Geometric Prequantization of the Moduli Space of the Vortex equations on a Riemann surface

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact K\"{a}hler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, K\"{a}hler) structures $\Omega_{\Psi_0}$ on the moduli space, parametrised by $\Psi_0$, a section of a line bundle on the Riemann surface. Next we show that corresponding to these there is a family of prequantum line bundles ${\mathcal P}_{\Psi_0}$on the moduli space whose curvature is proportional to the symplectic forms $\Omega_{\Psi_0}$.Comment: 8 page

Quantization Of Cyclotron Motion and Quantum Hall Effect

We present a two dimensional model of IQHE in accord with the cyclotron motion. The quantum equation of the QHE curve and a new definition of filling factor are also given.Comment: 13 Pages, Latex, 1 figure, to appear in Europhys. Lett. September 199

Time and Geometric Quantization

In this paper we briefly review the functional version of the Koopman-von Neumann operatorial approach to classical mechanics. We then show that its quantization can be achieved by freezing to zero two Grassmannian partners of time. This method of quantization presents many similarities with the one known as Geometric Quantization.Comment: Talk given by EG at "Spacetime and Fundamental Interactions: Quantum Aspects. A conference to honour A.P.Balachandran's 65th birthday

Effective Equations of Motion for Quantum Systems

In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture is developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.Comment: 31 pages; v2: a new example, new reference

Symplectic Cuts and Projection Quantization

The recently proposed projection quantization, which is a method to quantize particular subspaces of systems with known quantum theory, is shown to yield a genuine quantization in several cases. This may be inferred from exact results established within symplectic cutting.Comment: 12 pages, v2: additional examples and a new reference to related wor

Quantum-Mechanical Dualities on the Torus

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e., independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.Comment: New examples added, some precisions mad

Extended diffeomorphism algebras in (quantum) gravitational physics

We construct an explicit representation of the algebra of local diffeomorphisms of a manifold with realistic dimensions. This is achieved in the setting of a general approach to the (quantum) dynamics of a physical system which is characterized by the fundamental role assigned to a basic underlying symmetry. The developed mathematical formalism makes contact with the relevant gravitational notions by means of the addition of some extra structure. The specific manners in which this is accomplished, together with their corresponding physical interpretation, lead to different gravitational models. Distinct strategies are in fact briefly outlined, showing the versatility of the present conceptual framework.Comment: 20 pages, LATEX, no figure

Abelian BF theory and Turaev-Viro invariant

The U(1) BF Quantum Field Theory is revisited in the light of Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition function is related to the BF one and how the latter on its turn coincides with an abelian Turaev-Viro invariant. Significant differences compared to the non-abelian case are highlighted.Comment: 47 pages and 6 figure