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Mikhail Plyushchay

Publications and source records attributed to Mikhail Plyushchay.

At least 19 recordsLinked to original sources
arXiv2014

Nonlinear supersymmetry in the quantum Calogero model

It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g+1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order n(n-1)(2g-1)/2 in the momenta and squares to a polynomial in the Liouville charges. With a natural Z_2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.

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arXiv2004

Nonlinear supersymmetry: from classical to quantum mechanics

Quantization of the nonlinear supersymmetry faces a problem of a quantum anomaly. For some classes of superpotentials, the integrals of motion admit the corrections guaranteeing the preservation of the nonlinear supersymmetry at the quantum level. With an example of the system realizing the nonlinear superconformal symmetry, we discuss the nature of such corrections and speculate on their possible general origin.

hep-th↗
arXiv2004

Nonlinear superconformal symmetry

We discuss two different nonlinear generalizations of the osp(2|2) supersymmetry which arise in superconformal mechanics and fermion-monopole models.

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arXiv2003

Higher Spins from Tensorial Charges and OSp(N|2n) Symmetry

It is shown that the quantization of a superparticle propagating in an N=1, D=4 superspace extended with tensorial coordinates results in an infinite tower of massless spin states satisfying the Vasiliev unfolded equations for free higher spin fields in flat and AdS_4 N=1 superspace. The tensorial extension of the AdS_4 superspace is proved to be a supergroup manifold OSp(1|4). The model is manifestly invariant under an OSp(N|8) (N=1,2) superconformal symmetry. As a byproduct, we find that the Cartan forms of arbitrary Sp(2n) and OSp(1|2n) groups are GL(2n) flat, i.e. they are equivalent to flat Cartan forms up to a GL(2n) rotation. This property is crucial for carrying out the quantization of the particle model on OSp(1|4) and getting the higher spin field dynamics in super AdS_4, which can be performed in a way analogous to the flat case.

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arXiv2001

Nonlinear Supersymmetry, Quantum Anomaly and Quasi-Exactly Solvable Systems

The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is ``cured'' by the specific superpotential-dependent term of order $\hbar^2$. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems.

hep-th↗
arXiv1999

Supersymmetry of parafermions

We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal $Z_2$ structure.

hep-th↗
arXiv1999

Hidden nonlinear supersymmetries in pure parabosonic systems

The existence of intimate relation between generalized statistics and supersymmetry is established by observation of hidden supersymmetric structure in pure parabosonic systems. This structure is characterized generally by a nonlinear superalgebra. The nonlinear supersymmetry of parabosonic systems may be realized, in turn, by modifying appropriately the usual supersymmetric quantum mechanics. The relation of nonlinear parabosonic supersymmetry to the Calogero-like models with exchange interaction and to the spin chain models with inverse-square interaction is pointed out.

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arXiv1999

Zitterbewegung and reduction: 4D spinning particles and 3D anyons on light-like curves

We construct the model with light-like world-lines for the massive 4D spinning particles and 3D anyons. It is obtained via the formal bosonization of pseudoclassical model for the massive Dirac particle with subsequent reduction to the light-like curves. The peculiarity of the light-like trajectories produced due to the Zitterbewegung is explained from the viewpoint of reduction and reparametrization invariance.

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arXiv1998

Three aspects of bosonized supersymmetry and linear differential field equation with reflection

Recently it was observed by one of the authors that supersymmetric quantum mechanics (SUSYQM) admits a formulation in terms of only one bosonic degree of freedom. Such a construction, called the minimally bosonized SUSYQM, appeared in the context of integrable systems and dynamical symmetries. We show that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a nonlocal unitary transformation with a subsequent reduction to one of the eigenspaces of the total reflection operator. The transformation depends on the parity operator, and the deformed Heisenberg algebra with reflection, intimately related to parabosons and parafermions, emerges here in a natural way. It is shown that the minimally bosonized SUSYQM can also be understood as supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N=1 supersymmetry in 2 dimensions. The same special unitary transformation diagonalises the Hamiltonian operator of the 2D massive free Dirac theory. The resulting Hamiltonian is not a square root like in the Foldy-Wouthuysen case, but is linear in spatial derivative. Subsequent reduction to `up' or `down' field component gives rise to a linear differential equation with reflection whose `square' is the massive Klein-Gordon equation. In the massless limit this becomes the self-dual Weyl equation. The linear differential equation with reflection admits generalizations to higher dimensions and can be consistently coupled to gauge fields. The bosonized SUSYQM can also be generated applying the nonlocal unitary transformation to the Dirac field in the background of a nonlinear scalar field in a kink configuration.

hep-th↗
arXiv1998

Comment on "The relativistic particle with curvature and torsion of world trajectory"

Gogilidze and Surovtsev have claimed recently (hep-th/9809191) that the tachyonic sector can be removed from the spectrum of the relativistic particle with curvature and torsion by a proper gauge choice. We show that the mass-spin dependence obtained by them is incorrect and point out that their gauge surface does not cross all the gauge orbits. We discuss the nature of the tachyonic sector of the model and argue why it cannot be removed by any gauge fixing procedure.

hep-th↗
arXiv1998

Integrals of motion, supersymmetric quantum mechanics and dynamical supersymmetry

The class of relativistic spin particle models reveals the `quantization' of parameters already at the classical level. The special parameter values emerge if one requires the maximality of classical global continuous symmetries. The same requirement applied to a non-relativistic particle with odd degrees of freedom gives rise to supersymmetric quantum mechanics. Coupling classical non-relativistic superparticle to a `U(1) gauge field', one can arrive at the quantum dynamical supersymmetry. This consists in supersymmetry appearing at special values of the coupling constant characterizing interaction of a system of boson and fermion but disappearing in a free case. Possible relevance of this phenomenon to high-temperature superconductivity is speculated.

hep-th↗
arXiv1998

P,T-invariant system of Chern-Simons fields: Pseudoclassical model and hidden symmetries

We investigate hidden symmetries of P,T-invariant system of topologically massive U(1) gauge fields. For this purpose, we propose a pseudoclassical model giving rise to this field system at the quantum level. The model contains a parameter, which displays a quantization property at the classical and the quantum levels and demonstrates a nontrivial relationship between continuous and discrete symmetries. Analyzing the integrals of motion of the pseudoclassical model, we identify U(1,1) symmetry and S(2,1) supersymmetry as hidden symmetries of the corresponding quantum system. Representing the hidden symmetries in a covariant form, we show that one-particle states realize an irreducible representation of a non-standard super-extension of the (2+1)-dimensional Poincare group labelled by the zero eigenvalue of the superspin.

hep-th↗
arXiv1998

Pseudoclassical model for topologically massive gauge fields

A pseudoclassical model for P,T-invariant system of topologically massive U(1) gauge fields is analyzed. The model demonstrates a nontrivial relationship between continuous and discrete symmetries and reveals a phenomenon of ``classical quantization''. It allows one to identify SU(1,1) symmetry and S(2,1) supersymmetry as hidden symmetries of the corresponding quantum system. We show this P,T-invariant quantum system realizes an irreducible representation of a non-standard super-extension of the (2+1)-dimensional Poincare group.

hep-th↗
arXiv1998

Classical anomalies for spinning particles

We discuss the phenomenon of classical anomaly. It is observed for 3D Berezin-Marinov (BM), Barducci-Casalbuoni-Lusanna (BCL) and Cortes-Plyushchay-Velazquez (CPV) pseudoclassical spin particle models. We show that quantum mechanically these different models correspond to the same P,T-invariant system of planar fermions, but the quantum system has global symmetries being not reproducible classically in full in any of the models. We demonstrate that the specific U(1) gauge symmetry characterized by the opposite coupling constants of spin s=+1/2 and s=-1/2 states has a natural classical analog in CPV model but can be reproduced in BM and BCL models in an obscure and rather artificial form. We also show that BM and BCL models quantum mechanically are equivalent in any odd-dimensional space-time, but describe different quantum systems in even space-time dimensions.

hep-th↗
arXiv1997

Symmetries and classical quantization

A phenomenon of classical quantization is discussed. This is revealed in the class of pseudoclassical gauge systems with nonlinear nilpotent constraints containing some free parameters. Variation of parameters does not change local (gauge) and discrete symmetries of the corresponding systems, but there are some special discrete values of them which give rise to the maximal global symmetries at the classical level. Exactly the same values of the parameters are separated at the quantum level, where, in particular, they are singled out by the requirement of conservation of the discrete symmetries. The phenomenon is observed for the familiar pseudoclassical model of 3D P,T-invariant massive fermion system and for a new pseudoclassical model of 3D P,T-invariant system of topologically massive U(1) gauge fields.

hep-th↗
arXiv1997

R-deformed Heisenberg algebra, anyons and d=2+1 supersymmetry

A universal minimal spinor set of linear differential equations describing anyons and ordinary integer and half-integer spin fields is constructed with the help of deformed Heisenberg algebra with reflection. The construction is generalized to some d=2+1 supersymmetric field systems. Quadratic and linear forms of action functionals are found for the universal minimal as well as for supersymmetric spinor sets of equations. A possibility of constructing a universal classical mechanical model for d=2+1 spin systems is discussed.

hep-th↗
arXiv1997

Universality of the R-deformed Heisenberg algebra

We show that deformed Heisenberg algebra with reflection emerging in parabosonic constructions is also related to parafermions. This universality is discussed in different algebraic aspects and is employed for the description of spin-j fields, anyons and supersymmetry in 2+1 dimensions.

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