A new integrable system on the sphere and conformally equivariant quantization

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi–Moser system, we disclose a novel integrable system on the sphere Sⁿ$S^n$, namely the dual Moser system. The latter falls, along with the Jacobi–Moser and Neumann–Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann–Uhlenbeck and dual Moser systems is ensured by means of the conformally equivariant quantization procedure.     

Integrable quantum Stäckel systems

The Stäckel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. We consider a class of Stäckel separable systems where the entries of the Stäckel matrix are monomials in the separation variables. We show that the only systems in this class for which the integrals of motion arising from the Stäckel construction keep commuting after quantization are, up to natural equivalence transformations, the so-called Benenti systems. Moreover, it turns out that the latter are the only quantum separable systems in the class under study.     

Bihamiltonian structures and Stäckel separability

It is shown that a class of Stäckel separable systems is characterized in terms of a Gel’fand–Zakharevich bihamiltonian structure. This structure arises as an extension of a Poisson–Nijenhuis structure on phase space. It is also shown that the Casimir of the Gel’fand–Zakharevich bihamiltonian structure provides the family of commuting Killing tensors found by Benenti and that, because of Eisenhart’s theorem, characterize orthogonal separability. It is also shown that recently found properties of quasi-bihamiltonian systems are natural consequences of the geometry of the extension of the Poisson–Nijenhuis structure.     

Non-Hamiltonian systems separable by Hamilton–Jacobi method

We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.     

Generalized Stäckel systems

We consider the generalized Stäckel systems, the broadest class of integrable Hamiltonian systems that admit separation of variables and possess separation relations affine in the Hamiltonians. For these systems we construct in a systematic fashion hierarchies of basic separable potentials. Moreover, we show how the equations of motion for the systems under study are related through appropriately chosen reciprocal transformations and how the respective constants of motion are related through generalized Stäckel transforms.     

Separable quantizations of Stäckel systems

In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stäckel system (defined on 2n$2n$ dimensional Poisson manifold) for which Stäckel matrix consists of monomials in position coordinates there exist infinitely many quantizations–parametrized by n$n$ arbitrary functions–that turn this system into a quantum separable Stäckel system.     

Integration of the dirac equation,which does not presume complete separation of variables,in Stäckel spaces

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stäckel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stäckel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 31–37, January, 1996.     

Stäckel electrovacuum spaces with isotropic complete sets. Integration of field equations for the metric generalizing a space of type (1.1)

A system of Einstein-Maxwell equations is integrated for the metric that generalizes the metric of a space with full set of type (1.1). This is necessary for carrying out the classification of the corresponding Stäckel electrovacuum spaces. An algebraic classification of the solutions obtained is carried out.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 17–20, December, 1987.     

Classification of null-Stäckel electrovac metrics with cosmological constant

The complete classification of the null Stäckel electrovac spacetimes is realized. For these spacetimes it is possible to integrate the geodetic equations by the complete separation of variables in the Hamilton-Jacobi equation.     

Anomalous Energy Minima Surfaces

Previous publications^2–5 have examined the phenomenon of level crossings as predicted by the semiempirical techniques CNDO/2⁶, IMDO⁷, and MINDO/3⁸ for certain molecules when the total energy is viewed as a function of some conformational property. The purpose of this article is to examine this same phenomenon using the extended Hückel method(EHM)⁹, the iterated extended Hückel method(IEHM), and an ab-initio procedure with a minimal STO-3G basis set. The molecule chosen for this study is CO₂ due to its structural simplicity and since previous calculations^2–5 have demonstrated level crossing with this molecule.     

Separation of Variables for Bi-Hamiltonian Systems

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called N manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.     

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

Exact solutions of vacuum Brans-Dicke equations

Vacuum Stäckel type (2.1) spaces are classified in Brans-Dicke theory for the case when the scalar field depends on all the variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 86–90, January, 1992.     

Stäckel electrovacuum spaces of type (2.1)

Classification of Stäckel electrovacuum spaces of type (2.1) is carried out. All metrics and potentials associated with the given type of spaces are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 54–56, February, 1989.     

Stäckel electrovacuum spaces that admit full separation of variables in a diagonalized Dirac equation. Type (2.1)

We list all electrovacuum Stäckel spaces of the type (2.1) that admit diagonalization and full separation of variables in the Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 116–120, February, 1992.     

Quantum fluids in the Kähler parametrization

In this Letter we address the problem of the quantization of the perfect relativistic fluids formulated in terms of the Kähler parametrization. This fluid model describes a large set of interesting systems such as the power law energy density fluids, Chaplygin gas, etc. In order to maintain the generality of the model, we apply the BRST method in the reduced phase space in which the fluid degrees of freedom are just the fluid potentials and the fluid current is classically resolved in terms of them. We determine the physical states in this setting, the time evolution and the path integral formulation.     

Integrability Versus Separability for the Multi-Centre Metrics

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-Stäckel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors.     

Unitons of Harmonic Maps from R2 to U(p,q)

We calculate a second cohomology class which determines a deformation quantization up to equivalence for a deformation quantization with separation of variables on a Kähler manifold, following P. Deligne.     

Canonical Quantization of Field Theories with Boundaries

The problem of canonical quantization of singular systems in a finite volume is studied by analysing a non-relativistic field theory. Firstly, we take the boundary conditions (BCs) as primary Dirac constraints. The quantization is performed canonically using Dirac’s procedure. Then, we quantize this model canonically in the classical solution space. We show that these two different quantization schemes are equivalent although they start from different settings.     

Almost-Kähler Deformation Quantization

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure in order to construct a Fedosov-type deformation quantization on this manifold.