On the relation between the connection and the loop representation of quantum gravity
Open accessAbstract
Using Penrose's binor calculus for SU(2) (SL(2,C)) tensor expressions, a graphical method for the connection representation of Euclidean quantum gravity (real connection) is constructed. It is explicitly shown that: (i) the recently proposed scalar product in the loop-representation coincides with the Ashtekar - Lewandowski cylindrical measure in the space of connections; (ii) it is possible to establish a correspondence between the operators in the connection representation and those in the loop representation. The construction is based on an embedded spin network, the Penrose's graphical method of SU(2) calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations.
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Group-averaging techniques can produce invariant generalized states, but for the rotation group they do not give the desired results for this paper's purposes.
“One could use group-averaging techniques [11] to find invariant generalized states but, as will be discussed below for the rotation group, this does not lead to the desired results for the purposes of this paper.”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [11]
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Cites
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“Following [6], we choose the form of Ψ(j, i) by identifying Γ5 with the dual skeleton of a regular triangulation of the three-sphere.”
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- Authors
- Pietri, R. D.
- Affiliations
- University of Parma (IT)
- Corresponding author
- Pietri, Roberto De
- ORCID
- Pietri, Roberto De
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1997-01-01
- Volume
- 14
- Issue
- 1
- Pages
- 53-69
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2001159205
- Cited by
- 50
- References
- 8
- Citations by year
- 2023: 1 · 2021: 1 · 2016: 1 · 2015: 1 · 2012: 1
- FWCI
- 7.40
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- European Commission
- 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
- Euclidean quantum gravity, loop representation, SL(2,C), spin networks, cylindrical measures, inner product, gauge equivalence
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