171 research outputs found
Hidden symmetries in a gauge covariant approach, Hamiltonian reduction and oxidation
Hidden symmetries in a covariant Hamiltonian formulation are investigated
involving gauge covariant equations of motion. The special role of the
Stackel-Killing tensors is pointed out. A reduction procedure is used to reduce
the original phase space to another one in which the symmetries are divided
out. The reverse of the reduction procedure is done by stages performing the
unfolding of the gauge transformation followed by the Eisenhart lift in
connection with scalar potentials.Comment: 15 pages; based on a talk at QTS-7 Conference, Prague, August 7-13,
201
Twisted geometries: A geometric parametrisation of SU(2) phase space
A cornerstone of the loop quantum gravity program is the fact that the phase
space of general relativity on a fixed graph can be described by a product of
SU(2) cotangent bundles per edge. In this paper we show how to parametrize this
phase space in terms of quantities describing the intrinsic and extrinsic
geometry of the triangulation dual to the graph. These are defined by the
assignment to each triangle of its area, the two unit normals as seen from the
two polyhedra sharing it, and an additional angle related to the extrinsic
curvature. These quantities do not define a Regge geometry, since they include
extrinsic data, but a looser notion of discrete geometry which is twisted in
the sense that it is locally well-defined, but the local patches lack a
consistent gluing among each other. We give the Poisson brackets among the new
variables, and exhibit a symplectomorphism which maps them into the Poisson
brackets of loop gravity. The new parametrization has the advantage of a simple
description of the gauge-invariant reduced phase space, which is given by a
product of phase spaces associated to edges and vertices, and it also provides
an abelianisation of the SU(2) connection. The results are relevant for the
construction of coherent states, and as a byproduct, contribute to clarify the
connection between loop gravity and its subset corresponding to Regge
geometries.Comment: 28 pages. v2 and v3 minor change
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
Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity
We revise the use of 8-dimensional conformal, complex (Cartan) domains as a
base for the construction of conformally invariant quantum (field) theory,
either as phase or configuration spaces. We follow a gauge-invariant Lagrangian
approach (of nonlinear sigma-model type) and use a generalized Dirac method for
the quantization of constrained systems, which resembles in some aspects the
standard approach to quantizing coadjoint orbits of a group G. Physical wave
functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels
are explicitly calculated in and holomorphic picture in these Cartan domains
for both scalar and spinning quantum particles. Similarities and differences
with other results in the literature are also discussed and an extension of
Schwinger's Master Theorem is commented in connection with closure relations.
An adaptation of the Born's Reciprocity Principle (BRP) to the conformal
relativity, the replacement of space-time by the 8-dimensional conformal domain
at short distances and the existence of a maximal acceleration are also put
forward.Comment: 33 pages, no figures, LaTe
Dynamics of Void and its Shape in Redshift Space
We investigate the dynamics of a single spherical void embedded in a
Friedmann-Lema\^itre universe, and analyze the void shape in the redshift
space. We find that the void in the redshift space appears as an ellipse shape
elongated in the direction of the line of sight (i.e., an opposite deformation
to the Kaiser effect). Applying this result to observed void candidates at the
redshift z~1-2, it may provide us with a new method to evaluate the
cosmological parameters, in particular the value of a cosmological constant.Comment: 19 pages, 11 figure
Thermal Vacuum Radiation in Spontaneously Broken Second-Quantized Theories on Curved Phase Spaces of Constant Curvature
We construct second-quantized (field) theories on coset spaces of
pseudo-unitary groups U(p,q)$. The existence of degenerated quantum vacua
(coherent states of zero modes) leads to a breakdown of the original
pseudo-unitary symmetry. The action of some spontaneously broken symmetry
transformations destabilize the vacuum and make it to radiate. We study the
structure of this thermal radiation for curved phase spaces of constant
curvature: complex projective spaces CP^{N-1}=SU(N)/U(N-1) and open complex
balls CD^{N-1}=SU(1,N-1)/U(N-1). Positive curvature is related to generalized
Fermi-Dirac (FD) statistics, whereas negative curvature is connected with
generalized Bose-Einstein (BE) statistics, the standard cases being recovered
for N=2. We also make some comments on the contribution of the vacuum (dark)
energy to the cosmological constant and the phenomenon of inflation.Comment: 17 pages. This article deals with a sort of "curvature-statistics
connection". To appear in Int. J. Geom. Meth. Mod. Phy
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
On transversally elliptic operators and the quantization of manifolds with -structure
An -structure on a manifold is an endomorphism field
\phi\in\Gamma(M,\End(TM)) such that . Any -structure
determines an almost CR structure E_{1,0}\subset T_\C M given by the
-eigenbundle of . Using a compatible metric and connection
on , we construct an odd first-order differential operator ,
acting on sections of , whose principal symbol is of the
type considered in arXiv:0810.0338. In the special case of a CR-integrable
almost -structure, we show that when is the generalized
Tanaka-Webster connection of Lotta and Pastore, the operator is given by D
= \sqrt{2}(\dbbar+\dbbar^*), where \dbbar is the tangential Cauchy-Riemann
operator.
We then describe two "quantizations" of manifolds with -structure that
reduce to familiar methods in symplectic geometry in the case that is a
compatible almost complex structure, and to the contact quantization defined in
\cite{F4} when comes from a contact metric structure. The first is an
index-theoretic approach involving the operator ; for certain group actions
will be transversally elliptic, and using the results in arXiv:0810.0338,
we can give a Riemann-Roch type formula for its index. The second approach uses
an analogue of the polarized sections of a prequantum line bundle, with a CR
structure playing the role of a complex polarization.Comment: 31 page
Models for Modules
We recall the structure of the indecomposable sl(2) modules in the
Bernstein-Gelfand-Gelfand category O. We show that all these modules can arise
as quantized phase spaces of physical models. In particular, we demonstrate in
a path integral discretization how a redefined action of the sl(2) algebra over
the complex numbers can glue finite dimensional and infinite dimensional
highest weight representations into indecomposable wholes. Furthermore, we
discuss how projective cover representations arise in the tensor product of
finite dimensional and Verma modules and give explicit tensor product
decomposition rules. The tensor product spaces can be realized in terms of
product path integrals. Finally, we discuss relations of our results to brane
quantization and cohomological calculations in string theory.Comment: 18 pages, 6 figure
Reduction and approximation in gyrokinetics
The gyrokinetics formulation of plasmas in strong magnetic fields aims at the
elimination of the angle associated with the Larmor rotation of charged
particles around the magnetic field lines. In a perturbative treatment or as a
time-averaging procedure, gyrokinetics is in general an approximation to the
true dynamics. Here we discuss the conditions under which gyrokinetics is
either an approximation or an exact operation in the framework of reduction of
dynamical systems with symmetryComment: 15 pages late
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