On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation

Abstract

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this article we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints. For first class constraints with non trivial structure functions the equivalence can only be established by passing to Abelian(ised) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based.

43 pages

References in corpus (6)

• Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
• Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?
• Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order
• The Holst Spin Foam Model via Cubulations
• Solving the Problem of Time in General Relativity and Cosmology with Phantoms and k -- Essence
• Born--Oppenheimer decomposition for quantum fields on quantum spacetimes

Cited by in corpus (14)

• The Spin Foam Approach to Quantum Gravity
• 4-dimensional Spin-foam Model with Quantum Lorentz Group
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Operator Spin Foam Models
• Generalized Spinfoams
• Effective Dynamics from Coherent State Path Integral of Full Loop Quantum Gravity
• Critical Overview of Loops and Foams
• Loop Quantum Gravity
• The Holst Spin Foam Model via Cubulations
• Spin foams with timelike surfaces
• Semiclassical Limit of New Path Integral Formulation from Reduced Phase Space Loop Quantum Gravity
• 4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block
• Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity
• The new spin foam models and quantum gravity