Theory Of Canonical Quantum Gravity Research Articles

Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Three principles for canonical quantum gravity

We outline three principles that should guide us in the construction of a theory of canonical quantum gravity: (1) diffeomorphism invariance, (2) implementing the proper dynamics and related constraint algebra, (3) local Lorentz invariance. We illustrate each of them with its role in model calculations in loop quantum gravity.

TIME AND A TEMPORALLY STATISTICAL QUANTUM GEOMETRODYNAMICS

This paper is an exposition of the author's recent work on modeling M-theory vacua and quantum mechanical observers in the framework of a temporally statistical description of quantum geometrodynamics, including measurement processes based on the canonical theory of quantum gravity. In this paper, we deal with several fundamental issues of time: the timeless problem in canonical quantum gravity; the physical origin of state reductions; and time-reversal symmetry breaking. We first model the observers and consider the timeless problem by invoking the time reparametrization symmetry breaking in the quantum mechanical world as seen by the observers. We next construct the hidden time variable theory, using a model of the gauged and affinized S-duality symmetry in type IIB string theory, as the statistical theory of time and explain the physical origin of state reductions using it. Finally, by the extension of the time reparametrization symmetry to all of the temporal hidden variables, we treat the issue of time reversal symmetry breaking as the spontaneous breaking of this extended time reparametrization symmetry. The classification of unitary time-dependent processes and the geometrizations of unitary and nonunitary time evolutions using the language of the derived category are also investigated.

GENERALIZED QUANTIZATION PRINCIPLE IN CANONICAL QUANTUM GRAVITY AND APPLICATION TO QUANTUM COSMOLOGY

In this paper, a generalized quantization principle for the gravitational field in canonical quantum gravity, especially with respect to quantum geometrodynamics is considered. This assumption can be interpreted as a transfer from the generalized uncertainty principle in quantum mechanics, which is postulated as generalization of the Heisenberg algebra to introduce a minimal length, to a corresponding quantization principle concerning the quantities of quantum gravity. According to this presupposition there have to be given generalized representations of the operators referring to the observables in the canonical approach of a quantum description of general relativity. This also leads to generalized constraints for the states and thus to a generalized Wheeler–DeWitt equation determining a new dynamical behavior. As a special manifestation of this modified canonical theory of quantum gravity, quantum cosmology is explored. The generalized cosmological Wheeler–DeWitt equation corresponding to the application of the generalized quantization principle to the cosmological degree of freedom is solved by using Sommerfelds polynomial method.

Local and Non-Local Aspects of Quantum Gravity

The analysis of the measurement of gravitational fields leads to the Rosenfeld inequalities. They say that, as an implication of the equivalence of the inertial and passive gravitational masses of the test body, the metric cannot be attributed to an operator that is defined in the frame of a local canonical quantum field theory. This is true for any theory containing a metric, independently of the geometric framework under consideration and the way one introduces the metric in it. Thus, to establish a local quantum field theory of gravity one has to transit to non-Riemann geometry that contains (beside or instead of the metric) other geometric quantities. From this view, we discuss a Riemann–Cartan and an affine model of gravity and show them to be promising candidates of a theory of canonical quantum gravity.

Canonical quantum gravity in the Vassiliev invariants arena:I. Kinematical structure

We generalize the idea of Vassiliev invariants to the spin networkcontext, with the aim of using these invariants as a kinematical arenafor a canonical quantization of gravity. This paper presents adetailed construction of these invariants (both ambient and regularisotopic) requiring a significant elaboration based on the use ofChern-Simons perturbation theory which extends the work of Kauffman,Martin and Witten to four-valent networks. We show that this space ofknot invariants has the crucial property - from the point of view ofthe quantization of gravity - of being loop differentiable in thesense of distributions. This allows the definition of diffeomorphismand Hamiltonian constraints. We show that the invariants areannihilated by the diffeomorphism constraint. In a companion paper weelaborate on the definition of a Hamiltonian constraint, discuss theconstraint algebra and show that the construction leads to aconsistent theory of canonical quantum gravity.

Canonical quantum gravity in the Vassiliev invariants arena:II. Constraints, habitats and consistency of the constraint algebra

In a companion paper we introduced a kinematical arena for the discussionof the constraints of canonical quantum gravity in the spin networkrepresentation based on Vassiliev invariants. In this paper weintroduce the Hamiltonian constraint, extend the space of states tonon-diffeomorphism-invariant `habitats' and check that the off-shellquantum constraint commutator algebra reproduces the classical Poissonalgebra of constraints of general relativity without anomalies. Onecan therefore consider the resulting set of constraints and space ofstates as a consistent theory of canonical quantum gravity.

Quantum Gravity Experimental Physics?

Canonical quantum gravity theories predict apolymer-like structure for space time at the Plancksize. This granularity can be probed using gamma rayburst observations. Quantum gravity effects typically amount to corrections of Planck length size perwavelength. Because the distance to gamma ray burst isvery large as measured in the wavelength of gamma rays,the effects accumulate and are on the brink of being observable. These observations canconstrain certain aspects of the quantum stateunderlying our universe.

Existence of spinorial states in pure loop quantum gravity

We demonstrate the existence of spinorial states in a kinematical theory of canonical quantum gravity without matter. This should be regarded as evidence in support of the conjecture that bound states with particle properties appear in association with spatial regions of non-trivial topology. In asymptotically trivial general relativity the momentum constraint generates only a subgroup of the spatial diffeomorphisms. The remaining diffeomorphisms give rise to the mapping class group, which acts as a symmetry group on the phase space. This action induces a unitary representation on the loop state space of the Ashtekar formalism. Certain elements of the diffeomorphism group can be regarded as asymptotic rotations of space relative to its surroundings. We construct states that transform non-trivially under a 2-rotation: gravitational quantum states with fractional spin.

Unimodular theory of canonical quantum gravity.

Einstein's theory of gravity is reformulated so that the cosmological constant becomes an integration constant of the theory, rather than a "coupling" constant. However, in the Hamiltonian form of the theory, the Hamiltonian constraint is missing, while the usual momentum constraints are still present. Replacing the Hamiltonian constraint is a secondary constraint, which introduces the cosmological constant. The quantum version has a normal "Schr\"odinger" form of time development, and the wave function does not obey the usual "Wheeler-DeWitt" equation, making the interpretation of the theory much simpler. The small value of the cosmological constant in the Universe at present becomes a genuine question of initial conditions, rather than a question of why one of the coupling constants has a particular value. The key "weakness" of this formulation is that one must introduce a nondynamic background spacetime volume element.

Ground-state wave function of linearized gravity

The ground-state wave function for linearized gravity is calculated from the Euclidean functional-integral prescription. The result is identical to that obtained from the canonical theory of quantum gravity.

Indefinite-Metric Quantum Field Theory of General Relativity. IV: Background Curved Space-Time

In the framework of the manifestly-covariant canonical theory of quantum gravity proposed previously, a background curved space-time is introduced as the space-time-dependent vacuum expectation value of the gravitational field. The properties of the corresponding asymptotic fields are investigated in detail.