Francisco F. López-Ruiz

Harmonic states for the free particle

Different families of states, which are solutions of the time-dependent free Schrodinger equation, are imported from the harmonic oscillator using the quantum Arnold transformation introduced in Aldaya et al (2011 J. Phys. A: Math. Theor.44 065302). Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite–Gauss and Laguerre–Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.

A Quantizable Model of Massive Gauge Vector Bosons without Higgs

We incorporate the parameters of the gauge group G into the gauge theory of interactions through a nonlinear partial-trace σ-model Lagrangian on G/H. The minimal coupling of the new (Goldstone-like) scalar bosons provides mass terms to those intermediate vector bosons associated with the quotient G/H, without spoiling gauge invariance, while the H-vector potentials remain massless. The main virtue of a partial trace on G/H, rather than on the entire G, is that we can find an infinite-dimensional symmetry, with nontrivial Noether invariants, which ensures quantum integrability in a non-canonical quantization scheme. The present formalism is explicitly applied to the case G = SU(2)× U(1), as a Higgsless alternative to the Standard Model of electroweak interactions, although it can also be used in low-energy phenomenological models for strong interactions.

Symmetry group of massive Yang–Mills theories without Higgs and their quantization

We analyze the symmetry group of massive Yang–Mills theories and their quantization strongly motivated by an already proposed alternative to the standard model of electroweak interactions without Higgs. In these models the mass generation of the intermediate vector bosons is based on a non-Abelian Stueckelberg mechanism, where the dynamics of the Goldstone-like bosons is addressed by a partial-trace nonlinear-sigma piece of the Lagrangian. In spite of the high nonlinearity of the scalar sector, the existence of an infinite number of symmetries, extending the traditional gauge group, allows us to sketch a group-theoretical quantization algorithm specially suited to nonlinear systems, which departs from usual canonical quantization. On the quantum representation space of this extended symmetry group, a quantum Hamiltonian preserving the representation can be given, whose classical analog reproduces the equations of motion.

Coupling Nonlinear Sigma-Matter to Yang-Mills Fields: Symmetry Breaking Patterns

Abstract We extend the traditional formulation of Gauge Field Theory by incorporating the (non-Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinear-sigma-model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling prescription, which resembles the so-called Stueckelberg transformation [1], but for the non-Abelian case. Here we study the case of internal gauge symmetry groups, in particular, unitary groups U(N). We show how to couple standard Yang-Mills Theory to Nonlinear-Sigma Models on cosets of U(N): complex projective, Grassman and flag manifolds. These different couplings lead to distinct (chiral) symmetry breaking patterns and Higgs-less mass-generating mechanisms for Yang-Mills fields.

NON-CANONICAL PERTURBATION THEORY OF NONLINEAR SIGMA MODELS

We explore the O(N)-invariant Non-Linear Sigma Model (NLSM) in a different perturbative regime from the usual relativistic-free-field one, by using non-canonical basic commutation relations adapted to the underlying O(N) symmetry of the system, which also account for the nontrivial (nonflat) geometry and topology of the target manifold.

SYMMETRIES OF NON-LINEAR SYSTEMS: GROUP APPROACH TO THEIR QUANTIZATION

We report briefly on an approach to quantum theory entirely based on symmetry grounds which improves Geometric Quantization in some respects and provides an alternative to the canonical framework. The present scheme, being typically non-perturbative, is primarily intended for non-linear systems, although needless to say that finding the basic symmetry associated with a given (quantum) physical problem is in general a difficult task, which many times nearly emulates the complexity of finding the actual (classical) solutions. Apart from some interesting examples related to the electromagnetic and gravitational particle interactions, where an algebraic version of the Equivalence Principle naturally arises, we attempt to the quantum description of non-linear sigma models. In particular, we present the actual quantization of the partial-trace non-linear SU(2) sigma model as a representative case of non-linear quantum field theory.

Quantum Integrability of the Dynamics on a Group Manifold

Abstract We study the dynamics of a particle moving on the SU(2) group manifold. An exact quantization of this system is accomplished by finding the unitary and irreducible representations of a finite-dimensional Lie subalgebra of the whole Poisson algebra in phase space. In fact, the basic position and momentum operators, as well as the Hamiltonian, are found in the enveloping algebra of the anti-de Sitter group SO(3,2). The present algorithm mimics the one previously used in Ref. [1]. Our construction can be extended to more general semi-simple Lie groups. This framework would allow us to achieve the quantization of the geodesic motion in a symmetric pseudo-Riemannian manifold

UNFOLDING THE QUANTUM ARNOLD TRANSFORMATION

The recently proposed quantum Arnold transformation is revisited and extended and its relation with other methods is emphasized. Possible applications are also outlined.

A round trip from Caldirola to Bateman systems

For the quantum Caldirola-Kanai Hamiltonian, describing a quantum damped harmonic oscillator, a couple of constant of motion operators generating the Heisenberg algebra can be found. The inclusion in this algebra, in a unitary manner, of the standard time evolution generator , which is not a constant of motion, requires a non-trivial extension of this basic algebra and the physical system itself, which now includes a new dual particle. This enlarged algebra, when exponentiated, leads to a group, named the Bateman group, which admits unitary representations with support in the Hilbert space of functions satisfying the Schrodinger equation associated with the quantum Bateman Hamiltonian, either as a second order differential operator as well as a first order one. The classical Bateman Hamiltonian describes a dual system of a damped (losing energy) particle and a dual (gaining energy) particle. The classical Bateman system has a solution submanifold containing the trajectories of the original system as a submanifold. When restricted to this submanifold, the Bateman dual classical Hamiltonian leads to the Caldirola-Kanai Hamiltonian for a single damped particle. This construction can also be done at the quantum level, and the Caldirola-Kanai Hamiltonian operator can be derived from the Bateman Hamiltonian operator when appropriate constraints are imposed.

Group-quantization of nonlinear 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. 11.30.-j, 03.65.-w, 05.45.-a.