V. Aldaya

Quantization as a consequence of the symmetry group: An approach to geometric quantization

A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free‐particle dynamics and the harmonic oscillator.

Cohomology, central extensions, and (dynamical) groups

We analyze in this paper the process of group contraction which allows the transition from the Einstenian quantum dynamics to the Galilean one in terms of the cohomology of the Poincaré and Galilei groups. It is shown that the cohomological constructions on both groups do not commute with the contraction process. As a result, the extension coboundaries of the Poincaré group which lead to extension cocycles of the Galilei group in the “nonrelativistic” limit are characterized geometrically. Finally, the above results are applied to a quantization procedure based on a group manifold.

Higher-order Hamiltonian formalism in field theory

A Hamiltonian formalism is developed from a regular Lagrangian Lr depending on an arbitrary number of derivatives. The formalism leads to a set of Hamilton equations whose solutions are the same as those of the Euler-Lagrange equations derived from Lr. The Noether currents associated with a symmetry transformation of the Hamiltonian action are also derived.

Symmetry and quantization: Higher-order polarization and anomalies

The concept of (full) polarization subalgebra in a Group Approach to Quantization on a Lie group G as a generalization of the analogous concept in geometric or standard quantization is discussed. The lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition. A generalization of the subalgebra of (full) polarization is then provided, made out of higher‐order differential operators in the enveloping algebra of G. Higher‐order polarizations can also be used to quantize nonanomalous theories in different ‘‘representations.’’ Numerous examples are analyzed, including the finite‐dimensional dynamics associated with the Schrodinger group, which presents an anomaly, and an infinite‐dimensional anomalous system associated with the Virasoro group. In the last example, the operators in the higher‐order polarization are in one‐to‐one correspondence with the null vectors in the Verma module approach.

A note on the meaning of covariant derivatives in supersymmetry

We analyze in this paper the group theoretical meaning of the covariant derivatives, and show that they are horizontal left‐invariant vector fields on superspace obtained from a (super)Lie group which at the same time exhibits the structure of a principal bundle with a canonical connection. The geometrical construction is general and not restricted to the super‐Poincare group.

Group-theoretical construction of the quantum relativistic harmonic oscillator☆

We elaborate on a proposal for a quantum relativistic harmonic oscillator, based on a group-theoretical framework. The wave functions in both configuration and Bargmann-Fock-like space are explicitly given and a generalized Bargmann transform is also provided. The energy eigen-functions in configuration space are composed of a general weight function (the vacuum), which leads to the Gaussian one in the c → ∞ limit, a particular power of the vacuum reducing to unity in this limit and a polynomial leading to the corresponding non-relativistic Hermite polynomial as c → ∞.

Group manifold analysis of the structure of relativistic quantum dynamics

Abstract We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincare group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.

Quantization, symmetry, and natural polarization

We discuss the notion of polarization, as defined in a geometric quantization scheme recently introduced, in terms of the role played by the evolution operator of the quantum system. The analysis uses an integral transform representation of the group WSp(2,R). This clarifies the group theoretic origin of the natural polarizations and the meaning of the polarization changing transformations.

Higher-order polarization on the Poincare group and the position operator

The quantization of the free relativistic spinning particle is revised on the basis of a group approach to quantization. In momentum space, the wavefunctions provide the minimal canonical representation of mass m and spin j of the Poincare group P+up arrow . The quantization in configuration space requires, as in many other physical systems, polarizations of higher-order type. This higher-order polarization technique turns out to be a natural framework for studying localizability and to provide a position operator.

Symmetries of the Schrödinger bundle

We exhibit in this paper the invariance of the Schrodinger Lagrangian density under the eleven‐parameter group G(m), central extension of the Galilei group G. As a result, the quantum mechanical probability Fρ d3x turns out to be the conserved charge associated with the central generator of G(m), and the continuity equation is simply the expression of the conservation of the corresponding Noether current.