Constraint algebra in loop quantum gravity reloaded. I. Toy model of a U(1)3 gauge theory

Adam Henderson, Alok Laddha, and Casey Tomlin
Phys. Rev. D 88, 044028 – Published 20 August 2013

Abstract

We analyze the issue of anomaly-free representations of the constraint algebra in loop quantum gravity (LQG) in the context of a diffeomorphism-invariant U(1)3 theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the μ¯ scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework. Our work, along with the work done in [C. Tomlin and M. Varadarajan, Phys. Rev. D 87, 044039 (2013)], suggests a new approach to the construction of anomaly-free quantum dynamics in Euclidean LQG.

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  • Received 9 December 2012

DOI:https://doi.org/10.1103/PhysRevD.88.044028

© 2013 American Physical Society

Authors & Affiliations

Adam Henderson1, Alok Laddha1, and Casey Tomlin1,2

  • 1Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, Pennsylvania 16802-6300, USA
  • 2Raman Research Institute, Bangalore 560 080, India

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Issue

Vol. 88, Iss. 4 — 15 August 2013

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