Lattice knot theory and quantum gravity in the loop representation
Open accessAbstract
We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing, and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern-Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern-Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The orderings of the hamiltonian constraint studied so far generate a dynamical evolution that appears excessively 'local'; they preserve the general structure of the network on which they act, simply 'dressing' nodes.
“It has been observed [55] that the orderings of the hamiltonian constraint studied so far generate a dynamical evolution that appear to be excessively 'local'. They preserve the general structure of the network on which they act, simply 'dressing' nodes.”
From “Sum over surfaces” form of loop quantum gravity (1997) · cites this paper as [55]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
The citation lists haven’t been fetched yet
References 24 · Cited by 22
Research works printed but not linked: 1
- [22]L. Kauffman, S. Lins, 'Temperley-Lieb recoupling theory and invariants of three-manifolds', Annals of mathematics studies, no. 134., Princeton University Press (1994).
- Authors
- Fort, H., Gambini, R., & Pullin, J.
- Affiliations
- Universidad de la República de Uruguay (UY)
- Pennsylvania State University (US)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 1997-08-15
- Volume
- 56
- Issue
- 4
- Pages
- 2127-2143
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W1970261739
- Cited by
- 22
- References
- 24
- Citations by year
- 2024: 1 · 2022: 1 · 2017: 1 · 2015: 1 · 2013: 1 · 2012: 1
- FWCI
- 3.86
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Civil War Book Review (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 › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- loop representation, quantum gravity, square lattice, knot invariants, diffeomorphism constraint, Chern-Simons theory, framing, skein relations, kinks, continuum limit
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
