Central extensions and quantum physics

The unitary implementation of a symmetry group G of a classical system in the corresponding quantum theory entails unavoidable deformations ˜G of G, namely, central extensions by the typical phase invariance group U(1). The appearance of central charges in the corresponding Lie-algebra quantum commutators, as a consequence of non-trivial responses of the phase of the wave function under symmetry transformations, lead to a quantum generation of extra degrees of freedom with regard to the classical counterpart. In particular, symmetries of the Hall effect, Yang-Mills and conformally invariant classical field theories are affected when passing to the quantum realm.M. Calixto thanks the University of Granada for a Post-doctoral grant and the Department of Physics of Swansea for its hospitality

Group Quantization on Configuration Space: Gauge Symmetries and Linear Fields

A new, configuration-space picture of a formalism of group quantization, the GAQ formalism, is presented in the context of a previous, algebraic generalization. This presentation serves to make a comprehensive discussion in which other extensions of the formalism, principally to incorporate gauge symmetries, are developed as well. Both images are combined in order to analyse, in a systematic manner and with complete generality, the case of linear fields (abelian current groups). To ilustrate these developments we particularize them for several fields and, in particular, we carry out the quantization the abelian Chern-Simons models over an arbitrary closed surface in detail.M.N. is grateful to the Imperial College, where this paper has mainly been written, for its hospitality. M.N. is also grateful to the Spanish MEC, CSIC and IMAFF (Madrid) for a research contract. M.C. is grateful to the Spanish MEC for a FPI fellowship

Group-Theoretical Determination of the Mixing Angle in the Electroweak Gauge Group

The assumption that the Weinberg rotation between the gauge fields associated with the third component of the “weak isospin” (T3) and the hypercharge (Y ) proceeds in a natural way from a global homomorphism of the SU(2) U(1) gauge group in some locally isomorphic group (which proves to be U(2)), imposes strong restrictions so as to fix the single value sin2 W = 1/2. This result can be thought of only as being an asymptotic limit corresponding to an earlier stage of the Universe. It also lends support to the idea that e2/g2 and 1−M2W /M2Z are in principle unrelated quantities.V.A. is grateful to M. Asorey, J. Julve and A. Tiemblo, and all of us to J.M. Cerveró, J. Navarro-Salas, M. Navarro and A. Romero for valuable discussions

Global restrictions to the Mixing Angle W

In spite of its firm hold in particle physics, the gauge group SU(2)⊗U(1)Y proves not to be the most appropriate to describe the unification of weak and electromagnetic interactions. In fact, a look at the structure of SU(2)⊗U(1)Y tells us that both, the electric charge generator Q and its corresponding electromagnetic gauge field Aμ(x),are not basic constituents of this gauge group.This work was partially supported by the DGICYT. M. Calixto thanks the Spanish MEC for a FPI grant

Algebraic versus Topologic Anomalies

Within the frame of a Group Approach to Quantization anomalies arise in a quite natural way. We present in this talk an analysis of the basic obstructions that can be found when we try to translate symmetries of the Newton equations to the Quantum Theory. They fall into two classes: algebraic and topologic according to the local or global character of the obstruction. We present here one explicit example of each

Quantization on the Torus and modular invariance

The implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group-theoretical framework. Unlike the classical case, at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus. If it is of integer cohomology class n, then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if n is even, or antiperiodic if n is odd. If the symplectic form is of rational cohomology class n/r , a similar result holds –the wave functions must be either periodic or antiperiodic on a torus r times larger in both direccions, depending on the parity of nr. Application of these results to the Abelian Chern-Simons is discussed.J. Guerrero thanks the Spanish MEC for a Postdoctoral grant and the Department of Physics of Naples-INFN for its hospitality and financial support, and M. Calixto thanks the Spanish MEC for a FPI grant. Work partially supported by the DGICYT

Vacuum radiation in conformally invariant quantum field theory

Although the whole conformal group SO(4, 2) can be considered as a symmetry in a classical massless field theory, the subgroup of special conformal transformations (SCT), usually related to transitions to uniformly accelerated frames, causes vacuum radiation in the corresponding quantum field theory, in analogy to the Fulling-Unruh effect. The spectrum of the outgoing particles can be calculated exactly and proves to be a generalization of the Planckian one.Work partially supported by the DGICYT under contracts PB92-1055, PB92- 0302, PB95-1201 and PB95-094

Group-quantization of non-linear sigma models:particle on S2 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.Work partially supported by the Spanish MCYT, Junta de Andalucía and Fundación Séneca under projects FIS2005-05736-C03-01, P06-FQM-01951 and 03100/PI/05. F. F. López Ruiz would like to thank the C.S.I.C. for an I3P grant

Algebraic Quantization on the Torus and Modular Invariance

The aim of the Algebraic Quantization is the quantum description of a physical system by means of the unitary and irreducible representations of its symmetry group. Two cases have to be considered, corresponding to systems without constraints and to those with constraints, respectively. In the simplest case, the group e G of quantum symmetries will be a central extension by U(1) of the group G of classical symmetries.This work is partially suported by the DGICYT. J. Guerrero thanks the University of Granada for a postdoc grant