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CRC Press eBooks

Quantum Invariants and State System Generating Algebra

Vladimir I. Man’ko

Published Oct 31, 2024• 1 citation
Publisher

Abstract

About 20 years ago almost simultaneously a number of authors had introduced the notion of a dynamic group or dynamic symmetry for quantum systems [ 1 ] (also known as spectrum generating algebra [ 2 ] and non-invariance group [ 3 ]. Introducing this notion required a clearer understanding what is to be called the symmetry of an equation describing a physical system [ 4 ] and to investigate concrete examples of quantum systems. In [ 4–7 ] it had been shown that the dynamic symmetry of hydrogen atom is 0(4,2), as a dynamic group describing Landau levels of an electron in magnetic field can serve either non-homogeneous symplectic group ISp(4,R) or U(2,1) [ 8 , 9 ].

DOI
10.1201/9781003580850-3

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About 20 years ago almost simultaneously a number of authors had introduced the notion of a dynamic group or dynamic symmetry for quantum systems [ 1 ] (also known as spectrum generating algebra [ 2 ] and non-invariance group [ 3 ]. Introducing this notion required a clearer understanding what is to be called the symmetry of an equation describing a physical system [ 4 ] and to investigate concrete examples of quantum systems. In [ 4–7 ] it had been shown that the dynamic symmetry of hydrogen atom is 0(4,2), as a dynamic group describing Landau levels of an electron in magnetic field can serve either non-homogeneous symplectic group ISp(4,R) or U(2,1) [ 8 , 9 ].
In contrast with the case of symmetry groups, no unique definition for the dynamical group generators of a given quantum Hamiltonian has been adopted in the literature. Following some work by Malkin and Man'ko (1965), Lipkin (1969), Dothan (1970) proposed a general definition based on the symmetry group of the corresponding time-dependent quantum mechanical equation of motion for the system. The authors show that this definition arises naturally from the quantum version of time dependent Noether symmetry transformations and derive the dynamical algebras for some simple examples.
In this paper, we present various well-studied examples of quantum systems, and discuss how they might be analysed algebraically. We analyse the harmonic oscillator, hydrogenic atom, optical components, and particle spin using multiple algebraic methods, and highlight the central importance of algebraic relationships between operators in determining the physical character of the system. In particular, we show: the ubiquity of the Lie algebra so(2,1)≅su(1,1)\mathfrak{so}(2,1)\cong\mathfrak{su}(1,1)so(2,1)≅su(1,1) as the ``spectrum generating algebra''; the role of su(2)\mathfrak{su}(2)su(2) and su(1,1)\mathfrak{su}(1,1)su(1,1) in determining the action of optical components; that an elementary algebraic approach to particle spin offers novel insights into its nature versus the usual approaches; and how understanding the full (often hidden) symmetry of a problem enables it to be solved immediately without the need to solve differential equations. We hope this work will serve as a gentle introduction to algebraic methods in quantum mechanics for the interested reader --- we therefore signpost active areas of research within the community.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.