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Journal of the Physical Society of Japan

Co- and Contra-Variant Tensors of a General Rankin Quantum Matrix Algebrassuq(2) Expressedby Creation-Annihilation Operators. II

Masao Nomura

Published Dec 15, 1991• 7 citations
Publisher

Abstract

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 .

DOI
10.1143/jpsj.60.4060

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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 .
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrödinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate r1,r2,... and momentum p1, p2, . . . , boson creation and annihilation operators, b†iα, biα. The index i runs over the number of relevant degrees of freedom, while the index α runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i ≠ j, . . . [biα, b†jα´] = 0, [biα, bjα´] = 0,. . . . . .[bjα, b†iα´] = 0, [b†jα, b†iα´] = 0,. . . . . . [biα, b†iα´] = ẟαα´, [biα, b†iα´] = 0, [b†iα, b†iα´] = 0. . . .
Matrix elements for fermion and boson operators describing the creation or annihilation of particles with multiple angular-momentum quantum numbers can be factored into products of matrix elements of generalized operators, one for each quantum number. The generalized operators have nonzero matrix elements between states of arbitrary permutational symmetry. This factorization greatly increases the efficiency of shell-model calculations. We provide a physical interpretation for these generalized operators and introduce a simple operator-reordering procedure.
Eigenstates of the linear combinations a2+betaadagger2a^2+\\beta a^{\\dagger2}a2+betaadagger2 and $ab+\\beta a^\\dagger b^\\dagger$ of two boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett. {\\bf 72}, 1447, 1994] for constructing eigenstates of generalized annihilation operators.
The recent development of experimental techniques to implement the creation and the annihilation operators in the optical domain has found great interest for quantum information manipulation and processing. Quantum states of light obtained by the application of alternated sequences of the creation and annihilation operators a^†\hat{a}^\daggera^† and a^\hat{a}a^ to a thermal state of light depend on the order in which the two operations are performed. The non-commutativity of the bosonic operators, a^a^†≠a^†a^≠1\hat{a}\hat{a}^\dagger\ne \hat{a}^\dagger\hat{a}\ne 1a^a^†=a^†a^=1, has been experimentally verified in terms of density matrixes and Wigner functions, which are reconstructed by quantum homodyne tomography. Recently, a setup has been proposed to directly and completely prove the bosonic commutation relation where a single-photon interferometer is used to create coherent superpositions of two alternated sequences of operators, such as [a^a^†±a^†a^][\hat{a}\hat{a}^\dagger \pm \hat{a}^\dagger\hat{a}][a^a^†±a^†a^]. Some of the latest experimental results are discussed.