2,057 research outputs found

    Higher-Order Differential Operators on a Lie Group and Quantization

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    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

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    Relativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator x^\hat{x}, satisfying [x^,p^]=i1^[\hat{x},\hat{p}]=i\hbar\hat{1} with the ordinary momentum operator p^\hat{p}, 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 π^\hat{\pi}, satisfying the commutation relation [k^,π^]=i1^[\hat{k},\hat{\pi}]=i\hbar\hat{1} with the ordinary boost generator k^\hat{k}. 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

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    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

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    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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