Lee Smolin, Penn State

smolin@phys.psu.edu

In the old days of quantum gravity work was split between canonical and histories approaches. Since the invention of the Ashtekar formalism [1] and loop quantum gravity [2,3] more than ten years ago, much non-string theory work in quantum gravity was devoted to canonical approaches. The main results of this work have been the discovery that geometrical quantities such as areas and volumes [4,5] are represented by finite operators with discrete spectra, whose eigenstates give a basis of states which may be described in terms of spin networks [6,7]. There have even been rigorous theorems demonstrating that these results must be true of a large class of quantum theories of gravity [8].

In spite of these successes, in the last two years there has been a shift back to approaches that emphasize spacetime histories and path integrals. One reason for this has been the realization, following the work of Thiemann [9], that while quantum general relativity may be a finite and well defined theory at the Planck scale, there is evidence that the theory produced by the method canonical quantization does not have a continuum limit which reproduces classical general relativity [10,11]. As a result, the interest of many people turned to the possibility that the dynamics of the spin network states might be described in terms of a histories framework in a way that avoid the difficulties of the canonical approach.

In fact, even before these developments, the first formulation of a path integral framework to describe the dynamics of spin network states had been proposed by Reisenberger [12]. The connection of Reisenberger's proposal to the canonical formalism was then elucidated in papers with and by Rovelli [13,14]. In their proposal, and much subsequent work, the spacetime histories in the path integral are represented as discrete combinatorial structures, as is fitting as the basis states in the spin network representation are themselves largely combinatorial. These combinatorial structures can be often visualized as four dimensional triangulations, with labels associated with the discrete geometrical quantities attached to edges, surfaces and tetrahedra. However, unlike the Regge calculus and dynamical triangulation formulations, the triangulations are not meant as an approximation, but rather as a representation of the discrete structure of quantum geometry which was revealed by the results of the canonical formulation.

One strength of this approach is that it has merged with a set of developments in mathematics, which were also going on for several years. Since the early 1990's Louis Crane and collaborators have been working on extending topological quantum field theory (TQFT) from three to four dimensions [15]. The main goal of this work, so far unrealized, has been to find a combinatorial formulation of the Donaldson invariant. In three dimensions, TQFT relies on powerful and apparently deep connections between topology, combinatorics and representation theory, which are most succinctly expressed in terms of category theory. Crane, Frenkel, Yetter, Baez, Dolan and others have been exploring the idea that four dimensional TQFT's must involve still more intricate and subtle relations between the discrete and continuous, which they refer to as ``extended category theory." In all of these theories, a topological invariant is expressed as a sum over labels assigned to various parts of the triangulation of a manifold. Topological invariance is proven by showing that the sum is independent of the choice of triangulation. Such formulations are called ``state sum models".

The relationship of this work to quantum gravity came about because several people had noticed that classically general relativity can be understood as arising from a TQFT by the imposition of a constraint local in the fields [16]. Crane thus proposed a program of constructing quantum gravity as a discrete path integral by imposing an analogous constraint on a state sum model of a TQFT [17] This goal has apparently been realized by a recent proposal of Barrett and Crane [18], which has been studied in detail by Baez [19] and others [20]. Such models of quantum gravity were called by Baez, ``spin foam models" in homage to the spacetime foam of John Wheeler, and the name seems to have stuck.

One thing that is very impressive about these models is that the amplitudes for the histories do seem to reproduce, history by history, the Regge action for classical general relativity [21], while being derived by a particularly elegant restriction of an expression for a topological invariant. This is good because if such a theory is to have a continuum limit, it must be special so as to avoid the problems which prevent generic non-perturbatively-renormalizable theories from making sense. The fact that these theories are closely related to TQFT's, which by definition have continuum limits, suggest that there is reason to hope that this is the case.

Like the earlier Regge calculus and dynamical triangulation approaches, these new spin foam models of discrete quantum gravity are, so far, Euclidean, in that the histories represent discretizations of a four dimensional manifold of Euclidean signature. One expects that if such theories have continuum limits, they are related to second order equilibrium critical phenomena, of the type searched for in the older models.

An alternative approach to discrete spacetime histories which is intrinsically Lorentzian was then proposed by Markopoulou [22] and versions of it have been studied by several people [23,24,25,26]. In these theories spacetime is a discrete causal set, of the kind studied by Sorkin, Meyers and collaborators [27], and 't Hooft [28], with the additional structure that the causally unrelated sets (discrete spacelike slices) are closely related to the spin network states. Thus, the basic idea of Markopoulou is that spacetime is made of a discrete set of events which correspond to local changes in the spin network states.

The question of the existence of a continuum limit for these kinds of
theories has been studied by Abjorn and Loll in the **1+1** dimensional
case [24]. General considerations discussed in
[24]
suggest that
in such theories the continuum limit may be analogous to that found
in certain problems in non-equilibrium critical phenomena such as
directed percolation, where there are analogues of fluctuating causal
structure.
The relationship of these causal histories to the Euclidean histories
described by spin foam models is also being investigated [25].
The optimistic
expectation here is that there will be a non-perturbative analogue of
Euclidean continuation that will connect the two classes of theories.

Finally, it should be mentioned, that while new, these developments already show promising links to other approaches to quantum gravity. It is easy to extend the algebras that give rise to the labels so as to incorporate supersymmetry, perhaps giving rise to a non-perturbative formulation of string theory [26,29]. Perturbations of these discrete histories indeed look something like perturbative strings [30]. There are also possibilities that the special forms of the theory may allow the holographic principle to be formulated intrinsically, exploiting the fact that the TQFT's already naturally give finite dimensional state spaces associated with boundaries of spacetime [17] Finally, the categorical framework of the spin foam models may allow connections to be made [31] to the Gell-Mann Hartle, kind of histories formulations [32] as these have also been formulated categorically by Isham [33].

In general the most important feature of these theories may be that they depend on the deep connections between the continuous and discrete, and between the topological and algebraic, expressed in categorical terms in TQFT. These give a possibility of joining relativity and quantum theory at their roots, in a way that addresses both the technical and conceptual questions that have so far stymied all attempts to make a complete background independent formulation of quantum gravity.

References: [1] A. Ashtekar, Phys. Rev. Lett. 57 (1986) 2244; Phys. Rev. D36 (1987) 1587.

[2] T. Jacobson and L. Smolin, Nucl. Phys. B (1988)

[3] C. Rovelli and L. Smolin, Phys Rev Lett 61 (1988) 1155; Nucl Phys B133, 80 (1990)

[4] L Smolin: in * Quantum Gravity and
Cosmology*, eds J Pérez-Mercader * et al*, World Scientific,
Singapore 1992.

[5] C. Rovelli and L. Smolin Nuclear Physics B 442 (1995) 593. Erratum: Nucl. Phys. B 456 (1995) 734.

[6] R. Penrose,
* Theory of quantized directions* unpublished manuscript.;
in * Quantum theory and
beyond* ed T Bastin, Cambridge U Press 1971;
in * Advances in Twistor Theory*, ed. L. P. Hughston and R. S.
Ward,
(Pitman,1979) p. 301; in * Combinatorial Mathematics and
its Application* (ed. D. J. A. Welsh) (Academic Press,1971).

[7] C. Rovelli and L. Smolin, gr-qc/9505006, Physical Review D 52 (1995) 5743-5759.

[8] A Ashtekar J Lewandowski D Marolf J Mourão T Thiemann, gr-qc/9504018, JMP 36 (1995) 519; A. Ashtekar and J. Lewandowski, gr-qc/9602046; J. Lewandowski, gr-qc/9602035.

[9] T. Thiemann, gr-qc/9606089; gr-qc/9606090.

[10] L. Smolin, gr-qc/9609034.

[11] R. Gambini, J. Lewandowski, D. Marolf, J. Pullin, gr-qc/9710018.

[12] M. Reisenberger, gr-qc/9711052.

[13] M. Reisenberger and C. Rovelli, gr-qc/9612035, Phys.Rev. D56 (1997) 3490-3508.

[14] Carlo Rovelli, gr-qc/9806121

[15] L. Crane and D. Yetter, * On algebraic
structures implicit in topological quantum field theories*,
Kansas preprint, (1994); in * Quantum Topology*
(World Scientific, 1993) p. 120;
L. Crane and I. B. Frenkel, J. Math. Phys.
35 (1994) 5136-54; J. Baez,
q-alg/9705009.

[16] R. Capovilla, J. Dell and T. Jacobson, Phys. Rev. Lett. 21 (1989) 2325; Class. Quant. Grav. 8 (1991) 59; R. Capovilla, J. Dell, T. Jacobson and L. Mason, Class and Quant. Grav. 8 (1991) 41.

[17] L. Crane, * Clocks and Categories, is
quantum gravity algebraic?* to appear in the special issue of
J. Math. Phys. on quantum geometry, * Categorical Physics*,
in * Knot theory and
quantum gravity* ed. J. Baez, (Oxford University Press).

[18] J. Barrett and L. Crane, gr-qc/9709028

[19] J. Baez, gr-qc/9709052.

[20] L. Freidel, K. Krasnov, hep-th/9804185, hep-th/9807092.

[21] L. Crane, D.N. Yetter, gr-qc/9712087; L. Crane, gr-qc/9710108; J. Barrett and R. Williams, in preparation.

[22] F. Markopoulou, gr-qc/9704013.

[23] F. Markopoulou and L. Smolin gr-qc/9702025, Nucl.Phys. B508, 409-430 (1997).

[24] J. Ambjorn, R. Loll,hep-th/9805108; J. Ambjorn, J.L. Nielsen, J. Rolf, R. Loll, hep-th/9806241.

[25] S. Gupta; R. Borissov and S. Gupta, in preparation.

[26] F. Markopoulou and L. Smolin gr-qc/9712067; hep-th/9712148.

[27] L. Bombelli, J. Lee, D. Meyer and R. D. Sorkin, Phys. Rev. Lett. 59 (1987) 521.

[28] G. 't Hooft, * Quantum gravity: a fundamental
problem and some
radical ideas.* Cargèse Summer School Lectures 1978.
Publ. ``Recent
Developments in Gravitation''. Cargèse 1978.
Ed. by M. Lévy
and S. Deser. Plenum, New York/London, 323;
NATO lectures
gr-qc/9608037;
J. Mod. Phys. A11 (1996)
4623-4688
gr-qc/9607022.

[29] L. Crane, gr-qc/9806060.

[30] L. Smolin, * Strings from perturbations
of causally evolving spin networks* preprint, Dec. 1997.

[31] F. Markopoulou, * A universe of
partial observers*, (in preparation).

[32] Murray Gell-Mann, James B. Hartle, gr-qc/9404013; J. B. Hartle, gr-qc/9808070.

[33] C. J. Isham, Int.J.Theor.Phys. 36 (1997) 785-814, gr-qc/9607069.

Mon Sep 7 17:37:02 EDT 1998