Nonlinear Bogolyubov-Valatin transformations: Two modes

Journal Article · Tue Nov 15 00:00:00 EST 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2011.05.001· OSTI ID:21583296

Scharnhorst, K ^[1]; Holten, J.-W. van, ^[2]

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1. Vrije Universiteit Amsterdam, Faculty of Sciences, Department of Physics and Astronomy, De Boelelaan 1081, 1081 HV Amsterdam (Netherlands)
2. NIKHEF, P.O. Box 41882, 1009 DB Amsterdam (Netherlands)

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{sup n}=4) transformations, is isomorphic to SO(6;R)/Z{sub 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 Hamiltonians. 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 4x4 and 6x6) and their mutual relation. - Highlights: > Reveals the structure of nonlinear canonical transformations for two fermionic modes. > Studies nonlinear basis transformations in a Clifford algebra. > Focuses on methodological and structural aspects. > Presents a new approach to the diagonalization of 4x4 matrix Hamiltonians. > Presents the relation between explicit parametrizations of SU(4) and SO(6;R) matrices.

• OSTI ID: 21583296
• Journal Information: Annals of Physics (New York), Vol. 326, Issue 11; Other Information: DOI: 10.1016/j.aop.2011.05.001; PII: S0003-4916(11)00089-3; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CANONICAL TRANSFORMATIONS
CLIFFORD ALGEBRA
DIRAC OPERATORS
FERMIONS
HAMILTONIANS
NONLINEAR PROBLEMS
R MATRIX
SPIN
SU-4 GROUPS
SUPERSYMMETRY
TENSORS
ANGULAR MOMENTUM
LIE GROUPS
MATHEMATICAL OPERATORS
MATRICES
PARTICLE PROPERTIES
QUANTUM OPERATORS
SU GROUPS
SYMMETRY
SYMMETRY GROUPS
TRANSFORMATIONS