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Mathematical Notes

Algebra of Symmetries of Three-Frequency Resonance: Reduction of a Reducible Case to an Irreducible Case

Mihail Vladimirovich Karasev, E. M. Novikova

Published Nov 1, 2018• 4 citations
Publisher

Abstract

For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

DOI
10.1134/s0001434618110287

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For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.
All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para-Bose or para-Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three-dimensional rotation group in the para-Fermi case, and with the three generators of the three-dimensional Lorentz group in the para-Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.
In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys. 175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H\mathcal {H}HH of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H\mathcal {H}HH. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
Using mathematical tools developed by Hermann Weyl, the Wigner classification of group-representations and co-representations is clarified and extended. The three types of representation, and the three types of co-representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co-representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (qqq-numbers, qqq-analysis, qqq-oscillators, qqq-algebras), the suq_qq​(2) rotator model and its extensions, the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model), the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to molecular structure and an outlook are also given.