New scalar constraint operator for loop quantum gravity

TL;DR

This paper introduces a new scalar constraint operator for loop quantum gravity, constructed explicitly on a space of partially diffeomorphism invariant states, with properties ensuring anomaly freedom and potential symmetry.

Contribution

The paper presents a novel scalar constraint operator using special loops, preserving a specific state space, and demonstrates its anomaly-free algebra in loop quantum gravity.

Findings

Operator preserves partially diffeomorphism invariant states

The algebra of constraints is anomaly free

Potential for defining a symmetric scalar constraint operator

Abstract

We present a concrete and explicit construction of a new scalar constraint operator for loop quantum gravity. The operator is defined on the recently introduced space of partially diffeomorphism invariant states, and this space is preserved by the action of the operator. To define the Euclidean part of the scalar constraint operator, we propose a specific regularization based on the idea of so-called "special" loops. The Lorentzian part of the quantum scalar constraint is merely the curvature operator that has been introduced in an earlier work. Due to the properties of the special loops assignment, the adjoint operator of the non-symmetric constraint operator is densely defined on the partially diffeomorphism invariant Hilbert space. This fact opens up the possibility of defining a symmetric scalar constraint operator as a suitable combination of the original operator and its adjoint. We also show that the algebra of the scalar constraint operators is anomaly free, and describe the structure of the kernel of these operators on a general level.