The Phoenix Project: Master Constraint Programme for Loop Quantum Gravity
Thomas Thiemann
TL;DR
The paper proposes the Master Constraint Programme as a fundamentally new route to the quantum dynamics of general relativity in Loop Quantum Gravity, replacing the problematic Hamiltonian constraint algebra with a single positive constraint $\mathsf{M}$ whose vanishing encodes all constraints. It develops both a graph-changing and a non-graph-changing quantization strategy, laying out how $\mathsf{M}$ can be defined as a positive operator on the diffeomorphism-invariant Hilbert space and how the physical content can be extracted via direct integral decomposition and ergodic means to construct Dirac observables. The work connects canonical quantization with potential spin foam/path integral formulations, outlining how a generalized projector onto the physical subspace might be realized and how a path integral could be formulated from the MCP, while acknowledging significant mathematical and physical open questions (e.g., closability of the quadratic form, semiclassical states, and a proven classical limit). It presents a coherent framework to address anomaly-freeness, constraints, and observables in LQG, with clear avenues for testing in solvable models and potential extensions to lattice formulations.
Abstract
The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper we propose a solution to this set of problems based on the so-called {\bf Master Constraint} which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. If certain mathematical conditions, which still have to be proved, hold, then not only the problems with the commutator algebra could disappear, also chances are good that one can control the solution space and the (quantum) Dirac observables of LQG. Even a decision on whether the theory has the correct classical limit and a connection with the path integral (or spin foam) formulation could be in reach. While these are exciting possibilities, we should warn the reader from the outset that, since the proposal is, to the best of our knowledge, completely new and has been barely tested in solvable models, there might be caveats which we are presently unaware of and render the whole {\bf Master Constraint Programme} obsolete. Thus, this paper should really be viewed as a proposal only, rather than a presentation of hard results, which however we intend to supply in future submissions.