Quantum Spin Dynamics VIII. The Master Constraint

TL;DR

This paper proves that the Master Constraint in Loop Quantum Gravity is mathematically well-defined as a self-adjoint operator, enabling the construction of a physical Hilbert space for the theory.

Contribution

It establishes the closability and self-adjointness of the Master Constraint Operator in LQG, extending previous results to include matter coupling and arbitrary metric signatures.

Findings

Proved the quadratic form of the Master Constraint is closable.

Established the existence of a unique self-adjoint Master Constraint Operator.

Enabled spectral analysis to construct the physical Hilbert space.

Abstract

Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non -- Lie -- algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the Master Constraint Operator was derived. In this paper we close this gap and prove that the quadratic form is closable and thus stems from a unique self -- adjoint Master Constraint Operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis.