Canonical quantum gravity on noncommutative space–time

Abstract

In this paper canonical quantum gravity on noncommutative space-time is considered. The corresponding generalized classical theory is formulated by using the moyal star product, which enables the representation of the field quantities depending on noncommuting coordinates by generalized quantities depending on usual coordinates. But not only the classical theory has to be generalized in analogy to other field theories. Besides, the necessity arises to replace the commutator between the gravitational field operator and its canonical conjugated quantity by a corresponding generalized expression on noncommutative space-time. Accordingly the transition to the quantum theory has also to be performed in a generalized way and leads to extended representations of the quantum theoretical operators. If the generalized representations of the operators are inserted to the generalized constraints, one obtains the corresponding generalized quantum constraints including the Hamiltonian constraint as dynamical constraint. After considering quantum geometrodynamics under incorporation of a coupling to matter fields, the theory is transferred to the Ashtekar formalism. The holonomy representation of the gravitational field as it is used in loop quantum gravity opens the possibility to calculate the corresponding generalized area operator.

Black holes thermodynamics in a new kind of noncommutative geometry

“…It may be noted that various different choices for θ µν have been studied [80,81,82,83]. The noncommutativity is expected to arise because of background fluxes in string theory [84,85].…”

Quaternionic quantization principle in general relativity and supergravity

“…By using the representation of the momentum operator in position space given in (14) and the representation of the position operator in momentum operator given in (15), the commutation relations of the several components of the momentum operator with each other and of the several components of the position operator with each other can be determined. The position representation is only valid, if the algebra (7) is supplemented by the following commutation relations:…”

“…These concepts can be interpreted as fundamental properties of nature and thus they would belong to quantum theory itself and accordingly represent a generalization of the concept of quantization. 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.…”