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Quantum Spin Dynamics VIII. The Master Constraint

Thomas Thiemann

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.

Quantum Spin Dynamics VIII. The Master Constraint

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.

Paper Structure

This paper contains 9 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Derivation of the quadratic form of the master constraint
  3. Strategy
  4. New inner product on algebraic dual
  5. Derivation of the quadratic form
  6. The master constraint operator
  7. Physical inner product and Dirac observables
  8. The classical master equation for selecting weak Dirac observables
  9. Conclusions

Key Result

Theorem 3.1

i) Let $T$ be a symmetric operator ($D(T)\subset D(T^\ast)$, $T^\ast_{|D(T)}=T$). Then $T$ is closable, however, its closure may not be self -- adjoint ($D(\bar{T})\not=D(\bar{T}^\dagger$). ii) Let $Q$ be a semi -- bounded quadratic form. Then $Q$ may not be closable, but if it is and the closure is

Theorems & Definitions (6)

  • Definition 3.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Definition 4.1
  • Theorem 4.1