Quantum Constraint Algebra Research Articles

Quantization of Constrained Systems as Dirac First Class versus Second Class: A Toy Model and Its Implications

A toy model (suggested by Klauder) was analyzed from the perspective of first-class and second-class Dirac constrained systems. First-class constraints are often associated with the existence of important gauge symmetries in a system. A comparison was made by turning a first-class system into a second-class system with the introduction of suitable auxiliary conditions. The links between Dirac’s system of constraints, the Faddeev–Popov canonical functional integral method and the Maskawa–Nakajima procedure for reducing the phase space are explicitly illustrated. The model reveals stark contrasts and physically distinguishable results between first and second-class routes. Physically relevant systems such as the relativistic point particle and electrodynamics are briefly recapped. Besides its pedagogical value, the article also advocates the route of rendering first-class systems into second-class systems prior to quantization. Second-class systems lead to a well-defined reduced phase space and physical observables; an absence of inconsistencies in the closure of quantum constraint algebra; and the consistent promotion of fundamental Dirac brackets to quantum commutators. As first-class systems can be turned into well-defined second-class ones, this has implications for the soundness of the “Dirac quantization” of first-class constrained systems by the simple promotion of Poisson brackets, rather than Dirac brackets, to commutators without proceeding through second-class procedures.

New volume and inverse volume operators for loop quantum gravity

A new alternative volume operator is constructed for loop quantum gravity by using the so-called cotriad operators as building blocks. It is shown that the new volume operator shares the same qualitative properties with the standard volume operator. Moreover, a new alternative inverse volume operator is also constructed in the light of the construction of the alternative volume operator, which is possessed of the same qualitative properties as those of the alternative volume operator. The new inverse volume operator can be employed to construct the Hamiltonian operator of matter fields, which may lead to an anomaly-free on-shell quantum constraint algebra without any special restriction on the regularization procedure for gravity coupled to matter fields.

Spherically symmetric canonical quantum gravity

Canonical quantization of spherically symmetric space-times is carried out, using real-valued densitized triads and extrinsic curvature components, with specific factor ordering choices ensuring in an anomaly free quantum constraint algebra. Comparison with previous work [1] reveals that the resulting physical Hilbert space has the same form, although the basic canonical variables are different in the two approaches. As an extension, holonomy modifications from Loop Quantum Gravity are shown to deform the Dirac space-time algebra, while going beyond `effective' calculations.

Quantum spin dynamics (QSD): III. Quantum constraint algebra and physical scalar product in quantum general relativity

This paper deals with several technical issues of non-perturbative four-dimensional Lorentzian canonical quantum gravity in the continuum that arose in connection with the recently constructed Wheeler-DeWitt quantum constraint operator. The Wheeler-DeWitt constraint of quantum general relativity mixes the diffeomorphism superselection sectors for diffeomorphism-invariant theories of connections that were previously discussed in the literature. From it one can construct diffeomorphism-invariant operators which do not necessarily commute with the Hamiltonian constraint but which still mix those sectors and which, at the diffeomorphism-invariant level, encode physical information. Thus, if one adopts, as before in the literature, the strategy to solve the diffeomorphism constraint before the Hamiltonian constraint then those sectors become spurious. The inner product for diffeomorphism-invariant states can be fixed by requiring that diffeomorphism group averaging is a partial isometry. The established non-anomalous constraint algebra is clarified by computing commutators of duals of constraint operators. The full classical constraint algebra is faithfully implemented on the diffeomorphism-invariant Hilbert space in an appropriate sense. The Hilbert space of diffeomorphism-invariant states can be made separable if a natural new superselection principle is satisfied. We propose a natural physical scalar product for quantum general relativity by extending the group-average approach to the case of non-self-adjoint constraint operators like the Wheeler-DeWitt constraint. Equipped with this inner product, the construction of physical observables is straightforward.

Lorentz-invariant actions for chiralp-forms

We demonstrate how a Lorentz-covariant formulation of the chiral $p$-form model in $D=2(p+1)$ containing infinitely many auxiliary fields is related to a Lorentz-covariant formulation with only one auxiliary scalar field entering a chiral $p$-form action in a nonpolynomial way. The latter can be regarded as a consistent Lorentz-covariant truncation of the former. We make the Hamiltonian analysis of the model based on the nonpolynomial action and show that the Dirac constraints have a simple form and are all first class. In contrast with the Siegel model the constraints are not the square of second-class constraints. The canonical Hamiltonian is quadratic and determines the energy of a single chiral $p$-form. In the case of $D=2$ chiral scalars the constraint can be improved by use of a ``twisting'' procedure (without the loss of the property to be first class) in such a way that the central charge of the quantum constraint algebra is zero. This points to the possible absence of an anomaly in an appropriate quantum version of the model.

Canonical quantum supergravity in three dimensions

We discuss the canonical treatment and quantization of matter coupled supergravity in three dimensions, with special emphasis on N = 2 supergravity. We then analyze the quantum constraint algebra; certain operator ordering ambiguities are found to be absent due to local supersymmetry. We show that the supersymmetry constraints can be partially solved by a functional analog of the method of characteristics. We also consider extensions of Wilson loop integrals of the type previously found in ordinary gravity, but now with connections involving the bosonic and fermionic matter fields in addition to the gravitational connection. In a separate section of this paper, the canonical treatment and quantization of non-linear coset space sigma models are discussed in a self-contained way.

Dirac and BFV quantization methods in the 1-loop approximation: closure of the quantum constraint algebra and the conserved inner product

Operator realization of quantum constraints and the lowest-order structure functions is found in the 1-loop (linear in h) approximation of the Dirac and BFV quantization for the general gauge theory subject to first-class constraints. Semiclassical equivalence of the Dirac and BFV methods to the reduced quantization of physical variables is established in terms of the conserved inner product in the space of physical states. The global obstructions to this result are briefly discussed in connection with the ideas of the secondary (third) quantization of constrained systems.

AN OPERATOR REGULARIZATION FOR CHIRAL GAUGE THEORIES

We present a universal regularization of fermionic operators in the canonical formalism which respects all the symmetries which are manifest in that formalism. It allows an a priori discussion of the quantum constraint algebra and makes the formal steps leading to a path integral formula well-defined. We calculate the fermionic contributions to the renormalizations of the Gauss law and verify that the quantum constraint algebra contains Schwinger terms. We suggest a modification in the path integral formula which allows us to recover the classical degrees of freedom.

The Hamiltonian constraint in quantum gravity

We develop further the Rovelli-Smolin loop variable formulation of canonical quantum gravity. The main result is the construction of a hamiltonian operator which is well defined on all of loop state space. The hamiltonian operator acts as a generator of displacements which depend on the curvature of the loop and also reroutes the parametrisation of the loops at their intersections. Both a regularisation and a renormalisation are used in the construction. The quantum constraint algebra is worked out in some detail. We find that the diffeomorphism-hamiltonian commutator does not close. This can only be acceptable provided the physical state space is “sufficiently large”. We find a sector of state space annihilated by the diffeomorphism and hamiltonian operators which is also the full sector of a topological quantum field theory of the type studied by Horowitz. The topological sector itself is not large enough to allow non-closure of the quantum constraint algebra. We therefore make some initial attempts to find out whether the “large”, formal Rovelli-Smolin sector of physical states, namely the set of loop wavefunctionals having compact support on diffeomorphism equivalence classes of smooth, non-intersecting loops, is annihilated by the hamiltonian operator. In the final part of the paper we discuss the Bengtsson and Witten formulations of D = 2+1 quantum gravity in terms of the loop variables. It is argued that the Bengtsson formulation can be used as a toy model to study the loop variable approach to D = 3+1 gravity. In the conclusion we discuss further directions for research.

Hamiltonian formulation and quantum constraint algebra for a closed string moving in a curved background

The method of canonical quantization is used to derive the full algebra of constraints of a closed bosonic string moving in the most general background, to first order in α'. To this order the background field equations are obtained from the requirement that the algebra preserves the conformal invariance.

Quantum constraint algebra for a closed bosonic string in a gravitational and dilaton background

Abstract We derive the quantum constraint algebra for a closed bosonic string moving in a gravitational and dilaton background to first order in α '. The hamiltonian approach is used to directly compute the quantum constraint commutators and calculate the c-and q-number anomalies that arise at the quantum level. The requirement that the algebra preserves the conformal invariance leads to the known background field equations.

Quantum constraint algebra for an interacting superstring

We perform the hamiltonian BRST quantization of the type II superstrings interacting with a gravitational background. The quantum constraint algebra is derived to lowest (non-trivial) order in the space-time curvature, exhibiting the complete structure of c- and q-number anomalies.

Constraint algebra for interacting quantum systems

Abstract We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S -matrix.