Canonical form of a deformed Poisson bracket spacetime

TL;DR

This paper reformulates a gravity theory with modified Poisson brackets into a canonical, covariant form, enabling consistent dynamics with matter coupling.

Contribution

It constructs a Hamiltonian that restores covariance and canonical structure to a gravity theory originally defined by deformed Poisson brackets.

Findings

The theory becomes canonical and covariant after reformulation.

Scalar matter and dust can be coupled covariantly to the modified gravity.

The resulting equations of motion match those derived from the original distorted brackets.

Abstract

The general uncertainty principle applied to gravity can be implemented as a set of modified Poisson brackets in the canonical formalism. As such, the theory is not canonical and the resulting equations of motion do not lead to a covariant metric. We construct a Hamiltonian that when applying the usual canonical formalism gives a closed algebra and equations of motion that result in the original metric obtained by using distorted Poisson brackets. The resulting theory is thus rendered canonical and covariant. We then covariantly couple scalar matter and dust to the modified gravity to allow the study of dynamics.