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Theoretical and Mathematical Physics

Bogolyubov transformation in terms of creation and annihilation operators

O. D. Timofeevskaya

Published Mar 1, 1983• 2 citations
Publisher
DOI
10.1007/bf01018913

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In introducing second quantization for fermions, Jordan and Wigner (1927/1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H. For the first time, here we exploit this fact to study nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator.
In introducing second quantization for fermions, Jordan and Wigner (1927, 1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H . For the first time, here we exploit this fact to study nonlinear Bogolyubov–Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator.
Quantum matrix algebras s u q (2) are realized in terms of single-state boson operators and their conjugates, ( a + , a ), ( b + , b ), etc., specified as quantum analogs of symplectons. The boson operators obey the linear transformation of S U q (2) such as a ′ + = a + x + a v and a ′ = a + u + a y . Here, x , u , v , y are non-commutative elements which commute with a + and a . Commutation relations for creation and annihilation operators are determined so as to be invariant under the linear transformation. Exchange algebras with the universal R j ′ j ′′ matrix are realized in terms of P ( j ′ , m ′ ; b ), P ( j ′′ , m ′′ ; a ), etc., where P ( j ′ , m ′ ; b ) is the j z ′ = m ′ component of rank- j ′ tensor written as an order-2 j ′ polynomial in b + and b .