Canonical Quantum Gravity on Noncommutative Spacetime

TL;DR

This paper develops a framework for canonical quantum gravity on noncommutative spacetime using the Moyal star product, extending classical and quantum theories to incorporate noncommutativity and exploring implications for quantum constraints and loop quantum gravity.

Contribution

It introduces a generalized formalism for quantum gravity on noncommutative spacetime, including new operator representations and constraints, and applies it to loop quantum gravity concepts.

Findings

Generalized quantum constraints including Hamiltonian constraint.

Extended operator representations on noncommutative spacetime.

Potential for calculating generalized geometric operators like area.

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.