Affine Quantum Gravity Research Articles

Let Loop Quantum Gravity and Affine Quantum Gravity Examine Each Other

Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein's gravity.

Recent results regarding affine quantum gravity

Recent progress in the quantization of nonrenormalizable scalar fields has found that a suitable non-classical modification of the ground state wave function leads to a result that eliminates term-by-term divergences that arise in a conventional perturbation analysis. After a brief review of both the scalar field story and the affine quantum gravity program, examination of the procedures used in the latter surprisingly shows an analogous formulation which already implies that affine quantum gravity is not plagued by divergences that arise in a standard perturbation study. Additionally, guided by the projection operator method to deal with quantum constraints, trial reproducing kernels are introduced that satisfy the diffeomorphism constraints. Furthermore, it is argued that the trial reproducing kernels for the diffeomorphism constraints may also satisfy the Hamiltonian constraint as well.

An affinity for affine quantum gravity

The main principle of affine quantum gravity is the strict positivity of the matrix $$\{ \hat g_{ab} (x)\} $$ composed of the spatial components of the local metric operator. Canonical commutation relations are incompatible with this principle, and they can be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational constraint operators is formulated quite naturally as a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models.

OVERVIEW OF AFFINE QUANTUM GRAVITY

The basic principles of Affine Quantum Gravity are presented in a pedagogical style with a limited number of equations.

The utility of coherent states and other mathematical methods in the foundations of affine quantum gravity

Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accommodate first-and second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employed are (i) coherent-state representations, (ii) reproducing kernel Hilbert spaces, and (iii) functional-integral representations involving a continuous-time regularization. Of special importance is the profoundly different integration measure used for the Lagrange multiplier (shift and lapse) functions. These various concepts are first introduced on elementary systems to help motivate their application to affine quantum gravity.

Weak coherent state path integrals

Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise, e.g., in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.

AFFINE QUANTUM GRAVITY

A sketch of the affine quantum gravity program illustrates a different perspective on several difficult issues of principle: metric positivity; quantum anomalies; and nonrenormalizability.

Metric and curvature in gravitational phase space

At a fixed point in spacetime (say, x0), gravitational phase space consists of the space of symmetric matrices {Fab} (corresponding to the canonical momentum πab(x0)) and of symmetric matrices {Gab} (corresponding to the canonical metric gab(x0)), where 1 ⩽ a, b ⩽ n, and, crucially, the matrix {Gab} is necessarily positive definite, i.e. ∑uaGabub > 0 whenever ∑(ua)2 > 0. In an alternative quantization procedure, known as metrical quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase-space metrics, guided by a recent study of affine quantum gravity, leads to gravitational phase-space geometries which possess constant scalar curvature and may be regarded as higher-dimensional analogues of the Poincaré plane, which applies when n = 1. This result is important because phase spaces endowed with such symmetry lead naturally, via the procedures of metrical quantization, to acceptable Hilbert spaces of high dimension.

The affine quantum gravity programme

The central principle of affine quantum gravity is securing and maintaining the strict positivity of the matrix {ĝab(x)} composed of the spatial components of the local metric operator. On spectral grounds, canonical commutation relations are incompatible with this principle, and they must be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the recently developed projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. It is anticipated that skills and insight to study this formulation can be developed by studying special, reduced-variable models that still retain some basic characteristics of gravity, specifically a partial second-class constraint operator structure. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models. Finally, developing a procedure to pass to the genuine physical Hilbert space involves several interconnected steps that require careful coordination.

Ultralocal fields and their relevance for reparametrization-invariant quantum field theory

Reparametrization invariant theories have a vanishing Hamiltonian and enforce their dynamics through a constraint. We specifically choose the Dirac procedure of quantization before the introduction of constraints. Consequently, for field theories, and prior to the introduction of any constraints, it is argued that the original field operator representation should be ultralocal in order to remain totally unbiased toward those field correlations that will be imposed by the constraints. It is shown that relativistic free and interacting theories can be completely recovered starting from ultralocal representations followed by a careful enforcement of the appropriate constraints. In so doing all unnecessary features of the original ultralocal representation disappear. The present discussion is germane to a recent theory of affine quantum gravity in which ultralocal field representations have been invoked before the imposition of constraints.

Noncanonical quantization of gravity. I. Foundations of affine quantum gravity

The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method, which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the present paper, involves another reproducing kernel with which inner products are defined for the physical Hilbert space and which is obtained through a reduction of the original reproducing kernel. Several of the steps involved in this general analysis are illustrated by means of analogous steps applied to one-dimensional quantum mechanical models. These toy models help in motivating and understanding the analysis in the case of gravity.

Quantum gravity in the Eddington purely affine picture

It was shown by Kijowski and Tulczjew that pure gravity with a cosmological constant can be obtained by a covariant Legendre transformation of a purely affine Lagrangian the manner of constructed from a symmetric linear connection. In this paper I prove by explicit calculations that the Eddington Lagrangian is equivalent, in the sense which gives the same field equations, to a polynomial effective Lagrangian which turns out to be power-counting renormalizable. Then a formal proof of the unitarity of this theory is stated in the Kugo-Ojima formalism on the basis of the existence of two local Becchi-Rouet-Stora symmetries. These supertransformations are related to the algebra of the diffeomorphisms of the space-time, as well as to that of the volume-preserving space-time transformations which are not fixed by the gauge fixing used for the diffeomorphism group itself. Furthermore, I find that in the purely affine picture quantum gravity exhibits an infrared freedom. Since now the self-coupling constant is given by the cosmological constant, such a property could explain the observed almost zero value of the cosmological term at very large distances, i.e., to very low energies.