Hilbert space structure of covariant loop quantum gravity

TL;DR

This paper explores the Hilbert space structure in Lorentz covariant loop quantum gravity, proposing a sector where area operators are diagonalizable, leading to a basis of Lorentz Wilson line-based spin network states that preserve covariance.

Contribution

It introduces a sector with diagonalizable area operators, constructs a basis of Lorentz covariant spin network states, and addresses the noncommutativity issue via simple representations.

Findings

Spin network states form an orthonormal basis in the studied sector.

Projection onto simple representations preserves Lorentz covariance.

The structure resembles the spin foam approach.

Abstract

We investigate the Hilbert space in the Lorentz covariant approach to loop quantum gravity. We restrict ourselves to the space where all area operators are simultaneously diagonalizable, assuming that it exists. In this sector quantum states are realized by a generalization of spin network states based on Lorentz Wilson lines projected on irreducible representations of an SO(3) subgroup. The problem of infinite dimensionality of the unitary Lorentz representations is absent due to this projection. Nevertheless, the projection preserves the Lorentz covariance of the Wilson lines so that the symmetry is not broken. Under certain conditions the states can be thought as functions on a homogeneous space. We define the inner product as an integral over this space. With respect to this inner product the spin networks form an orthonormal basis in the investigated sector. We argue that it is the only relevant part of a larger state space arising in the approach. The problem of the noncommutativity of the Lorentz connection is solved by restriction to the simple representations. The resulting structure shows similarities with the spin foam approach.