A new type of phase-space path integral

Marinov, M. S.

doi:10.1016/0375-9601(91)90352-9

Cited by 41 publications

(47 citation statements)

References 12 publications

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“…The imaginary time formalism combined with the Monte-Carlo method is effective for finding binding energies of the lowest states, but not for scattering problems. The path-integral method in the phase space [60,61] represents perhaps more promising tool, since the Green function is a real function, it is positive definite in the classical limit, although it is not positive definite in general. The phase-space path integral method is used in [66] to find the evolution of a quantum state as an illustrative example (see also [67]).…”

Section: Monte Carlo Methods For Averaging Over the Wigner Functionmentioning

confidence: 99%

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Semiclassical expansion of quantum characteristics for many-body potential scattering problem

2007

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In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the starproduct operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in . We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as nonlocality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p-forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.

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Section: Monte Carlo Methods For Averaging Over the Wigner Functionmentioning

confidence: 99%

“…We wish to establish a connection of the phase-space Green function [60,61] with quantum characteristics and to derive eqs. (3.11) using the Green function method.…”

Section: Green Function In Phase Space In Terms Of Characteristicsmentioning

confidence: 99%

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

2007

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“…An alternate characterization of Heisenberg evolution in symbol space is found in Marinov's [31] path integral representation of ρ(t, x). This path integral has the following structure.…”

Section: Discussionmentioning

confidence: 99%

Heisenberg evolution WKB and symplectic area phases

Osborn

Kondratieva

2002

J. Phys. A: Math. Gen.

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The Schrödinger and Heisenberg evolution operators are represented in phase space T * R n by their Weyl symbols. Their semiclassical approximations are constructed in the short and long time regimes. For both evolution problems, the WKB representation is purely geometrical: the amplitudes are functions of a Poisson bracket and the phase is the symplectic area of a region in T * R n bounded by trajectories and chords. A unified approach to the Schrödinger and Heisenberg semiclassical evolutions is developed by introducing an extended phase space χ 2 ≡ T * (T * R n ). In this setting Maslov's pseudodifferential operator version of WKB analysis applies and represents these two problems via a common higher dimensional Schrödinger evolution, but with different extended Hamiltonians. The evolution of a Lagrangian manifold in χ 2 , defined by initial data, controls the phase, amplitude and caustic behavior. The symplectic area phases arise as a solution of a boundary condition problem in χ 2 . Various applications and examples are considered. Physically important observables generally have symbols that are free of rapidly oscillating phases. The semiclassical Heisenberg evolution in this context has been traditionally realized as an power series expansion whose leading term is classical transport. The extended Heisenberg Hamiltonian has a reflection symmetry that ensures this behavior. If the WKB initial phase is zero, it remains so for all time, and semiclassical dynamics reduces to classical flow plus finite order corrections.

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“…Formula (7.19) is dual to the Feynman path-integral formula [45,46,47,48,21]. The difference between (7.19) and the path integral is the same as between (3.5) and (3.7).…”

Section: Theoremmentioning

confidence: 99%

Symplectic areas, quantization, and dynamics in electromagnetic fields

Karasev

Osborn

2002

Journal of Mathematical Physics

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A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

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A new type of phase-space path integral

Cited by 41 publications

References 12 publications

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

Heisenberg evolution WKB and symplectic area phases

Symplectic areas, quantization, and dynamics in electromagnetic fields

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