arXiv subjects
Publications and source records attributed to Armen Nersessian.
In these lectures we discuss some basic aspects of Hamiltonian formalism, which usually do not appear in standard texbooks on classical mechanics for physicists. We pay special attention to the procedure of Hamiltonian reduction illustrating it by the examples related to Hopf maps. Then we briefly discuss the supergeneralisation(s) of the Hamiltonian formalism and present some simple models of supersymmetric mechanics on Kähler manifolds.
We suggest the exactly solvable model of the oscillator on a four-dimensional hyperboloid which interacts with a SU(2) instanton. We calculate its wavefunctions and spectrum.
We show that the nonlinear chiral supermultiplet allows one to construct, over given two-dimensional bosonic mechanics, the family of two-dimensional ${\cal N}=4$ supersymmetric mechanics parameterized with the holomorphic function $λ(z)$. We show, that this family includes, as a particular case, the ${\cal N}=4$ superextensions of two-dimensional mechanics with magnetic fields, which have factorizable Schroedinger equations.
We suggest the exactly solvable model of oscillator on the four-dimensional sphere interacting with the SU(2) Yang monopole. We show, that the properties of the model essentially depend on the monopole charge.
We define the "maximally integrable" isotropic oscillator on CP(N) and discuss its various properties, in particular, the behaviour of the system with respect to a constant magnetic field. We show that the properties of the oscillator on CP(N) qualitatively differ in the N>1 and N=1 cases. In the former case we construct the ``axially symmetric'' system which is locally equivalent to the oscillator. We perform the Kustaanheimo-Stiefel transformation of the oscillator on CP(2) and construct some generalized MIC-Kepler problem. We also define a N=2 superextension of the oscillator on CP(N) and show that for N>1 the inclusion of a constant magnetic field preserves the supersymmetry of the system.
We propose the Hamiltonian model of ${\cal N}=8$ supersymmetric mechanics on $n-$dimensional special Kähler manifolds (of the rigid type).
We equip the whole space of fields of the triplectic formalism of Lagrangian quantization with an even supersymplectic structure and clarify its geometric meaning. We also discuss its relation to a closed two-form arising naturally in the superfield approach to the triplectic formalism.
We construct the quantum oscillator interacting with a constant magnetic field on complex projective spaces $\DC P^N$, as well as on their non-compact counterparts, i. e. the $N-$dimensional Lobachewski spaces ${\cal L}_N$. We find the spectrum of this system and the complete basis of wavefunctions. Surprisingly, the inclusion of a magnetic field does not yield any qualitative change in the energy spectrum. For $N>1$ the magnetic field does not break the superintegrability of the system, whereas for N=1 it preserves the exact solvability of the system. We extend this results to the cones constructed over $\DC P^N$ and ${\cal L}_N$, and perform the (Kustaanheimo-Stiefel) transformation of these systems to the three-dimensional Coulomb-like systems.
We propose an exactly-solvable model of the quantum oscillator on the class of Kähler spaces (with conic singularities), connected with two-dimensional complex projective spaces. Its energy spectrum is nondegenerate in the orbital quantum number, when the space has non-constant curvature. We reduce the model to a three-dimensional system interacting with the Dirac monopole. Owing to noncommutativity of the reduction and quantization procedures, the Hamiltonian of the reduced system gets non-trivial quantum corrections. We transform the reduced system into a MIC-Kepler-like one and find that quantum corrections arise only in its energy and coupling constant. We present the exact spectrum of the generalized MIC-Kepler system. The one-(complex) dimensional analog of the suggested model is formulated on the Riemann surface over the complex projective plane and could be interpreted as a system with fractional spin.
We propose the notion of the oscillator on Kähler space and consider its supersymmetrization in the presence of a constant magnetic field.
We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kahler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kahler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid are originated. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to monopole, and find their four-dimensional origins. In the latter case the metric of configuration space is non-Kahler one. Finally, we extend these results to the family of Kahler spaces with conic singularities.
In order to establish an explicit connection between four-dimensional Hall effect on $S^4$ and six-dimensional Hall effect on $\DC P^3$, we perform the Hamiltonian reduction of a particle moving on $\DC P^3$ in a constant magnetic field to the four-dimensional Hall mechanics (i.e. a particle on $S^4$ in a SU(2) instanton field). This reduction corresponds to fixing the isospin of the latter system.
The linear Stark effect in the MIC-Kepler problem describing the interaction of charged particle with Dirac's dyon is considered. It is shown that constant homogeneous electric field completely removes the degeneracy of the energy levels on azimuth quantum number
We compare the non-commutative quantum mechanics (NCQM) on sphere and the discrete part of the spectrum of NCQM on pseudosphere (Lobachevsky plane, or $AdS_2$) in the presence of a constant magnetic field $B$ with planar NCQM. We show, that (pseudo)spherical NCQM has a ``critical point'', where the system becomes effectively one-dimensional, and two different ``phases'', which the phases of the planar system originate from, specified by the sign of the parameter $κ=1-Bθ$. The ``critical point'' of (pseudo)spherical NCQM corresponds to the $κ\to\infty $ point of conventional planar NCQM, and to the ``critical point'' $κ=0$ of the so-called ``exotic'' planar NCQM, with a symplectic coupling of the (commutative) magnetic field.
We consider quantum mechanics on the noncommutative plane in the presence of magnetic field $B$. We show, that the model has two essentially different phases separated by the point $Bθ=c\hbar^2/e$, where $θ$ is a parameter of noncommutativity. In this point the system reduces to exactly-solvable one-dimensional system. When $κ=1-eBθ/c\hbar^2<0$ there is a finite number of states corresponding to the given value of the angular momentum. In another phase, i.e. when $κ>0$ the number of states is infinite. The perturbative spectrum near the critical point $κ=0$ is computed.
We propose an extended BRST invariant Lagrangian quantization scheme of general gauge theories based on an explicit realization of the modified triplectic algebra that was announced in our previous investigation (hep-th/0104189). The algebra includes, besides the odd operators $V^a$ appearing in the triplectic formalism, also the odd operators $U^a$ introduced within modified triplectic quantization, both of which being anti-Hamiltonian vector fields. We show that some even supersymplectic structure defined on the space of fields and antifields provides the extended BRST path integral with a well-defined integration measure. All the known Lagrangian quantization schemes based on the extended BRST symmetry are obtained by specifying the (free) parameters of that method.
We show that the specific operators V^a appearing in the triplectic formalism can be viewed as the anti-Hamiltonian vector fields generated by a second rank irreducible Sp(2) tensor. This allows for an explicit realization of the triplectic algebra being constructed from an arbitrary Poisson bracket on the space of the fields only, equipped by the flat Poisson connection. We show that the whole space of fields and antifields can be equipped with an even supersymplectic structure when this Poisson bracket is non-degenerate. This observation opens the possibility to provide the BRST/antiBRST path integral by a well-defined integration measure, as well as to establish a direct link between the Sp(2) symmetric Lagrangian and Hamiltonian BRST quantization schemes.
We present two examples of SUSY mechanics related with Kähler geometry. The first system is the N=4 supersymmetric one-dimensional sigma-model proposed in hep-th/0101065. Another system is the N=2 SUSY mechanics whose phase space is the external algebra of an arbitrary Kähler manifold. The relation of these models with antisymplectic geometry is discussed.