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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13, 599, 1965.
2. Smorodinsky Ya.A. Soviet Physics JETP, 16, 1566, 1963.
3. Vilenkin N. Ya., Smomdinsky Ya.A. Soviet Physics JETP, 19, 1209,
1964.
4. Wmternitz P., Smomdinsky Ya.A., Sheftel M. B. Sov. J. Nucl. Phys.,
7, 785, 1968.
5. Wmternitz P., Smorodinsky Ya.A., Sheftel M. B. Sov. J. Nucl. Phys.,
8, 485, 1969.
6. Wmternitz P., Smorodinsky Ya.A., Uhlir M. Soviet J. Nucl. Phys.,
1, 113, 1965.
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Phys., 2, 645, 1966.
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11. Liberman M.A., Smorodinsky Ya.A., Sheftel M.B. Sov. J. Nucl. Phys., 7,
146, 1968.
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636, 1965.
19. Winternitz P. In: Integrable Systems, Quantum Groups and Quantum Field
Theory. Kluwer, Dordrecht, 1993. P. 429-495. (Contains numerous references
to the original work).
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1968.
21. Patera J, Winternitz P.J. Math. Phys., 14, 1130, 1973.
22. Miller W., Jr., Patera J., Winternitz P. J. Math. Phys., 22, 251, 1981;
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1977.
23. Kalnins E.G. Separation of Variables for Riemannian Spaces of Constant
Curvature. Longman Scientific, Essex, 1986.
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26. Smorodinsky Ya.A. Izvestiya vuzov. Ser. Radiofizika, 19, 932, 1976.
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28. Kuznetsov G.I., Moskalyuk S.S., Yu. Smimov F., Shelest V. P. Graph Theory
for Representations' of Orthogonal and Unitary Groups. Naukova Dumka, Kiev,
1992. (In Russian.)
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4, 444, 1967.
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34. Pogosyan G.S., Smorodinsky Ya.A., Ter-Antonyan V.M. Multi-Dimensional
Isotropic Oscillator: Transition from the Carthesian Basis to Hyperspherical
Bases. Dubna, Preprint R2-82-118, 1982.
35. Vinet L. Ann. Phys., 243, 144, 1995.
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Quarks. Gordon & Breach, New York, 1969.
40. Grosche C. Path Integrals, Hyperbolic Spaces and Selberg Trace Formulas.
World Scientific, Singapore, 1995.
41. Perelomov A. M. Integrable Systems of Classical Mechanics and Lie Algebras.
Birkhausen, Basel, 1990.
42. Kalnins E. G., Miller W., Jr., Pogosyan G.S. J. Math. Phys., 37, 6439,
1996.