Operators In Quantum Gravity Research Articles

Relational observables, reference frames, and conditional probabilities

We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum reference frames. After presenting the general formalism, we analyze a recollapsing cosmological model, for which we construct unitarily evolving quantum relational observables. We conclude with some remarks about the relevance of these results for the construction and interpretation of diffeomorphism-invariant operators in quantum gravity.

Affine connections in quantum gravity and new scalar fields

In this manuscript, we will discuss the construction of covariant derivative operator in quantum gravity. We will find it is more perceptive alternative to use affine connections more general than metric compatible connections in quantum gravity. We will demonstrate this using the canonical quantization procedure. This is valid irrespective of the presence and nature of sources. The conventional Palatini and metric-affine formalisms, where the actions are linear in the scalar curvature with metric and affine connections being the the independent variables, are not much suitable to construct a source-free theory of gravity with general affine connections. This is also valid for many minimally coupled interacting theories where sources only couple with metric by using the Levi-Civita connections exclusively. We will discuss potential formalism of affine connections to introduce affine connections more general than metric compatible connections in gravity. We will also discuss possible extensions of the actions for this purpose. General affine connections introduce new fields in gravity besides metric. In this article, we will consider a simple potential formalism with symmetric affine connections and symmetric Ricci tensor. Corresponding affine connections introduce two massless scalar fields. One of these fields contributes a stress-tensor with opposite sign to the sources of Einstein's equation when we state the equation using the Levi-Civita connections. This means we have a massless scalar field with negative stress-tensor in Einstein's equation. This field brings us beyond strict local Minkowski geometries. These scalar fields can be useful to explain inflation and dark energy.

Lorentz invariant mass and length scales

We show that the standard Lorentz transformations admit an invariant mass (length) scale, such as the Planck scale. In other words, the frame independence of such a scale is built-in within those transformations, and one does not need to invoke the principle of relativity for their invariance. This automatically ensures the frame-independence of the spectrum of geometrical operators in quantum gravity. Furthermore, we show that the above predicts a small but measurable difference between the inertial and gravitational mass of any object, regardless of its size or whether it is elementary or composite.

ERRATUM: "GRAVITATIONAL WAVES ABOUT CURVED BACKGROUNDS: A CONSISTENCY ANALYSIS IN DE SITTER SPACETIME"

Gravitational waves are considered as metric perturbations about a curved background metric, rather than the flat Minkowski metric since several situations of physical interest can be discussed by this generalization. In this case, when the de Donder gauge is imposed, its preservation under infinitesimal spacetime diffeomorphisms is guaranteed if and only if the associated covector is ruled by a second-order hyperbolic operator which is the classical counterpart of the ghost operator in quantum gravity. In such a wave equation, the Ricci term has opposite sign with respect to the wave equation for Maxwell theory in the Lorenz gauge. We are, nevertheless, able to relate the solutions of the two problems, and the algorithm is applied to the case when the curved background geometry is the de Sitter spacetime. Such vector wave equations are studied in two different ways: (i) an integral representation, (ii) through a solution by factorization of the hyperbolic equation. The latter method is extended to the wave equation of metric perturbations in the de Sitter spacetime. This approach is a step towards a general discussion of gravitational waves in the de Sitter spacetime and might assume relevance in cosmology in order to study the stochastic background emerging from inflation.

GRAVITATIONAL WAVES ABOUT CURVED BACKGROUNDS: A CONSISTENCY ANALYSIS IN DE SITTER SPACETIME

Gravitational waves are considered as metric perturbations about a curved background metric, rather than the flat Minkowski metric since several situations of physical interest can be discussed by this generalization. In this case, when the de Donder gauge is imposed, its preservation under infinitesimal spacetime diffeomorphisms is guaranteed if and only if the associated covector is ruled by a second-order hyperbolic operator which is the classical counterpart of the ghost operator in quantum gravity. In such a wave equation, the Ricci term has opposite sign with respect to the wave equation for Maxwell theory in the Lorenz gauge. We are, nevertheless, able to relate the solutions of the two problems, and the algorithm is applied to the case when the curved background geometry is the de Sitter spacetime. Such vector wave equations are studied in two different ways: (i) an integral representation, (ii) through a solution by factorization of the hyperbolic equation. The latter method is extended to the wave equation of metric perturbations in the de Sitter spacetime. This approach is a step towards a general discussion of gravitational waves in the de Sitter spacetime and might assume relevance in cosmology in order to study the stochastic background emerging from inflation.

Partial observables

We discuss the distinction between the notion of partial observable and the notion of complete observable. Mixing up the two is frequently a source of confusion. The distinction bears on several issues related to observability, such as (i) whether time is an observable in quantum mechanics, (ii) what are the observables in general relativity, (iii) whether physical observables should or should not commute with the Wheeler-DeWitt operator in quantum gravity. We argue that the extended configuration space has a direct physical interpretation, as the space of the partial observables. This space plays a central role in the structure of classical and quantum mechanics and the clarification of its physical meaning sheds light on this structure, particularly in context of general covariant physics.

Further results on geometric operators in quantum gravity

We investigate some properties of geometric operators in canonical quantum gravity in the connection approach à la Ashtekar, which are associated with the volume, area and length of spatial regions. We give the motivations for the construction of analogous discretized lattice quantities, compute various quantum commutators of the type [area, volume], [area, length] and [volume, length], and find that they are generally non-vanishing. Although our calculations are performed mostly within a lattice-regularized approach, some are - for special, fixed spin-network configurations - identical to corresponding continuum computations. Comparison with the structure of the discretized theory leads us to conclude that anomalous commutators may be a general feature of operators constructed along similar lines within a continuum loop representation of quantum general relativity; the validity of the lattice approach remains unaffected.

Scaling behavior of Ricci curvature at short distance near two dimensions

We study the renormalization of the Ricci curvature as an example of generally covariant operators in quantum gravity near two dimensions. We find that it scales with a definite scaling dimension at short distance. The Ricci curvature singularity at the big bang can be viewed as such a scaling phenomenon. The problem of the spacetime singularity may be resolved by the scale invariance of the spacetime at short distance.

Spectrum of the volume operator in quantum gravity

The volume operator is an important kinematical quantity in the non-perturbative approach to four-dimensional quantum gravity in the connection formulation. We give a general algorithm for computing its spectrum when acting on four-valent spin network states, evaluate some of the eigenvalue formulae explicitly, and discuss the role played by the Mandelstam constraints.

Volume operator in discretized quantum gravity.

We investigate the spectral properties of the volume operator in quantum gravity in the framework of a previously introduced lattice discretization. The presence of a well-defined scalar product in this approach permits us to make definite statements about the hermiticity of quantum operators. We find that the spectrum of the volume operator is discrete, but that the nature of its eigenstates differs from that found in an earlier continuum treatment.