## Knots and quantum gravity: progress and prospects, to appear in the proceedings of the Seventh Marcel Grossman Meeting on General Relativity, University of California at Riverside preprint available as gr-qc/9410018. 38

Citations: | 19 - 11 self |

### BibTeX

@MISC{Baez_knotsand,

author = {John C. Baez},

title = {Knots and quantum gravity: progress and prospects, to appear in the proceedings of the Seventh Marcel Grossman Meeting on General Relativity, University of California at Riverside preprint available as gr-qc/9410018. 38},

year = {}

}

### OpenURL

### Abstract

to appear in proceedings of the

### Citations

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Citation Context ...ately related to conformal field theory in 2 dimensions. Atiyah [9], however, conjectured that there should be an intrinsically 3-dimensional definition of these invariants using gauge theory. Witten =-=[53]-=- gave a heuristic proof of Atiyah's conjecture by deriving the Jones polynomial and its generalizations from Chern-Simons theory. The basic idea is simply that the vacuum expectation values of Wilson ... |

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Citation Context ...ory has been extensively studied in dimensions 2, 3, and 4. In dimension 2, it is closely related to Yang-Mills theory [54]. In dimension 3, it has gravity in the Palatini formalism as a special case =-=[5, 52]-=-. In dimension 4, it is also known as `topological gravity' when we take G = SL(2; C) and take P to be the bundle P+ used in the self-dual formulation of general relativity [29, 30]. Mathematically, B... |

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Citation Context ...on value For the groups SU(n) one obtains a link invariant generalizing the Kauffman bracket known as the HOMFLY polynomial, while for SO(n) one obtains yet another invariant, the Kauffman polynomial =-=[35, 43]-=-. All of these link invariants are also defined by skein relations. Since / CS may be defined as the unique function on A annihilated by the constraint C a ij , while / CS is determined (at least on l... |

217 |
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Citation Context ...by a generalized measure. 1 Introduction The relation between knots and quantum gravity was discovered in the course of a fascinating series of developments in mathematics and physics. In 1984, Jones =-=[34]-=- announced the discovery of a new link invariant, which soon led to a bewildering profusion of generalizations. It was clear early on that these new invariants were intimately related to conformal fie... |

174 |
quantum gauge theories in two dimensions
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Citation Context ...was drawn to it by the work of Blau and Thompson [22] and Horowitz [33], this theory has been extensively studied in dimensions 2, 3, and 4. In dimension 2, it is closely related to Yang-Mills theory =-=[54]-=-. In dimension 3, it has gravity in the Palatini formalism as a special case [5, 52]. In dimension 4, it is also known as `topological gravity' when we take G = SL(2; C) and take P to be the bundle P+... |

142 | On the Vassiliev knot invariants
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Citation Context ...eed, the HOMFLY and Kauffman polynomials can all be expanded as power series in , with coefficients being link invariants of a special sort known as invariants of finite type, or Vassiliev invariants =-=[17]-=-. If we accept the assumption that the Chern-Simons state for SL(2; C) corresponds to the Kauffman bracket, at least on S 3 , we obtain a fascinating relation between quantum gravity and Vassiliev inv... |

107 | New points of view in knot theory
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Citation Context ...um gravity, since the Hamiltonian constraint is also sensitive to self-intersections [46]. Also, within knot theory itself, more and more attention is being paid to multiloops with self-intersections =-=[17, 20]-=-. It is typical that the measures appearing in quantum field theory (either as pathintegrals or as states in the canonical formalism) are not supported on the space of smooth fields[40], but on a larg... |

76 |
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Citation Context ...ant measures on the space of connections correspond to isotopy invariants of links (or more generally, `multiloops'). In Section 2 we review recent work by Ashtekar, Isham, Lewandowski and the author =-=[6, 7, 13, 14, 15, 38]-=- on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles [2, 3, 25, 39, 48] and books [4]. 2 2... |

55 |
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Citation Context ...n-Simons theory and a conformal field theory known as the Wess-Zumino-Witten (or WZW) model. 1 In parallel to this work, a new approach to quantum gravity was being developed, initiated by Ashtekar's =-=[1]-=- discovery of the `new variables' for general relativity. In this approach, the classical configuration space is a space of connections, and states of the quantum theory are (roughly speaking) measure... |

47 | Generalized measures in gauge theory
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Citation Context ...ant measures on the space of connections correspond to isotopy invariants of links (or more generally, `multiloops'). In Section 2 we review recent work by Ashtekar, Isham, Lewandowski and the author =-=[6, 7, 13, 14, 15, 38]-=- on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles [2, 3, 25, 39, 48] and books [4]. 2 2... |

40 | On the support of the Ashtekar-Lewandowski measure
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Citation Context ...ersections [17, 20]. It is typical that the measures appearing in quantum field theory (either as pathintegrals or as states in the canonical formalism) are not supported on the space of smooth fields=-=[40]-=-, but on a larger space of `distributional fields'. And indeed, generalized measures on A can alternatively be described as honest measures on a space A of `generalized connections' containing A as a ... |

39 |
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Citation Context ...te appears to represent a `quantized deSitter universe' (or anti-deSitter, depending on the sign of ). Smolin and Soo have recently done some fascinating work on the `problem of time' using this idea =-=[49]-=-. Similarly, if S = R 3 it appears that the single flat state represents a `quantized Minkowski space' ! However, there has been some debate over whether the Chern-Simons state is normalizable, and th... |

35 |
Topological quantum field theories. Inst. Hautes Études Sci
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Citation Context ...far from clear: Problem 2. Describe Chern-Simons theory with noncompact gauge group (in particular,sSL(2; C)) as a topological quantum field theory satisfying axioms similar to those listed by Atiyah =-=[10]-=-, and compute the vacuum expectation values of Wilson loops in this theory. (Hint: see the work of Bar-Natan and Witten [19, 55].) 10 Let us now turn to how one computes the link invariant correspondi... |

34 |
Quantization of Chern-Simons gauge theory with complex gauge group
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Citation Context ...al quantum field theory satisfying axioms similar to those listed by Atiyah [10], and compute the vacuum expectation values of Wilson loops in this theory. (Hint: see the work of Bar-Natan and Witten =-=[19, 55]-=-.) 10 Let us now turn to how one computes the link invariant corresponding to /CS . For reasons of space we will be very sketchy here. First, writing / CS (A) = e \Gamma 6sSCS (A) ; the quantity SCS (... |

33 |
Derivation Of The Verlinde Formula From Chern-Simons Theory And The G/G Model,” Nucl. Phys. B408
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Citation Context ...group. However, more recently it has become clear that the more fundamental relation is that between Chern-Simons theory and a 2-dimensional topological quantum field theory, the G=G gauged WZW model =-=[23]-=-. From this point of view, the WZW model itself serves mainly as a computational tool. What is the real meaning of the dimensional ladder? Most importantly, one climbs down it by considering `boundary... |

32 | Coherent state transforms for spaces of connections
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Citation Context ...eneral relativity, and some extra work is needed to restrict to real-valued metrics. (In what follows we will gloss over these very important `reality conditions', on which progress is just beginning =-=[3, 8]-=-.) The idea is to work with a complex-valued soldering form, that is, 1-form on M with values in the complexified bundle CT , and a self-dual connection A+ . To understand this concept of self-duality... |

30 | Diffeomorphism-invariant Generalized Measures on the Space of Connections Modulo Gauge Transformations
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Citation Context ...ant measures on the space of connections correspond to isotopy invariants of links (or more generally, `multiloops'). In Section 2 we review recent work by Ashtekar, Isham, Lewandowski and the author =-=[6, 7, 13, 14, 15, 38]-=- on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles [2, 3, 25, 39, 48] and books [4]. 2 2... |

30 | The basis of the Ponzano–Regge–Turaev–Viro–Ooguri model is the loop representation basis”, Phys
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Citation Context ...nd 3-dimensional topology. For example, Rovelli has drawn inspiration from the Turaev-Viro theory, a topological quantum field theory in 3 dimensions, to give a formula for the physical inner product =-=[45]-=-, which unfortunately is purely formal at present. An alternative strategy, which is mathematically rigorous but physically more radical, is to split S into two manifolds with boundary, and to use the... |

30 |
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Citation Context ...sures on the space of connections which satisfy certain constraints: the Gauss law, the diffeomorphism constraint, and the Hamiltonian constraint. In an effort to find such states, Rovelli and Smolin =-=[46]-=- used a `loop representation' in which one works, not with the measures per se, but with the expectation values of Wilson loops with respect to these measures. In these terms, the diffeomorphism const... |

25 | Recent Developments in non Perturbative Quantum Gravity; in Quantum Gravity and Cosmology
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Citation Context ...and the author [6, 7, 13, 14, 15, 38] on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles =-=[2, 3, 25, 39, 48]-=- and books [4]. 2 2 The New Variables and the Dimensional Ladder Traditionally, general relativity has been viewed as a theory in which a metric is the basic field. In these terms, the Einstein-Hilber... |

24 |
invariants of finite type and perturbation
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Citation Context ...(2; C) corresponds to the Kauffman bracket, at least on S 3 , we obtain a fascinating relation between quantum gravity and Vassiliev invariants. For more on this, we urge the reader to the references =-=[11, 27, 32, 36, 42]-=-. 3 Multiloop Invariants and Generalized Measures In the previous section, much of the discussion of quantum gravity in 4 dimensions was heuristic in character. In particular, we imagined starting wit... |

17 | Mathematical Problems of Non-perturbative Quantum General Relativity
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Citation Context ...and the author [6, 7, 13, 14, 15, 38] on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles =-=[2, 3, 25, 39, 48]-=- and books [4]. 2 2 The New Variables and the Dimensional Ladder Traditionally, general relativity has been viewed as a theory in which a metric is the basic field. In these terms, the Einstein-Hilber... |

17 | c.. Ashtekar Formulation of General Relativity and the Loop Space NonPerturbative Quantum Gravity: A Repport. Class. Quantum Grav - Rovelli - 1991 |

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Citation Context ...heory in dimension 4, Chern-Simons theory in dimension 3, and the WZW model in dimension 2. The concept of a ladder of field theories has appeared in other contexts and appears to be an important one =-=[9, 49]-=-. In Section 1, after an introduction to the ‘new variables’, we review this ladder of field theories and its relation to the new knot invariants. In addition to understanding the Chern-Simons state a... |

14 |
Topological Field Theory, Phys. Rep
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Citation Context ...heory is closely related to moduli spaces of flat connections, and thereby to the RaySinger torsion, the Alexander-Conway polynomial invariant of links, and the Casson invariant of homology 3-spheres =-=[21, 22, 28, 47]-=-. In what follows we will focus on dimension 4, and consider a variant of the BF action that includes a BsB term: SBF (A; B) = Z M tr(BsF + 12 BsB): (4) 5 Ignoring boundary terms, the variation of the... |

14 |
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11 | Perturbative expansion of Chern–Simons theory with noncompact gauge group
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Citation Context ...al quantum field theory satisfying axioms similar to those listed by Atiyah [10], and compute the vacuum expectation values of Wilson loops in this theory. (Hint: see the work of Bar-Natan and Witten =-=[19, 55]-=-.) 10 Let us now turn to how one computes the link invariant corresponding to /CS . For reasons of space we will be very sketchy here. First, writing / CS (A) = e \Gamma 6sSCS (A) ; the quantity SCS (... |

11 |
Knot invariants as nondegenerate quantum geometries”, Phys
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Citation Context ...nk invariants. No formalism for quantum gravity has been worked out to the point where we can feel full confidence in these results, but the work of various authors using the connection [37] and loop =-=[27]-=- representations, as well as the BRST formalism [30], all seems to point in the same direction. In what follows we will describe these results in terms of Dirac's approach to canonical quantization of... |

10 | Three-dimensional BF Theories and the Alexander–Conway Invariant of Knots Nucl
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Citation Context ...heory is closely related to moduli spaces of flat connections, and thereby to the RaySinger torsion, the Alexander-Conway polynomial invariant of links, and the Casson invariant of homology 3-spheres =-=[21, 22, 28, 47]-=-. In what follows we will focus on dimension 4, and consider a variant of the BF action that includes a BsB term: SBF (A; B) = Z M tr(BsF + 12 BsB): (4) 5 Ignoring boundary terms, the variation of the... |

9 | Recent mathematical developments in quantum general relativity, preprint gr-qc/9411055. C. Rovelli, Ashtekar Formulation of general relativity and loop-space nonpertubative quantum gravity: a report, Classical and Quantum Gravity 8
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Citation Context ...and the author [6, 7, 13, 14, 15, 38] on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles =-=[2, 3, 25, 39, 48]-=- and books [4]. 2 2 The New Variables and the Dimensional Ladder Traditionally, general relativity has been viewed as a theory in which a metric is the basic field. In these terms, the Einstein-Hilber... |

9 |
New invariants of 3- and 4-dimensional manifolds
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Citation Context ...link invariant, which soon led to a bewildering profusion of generalizations. It was clear early on that these new invariants were intimately related to conformal field theory in 2 dimensions. Atiyah =-=[9]-=-, however, conjectured that there should be an intrinsically 3-dimensional definition of these invariants using gauge theory. Witten [53] gave a heuristic proof of Atiyah's conjecture by deriving the ... |

9 |
A new class of topological field theories and the Ray–Singer torsion, Phys
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Citation Context ...idences. Some of the mystery is removed when we note that the `Chern-Simons state' of quantum gravity is the only state of a simpler diffeomorphism-invariant theory in 4 dimensions known as BF theory =-=[22, 33]-=-. However, a truly systematic explanation would require understanding the following `ladder' of field theories as a unified structure: general relativity and BF theory in dimension 4, Chern-Simons the... |

9 | Actions for gravity, with generalizations: a review
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Citation Context ...ally, the first step towards viewing general relativity as a gauge theory was the Palatini formalism. (For a discussion of various Lagrangians for general relativity, see the review article by Peldan =-=[41]-=-.) In this approach, we fix an oriented bundle T over M (usually called the `internal space') that is isomorphic to TM and equipped with a Lorentzian metric j, and we assume that the spacetime metric ... |

8 |
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Citation Context ...'s original paper they gave a heuristic construction assigning to each isotopy class of unoriented links a solution of all the constraints of quantum gravity in the loop representation. Later, Kodama =-=[37]-=- showed how to obtain another sort of solution using Chern-Simons theory. From Witten's work it is clear that in the loop representation this solution is just the Jones polynomial --- or more precisel... |

7 | invariants, holonomy algebras, and functional integration - Baez, Link - 1995 |

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Citation Context ...ribe the flat states as functions on the moduli space of flat connections on P j S , which has the advantage over A of being finite-dimensional. (For more on the flat states, see the work of Blencowe =-=[24]-=-.) One can attempt to quantize gravity in a similar fashion, defining operators on H kin by ( A a i (x)/)(A) = A a i (x)/(A); (s~ E i a (x)/)(A) = ffi/ ffiA a i (x) (A); and seeking solutions of the c... |

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Citation Context ...ory has been extensively studied in dimensions 2, 3, and 4. In dimension 2, it is closely related to Yang-Mills theory [54]. In dimension 3, it has gravity in the Palatini formalism as a special case =-=[5, 52]-=-. In dimension 4, it is also known as `topological gravity' when we take G = SL(2; C) and take P to be the bundle P+ used in the self-dual formulation of general relativity [29, 30]. Mathematically, B... |

5 | Quantum gravity and the algebra of tangles - Baez - 1993 |

5 |
BRST cohomology and invariants of four-dimensional gravity in Ashtekar’s variables Phys. Rev. D46
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Citation Context ...m as a special case [5, 52]. In dimension 4, it is also known as `topological gravity' when we take G = SL(2; C) and take P to be the bundle P+ used in the self-dual formulation of general relativity =-=[29, 30]-=-. Mathematically, BF theory is closely related to moduli spaces of flat connections, and thereby to the RaySinger torsion, the Alexander-Conway polynomial invariant of links, and the Casson invariant ... |

5 |
Knot theory and quantum gravity: A primer.”, University of Utah preprint UU-REL-93/1/9
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Citation Context ...(2; C) corresponds to the Kauffman bracket, at least on S 3 , we obtain a fascinating relation between quantum gravity and Vassiliev invariants. For more on this, we urge the reader to the references =-=[11, 27, 32, 36, 42]-=-. 3 Multiloop Invariants and Generalized Measures In the previous section, much of the discussion of quantum gravity in 4 dimensions was heuristic in character. In particular, we imagined starting wit... |

4 |
and invited contributors, New Perspectives in Canonical Gravity, Bibliopolis
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Citation Context ... 15, 38] on this problem. In what follows we will not concentrate on the loop representation per se, as it is already the subject of a number of excellent review articles [2, 3, 25, 39, 48] and books =-=[4]-=-. 2 2 The New Variables and the Dimensional Ladder Traditionally, general relativity has been viewed as a theory in which a metric is the basic field. In these terms, the Einstein-Hilbert action with ... |

4 |
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Citation Context ...idences. Some of the mystery is removed when we note that the `Chern-Simons state' of quantum gravity is the only state of a simpler diffeomorphism-invariant theory in 4 dimensions known as BF theory =-=[22, 33]-=-. However, a truly systematic explanation would require understanding the following `ladder' of field theories as a unified structure: general relativity and BF theory in dimension 4, Chern-Simons the... |

3 | Vassiliev invariants and the loop states in quantum gravity
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Citation Context ...(2; C) corresponds to the Kauffman bracket, at least on S 3 , we obtain a fascinating relation between quantum gravity and Vassiliev invariants. For more on this, we urge the reader to the references =-=[11, 27, 32, 36, 42]-=-. 3 Multiloop Invariants and Generalized Measures In the previous section, much of the discussion of quantum gravity in 4 dimensions was heuristic in character. In particular, we imagined starting wit... |

3 |
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Citation Context ...heory is closely related to moduli spaces of flat connections, and thereby to the RaySinger torsion, the Alexander-Conway polynomial invariant of links, and the Casson invariant of homology 3-spheres =-=[21, 22, 28, 47]-=-. In what follows we will focus on dimension 4, and consider a variant of the BF action that includes a BsB term: SBF (A; B) = Z M tr(BsF + 12 BsB): (4) 5 Ignoring boundary terms, the variation of the... |

3 |
Topological field theories, Phys. Rep. 209
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Citation Context ...heory is closely related to moduli spaces of flat connections, and thereby to the RaySinger torsion, the Alexander-Conway polynomial invariant of links, and the Casson invariant of homology 3-spheres =-=[21, 22, 28, 45]-=-. In what follows we will focus on dimension 4, and consider a variant of the BF action that includes a B ∧ B term: ∫ SBF(A, B) = M tr(B ∧ F + Λ B ∧ B). (4) 12 5Ignoring boundary terms, the variation... |

2 |
On a geometric derivation of Witten’s identity for Chern–Simons theory, preprint MPI-PH-93-107
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Citation Context ...and HOMFLY polynomials as directly as possible from the corresponding SU(n) and SO(n) BF theories in 4 dimensions. (Hint: study the existing work on deriving the skein relations via loop deformations =-=[26]-=-, and consider the possibility of a relation to the theory of surfaces immersed in 4-manifolds [29].) It is often tacitly assumed that the Chern-Simons state for SL(2; C), which is the one relevant to... |