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Cited by 40Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

On the constraints of quantum gravity in the loop representation

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Brügmann, Bernd, Pullin, Jorge·1993·Nuclear Physics B
DOI 10.1016/0550-3213(93)90462-x
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Abstract

Based on Ashtekar's new variables there are two different representation for canonical quantum gravity, the connection representation and the loop representation. In contrast to the situation in the connection representation, it is not straightforward to obtain the constraints of quantum gravity in the loop representation. Classically, the constraints expressed in terms of the Ashtekar variables can be obtained from the loop variables in the limit where loops are shrunk to a point. We give precise meaning to this shrinking process at the classical and quantum level and find as the corresponding operator the area derivative on the space of loop functionals. This enables us to derive the diffeomorphism and hamiltonian constraints of quantum gravity in the loop representation in terms of the area derivative. In particular, the hamiltonian constraint arises in exactly the form already obtained by Gambini, but we do not resort to a formal transform between the connection and the loop representation. We interpret the hamiltonian constraint in terms of a generalized version of the shift operator introduced by Rovelli and Smolin.

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  • The same technique gives rigorous meaning to other operators, including the length of a curve, Poincaré generators for asymptotically flat topologies, and matter contributions to the Hamiltonian constraint.

    “Using the same technique one can give rigorous meaning to a whole bunch of other operators in a representation where the metric is not diagonal, including but not exhausting: (i) the operator corresponding to the length of a curve [19]; (ii) the generators of the Poincaré group for asymptotically flat topologies [22] and (iii) matter contributions to the Hamiltonian constraint [23].”

    From Quantum spin dynamics (QSD) (1998) · cites this paper as [23]

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Authors
Brügmann, B., & Pullin, J.
Affiliations
  • Syracuse University (US)
  • University of Utah (US)
ORCID
Venue
Nuclear Physics B
Publisher
Elsevier BV
Published
1993-02
Volume
390
Issue
2
Pages
399-438
ISSN
0550-3213 (ISSN-L)
DOI
10.1016/0550-3213(93)90462-x
OpenAlex
W2080159062
Cited by
40
References
16
Citations by year
2019: 1 · 2012: 1
FWCI
4.80
Type
journal article
Language
en
Open access
Licence
Locations
  • Nuclear Physics B (publishedVersion, cc-by-nc-nd)
  • Civil War Book Review (submittedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
loop representation, canonical quantum gravity, Ashtekar variables, loop variables, diffeomorphism constraint, Hamiltonian constraint, shift operator

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