Symmetries in Physics

Symmetries in Physics

Pavel Winternitz
Montreal University, Montreal, Canada

A typical feature of the research and even of the personality of Yakov Abramovich Smorodinsky was the broadness of his scientific interests, covering virtually all of theoretical physics (and not only theoretical physics). Probably all of us, who have worked with Ya. A. on some problem, came away with the impression that precisely that field is Ya. A.'s main interest. I was fortunate enough to have worked in Ya. A.'s group in Dubna for several years, mainly on applications of group theory in physics. Not surprisingly, my (biased) opinion is that symmetry theory was his main love in science.
The articles on symmetries in physics, in particular those included in this book, can be subdivided into several categories.
One of them could be called SU(3) and beyond. Ya. A. was one of the first physicists who recognized the importance of unitary symmetry in elementary particle theory, i. e. the necessity of going beyond Wigner multiplets, related to isotopic spin, to the «supermultiplets» of Gell-Mann's «eightfold way». Ya. A.'s own contributions concerned the representation theory of SU(3) in specific bases and mainly the unification of internal SU(3) symmetries with space-time ones (leading to the sequence of Lie groups SU(3), SU(6), SL(6, C) and SU(6,6)). This line of work is represented by an influential review article [1]. A second major series of papers could be grouped under the title Lobachevsky geometry, representations of the Lorentz group and invariant expansions of scattering amplitudes. This started out from the realization that the Lobachevsky geometry of velocity space provides very efficient tools for describing relativistic kinematics [2]. A seminal paper with N. Ya. Vilenkin [3] followed and laid the basis for several new directions in mathematical physics that are actively pursued to this day. The original purpose of these papers was to develop a mathematical basis for relativistic expansions of scattering amplitudes., e. g. a relativistic energy dependent phase shift analysis. The necessary tools involved harmonic analysis on homogeneous manifolds of the Lorentz group O(3,1), and later on the group manifold itself. A crucial observation was that reductions of O(3,1) to different subgroups lead to different bases for group representations, different special functions and hence different expansions. Moreover, different subgroups yield different physical applications. The O(3,1)

O(3), O(3,1)

O(2,1), and O(3,1)

E(2) two-variable expansions incorporated partial wave analysis, Regge pole expansions and eikonal expansions, respectively [4, 5]. For further physical and mathematical developments of this theme see e.g. Ref. [6-12]. Two-variable expansions were later generalized to the case of reactions involving four particles with arbitrary spins [13, 14] and applied to analyze distributions on Dalitz plots and nucleon-nucleon scattering data [15].
The group theoretical and special function theory aspects of this research program induced several lines of further mathematical research. One was a series of papers [16, 17] on basis functions of representations of O(p, q), SU(p, q) and other groups. Others are ongoing series of articles on classifying subgroups of Lie groups [1, 18, 19] and on the separation of variables in Hamilton—Jacobi and Laplace—Beltrami equations [3, 18-23]. The «method Introduction of trees» proposed by Vilenkin [24] and further developed by Ya. A. Smorodinsky and collaborators [25] is now being applied in areas varying from nuclear physics to quantum groups [28]. The third major topic included in this Section is that of finite-dimensional integrable and «superintegrable» systems [29]. The problem originally posed was to find all potentials in two and three dimensions that can be inserted into the Schrodinger, or Hamilton—Jacobi equation and that would allow a «dynamical» symmetry group. A useful way of distinguishing between kinematical and dynamical symmetries for the Schrodinger equation was proposed. Namely, kinematical symmetries correspond to Lie point ones and are generated by sets of first order differential operators, commuting with the Hamiltonian. Dynamical symmetries, on the other hand, are generated by sets of higher order operators, commuting with the Hamiltonian. They correspond to generalized symmetries, or Lie—Backhand symmetries, in the theory of differential equations. Second order operators in quantum mechanics, or integrals of motion in classical mechanics that are quadratic in the momenta, turned out to be specially interesting. Indeed, Hamiltonians in n dimensions, allowing n commuting quadratic integrals of motion, will also allow the separation of variables in the Schrodinger and Hamilton—Jacobi equation.
Such a Hamiltonian system will be integrable, both in the classical and in the quantum mechanical sense. A further question that was posed was that of finding «superintegrable» systems with more than n integrals of motion. In classical mechanics a system can have up to 2n - 1 functionally independent integrals of motion. In quantum mechanics,they all commute with the Hamiltonian, but amongst each other they generate some algebraic structure. This may be a finite-dimensional Lie algebra, an infinite-dimensional one, or for instance a quadratic algebra [35]. If the integrals of motion are second-order operators, then superintegrable systems are those that allow a separation of variables in at least two systems of coordinates. A complete classification of superintegrable systems was given for n = 2 [29], a partial one for n = 3 [30]. The n = 3 case was actually completed more than 20 years later by W. Evans [36]. The common features of all. known maximally superintegrable systems is the existence of dense sets of periodic orbits in the classical case, degenerate energy levels in the quantum one, and the presence of a nonabelian dynamical symmetry algebra, explaining the degeneracy of energy levels.
When articles [29] and [30] were written, only two «superintegrable» (in the above sense) systems were known: the hydrogen atom, with its 0(4) symmetry, discovered by Bargmann [37] and Fock [38] and the harmonic oscillator, with its SU(3) symmetry [39]. This very brief list was significantly extended in Ref. [29] and [30]. By now the theory of integrable systems has been greatly developed and generalized and pervades much of physics. Finite-dimensional systems with more than their proper share of integrals of motion, include such systems as the Calogero—Moser one. For some recent reviews and new results on superintegrable finite dimensional systems, see e. g. Ref. [40-42]. Infinite-dimensional systems with infinitely many integrals of motion describe all of soliton theory. In short, this particular scientific program, started by Ya. A. Smorodinsky and his collaborators in the early sixties, has been incorporated into a vast and extremely active field of research.

References

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