On the geometric prequantization of Maxwell equations

The geometric prequantization of Maxwell equations in a vacuum is described and its relation with geometric prequantization of the extended phase space is pointed out.     

On the reduced phase space of a cotangent bundle

The geometric prequantization of a reduced phase space of a cotangent bundle is described and its relation with the geometric prequantization of the cotangent bundle is pointed out.     

Geometric BRST quantization,I: Prequantization

This is the first part of a two-part paper dedicated to the definition of BRST quantization in the framework of geometric quantization. After recognizing prequantization as a manifestation of the Poisson module structure of the sections of the prequantum line bundle, we define BRST prequantization and show that it is the homological analog of the symplectic reduction of prequantum data. We define a prequantum BRST cohomology theory and interpret it in terms of geometric objects. We then show that all Poisson structures correspond under homological reduction. This allows to prove, in the BRST context, that prequantization and reduction commute.     

On the geometric prequantization of Poisson manifolds

The geometric prequantization of Poisson manifolds is described using the Weinstein theory of local symplectic groupoids.     

Condition of Prequantization to Two—Dimensional Manifolds with Boundary

The Weil‘s integrality condition of prequantization is generalized to two-dimensional phase space with boundaries.It is shown that in the prequantization condition a term related to the symplectic potential on the boundary appears.The necessity of the generalized condition is proved by analyzing the isolated singularities of the Hermitian bundle while the sufficiency of the condition is proved via geometric construction on the space of equivalence class.     

Prequantization and KMS structures

The properties of the representations of the canonical commutation relations, obtained in the prequantization program, are investigated with special attention to the relevance of the KMS structures in this context. In particular, we show how these structures provide a natural way to pass from the prequantization representation of the CCR to the Schrödinger representation.     

Equivariant Differential Characters and Symplectic Reduction

We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction. Supported in part by NSF grant DMS-0603892. Supported in part by NSF grant DMS-0456714.     

Functorial geometric quantization and Van Hove's theorem

A classical theorem of Van Hove in conjunction with a formalism developed by Weinstein is used to prove that a quantization functor does not exist. In the proof a category of exact transverse Lagrangian submanifolds is introduced which provides a functorial link between Schrödinger quantization and the prequantization/polarization theory of Kostant and Souriau.     

Closed 2-Form of 2D Black Holes from Geometric Prequantization Method

In this paper, we consider effective Hamiltonian of 2D dilatonic black hole adding Axion field and calculate closed 2-form by using geometric prequantization method. It yields to the Schrödinger equation which may be solved to obtain wave function. We obtained a condition on the cosmological constant to obtain appropriate Hamilton equation of motions.     

Constraints and polarizations

Geometric quantization is applied to first-class constrained systems. It is shown how certain inconsistencies associated with the prequantization of open gauge algebras and observables can be cured at the quantum level by the introduction of suitable polarizations. Furthermore it is shown a large Poisson subalgebra of the ideal of constrait functions can be consistently implemented at the quantum level and how reparametrization invariance of the constraint surface relates to a partially flat local Kähler structure on phase space.     

Prequantum bundles and projective Hilbert geometries

Principal circle bundles with connection and symplectic curvature over Banach manifolds are investigated. Using results on contact manifolds alternate proofs for some results of B. Kostant are given and a symplectic structure for the total space of the corresponding principal \ {0} bundle is constructed. As an example, these results are applied to the projective fibration of a complex Hilbert space. This gives close relations between the geometric formulation of classical and quantum dynamical systems. As another application, a functorial construction of the prequantization procedure of B. Kostant is given.     

Gravitational Descendants in Symplectic Field Theory

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown using the ideas in Okounkov and Pandharipande (Ann Math 163(2):517–560, 2006) that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we follow the ideas in Cieliebak and Latschev ( [math.s6], 2007) to compute the corresponding sequence of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold.     

Note on duality on polarized symplectic manifolds

In this note we use some of the results of [3] to derive a general duality theorem for the cohomologies of foliated structures on a manifold. The result is applied to the special case of a symplectic manifold M on which the foliation is given by a complex polarization F in the sense of geometric quantization. We obtain, for example, a rigorous proof of the fact that for a smooth function ƒ on M whose Hamiltonian vector field leaves F invariant, the spectrum of the corresponding prequantization operator v(ƒ) coincides with the spectrum of its transpose, under the above duality. This latter result was obtained by Simms in [12] under certain hypotheses. Proofs of the validity of those hypotheses are now available in the literature; cf. [3] and [7].     

An associative algebra of functions on the orbits of nilpotent Lie groups

We introduce an associative bilinear product F * G on the space of Schwartz functions of the coadjoint orbits of a large family of nilpotent Lie groups. We can then introduce an operator , for all . When the nilpotent group is the Heisenberg group N(3), the correspondence is a canonical prequantization of functions F(q, p) in phase space. Part of this work was done at the University of California, Los Angeles.     

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold P$P$, there are two approaches to its geometric quantization: one involves a circle bundle Q$Q$ over P$P$ endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of P$P$. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P$P$ is obtained from the Lie groupoid of Q$Q$ via an S¹$S^1$ reduction that preserves both the Lie groupoid and the geometric structures.     

Quantum electrodynamics in a laser and the electron laser collision

Quantum electrodynamics in a laser is formulated, in which the electron–laser interaction is exactly considered, while the interaction of an electron and a single photon is considered by perturbation. The formulation is applied to the electron– laser collisions. The effect of coherence between photons in the laser is therefore fully considered in these collisions. The possibility of γ-ray laser generation by use of this kind of collision is discussed.     

Influence of nonlinear interactions on the development of instability in hydrodynamic wave systems

The problem of the development of shear instability in a three-layer medium simulating the flow of a stratified incompressible fluid is considered. The hydrodynamic equations are solved by expanding the Hamiltonian in a small parameter. The equations for three interacting waves, one of which is unstable, have been derived and solved numerically. The three-wave interaction is shown to stabilize the instability. Various regimes of the system’s dynamics, including the stochastic ones dependent on one of the invariants in the problem, can arise in this case. It is pointed out that the instability development scenario considered differs from the previously considered scenario of a different type, where the three-wave interaction does not stabilize the instability. The interaction of wave packets is considered briefly.     

Semi-statistical model for metals

Calculations of the cohesive energy, Wigner-Seitz radius and compressibility of some metals are performed by a semi-statistical method. Two regions are considered in the Wigner-Seitz cell, taken in the spherical approximation. In the internal region the electronic density is considered the same as in the free atom, while in the external region it is computed by means of the Thomas-Fermi-Dirac equation. In the calculation of energies, inhomogeneity and correlation corrections are considered.     

线性聚焦和线性散焦效应对空间光孤子间相互作用的影响

考虑非均匀一维自聚焦介质的横向不均匀性,利用非线性薛定谔方程满足的守恒律给出了相邻空间孤子间隔的解析式,并对空间孤子之间的相互作用进行了数值模拟.结果表明,线形聚焦效应增强了空间孤子之间的相互作用;而线形散焦效应减弱了空间孤子之间的相互作用.当不考虑介质横向不均匀时,空间孤子之间发生周期性的碰撞.线性散焦效应使相邻空间孤子之间的间隔随传输距离发生周期性的变化,但孤子之间并不发生碰撞.线性聚焦效应使相邻空间孤子随传输距离发生周期性的碰撞,线性聚焦效应具有压制损耗使相邻空间孤子间隔变大的作用.     

Field ionization processes in field ion microscopy

The processes of field ionization are considered by the time dependent perturbation theory. The metal wave function to be occupied by a tunneling electron from a monovalent atom is approximated by the tight-binding method. The system considered is hydrogen imaging a tungsten-type metal. The (001) crystallographic plane with an interatomic spacing of 3.16 Å and an externally applied electric field of 2.3 V/Å is considered for numerical results. Image formation mechanisms of field ion microscopy are qualitatively discussed.