Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, Thomas Thiemann
TL;DR
This work develops a nonperturbative, background-free program to quantize diffeomorphism-invariant theories of connections by constructing the quantum configuration space $\overline{\cal A/\cal G}$, building the auxiliary Hilbert space $\mathcal H_{aux}$, and solving the diffeomorphism constraint via group averaging to obtain a physical Hilbert space. Central tools include holonomy/loop variables, the Ashtekar–Lewandowski measure, and a projective-limit framework that underpins a diffeomorphism-invariant differential calculus and a spin-network basis. The authors provide complete solutions for the Gauss and diffeomorphism constraints in the Husain–Kuchař model and outline a viable path to include the Hamiltonian constraint, with clear implications for quantum geometry and loop quantum gravity. The approach offers a mathematically rigorous avenue for quantizing gravity-like theories and clarifies the role of diffeomorphism invariance in defining physical states and observables.
Abstract
Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchař model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.