂. 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.

• Electronic copy of the article: Annals of Physics (New York)
• Electronic copy of the preprint: arXiv:1002.2737
• Citations

• Misprints, errata, addenda:
  • P. 2871, in line 13 from top the sequence "is isomorphic to the twofold tensor product of the algebra of quaternions" should read: is isomorphic to the real algebra of 2x2 matrices over the algebra of quaternions.
  • P. 2874, in line 7 from top the sequence "[31,35,127]" should read: [127,31,35].
  • P. 2925, in ref. [54] the names of the second and the third author should read correctly: J. da Providência, C. Providência.
  • P. 2927, in ref. [127] the author name should read correctly: B.G. Wybourne.
  • P. 2929, in ref. [218] the author name should read correctly: Ю.П. Степановський [Yu.P. Stepanovs'kiĭ].
  • P. 2929, in the fourth line of [221] the sequence "[100,230] (also note [110]),"should read correctly: [230], [100] (also note [110]),.
  • P. 2931, in ref. [263] title sequence "1965 - 25 Mei - 1935" should read correctly: 1865 - 25 Mei - 1935.
  • P. 2931, in the first line of [267] the sequence "[105,268], p. 37" should read correctly: [268] , [105], p. 37.
  • P. 2931, in the first line of ref. [288] the word "Metereol." should read correctly: Meteorol..

© K. Scharnhorst . Document last modified: June 8, 2023