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## and Category: is quantum gravity algebraic (1995)

Venue: | Journal of Mathematical Physics |

Citations: | 56 - 3 self |

### Citations

1317 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...l discussed here creates a rather difficult communication problem, since vanishingly few theoretical physicists study category theory. An ambitious reader might want to study the subject from Maclane =-=[7]-=-. It may also be possible to wing it and just follow my development as it stands. Although it was clearly impossible to make it self contained, I have interspersed my discussion with thumbnail definit... |

765 |
Topological quantum field theory,
- WITTEN
- 1988
(Show Context)
Citation Context ...is so extreme and so jarring that most workers in the field tend to blot it out of their minds. Topological quantum field theories (TQFTs), were originally defined formally in terms of path integrals =-=[1,2]-=-. Subsequently, they were rigorously constructed in low dimensions by various authors, using algebraic tools [3,4,5]. Among the algebraic tools which appear in this process, the techniques of category... |

412 |
Invariants of 3-manifolds via link polynomials and quantum groups,
- ikhin, Turaev
- 1991
(Show Context)
Citation Context ...m field theories (TQFTs), were originally defined formally in terms of path integrals [1,2]. Subsequently, they were rigorously constructed in low dimensions by various authors, using algebraic tools =-=[3,4,5]-=-. Among the algebraic tools which appear in this process, the techniques of category theory play a special role. The purpose of this paper is to propose that the same mathematical tools which put TQFT... |

308 |
Classical and quantum conformal field theory,
- Moore, Seiberg
- 1989
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Citation Context ...ar tensor category in [5], and the topological state sum, also from a modular tensor category in [4], but since generalized to any tensor category with only a weak condition on its duals in [9]. (See =-=[10]-=- for the definition of a modular tensor category.) The construction in [3] is also elegant and historically important, but does not play as big of a role in what follows. 4The point which is most imp... |

274 |
State sum invariants of 3-manifolds and quantum 6j-symbols,
- Viro
- 1992
(Show Context)
Citation Context ...m field theories (TQFTs), were originally defined formally in terms of path integrals [1,2]. Subsequently, they were rigorously constructed in low dimensions by various authors, using algebraic tools =-=[3,4,5]-=-. Among the algebraic tools which appear in this process, the techniques of category theory play a special role. The purpose of this paper is to propose that the same mathematical tools which put TQFT... |

242 | Discreteness of area and volume in quantum gravity.
- Rovelli, Smolin
- 1995
(Show Context)
Citation Context ...logical constant. Now the similarities of the 3D and 4D formulas give us reason to hope. Formula 4A can be thought of as putting spins on surfaces in 4D, which is reminiscent of the recent suggestion =-=[31]-=- that the observables for the loop variables are areas of surfaces, not lengths. In fact, it is possible, up to a small ambiguity, to define the geometry of a triangulated 4-manifold by assigning area... |

177 |
Topological quantum field theories,
- Atiyah
- 1989
(Show Context)
Citation Context ...is so extreme and so jarring that most workers in the field tend to blot it out of their minds. Topological quantum field theories (TQFTs), were originally defined formally in terms of path integrals =-=[1,2]-=-. Subsequently, they were rigorously constructed in low dimensions by various authors, using algebraic tools [3,4,5]. Among the algebraic tools which appear in this process, the techniques of category... |

149 | Angular momentum; an approach to combinatorial spacetime
- Penrose
- 1971
(Show Context)
Citation Context ...sor operators). C. CSW THEORY AND DEFORMED SPIN NETWORKS The invariants of embedded labelled graphs at the heart of the CSW TQFT are closely related to the evaluations of spin networks due to Penrose =-=[11]-=-. Spin networks are not sensitive to embeddings, but that is due precisely to the fact that Penrose did not have quantum groups to work with, and used representations of ordinary SU(2) instead [12]. S... |

146 |
Lectures on Non-Perturbative Canonical Gravity (World Scientific
- Ashtekar
- 1991
(Show Context)
Citation Context ...chanics, and is still unclear. Nevertheless, the resonances between the mathematical structure presented here and the development of the Ashtekar/loop variables approach to quantum general relativity =-=[6]-=- are so strong that the suggestion that they are related deserves to be pursued. In a certain sense, I am proposing that a particular three dimensional TQFT, namely the CSW theory, already contains th... |

130 |
Four-dimensional topological quantum field theory, Hopf categories and the canonical bases
- Crane, Frenkel
- 1994
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Citation Context ...gorical levels. The structure on surfaces in a 3D-TQFT, for example, is a categorical analog of the structure of a 2D-TQFT. To make this notion precise, we have defined the idea of a categorification =-=[23]-=-. The categorification of any type of algebraic structure is a type of structure with analogous operations one categorical level higher. A categorification of a particular algebraic structure is an ex... |

80 |
Semiclassical limit of Racah coefficients, Spectroscoplc and group theoretical methods in physics, North Holland
- Ponzano, Regge
- 1968
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Citation Context ...n effort to bring Penrose’s program up one dimension by adding better algebra. The best way to understand how spin networks apply to quantizing geometry is to follow the approach of Regge and Ponzano =-=[13]-=-. What they do is to triangulate a region of space surrounded by the embedded graph, and prove by an inductive argument that the evaluation of the graph is given by a formula involving a sum over labe... |

64 | Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories, - Connes, Rovelli, et al. - 1994 |

52 | On algebraic structures implicit in topological quantum field theories
- Crane, Yetter
- 1999
(Show Context)
Citation Context ... categorical analogue of a dual pairing!). The definition of a TQFT with factorizability is similar in higher dimension, except that there are successive categorical layers at higher codimension. See =-=[20]-=- for the general definition. We shall give the 3D definition in its full glory in a minute. The importance of factorizability is that it allows us to obtain the objects on manifolds with boundary by c... |

50 |
The Chern-Simons Invariant as the Natural Time Variable for Classical and Quantum Cosmology
- Smolin, Soo
- 1995
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Citation Context ...at this procedure produces a thermal state; much of Hawking’s work on black hole radiation is based on this idea. Thus, the idea that CSW is recovered by tracing out is consistent with the suggestion =-=[29]-=- that CSW is a thermal state for quantum gravity. The suggestion in [30] that the CSW lagrangian provides a time parameter for quantum gravity is also closely related. How likely then, is it that our ... |

27 |
Topological quantum field theories associated to finite groups and crossed G-sets
- Yetter
- 1992
(Show Context)
Citation Context ...cal algebra. For the purposes of this work, we are especially interested in the construction of 4D TSSs. The literature on this is quite thin. Aside from the construction related to a finite group in =-=[27]-=-, which seems far removed from realistic physics, 14(although it does reproduce Dijkgraf-Witten theory), there are only two formulas proposed: the one in [28], which is closely connected to CSW theor... |

15 |
State-sum invariants of 3-manifolds associated to artinian semisimple tortile categories, Topology Appli. 58
- Yetter
- 1994
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Citation Context ...om a modular tensor category in [5], and the topological state sum, also from a modular tensor category in [4], but since generalized to any tensor category with only a weak condition on its duals in =-=[9]-=-. (See [10] for the definition of a modular tensor category.) The construction in [3] is also elegant and historically important, but does not play as big of a role in what follows. 4The point which ... |

15 |
A categorical construction of 4d TQFTs
- Crane, Yetter
- 1993
(Show Context)
Citation Context ...onstruction related to a finite group in [27], which seems far removed from realistic physics, 14(although it does reproduce Dijkgraf-Witten theory), there are only two formulas proposed: the one in =-=[28]-=-, which is closely connected to CSW theory, and also uses a modular tensor category, and the one in [23], which is part of the dimensional ladder picture, and utilizes the categorification of a Hopf a... |

12 |
2-d Physics and 3-d
- Crane
- 1991
(Show Context)
Citation Context ...m field theories (TQFTs), were originally defined formally in terms of path integrals [1,2]. Subsequently, they were rigorously constructed in low dimensions by various authors, using algebraic tools =-=[3,4,5]-=-. Among the algebraic tools which appear in this process, the techniques of category theory play a special role. The purpose of this paper is to propose that the same mathematical tools which put TQFT... |

7 | Conformal field theory, spin geometry, and quantum gravity,” Phys - Crane - 1991 |

6 |
Gravity as an Exactly Soluble System,
- Witten
(Show Context)
Citation Context ...CSW theory gives us 2+1 gravity, the argument would be quite straightforward. Indeed , it would also not be terribly original. A number of authors have made just such a connection in a number of ways =-=[17]-=-. Nobody expects 2+1 dimensional gravity to behave very differently from a TQFT. Why then is the suggestion in this paper so novel? The physical objection to the idea that 3+1 gravity has a similar fo... |

6 |
Einstein and quantummechanics,” in Conceptual Problems of Quantum Gravity,
- Stachel
- 1991
(Show Context)
Citation Context ...are naturally related to four dimensional geometry creates our world has a strange appealing simplicity. 16VI. ANCESTRAL VOICES To my great surprise, I recently learned from an essay by John Stachel =-=[32]-=-, that the ideas explored here are very similar to the ones that Einstein was exploring near the end of his career. Apparently, these ideas remained obscure because Einstein did not have the mathemati... |

5 |
Involutory Hopf Algebras and Three-manifold
- Kuperberg
- 1989
(Show Context)
Citation Context ...invariant by summing over the entire manifold. (It would be very attractive to find a new formula of a similar form which bridged this gap, ie gave CSW when computed on the entire manifold. Kuperberg =-=[15]-=- has given a candidate for such a formula, which reproduces CSW for embedded graphs, but then gives zero on most manifolds. It is not yet clear if his work admits a modification to reproduce CSW on th... |

4 |
Categorical linear algebra—a setting for questions from physics and low-dimensional topology, Kansas State U. preprint. Available at http://math.ucr.edu/home/baez/yetter.pdf
- Yetter
(Show Context)
Citation Context ...he sense of [21] of the semisimple categories attached to the parts. The empty 1-manifold will be assigned VECT. 2. The 1-manifold with opposite orientation is assigned the dual category. (cf. Yetter =-=[22]-=-) 3. The disjoint union of surfaces is assigned the tensor product of the vector spaces on the surfaces. 4. The surface with opposite orientation is assigned the dual vector space. 5. If we cut an ori... |

4 | Lusztig , Introduction to Quantum Groups , Birkhauser - unknown authors - 1993 |

2 |
Lattice Topological Field Theory , preprint
- Fukama, Hosong, et al.
- 1993
(Show Context)
Citation Context ...s analogy to a path integral, so it is no surprise that a TQFT can be constructed from it. In fact, interesting examples of 2D and 3D TSSs were constructed by physicists interested in lattice QFTs in =-=[25,26]-=-. They found that 2D theories were related to semisimple algebras, while 3D theories were related to Hopf algebras. The latter work is equivalent to the work of Kuperberg in [15], although the notatio... |

1 |
Quantum gravity and the Akgebra of Tangles
- Baez
- 1992
(Show Context)
Citation Context ...osition of linear maps, the disjoint union of manifolds gets the tensor product of vector spaces, and the manifold with opposite orientation gets assigned the dual space. As John Baez has pointed out =-=[8]-=-, there is a natural inner product on the vector spaces whenever they are associated to a bounding 3-manifold, generated by gluing pairs of links along the boundary. Thus, we can think of them as Hilb... |