J. Guerrero

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.

Group approach to the quantization of the Pöschl–Teller dynamics*

The quantum dynamics of a particle in the modified Poschl-Teller potential is derived from the group SL (2, R) by applying a group approach to quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the SL(2, R) group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU(2) where all the states are physical.

Symmetries of the quantum damped harmonic oscillator

For the non-conservative Caldirola–Kanai system, describing a quantum damped harmonic oscillator, a couple of constant-of-motion operators generating the Heisenberg–Weyl algebra can be found. The inclusion of the standard time evolution generator (which is not a symmetry) as a symmetry in this algebra, in a unitary manner, requires a non-trivial extension of this basic algebra and hence of the physical system itself. Surprisingly, this extension leads directly to the so-called Bateman dual system, which now includes a new particle acting as an energy reservoir. In addition, the Caldirola–Kanai dissipative system can be retrieved by imposing constraints. The algebra of symmetries of the dual system is presented, as well as a quantization that implies, in particular, a first-order Schrodinger equation. As opposed to other approaches, where it is claimed that the spectrum of the Bateman Hamiltonian is complex and discrete, we obtain that it is real and continuous, with infinite degeneracy in all regimes.

Gauge principle revisited: 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 first consider the case of the electromagnetic potential; the Galilei and/or Poincare group is (noncentrally) extended by the “local” U(1) group. The resulting 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 making the space–time translations “local.” 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 electromagneticlike force of pure gravitational origin. This is a consequence...

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

Modular invariance on the torus and Abelian Chern–Simons theory

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 directions, depending on the parity of nr. Application of these results to the Abelian Chern–Simons theory is discussed.

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.

Inconsistencies in the description of a quantum system with a finite number of bound states by a compact dynamical group

In a recent paper (Aldaya V and Guerrero J 2005 J. Phys. A: Math. Gen. 38 6939) it is shown that the bound states of the Modified Poschl–Teller potential should be described by the non-compact dynamical group SU(1, 1) instead of the usual compact group SU(2). Here we prove that SU(2) cannot be the dynamical group for a potential with a finite number of bound states on the basis of the Modified Poschl–Teller potential and the Morse potential. This suggests that a quantum system with a continuum spectrum and a finite number of bound states should be described, both in the continuum and the discrete spectrum, by a non-compact dynamical group instead of a compact one.

New insights in particle dynamics from group cohomology

The dynamics of a particle moving in background electromagnetic and gravitational fields is revisited from a Lie group cohomological perspective. Physical constants characterizing the particle appear as central extension parameters of a group which is obtained from a centrally extended kinematical group (Poincare or Galilei) by making some subgroup local. The corresponding dynamics is generated by a vector field inside the kernel of a pre-symplectic form which is derived from the canonical left-invariant 1-form on the extended group. A non-relativistic limit is derived from the geodesic motion via an Inonu–Wigner contraction. A deeper analysis of the cohomological structure reveals the possibility of a new force associated with a non-trivial mixing of gravity and electromagnetism leading to, in principle, testable predictions.

SU(2) particle sigma model: the role of contact symmetries in global quantization

In this paper we achieve the quantization of a particle moving on the SU(2) group manifold, that is, the three-dimensional sphere S 3, by using group-theoretical methods. For this purpose, a fundamental role is played by contact symmetries, i.e., symmetries that leave the Poincare–Cartan form semi-invariant at the classical level, although not necessarily the Lagrangian. Special attention is paid to the role played by the basic quantum commutators, which depart from the canonical, Heisenberg–Weyl ones, as well as the relationship between the integration measure in the Hilbert space of the system and the non-trivial topology of the configuration space. Also, the quantization on momentum space is briefly outlined.