SYMMETRIES AND QUANTUM MECHANICS

Abstract

Symmetries play a key role in investigation of the laws of Nature. Sometimes, unusual properties of a physical system indicate on the presence of a hidden symmetry. The analysis of the latter helps to understand better the system itself. Perhaps the most known example of such a situation is provided by the Kepler problem, in which the closed character of classical bounded orbits and peculiarities of the quantum (hydrogen atom) spectrum are explained by the presence of the hidden, nonlinear symmetry associated with Laplace-Runge-Lenz vector. A special interest attracts the case of unusual, exotic symmetries, to investigation of various aspects of which the present Project has been devoted. These include a hidden bosonized supersymmetry, nonlinear supersymmetry, and exotic Galilei and Newton-Hooke symmetries associated with non-commutative geometry. In 1996, the author made an observation that supersymmetry in exotic, bosonized form can be realized in spinless, fermion-free quantum mechanical systems, where the role of the grading operator is played by a reflection (parity). However, such unusual supersymmetry was found initially in nonlocal quantum systems. Therefore, the question emerged whether bosonized supersymmetry could be present in quantum systems with local Hamiltonians, and if so, what is its impact on the physical properties of corresponding systems. In developing the present Project, surprisingly, hidden bosonized supersymmetry of a linear and nonlinear (polynomial) form was revealed in such very well studied quantum mechanical systems like the bound-state AharonovBohm effect (particle confined to move on a circle pierced by a singular magnetic flux), Dirac delta potential problem, and reflectionless Poschl-Teller system. The list of systems was extended subsequently to include all the finite-gap periodic quantum systems with parity-even potentials. The revealed hidden supersymmetry reflects special properties of the corresponding systems, like degeneracies of the spectra, and peculiarities of the quantum scattering and band structure in non-periodic and periodic cases, respectively. It was showed that when corresponding systems are extended by inclusion of the spin degrees of freedom, hidden supersymmetry leads to a very reach, earlier unknown, supersymmetric structure called tri-supersymmetry, which is related to a nontrivial Lax operator of the associated nonlinear integrable system. A simple physical system where tri-supersymmetry was observed corresponds to a generalization of Landau problem for the case of electron moving in periodic magnetic and electric fields of a special form, that produce a 1D crystal for two spin components separated by a half-period spacing. Another interesting result of the Project is the explanation of the hidden bosonized nonlinear supersymmetry of reflectionless Poschl-Teller system in terms of Aharonov-Bohm effect for non-relativistic particle on the AdS(2), and observation of the hidden nonlinear superconformal symmetry in it. It was showed that bosonized supersymmetry can be realized also in relativistic field systems. For the purpose, we used a construction based on the infinite-component Majorana equation [development of the concept of the infinite-component fields introduced by Majorana in 1932 culminated in 1960-70’s in the construction of the dual resonance models and the origin of the superstring theory]. The resulting theory was shown is described by a nonlinear symmetry superalgebra, that in the large-spin limit reduces to the super-Poincaré algebra with or without tensorial central charge. In the case of two spatial dimensions, Galilei and Newton-Hooke symmetries admit the exotic deformation characterized by non-commutativity of the boost generators and by related second central charge in the algebra. Such symmetries appear in a natural way as symmetries of a free particle and Landau problem on non-commutative plane (observation of the indicated exotic symmetry in the second system is one of the original results obtained within the Project). Various aspects of such exotic symmetries were investigated. For instance, it was showed that the appropriate non-relativistic limit applied to bosonized supersymmetry realized in (2+1)- dimensional system of anyons, produces a non-relativistic supersymmetric particle on a noncommutative plane, which is described by a supersymmetric extension of the exotic Galilei symmetry. On the other hand, it was found that quantum systems with exotic Galilei and Newton-Hooke symmetries appear naturally under dimensional reduction of field theories with higher-derivative Chern-Simons terms [such theories were considered by Hagen, and by Deser and Jackiw (1987, 1999)]. Another interesting observation made is that in a generic case a system with exotic Newton-Hooke symmetry possesses three essentially different phases. It is these different phases which were revealed also under investigation of conformal symmetry in super-extended non-commutative Landau problem.