2,113 research outputs found
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 on the moduli space,
parametrised by , 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 on the moduli space whose curvature is
proportional to the symplectic forms .Comment: 8 page
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
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
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
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
Quantized Nambu-Poisson Manifolds and n-Lie Algebras
We investigate the geometric interpretation of quantized Nambu-Poisson
structures in terms of noncommutative geometries. We describe an extension of
the usual axioms of quantization in which classical Nambu-Poisson structures
are translated to n-Lie algebras at quantum level. We demonstrate that this
generalized procedure matches an extension of Berezin-Toeplitz quantization
yielding quantized spheres, hyperboloids, and superspheres. The extended
Berezin quantization of spheres is closely related to a deformation
quantization of n-Lie algebras, as well as the approach based on harmonic
analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms
of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative
hyperplanes. Some applications to the quantum geometry of branes in M-theory
are also briefly discussed.Comment: 43 pages, minor corrections, presentation improved, references adde
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
Finite-Difference Equations in Relativistic Quantum Mechanics
Relativistic Quantum Mechanics suffers from structural problems which are
traced back to the lack of a position operator , satisfying
with the ordinary momentum operator
, in the basic symmetry group -- the Poincar\'e group. In this paper
we provide a finite-dimensional extension of the Poincar\'e group containing
only one more (in 1+1D) generator , satisfying the commutation
relation with the ordinary boost generator
. The unitary irreducible representations are calculated and the
carrier space proves to be the set of Shapiro's wave functions. The generalized
equations of motion constitute a simple example of exactly solvable
finite-difference set of equations associated with infinite-order polarization
equations.Comment: 10 LaTeX pages, final version, enlarged (2 more pages
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