arXiv subjects

J. Guerrero

Publications and source records attributed to J. Guerrero.

31 records · Page 2Linked to original sources

arXiv1998

Higher-Order Quantization on a Lie Group

In this paper we are mainly concerned with the study of polarizations (in general of higher-order type) on a connected Lie group with a U(1)-principal bundle structure. The representation technique used here is formulated on the basis of a group quantization formalism previously introduced which generalizes the Kostant-Kirillov co-adjoint orbits method for connected Lie groups and the Borel-Weyl-Bott representation algorithm for semisimple groups. We illustrate the fundamentals of the group approach with the help of some examples like the abelian group $R^k$ and the semisimple group SU(2), and the use of higher-order polarizations with the harmonic oscillator group and the Schrödinger group, the last one constituting the simplest example of an anomalous group. Also, examples of infinite-dimensional anomalous groups are briefly considered.

math-ph↗

arXiv1998

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.

hep-th↗

arXiv1998

A Perturbative Approach to the Relativistic Harmonic Oscillator

A quantum realization of the Relativistic Harmonic Oscillator is realized in terms of the spatial variable $x$ and ${\d\over \d x}$ (the minimal canonical representation). The eigenstates of the Hamiltonian operator are found (at lower order) by using a perturbation expansion in the constant $c^{-1}$. Unlike the Foldy-Wouthuysen transformed version of the relativistic hydrogen atom, conventional perturbation theory cannot be applied and a perturbation of the scalar product itself is required.

math-ph↗

arXiv1997

Quantization on a Lie group: Higher-order Polarizations

Contents * Introduction -- Why $S^1$-extended phase space? -- Why central extensions of classical symmetries? * Central extension \Gt of a group $G$ -- Group cohomology -- Cohomology and contractions: Pseudo-cohomology -- Principal bundle with connection $(\Gtm,Θ)$ * Group Approach to Quantization -- $U(1)$-quantization -- Non-horizontal polarizations * Simple examples -- The abelian group $R^{k}$ -- The semisimple group $SU(2)$ * Algebraic anomalies -- Higher-order polarizations -- The Schrödinger group and Quantum Optics -- The Virasoro group and String Theory

math-ph↗

arXiv1997

Generalized Conformal Symmetry and Extended Objects from the Free Particle

The algebra of linear and quadratic functions of basic observables on the phase space of either the free particle or the harmonic oscillator possesses a finite-dimensional anomaly. The quantization of these systems outside the critical values of the anomaly leads to a new degree of freedom which shares its internal character with spin, but nevertheless features an infinite number of different states. Both are associated with the transformation properties of wave functions under the Weyl-symplectic group $WSp(6,\Re)$. The physical meaning of this new degree of freedom can be established, with a major scope, only by analysing the quantization of an infinite-dimensional algebra of diffeomorphisms generalizing string symmetry and leading to more general extended objects.

hep-th↗

arXiv1997

Global restrictions to the Mixing Angle $θ_W$

We provide the theoretic value 1/2 for the quantity $\sin**2θ_W=e**2/g**2$, as opposite to the experimental value of $\sin**2θ_W=1-M_W**2/M_Z**2 \sim 0.23$, in the electro-weak gauge group $SU(2)\otimes U(1)_Y$. The arguments we use are of topological nature and have to do with natural requirements of globality in the transformations of the gauge algebra. This results supports the idea that the two definitions of $\sin**2θ_W$ cannot be simultaneously identified except, maybe, in an asymptotic limit corresponding to an earlier stage of the Universe, before the spontaneous symmetry breaking had taken place.

hep-th↗

arXiv1997

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.

hep-th↗

arXiv1997

Algebraic Quantization on the Torus and Modular Invariance

New features of systems with non-trivial topology such as fractional quantum numbers, inequivalent quantizations, good operators, topological anomalies, etc. are described in the framework of an algebraic quantization procedure on a group. Modular invariance naturally appears as a subgroup of good operators in the particular case of the torus.

hep-th↗

arXiv1995

Algebraic Quantization, Good Operators and Fractional Quantum Numbers

The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the ``failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of {\it good} (and {\it bad}) operators. The analysis of ``constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring ``anomalous" operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively.

hep-th↗

arXiv1995

Canonical Coherent States for the Relativistic Harmonic Oscillator

In this paper we construct manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given $SL(2,R)$ representation, once a change of variables $z\in C\rightarrow z_D \in $ unit disk is performed. We also introduce higher-order, relativistic creation and annihilation operators, $\C,\Cc$, with canonical commutation relation $[\C,\Cc]=1$ rather than the covariant one $[\Z,\Zc]\approx$ Energy and naturally associated with the $SL(2,R)$ group. The canonical (relativistic) coherent states are then defined as eigenstates of $\C$. Finally, we construct a canonical, minimal representation in configuration space by mean of eigenstates of a canonical position operator.

hep-th↗

arXiv1995

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 $\hat{x}$, satisfying $[\hat{x},\hat{p}]=i\hbar\hat{1}$ with the ordinary momentum operator $\hat{p}$, in the basic symmetry group -- the Poincaré group. In this paper we provide a finite-dimensional extension of the Poincaré group containing only one more (in 1+1D) generator $\hatπ$, satisfying the commutation relation $[\hat{k},\hatπ]=i\hbar\hat{1}$ with the ordinary boost generator $\hat{k}$. 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.

hep-th↗

arXiv1994

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" ($T_3$) and the hypercharge ($Y$) proceeds in a natural way from a global homomorphism of the $SU(2)\otimes U(1)$ gauge group in some locally isomorphic group (which proves to be $U(2)$), imposes strong restrictions so as to fix the single value $\sin^2θ_W=1/2$. 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 $e^2/g^2$ and $1-M_W^2/M_Z^2$ are in principle unrelated quantities.

hep-th↗

arXiv1994

Optics, Mechanics and Quantization of Reparametrization Systems

In this paper we regard the dynamics obtained from Fermat principle as begin the classical theory of light. We (first-)quantize the action and show how close we can get to the Maxwell theory. We show that Quantum Geometric Optics is not a theory of fields in curved space. Considering Classical Mechanics to be on the same footing, we show the parallelism between Quantum Mechanics and Quantum Geometric Optics. We show that, due to the reparametrization invariance of the classical theories, the dynamics of the quantum theories is given by a Hamiltonian constraint. Some implications of the above analogy in the quantization of true reparameterization invariant systems are discussed.

hep-th↗