Antibrackets and localization of (path) integrals (original) (raw)
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On Exact Evaluation of Path Integrals
Annals of Physics, 1994
We develop a general method to evaluate exactly certain phase space path integrals. Our method is applicable to hamiltonians which are functions of a classical phase space observable that determines the action of a circle on the phase space. Our approach is based on the localization technique, originally introduced to derive the Duistermaat-Heckman integration formula and its path integral generalizations. For this, we reformulate the phase space path integral in an auxiliary field representation that corresponds to a superloop space with both commuting and anticommuting coordinates. In this superloop space, the path integral can be interpreted in terms of a model independent equivariant cohomology, and evaluated exactly in the sense that it localizes into an integral over the original phase space. The final result can be related to equivariant characteristic classes. Curiously, our auxiliary field representation and the corresponding superloop space equivariant cohomology interpretation of the path integral essentially coincides with a superloop space formulation of ordinary Poincare supersymmetric quantum field theories. † Permanent Address
Path Integral Quantization of Doubly Supersymmetric Model
Journal of Applied Mathematics and Physics, 2022
The Hamilton-Jacobi formalism is used to discuss the path integral quantization of the double supersymmetric models with the spinning superparticle in the component and superfield form. The equations of motion are obtained as total differential equations in many variables. The equations of motion are integrable, and the path integral is obtained as an integration over the canonical phase space coordinates.
2013
Abstract. We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n), so(2n)) or (su(p, q), so(2p, 2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness. Key words: superintegrable systems; intertwining operators; dynamical algebras 2000 Mathematics Subject Classification: 17B80; 81R12; 81R15 1
Quantum, classical symmetries and action-angle variables by factorization of superintegrable systems
arXiv (Cornell University), 2023
The purpose of this work is to present a method based on the factorizations used in one dimensional quantum mechanics in order to find the symmetries of quantum and classical superintegrable systems in higher dimensions. We apply this procedure to the harmonic oscillator and Kepler-Coulomb systems to show the differences with other more standard approaches. We have described in detail the basic ingredients to make explicit the parallelism of classical and quantum treatments. One of the most interesting results is the finding of action-angle variables as a natural component of the classical sysmmetries within this formalism.
Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures
Mathematics, 2021
In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expressi...
Antibrackets and Supersymmetric Mechanics
NATO ASI Series, 1994
Using odd symplectic structure constructed over tangent bundle of the symplectic manifold, we construct the simple supergeneralization of an arbitrary Hamiltonian mechanics on it. In the case, if the initial mechanics defines Killing vector of some Riemannian metric, corresponding supersymmetric mechanics can be reformulated in the terms of even symplectic structure on the supermanifold.
Integrals of motion, supersymmetric quantum mechanics and dynamical supersymmetry
Lecture Notes in Physics
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. * Based on invited talk given at the International Seminar "Supersymmetries and Quantum Symmetries" dedicated to the
Complexified path integrals, exact saddles and supersymmetry
In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of p in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.
Symplectic integration of Hamiltonian systems
We survey past work and present new algorithms to numerically integrate the trajectories of Hamiltonian dynamical systems.These algorithms exactly preserve the symplectic 2-form, i.e. they preserve all the Poincar6 invariants. The algorithms have been tested on a variety of examples and results are presented for the Fermi-Pasta-Ulam nonlinear string, the Henon-Heiles system, a four-vortex problem, and the geodesic flow on a manifold of constant negative curvature. In all cases the algorithms possess long-time stability and preserve global geometrical structures in phase space.