Jones polynomials for intersecting knots as physical states of quantum gravity
Open accessAbstract
We find a consistent formulation of the constraints of quantum gravity with a cosmological constant in terms of the Ashtekar new variables in the connection representation, including the existence of a state that is a solution to all the constraints. This state is related to the Chern-Simons form constructed from the Ashtekar connection and has an associated metric in space-time that is everywhere nondegenerate. We then transform this state to the loop representation and find solutions to all the constraint equations for intersecting loops. These states are given by suitable generalizations of the Jones knot polynomial for the case of intersecting knots. These are the first physical states of quantum gravity for which an explicit form is known both in the connection and loop representations. Implications of this result are also discussed.
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It seems feasible to determine the space of diffeomorphism invariant solutions to the Wheeler-DeWitt constraint as well as a physical inner product thereon by the group averaging method.
“it seems feasable to determine the space of diffeomorphism invariant solutions to the Wheeler-DeWitt constraint as well as a physical inner product thereon by the group averaging method [16, 7].”
From Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity (1996) · cites this paper as [16]
The normalization factor 'norm' of Brink and Satcheler corresponds to the choice given in the paper, which is also the standard normalization of the Clebsh-Gordon coefficient giving the usual normalization of the Wigner 3nJ-Symbol.
“where the normalization factor 'norm' of [52] corresponds to the choice”
From Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity (1996) · cites this paper as [52]
The most used graphical methods of computation in su(2) representation theory are those of Levinson, developed by Yutsis, Levinson and Vanagas, and the modified version of Brink and Satcheler.
“To our knowledge the most used graphical method of computation in the representation theory of su (2) are the one due to Levinson [62] and developed by Yutsis, Levinson and Vanagas [63] and the slightly modified version of Brink and Satcheler [52].”
From Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity (1996) · cites this paper as [52]
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References 20 · Cited by 98
Research works printed but not linked: 5
- [17][17] R. Gambini 'Link invariant polynomials for intersecting loops', Preprint IFFI, Montevideo (1992). -
- [18][18] J. Birman Ann. Math. Stud. 82, 321 (1974). Can. J. Math. 28, 264 (1976). -
- [23][23] B. Br¨ ugmann, J. Pullin, Syracuse University Preprint SU-GP-91/8-5 (1991). -
- [27][27] R. Gambini, L. Leal, Preprint IFFI 91.01 (1991) (Montevideo) -
- [28][28] C. Di Bartolo, R. Gambini, J. Griego, L. Leal, Preprint IFFI (1991) (Montevideo). -
- Authors
- Brügmann, B., Gambini, R., & Pullin, J.
- Affiliations
- Syracuse University (US)
- Universidad de la República de Uruguay (UY)
- University of Utah (US)
- Venue
- Nuclear Physics B
- Publisher
- Elsevier BV
- Published
- 1992-10
- Volume
- 385
- Issue
- 3
- Pages
- 587-603
- ISSN
- 0550-3213 (ISSN-L)
- OpenAlex
- W2068017004
- Cited by
- 98
- References
- 20
- Citations by year
- 2017: 1 · 2016: 3 · 2015: 5 · 2014: 2 · 2013: 1 · 2012: 1
- FWCI
- 7.84
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Nuclear Physics B (publishedVersion, cc-by-nc-nd)
- 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 › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- quantum gravity, Jones polynomial, Ashtekar variables, Chern-Simons form, cosmological constant, loop representation, physical states, constraint equations
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