Quantization of Constrained Systems

Abstract

The present article is primarily a review of the projection-operator approach to quantize systems with constraints. We study the quantization of systems with general first-and second-class constraints from the point of view of coherent-state, phase-space path integration, and show that all such cases may be treated, within the original classical phase space, by using suitable path-integral measures for the Lagrange multipliers which ensure that the quantum system satisfies the appropriate quantum constraint conditions. Unlike conventional methods, our procedures involve no δ-functionals of the classical constraints, no need for dynamical gauge fixing of first-class constraints nor any average thereover, no need to eliminate second-class constraints, no potentially ambiguous determinants, as well as no need to add auxiliary dynamical variables expanding the phase space beyond its original classical formulation, including no ghosts. Besides several pedagogical examples, we also study: (i) the quantization procedure for reparameterization invariant models, (ii) systems for which the original set of Lagrange mutipliers are elevated to the status of dynamical variables and used to define an extended dynamical system which is completed with the addition of suitable conjugates and new sets of constraints and their associated Lagrange multipliers, (iii) special examples of alternative but equivalent formulations of given first-class constraints, as well as (iv) a comparison of both regular and irregular constraints. *

The marginal Fermi Liquid – An exact derivation based on Dirac's first class constraints method

“…The integration with respect to the unknown Φ β constraints will modify the integration measure from dλ α to a non − linear integration measure Dλ α (T = t i − t f ). This allows to represent the physical evolution operator in a form which is similar with to the result given by [36]:…”

Canonical quantization of motion on submanifolds

“…It was concluded that the Abelian conversion method is preferable because it involves no extrinsic geometry in the results. For this and other reasons it was used in the beautiful projection operator approach to path integral quantization of constrained systems [20] aiming at quantizing the gravity [21]. But is it possible to generalize the above result to other surfaces?…”

Relating covariant and canonical approaches to triangulated models of quantum gravity

“…Covariant quantum gravity. The projector can be expressed as a phase space path integral as is demonstrated by Klauder [12] for general constrained systems. This phase space integral can also be derived from a sum over geometries approach to quantum gravity, as has been investigated in detail by Teitelboim [29,31], forging the link between covariant and canonical frameworks.…”

“…12 The reason that we chose to include a sum over triangulations initially is that it will play an important role in the following. The above procedure is also necessary for gravity in higher dimensions where we no longer have triangulation independence.…”