A canonical rank-three tensor model with a scaling constraint

TL;DR

This paper introduces a new canonical rank-three tensor model with a scaling constraint to eliminate degeneracies, resulting in a compact configuration space and more controlled classical solutions, advancing the understanding of fuzzy space dynamics.

Contribution

The paper proposes a novel canonical tensor model with a scaling symmetry constraint, addressing degeneracies and establishing a unique minimal model with improved classical solution behavior.

Findings

Classical solutions tend toward fixed points or cyclic orbits.

Configurations with group symmetries may represent stationary fuzzy spaces.

The model's configuration space is compact, unlike previous models.

Abstract

A rank-three tensor model in canonical formalism has recently been proposed. The model describes consistent local-time evolutions of fuzzy spaces through a set of first-class constraints which form an on-shell closed algebra with structure functions. In fact, the algebra provides an algebraically consistent discretization of the Dirac-DeWitt constraint algebra in the canonical formalism of general relativity. However, the configuration space of this model contains obvious degeneracies of representing identical fuzzy spaces. In this paper, to delete the degeneracies, another first-class constraint representing a scaling symmetry is added to propose a new canonical rank-three tensor model. A consequence is that, while classical solutions of the previous model have typically runaway or vanishing behaviors, the new model has a compact configuration space and its classical solutions asymptotically approach either fixed points or cyclic orbits in time evolution. Among others, fixed points contain configurations with group symmetries, and may represent stationary symmetric fuzzy spaces. Another consequence on the uniqueness of the local Hamiltonian constraint is also discussed, and a minimal canonical tensor model, which is unique, is given.