Hilbert space structure of covariant loop quantum gravity

TL;DR

The paper develops a Lorentz-covariant Hilbert space for loop quantum gravity by restricting to a sector where spacelike area operators are simultaneously diagonalizable, and building Lorentz spin-network states from projected Wilson lines onto an SO(3) subgroup. It shows that, by employing simple Lorentz representations $(0,i\rho)$ and introducing an auxiliary variable $\varphi$, one can define a consistent inner product on a state space realized as functions on the homogeneous space $[SO(3)\times\mathbb{R}]^n\times[SO(3,1)/SO(3)]^m$, with gauge-invariant sectors reducing to a tractable subspace. The resulting framework yields an orthonormal basis of Lorentz spin networks and a continuous area spectrum $S = 8\pi \hbar G \sum_i \sqrt{ j_i(j_i+1) + \rho_i^2 + 1 }$, showcasing intriguing parallels with spin-foam models while highlighting essential differences from SU(2) quantization. The work provides a foundation for covariant quantization, clarifying how noncommutativity is resolved and pointing toward a covariant bridge to spin foams and potential implications for black-hole entropy and Hamiltonian constraints.

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.