Generalized Quantization Principle in Canonical Quantum Gravity and Application to Quantum Cosmology

Abstract

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.

Canonical quantum gravity on noncommutative space–time

“…In this approximation the generalized area operator in loop quantum gravity on noncommutative space-time has finally been determined. The generalized quantization principle as basic constituent of the presented theory, which is implied by the direct combination of the assumption of noncommutative geometry and field quantization, is a different version of a generalized quantization principle as it has been considered with respect to the variables of quantum mechanics as generalized uncertainty principle [56], [57], [58] and in quantum gravity [36], [37], [38], [39], [40]. Usually, noncommutative geometry is mainly interpreted as a possible extension of the structure of space-time, which could possibly cure the appearance of divergencies in quantum field theory, since the presupposed noncommutative geometry implies the existence of a minimal length.…”

“…These constraints can in complete analogy to the case of the gravitational field, (32) and (33), be converted to the corresponding quantum constraints by using the quantization rule (20) and the corresponding representation of the operators (21) and inserting them into (36). This leads to the following expressions for the quantum constraints referring to the scalar field:…”

“…To formulate the generalized quantum constraints, the three metric representation of the generalized operators describing the gravitational field is used. Such a generalization of the quantization principle in canonical quantum gravity as it is implied by the noncommutativity of space-time presupposed in this paper, has in other manifestations been considered in [36], [37], [38], [39], [40]. A canonical noncommutativity algebra between the components of the gravitational field and the corresponding generalization of general relativity has been considered in [41].…”

Interacting dark side of universe through generalized uncertainty principle

“…6 Also, the quantum commutator of GUP is applied for a variety of quantum cosmological models. 7 We should note that GUP deformation of Poisson brackets, for a point particles, implies a clear violation of the equivalence principle (EP). 8 Consistency between DPB and EP can be partially recovered only for composite systems 9 and at the price of defining different β parameters for different species of particles.…”

Quaternionic quantization principle in general relativity and supergravity

“…If this is postulated, then these generalized quantization principles have also to be transferred to the quantization of general relativity an thus the gravitational field what differs from the formulation of clasical general relativity on noncommutative space-time [10] or even usual quantum general relativity on noncommutative space-time [11], [12], [13], [14]. In [15], [16], [17], [18], [19] ideas to transfer the concept of a generalized uncertainty principle to gravity can be found, but in [20] and [21] the generalized uncertainty principle principle has really been transferred to the variables of canonical quantum gravity and quantum cosmology, whereas in [22] the concept of noncommutative geometry has been transferred to the components of the tetrad field. An extension of the field theoretic quantization principle to a nonlocal quantization principle has been considered in [23].…”