2,057 research outputs found
Higher-Order Differential Operators on a Lie Group and Quantization
This talk is devoted mainly to the concept of higher-order polarization on a
group, which is introduced in the framework of a Group Approach to
Quantization, as a powerful tool to guarantee the irreducibility of
quantizations and/or representations of Lie groups in those anomalous cases
where the Kostant-Kirilov co-adjoint method or the Borel-Weyl-Bott
representation algorithm do not succeed.Comment: 9 pages, latex, no figures, uses IJMPB.sty (included). New version
partially rewritten (title changed!), presented to the II Int. Workshop on
Class. and Quant. Integrable Systems, Dubna (Rusia) 1996, and published in
Int. J. Mod. Phys.
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
Revisited gauge principle: towards a unification of space-time and internal gauge interactions
The minimal coupling principle is revisited under the quantum perspectives of
the space-time symmetry. This revision is better realized on a Group Approach
to Quantization (GAQ) where group cohomology and extensions of groups play a
preponderant role. We firstly consider the case of the electromagnetic
potential; the Galilei and/or Poincare group is (non-centrally) extended by the
"local" U(1) group. This group can also be seen as a central extension,
parametrized by both the mass and the electric charge, of an
infinite-dimensional group, on which GAQ leads to the dynamics of a particle
moving in the presence of an electromagnetic field. Then we try the
gravitational interaction of a particle by turning into "local" the space-time
translations. However, promoting to "local" the space-time subgroup of the true
symmetry of the quantum free relativistic particle, i.e. the centrally extended
by U(1) Poincare group, results in a new electromagnetic-like force of pure
gravitational origin. This is a consequence of the space-time translations not
being an invariant subgroup of the extended Poincare group and constitutes a
preliminary attempt to a non-trivial mixing of space-time and internal gauge
interactions.Comment: 22 pages, LATEX, no figure
Group-quantization of non-linear sigma models: particle on S^2 revisited
We present the quantum mechanics of "partial-trace" non-linear sigma models,
on the grounds of a fully symmetry-based procedure. After the general scheme is
sketched, the particular example of a particle on the two-sphere is explicitly
developed. As a remarkable feature, no explicit constraint treatment is
required nor ordering ambiguities do appear. Moreover, the energy spectrum is
recovered without extra terms in the curvature of the sphere.Comment: 8 page
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