Hermitian operators on quantum general relativity loop space
Abstract
The author discusses a more precise mathematical formulation of the Rovelli and Smolin loop space representation of quantum general relativity. Their representation space based on wavefunctions defined on sets of loops can be made more definitive by using either subsets or Cartesian products of the loop space of the 3-manifold Sigma . This enables him to consider the problem of defining an inner product on the wavefunctions. He constructs a 'model' inner product on a subspace of the representation space using a discrete sum instead of an integral over elements of the domain space of the wavefunctions. With respect to this inner product, Hermitian conjugates to the T operators can be constructed. The Hermitian operators he obtains are significant from the point of view of their action on the sets rather than the individual loops.
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Any actual volume or area measurement defines a 4-dimensional diffeomorphism invariant observable, because any region whose volume or area we measure is determined by physical fields and these fields are necessarily coupled to the gravitational field.
“A very important observation is that any actual volume or area measurement that we may concretely perform defines such a 4-dimensional diffeomorphism invariant observable, because any region whose volume or area we measure is determined by physical fields and, in the context of relativistic gravitation, these fields are necessarily coupled to the gravitational field [12].”
From Discreteness of area and volume in quantum gravity (1995) · cites this paper as [12]
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- Authors
- Rayner, D.
- Affiliations
- Imperial College London (GB)
- Corresponding author
- Rayner, D
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1990-04-01
- Volume
- 7
- Issue
- 4
- Pages
- 651-661
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2030684107
- Cited by
- 17
- References
- 6
- FWCI
- 1.83
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Classical and Quantum Gravity (publishedVersion)
- 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 › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Keywords
- loop quantum gravity, Hermitian operators, 3-manifolds, T-operators, representation space
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