A combinatorial approach to diffeomorphism invariant quantum gauge theories
Open accessAbstract
Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.
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Spin foam models of gravity incorporate the discreteness of the geometry of space first uncovered in loop quantized canonical theory into a spacetime sum over histories formalism naturally suited to a 4-diffeo invariant theory.
“Spin foam models of gravity incorporate the discreteness of the geometry of space first uncovered [20, 21, 22, 23, 24] in loop quantized canonical theory into a spacetime sum over histories formalism naturally suited to a 4-diffeo invariant theory.”
From Spacetime as a Feynman diagram: the connection formulation (2001) · cites this paper as [22]
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Self-citation
- Authors
- Zapata, J. A.
- Affiliations
- Center for Gravitational Physics and Geometry, Department of Physics, The Pennsylvania State University, 104 Davey Laboratory, University Park, Pennsylvania 16802
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1997-11-01
- Volume
- 38
- Issue
- 11
- Pages
- 5663-5681
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W1973160176
- Cited by
- 32
- References
- 31
- Citations by year
- 2019: 1 · 2018: 2 · 2016: 1 · 2014: 1 · 2013: 1 · 2012: 2
- FWCI
- 2.90
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- quantum gauge theory, diffeomorphism invariance, holonomy, physical states, gauge invariance, simplicial complexes, unitary equivalence
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