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arXiv:2301.10747

$h(1) \oplus su(2)$ vector algebra eigenstates with eigenvalues in the matrix domain

Nibaldo-Edmundo Alvarez-Moraga

Published Jan 25, 2023• 0 citations
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Abstract

A new set of $ h(1) \oplus su(2)$ vector algebra eigenstates on the matrix domain is obtained by defining them as eigenstates of a generalized annihilation operator formed from a linear combination of the generators of this algebra which eigenvalues are distributed as the elements of a square complex normal matrix. A combined method is used to compute these eigenstates, namely, the method of exponential operators and that of a system of first-order linear differential equations. We compute these states for all possible combination of generators and classify them in different categories according to a generalized commutation relation as well as according to the value of a characteristic parameter related to the $su(2)$ algebra eigenvalues. Proceeding in this way, we found a subset of generalized vector coherent states in the matrix domain which can be easily separated from the general set of Schrödinger-Robertson minimum uncertainty intelligent states. In particular, for a special choice of the matrix eigenvalue parameters we found the so-called vector coherent states with matrices associated to the Heisenberg-Weyl group as well as a generalized version of them, and also a direct connection with the coherent state quantization of quaternions.

DOI
10.48550/arxiv.2301.10747

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h(1)⊕su(2)h(1) \oplus su(2)h(1)⊕su(2) vector algebra eigenstates with eigenvalues in the matrix domain
A new set of h(1)⊕su(2) h(1) \oplus su(2)h(1)⊕su(2) vector algebra eigenstates on the matrix domain is obtained by defining them as eigenstates of a generalized annihilation operator formed from a linear combination of the generators of this algebra which eigenvalues are distributed as the elements of a square complex normal matrix. A combined method is used to compute these eigenstates, namely, the method of exponential operators and that of a system of first-order linear differential equations. We compute these states for all possible combination of generators and classify them in different categories according to a generalized commutation relation as well as according to the value of a characteristic parameter related to the su(2)su(2)su(2) algebra eigenvalues. Proceeding in this way, we found a subset of generalized vector coherent states in the matrix domain which can be easily separated from the general set of Schrödinger-Robertson minimum uncertainty intelligent states. In particular, for a special choice of the matrix eigenvalue parameters we found the so-called vector coherent states with matrices associated to the Heisenberg-Weyl group as well as a generalized version of them, and also a direct connection with the coherent state quantization of quaternions.
We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.
Robertson intelligent states which minimize the Schr\" odinger-Robertson uncertainty relation are constructed as eigenstates of a linear combination of Weyl generators of the su(3)su(3)su(3) algebra. The construction is based on the analytic representations of su(3)su(3)su(3) coherent states. New classes of coherent and squeezed states are explicitly derived.
Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
In this article a study was made of the conditions under which a Hamiltonian which is an element of the complex $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $ Lie algebra admits ladder operators which are also elements of this algebra. The algebra eigenstates of the lowering operator constructed in this way are computed and from them both the energy spectrum and the energy eigenstates of this Hamiltonian are generated in the usual way with the help of the corresponding raising operator. Thus, several families of generalized Hamiltonian systems are found, which, under a suitable similarity transformation, reduce to a basic set of systems, among which we find the 1:1, 2:1, 1:2, su(2)su(2)su(2) and some other non-commensurate and commensurate anisotropic 2D quantum oscillator systems. Explicit expressions for the normalized eigenstates of the Hamiltonian and its associated lowering operator are given, which show the classical structure of two-mode separable and non-separable generalized coherent and squeezed states. Finally, based on all the above results, a proposal for new ladder operators for the p:qp:qp:q coprime commensurate anisotropic quantum oscillator is made, which leads us to a class of Chen SU(2)SU(2)SU(2) coherent states.
We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.