%Paper: hep-th/9308126
%From: crane@math.ksu.edu (Louis Crane)
%Date: Thu, 26 Aug 93 13:24:50 CDT

\documentstyle{report}

\begin{document}

{\large TOPOLOGICAL FIELD THEORY AS THE KEY TO QUANTUM GRAVITY}

\smallskip

by: Louis Crane, mathematics department,KSU

\smallskip

presented to the conference on knot theory and quantum gravity
Riverside California

\smallskip



{\bf Abstract}: {\small Motivated by the similarity between CSW theory and the
Chern Simons state for General Relativity in the Ashtekar variables,
we explore what a universe would look like if it were in a state
corresponding to a 3D TQFT. We end up with a construction of
propagating states for parts of the universe and a Hilbert space
corresponding to a certain approximation. The construction avoids
path integrals, using instead recombination diagrams in a very
special tensor category.}

\bigskip

1. INTRODUCTION

\smallskip


What I wish to propose is that the quantum theory of gravity can be
constructed from a topological field theory. Notice, I do not say that
it {\bf is} a topological field theory. Physicists will be quick to point
out that there are local excitations in gravity, so it cannot be
topological.

Nevertheless, the connections between general relativity in the loop
formulation and the CSW TQFT are very suggestive. Formally, at least,
the CSW action term

\bigskip

*

\bigskip

gives a state for the Ashtekar variable version of quantized general
relativity, as we have heard at this conference.

That is, we can think of this ``measure'' on the space of connections
on a 3 manifold either as defining a theory in 3 dimensions, or as a
state in a 4-D theory, which is GR with a cosmological constant.

Furthermore, states for the Ashtekar or loop variable form of GR
are very scarce, unless one is prepared to consider states of zero
volume. Finally, one should note that a number of very talented
workers in the field have searched in vain for a Hilbert space of
states in either representation.

If one attempts to use the loop variables to quantize subsystems of
the universe, one ends up looking at functionals on the set of links
in a 3 manifold with boundary with ends on the boundary.  It turns out in
regularizing
that the links need to be framed, and one can trace them in any
representation of SU(2), so they need to be labeled. The coincidence
of all this with the picture of 3D TQFT was the initial motivation for
this work.

Another point, which seems important, is that although path integrals
in general do not exist, the CSW path integral can be thought of as
a symbolic shorthand for a 3-D TQFT which can be rigorously defined
by combining combinatorial topology with some very new algebraic
structures, called modular tensor categories. Thus, if we want to
explore * as a state for GR, we have the possibility of switching
from the language of path integrals to a finite, discrete, picture,
where physics is described by variables on the edges of a triangulation.

Physics on a lattice is nothing new. What is new here is that the
choice of lattice is unimportant, because the coefficients attached to
simplices are algebraically special. Thus we do not have to worry
about a continuum limit, since a finite result is exact.

Motivated by these considerations, I propose to consider the
hypothesis:

\smallskip

{\large CONJECTURE: The universe is in the CSW state.}

\smallskip

This idea is similar to the suggestion of Witten that the CSW theory
is the topological phase of quantum gravity [2]. My suggestion differs
from his in assuming that the universe is still in the topological
phase, and that the concrete geometry we see is not the result of a
fluctuation of the state of the universe as a whole, but rather due to
the collapse of the wave packet in the presence of classical
observers. As I shall explain below, the machinery of CSW theory
provides spaces of states for parts of the universe, which I interpret
as the relative states an observer might see. This suggestion is
closely related to the many worlds interpretation of quantum
mechanics:
the states on surfaces are data in one particular world. The state of
the universe is the sum of all possible worlds, which takes the
elegant
form of the CSW invariant of links, or more concretely, of the Jones
polynomial
or one of its generalisations.

Durring the talk I gave at the conference one questioner pointed out,
quite correctly, that it is not necessary to make this conjecture,
since other states exist. Let me emphasize that it is only a
hypothesis. I think it is fair to say that it leads to more beautiful
mathematics than any other I know of. Whether that is a good guide
for physics only time will tell.

Topological quantum field theory in dimensions 2,3, and 4 is leading
into an extraordinarily rich mathematical field. In the spirit of the
drunkard who looks for his keys near a street light, I have been
thinking for several years about an attempt to unite the structures of
TQFT with a suitable reinterpretation of quantum mechanics for the
entire universe. I believe that the two subjects resemble one another
strongly enough that the program I am outlining becomes natural.


\smallskip

2. QUANTUM MECHANICS OF THE UNIVERSE. CATEGORICAL PHYSICS.

\smallskip

I want to propose that a quantum field theory is the wrong structure
to describe the entire universe. Quantum field theory, like quantum
mechanics generally, presupposes an external classical observer.
There can be no probability interpretation for the whole universe.
In fact, the state of the universe cannot change, because there is no
time, except in the presence of an external clock.
Since the universe as a whole is in a fixed state, the quantum theory
of the gravitational field as a whole is not a physical theory. It
describes all possible universes, while the task of a physical theory
is to explain measurements made in {\bf this} universe in {\bf this} state.
Thus,
what
is needed is a theory which describes a universe in a particular
state. This is similar to what some researchers, such as Penrose, have
suggested: that the initial conditions of the universe are determined
by the laws. What I am proposing is that the initial conditions become
part of the laws. Naturally, this requires that the universe be in a
very special state.

Philosophically, this is similar to the idea of a wave function of the
universe. One can phrase this proposal as the suggestion that the wave
function of the universe is the CSW functional. There is also a good
deal here philosophically in common with the recent paper of Rovelli
and Smolin [23]. Although they do not assume the universe is in the
CSW state, they couple the gravitational field to a particular matter
field,
which acts as a clock, so that the gravitational field is treated as
only part of a universe.

What replaces a quantum field theory is a family of quantum theories
corresponding to
parts of the universe. When we divide the universe into two parts,
we obtain a Hilbert space which is associated to the boundary between
them. This is another departure from quantum field theory; it amounts
to abandonment of observation at a distance.

The quantum theories of different parts of the universe are not, of
course, independent. In a situation where one observer watches another
there must be maps from the space of states which one observer sees to
the other. Furthermore, these maps must be consistent. If A watches B
watch C watch the rest of the universe, A must see B see C see
what A sees C see.

Since, in the Ashtekar/loop variables the states of quantum gravity
are invariants of embedded graphs, we allow labeled punctures on the
boundaries between parts of the universe, and include embedded graphs
in the parts of the universe we study, ie. in 3 manifolds with
boundary.

The structure we arrive at has a natural categorical flavor. Objects
are places where observations take place, ie boundaries; and morphisms
are 3D cobordisms, which we think of as A observing B.

Let us formalize this as follows:

Definition: {\bf An \underbar{Observer} is an oriented  $3$
manifold with boundary containing an embedded labeled framed graph which
intersects the boundary in isolated labeled points.}

(The labeling sets for the edges and vertices of the graph are finite,
and need to be chosen for all of what ensues.)

Definition: {\bf A \underbar{skin of observation} is a closed oriented
surface with labeled punctures.}

Definition: {\bf If A and B are skins of observation an
\underbar{inspection, $\alpha$, of B by A} is an observer whose boundary
is identified with  $\overline{A} \cup B$  (i.e.\ reverse the
orientation on A) such that the labelings of the components of the graph
which reach the boundary of  $\alpha$  match the labelings in
$\overline{A} \cup B$.}

This definition requires that the set of labelings possess an involution
corresponding to reversal of orientation on a surface.

To every inspection  $\alpha$  of B by A there corresponds a dual
inspection of  $\overline{A}$  by  $\overline{B}$, given by reversing
orientation and dualizing on  $\alpha$.

Definition: {\bf The \underbar{category of observation} is the category
whose objects are skins of observation and whose morphisms are
inspections.}

Definition: {\bf If  $M^3$  is a closed oriented  $3$-manifold the
\underbar{category of observation in  $M^3$} is the relative (i.e.\
embedded) version of the above.}

Of course, we can speak of observers, etc. in  $M^3$.

Definition: {\bf A state for quantum gravity (in  $M^3$)  is a functor
from the category of observation (in  $M^3$)  to the category of vector
spaces.}

Nowhere in any of this do we assume that these boundaries are
connected. In fact, the most important situation to study may be one
in which the universe is crammed full of a ``gas'' of classical observers.
We shall discuss this situation below as a key to a physical
interpretation of our theory.


If we examine the mathematical structure necessary to produce a
``state'' for the universe in this sense, we find that it is identical
to a 3D TQFT. Thus, the CSW state produces a state in this new sense
as well.

Another way to look at this proposal is as follows: The CSW state for the
Ashtekar variables is a very
special state, in that it factorizes when we cut the 3 manifold along
a surface with punctures, so that a finite dimensional Hilbert space is
attached to
each such surface, and the invariant of any link cut by the surface can be
expressed as
the inner product of two vectors corresponding to the two half links. (John
Baez has pointed out that these finite
dimensional spaces really do possess natural inner products). Thus, it
produces a ``state'' also in the sense we have defined above.

I interpret this as saying that the state of the universe is
unchanging, but that because the universe is in a very special state,
it can contain a classical world, ie. a family of classical observers
with consistent mutual observations. The states on pieces of the
universe (ie. links with ends in manifolds with boundary) can be
interpreted as changing, once we learn to interpret vectors in the
Hilbert spaces as clocks.

So far, we have a net of finite dimensional Hilbert spaces, rather
than one big one, and no
idea how to reintroduce time in the presence of observers.
There are natural things to try for both of these problems in the
mathematical context of TQFT.

Before going into a program for solving these problems (and thereby
opening the possibility of computing the results of real experiments)
let us make a survey of the mathematical toolkit we inherit from TQFT.

\smallskip

3. IDEAS FROM TQFT

\smallskip

As we have indicated, the notion of a ``state of quantum gravity'' we defined
above is
equivalent to the notion of a 3D TQFT which is currently prevalent in
the literature.

The simplest definition of a 3D TQFT is that it is a machine which
assigns a vector space to a surface, and a linear map to a cobordism
between 2 surfaces. The empty surface receives a 1 dimensional vector
space, so a closed 3 manifold gets a numerical invariant. Composition
of cobordisms corresponds to composition of the linear maps.
A more abstract way to phrase this is that a TQFT is a functor from
the cobordism category to the category of vector spaces.

The TQFTs which have appeared lately are richer than this. They assign
vector spaces to surfaces with labeled punctures, and linear maps to
cobordisms containing links or knotted graphs with labelled edges.
The labels correspond to representations. Since it is easy
to extend the loop variables to allow traces in arbitrary
representations (characters), there is a great deal of coincidence in
the pictures of CSW TQFT and the loop variables. (That was a lot of
the initial motivation for this program).
We can describe this as a functor from a richer cobordism category to
VECT. The objects in the richer cobordism category are surfaces with
labelled punctures, and the morphisms are cobordisms containing
labelled links with ends on the punctures.

There are several ways to construct TQFTs in various dimensions.
Let us here discuss the construction of a TQFT from a triangulation.
We assign labels to edges in the triangulation, and some sort of
factors combining the edge labels to different dimensional simplices
in the triangulation, multiply the factors together, and sum over
labelings.

In order to obtain a topologically invariant theory, we need the
combination factors to satisfy some equations. The equations they
need to satisfy are very algebraic in nature; as we go through
different classes of theories in different low dimensions we first
rediscover most of the interesting classes of associative algebras,
then of tensor categories [19].

A simple example in two dimensions may explain why the equations for a
topological theory have a fundamental algebraic flavor. For a 2D TQFT
defined from a triangulation, we need invariance under the move:

\bigskip

\bigskip

\bigskip

                                                             .
Now, if we think of the coefficients which we use to combine the
labels around a triangle as the structure coefficients of an algebra,
this is exactly the associative law. Much of the rest of classical abstract
algebra makes an appearance here too.

As has been described elsewhere [3], a 3D TQFT can be constructed from
a modular tensor category. The interesting examples of MTCs can be
realized as quotients of the categories of representations of quantum
groups.

If we try to use the modular tensor categories to construct a TQFT
on a triangulation, we obtain, not the CSW theory, but the weaker
TQFT of Viro and Turaev [20]. Here we label edges, not from a basis
for
an algebra, but with irreducible objects from a tensor category.
The combinations we attach to tetrahedra (3 simplices) are the quantum
6J symbols, which come from the associativity isomorphisms of the
category.

We refer to this sort of formula as a state summation.

The full CSW theory can also be reproduced from a modular tensor
category by a slightly subtler construction which uses a Heegaard
splitting (which can be easily produced from a triangulation) [3].

The quantum 6J symbols satisfy some identities, which imply that the
summation formula for a 3 manifold is independent under a change of
the triangulation. The most interesting of these is the
Eliot-Biedenhorn identity. This identity is the consistency relation
for the associativity isomorphism of the MTC. this is another example
of the marriage between algebra and topology which underlies TQFT.

The suggestion that this sort of summation could be related to the
quantum theory of gravity is older than one might think. If we use the
representations of an ordinary lie group instead of a quantum group,
then the Viro-Turaev formula becomes the Regge-Ponzano formula for the
evaluation of a spin network.


Regge and Ponzano [9], were able to interpret the evaluation as a sort
of discrete path integral for  $3-d$  euclidean quantum gravity.  The
formula which Regge and Ponzano found for the evaluation of a graph
is the analog for a lie group of the state sum of Viro and Turaev for
a quantum group.
Thus,
the evaluation of a tetrahedron for a spin network is a  6J  symbol
for the group  SU(2).


\bigskip

\bigskip

\bigskip

(1)

\bigskip

\bigskip


The topological invariance of this formula
follows from some elementary properties of representation theory.  In
(1), we have placed our trivalent labeled graph on the boundary of a
3-manifold with boundary, then cut the interior up into tetrahedra,
labeling the edges of all the internal lines with arbitrary spins.  (The
Clebsch-Gordon Relations imply that only finitely many terms in (1) are
non-zero.)

Regge and Ponzano then proceeded to interpret (1) as a discretized
path integral for 3D quantum GR. The geometric interpretation
consisted inthinking of the Casimirs of the representations as lengths
of edges. Representation theory implied that the summation was
dominated by flat geometries [9].

Another way to think of the program in this paper is an attempt to find
the
appropriate algebraic structure for extending the spin network
approach to quantum gravity from D=3 to D=4.

Another idea from TQFT, which seems to have relevance for this
physical program is the ladder of dimensions [19]. TQFTs in adjacent
dimensions
seem to be related algebraically. The spirit of the relationship is
like the relationship of a tensor category to an algebra. Tensor
categories look just like algebras with the operation symbols in circles.
The identities of an algebra correspond to isomorphisms in a tensor
category.  These isomorphisms then satisfy higher order consistency
or''coherence'' relations, which relate to topology up one dimension.

The program of construction in TQFT, which I hope will yield the tools
for the quantization of general relativity, is not yet completed, but
is progressing rapidly. There are two outstanding
problems, which are mathematically natural, and which I believe are
crucial to the physical problem as well.They are:

Topological problem 1: find a triangulation version of CSW theory (not
merely its  absolute square as in [20])

Topological problem 2: Construct a 4D TQFT related via the dimensional
ladder to CSW theory.

The solution of these two problems will be in reach, if the
program in [19] succeeds. The program of [19] also suggests that the
solutions of these two topological problems are closely related, both
being constructed from state sums involving the same new algebraic
tools.


In this context we should also note that a 4D TQFT has been
constructed in [21], out of a modular tensor category. This
construction begins with a triangulation of a 4 manifold, and picks a
Heegaard splitting for the boundary of each 4-simplex of the
triangulation.
Thus the theory in [21] can be thought of as producing a 4D theory by
filling the 4 space with a network of 3D subspaces containing observers.
This connection between TQFTs in 3D and 4D is the sort of
thing I believe we will need to solve the problem of reintroducing
time
in a universe in a TQFT state. What I expect is that the program of
[19] will provide richer examples of such a connection, which will
prove to be relevant to quantizing General Relativity.

What we conjecture in [19], is that a new algebraic structure will
give us a new state summation, similar to the one we discussed above,
which will give us CSW in three dimensions. If we use the new
summation in four dimensions, we should obtain a discrete version of
$F\wedge F$ theory. This combination seems very suggestive, since we are
supposed
to be obtaining our relative states from CSW theory on a boundary,
while $ F\wedge F$ theory is a lagrangian for the topological sector for the
quantum theory of GR [22].
Also. the algebraic structure we need to construct seems to be an
expanded quantum version of the lorentz group. It is this new, not
yet understood summation, which comes from the representation theory
of  the new structure which we call an F algebra in [19], which I
believe should give us a discretized version of a path integral for
quantum gravity.

\smallskip

4.HILBERT SPACE IS DEAD. LONG LIVE HILBERT SPACE. or THE OBSERVER GAS
APPROXIMATION.

\smallskip

Let us assume that we have found a suitable state summation formula to
solve our 2 topological problems (so that what we suggest will be
predicated on the success of the program in [19], although one could
also try to use the state sum in [21] together with Viro-Turaev
theory). There is a natural proposal to make
a physical interpretation within it in a 4 dimensional setting.
In the scheme I am suggesting, we can recover a Hilbert space in
a certain sort of classical limit, in which the universe is full of
a ``gas,'' of classical observers. The idea is that if we choose a
triangulation for the 4 manifold and a Heegaard splitting for the
boundary of each 4 simplex, then we obtain a family of 2-surfaces
which can be thought of as filling up the 4 manifold.These surfaces
should be thought of as skins of observation for a family of observers
who are crowding the spacetime.
Now let us think of a 4 manifold with boundary. In a three dimensional
boundary component,
we can then
combine
the vector spaces which we assign to the surfaces
which cut the boundaries of those 4 simplices lying on
the boundary into a larger Hilbert space, which
is a quotient of their tensor product. (It is a quotient because the
surfaces overlap.) Now, if we pass to a finer triangulation, we will
obtain a larger space, with a linear map to the smaller one. The
linear map comes from the fact that we are using a 3D TQFT, and the
existence of 3D cobordisms connecting pieces of the surfaces of the
two sets of Heegaard splittings.

We end up with a large vector space for each triangulation of the
boundary 3 manifold, and a
linear map when one triangulation is a refinement of another.
This produces a directed graph of vector spaces and linear maps
associated to a 3 manifold. The 4D state sum can be extended to act on
the vectors in these vector spaces by extending a triangulation for
the boundary 3 manifold to
one for the 4 manifold.

Whenever we have a directed set of vector spaces and consistent linear
maps, there is a construction called the inverse limit which can
combine them into a single vector space. The vectors are vectors in any
of the spaces, with images under the maps identified.
Since the 4D state sum is assumed to be topologically invariant, we
can extend the linear map it assigns to a 4D cobordism to a map on the
inverse limit spaces. I propose that it is these inverse limit spaces
which play the role of the physical Hilbert spaces of the theory.

The thought here is that physical Hilbert space is the union of the
finite dimensional spaces which a gas of observers can see, in the
limit of an infinitely dense gas.

The effect of this suggestion is to reverse the relationship between
the finite dimensional spaces of states which each observer can
actually see and the global hilbert space. We are regarding the states
which can be observed at one skin of observation as primary. This
``relational''  approach bears some resemblance to the physical ideas
of Leibnitz, although that is not an argument for it.

5. THE PROBLEM OF TIME

\smallskip


The test of this proposal is whether it can reproduce Einstein's
equations in some classical limit. What I am proposing is that the 4D
state sum which I hope to construct from the representation theory of
the F algebra can be interpreted as a discretized version of the path
integral for general relativity. The key question is whether the state sum is
dominated in the limit of large spins on edges by assignments of spins
whose geometric interpretation would give a discrete approximation to
an Einstein manifold. This would mean that the theory was a quantum
theory whose classical limit was general relativity.

The analogy with the work of Regge and Ponzano is the first suggestion
that this program might work. In [9], they interpreted the summation
we wrote down above as a discretized path integral for 3D quantum
gravity. The geometric interpretation consisted in thinking of the
Casimirs of representations as lengths. It was somewhat easier to get
Einstein's equations in D=3, since the solutions are just flat
metrics.

Should this program work? One objection, which was raised in the
discussion, is that the $F\wedge F$ model may only be a sector of
quantum gravity in some weak sense. (I am restating this objection
somewhat in the absence of the questioner). It can be replied that
since the theory we are writing has the right symmetries, the action
of the renormalization group should fix the lagrangian. I do not think
that either the objection or the reply is decisive in the abstract.
If the state sum suggested in [19] can be defined, then we can
investigate whether we recover Einstein's equation or not.

If this does work, one could do the state sum on a 4 manifold with
corners, ie, fix states on surfaces in the three dimensional boundaries
of a 4D cobordism. One could interpret these states as initial
conditions for an experiment, and investigate the results by studying
the propagation via the 4D state sum.

One could conjecture that this would give a spacetime picture for the
evolution of relative states within the framework of the CSW state of
the universe.

\smallskip

6. MATTER AND SYMMETRY

\smallskip

It is natural to ask whether this
picture could be expanded to include matter fields. The obvious thing
to try is to pass from SU(2) to a larger group. It may be noted that
Peldan [12] has observed that the CSW state also satisfies the Yang
Mills equation. If we really get a picture of gravity from the state
sum associated to SU(2), it would not be a great leap to try a larger
gauge group.

As I was writing this I became aware of [24], in which Gambini and
Pullin point out that the CSW state is also a state for GR coupled to
electromagnetism. They also say that the sources for such a theory,
thought of as places where links have ends, are necessarily fermionic.
These results seem to further strengthen the program set forth here.

In general, the constructions which lead up to the F  algebras in [19]
can be thought of as expressions of ``quantum symmetry'' ie as
relatives of lie groups in a quantum context. For example, the modular
tensor categories can be described as deformed Clebsch Gordon
coefficients. The F algebras arise as a result of pushing this process
further, recategorifying the categories into 2-categories. These
constructions can also be done from extremely simple starting points,
looking at representations of Dynkin diagrams as ``quivers.''
The idea that theories in physics should be reconstructed from
symmetries actually originated with Einstein, who was thinking of
general relativity. It would be fitting somehow, to reconstruct truly
fundamental theories of physics from fundamental mathematical
expressions of symmetries.

\smallskip

{\bf Acknowledgements:} the author wishes to thank Lee Smolin and Carlo
Rovelli for many years of conversation on this extremely elusive
topic. The mathematical ideas here are largely the result of
collaborations with David Yetter and Igor Frenkel. Louis Kauffman has
provided many crucial insights. The author is supported by NSF grant
DMS-9106476 .

\bigskip

References.

\smallskip


1.C. Rovelli and L. Smolin, Loop Space Representations of Quantum
General Relativity, preprint IV 20 Phys. Dept. V. di Roma ``la
Sapienza'' 1988

2. E. Witten, Quantum Field Theory and The Jones Polynomial, IAS
preprint
HEP 88/33

3. L. Crane, 2-d Physics and 3-d Topology, Commun. Math. Phys.
135 615-640 (1991)

4. G. Moore and N. Seiberg, Classical and Quantum Conformal Field
Theory, Commun Math. Phys. 123 177-254 (1989)

5. B. Brugman, R Gambini and J. Pullin, Gen. Rel. Grav. in press

6. L. Crane, Quantum Symmetry, Link Invariants and Quantum Geometry,
in Proceedings XX International Conference on differential Geometric
Methods in Theoretical Physics Baruch College (1991)

7. L. Crane, Conformal Field Theory, Spin Geometry, and Quantum Gravity,
Phys.  Lett. B v259 #3 (1991)

8. R. Penrose, Angular Momentum; an Approach to Combinatorial Space
Time, in Quantum Theory and Beyond ed T.Bastin(Cambridge)

9. G. Ponzano and T. Regge, Semiclassical Limits of Racah Coefficients,
in Spectroscopic and Group Theoretical Methods in Physics, ed. F.
Bloch (North-Holland, Amsterdam)

10. C. Rovelli, Class. Quan. Grav. 8 297 (1991)

11. J. Moussouris, Quantum Models of Space Time Based on Recoupling
Theory, Thesis, Oxford University (1983)

12. P. Peldan, Phys. Rev. D46 R2279 (1992)

13. L. Smolin, Personal communication

14. V. Moncrief, personal communication

15. K. Walker, On Wittens 3 Manifold Invariants, unpublished

16. L. Crane and Igor Frenkel, Hopf Categories and Their
Representations, to appear.

17. M. Kapranov and V. Voevodsky, Braided Monoidal 2-Categories,
2-Vector Spaces and Zamolodchikov's Tetrahedra Equation,
preprint

18.C. Rovelli and L. Smolin, Phys. Rev. Lett. 61 1155 (1988)

19. L. Crane and I. Frenkel, A Representation theoretic Approach to
4-Dimensional Topological Quantum Field Theory, to appear in
proceedings of AMS conference, Mt. Holyoke MA., June 1992

20 V. Turaev and O. Viro, State Sum Invariants of 3-Manifolds and
Quantum 6J Symbols Topology 31 (1992) 865-902

21 L. Crane and D. Yetter, A Categorical Construction of 4D TQFT's To
appear in Proceedings of AMS conference, Dayton, Ohio, October, 1992


22 L. N. Chang and C. Soo, BRST Cohomology and Invariants of 4D
Gravity in Ashtekar Variables VPI preprint


23 C. Rovelli and L. Smolin The Physical Hamiltonian in Non
Perturbative Quantum Gravity. preprint.


24 R. Gambini and J. Pullin Quantum Einstein-Maxwell Fields:
A Unified Viewpoint From The Loop Representation. preprint.


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\title  Categorical Physics  \endtitle

\author Louis Crane \\
        Department of Mathematics \\
        Kansas State University \\
        Manhattan, KS   66506 \endauthor

\subheading{I. Introduction}

The purpose of this letter is to outline a rather novel suggestion as to
what form the quantum theory of gravity should take.  The suggestion has
a certain mathematical elegance, but that was not the motivation which
brought it into being.  Rather, it was motivated by the juxtaposition of
two ideas:

(A) the work of Rovelli and Smolin on the loop representation
(1) which showed that a state for the quantum theory of gravity could be
described by an invariant of links in a $3$-manifold, given by
integrating a measure on the space of connections on the manifold
against the traced holonomy around the link.

(B) the work of E.\ Witten (2) which recovered a generalized Jones
polynomial as precisely such an integral; using the Chern-Simons
invariant to create a measure.

The ``measures'', in A and B above are, of course, formal.  It was in
order to make mathematical sense of Witten's Chern-Simons invariant that
I showed in (3) that any modular tensor category gave rise to a  $3-D$
Topological Quantum Field Theory, i.e.\ allows us to reproduce all the
results of Witten's path integral by rigorous methods.

The juxtaposition of ideas (A) and (B) tells us that the Chern Simons
measure gives us a state for quantum gravity on a closed  $3$-manifold,
which can be reproduced entirely from an algebraic structure called a
modular tensor category, first described in (4).  The construction gives
us more than a state (i.e.\ link invariant) on a closed  $3$-manifold;
it also gives us vector valued invariants of links in  $3$-manifolds
with boundary, and a nice factorization when we join along boundary
components.  This is what we mean when we say that we can construct a
topological quantum field theory.

What I want to propose is that the universe as a whole is in a
Chern-Simons state (presumably for a larger group than  $SU(2)$, in
order to include matter - more on this latter).

The connection of the Chern Simons state to a TQFT is what allows us to
experience a world of changing phenomena in a universe which is not
changing as a whole.  Manifolds with boundary correspond to observers,
and observations take place in the Hilbert spaces which our TQFT assigns
to boundaries.  Thus, in order to quantize gravity (or gravity plus
matter) we have to abandon the idea of measurement at a distance, and
create a relative form of the probability interpretation of quantum
mechanics; with a Hilbert space for each observer, and consistency
relations which are best expressed in categorical language.  We can say
that a state for quantum gravity is given by a functor from the category
of observers to the category of vector spaces; this coincides with the
notion of a TQFT.  Time evolution must be given by a natural
transformation between functors.

This is obviously an ambitious program.  As yet, I am not able to
compute the result of any experiment in this framework, but I can see a
path which could lead to that goal.  In the rest of this letter, I
summarize what I can understand of various aspects of this program, and
suggest various directions for further research.

\vskip .2in

\subheading{II. The Category of Observers}

In this section, I make a series of fundamental definitions.  On the one
hand, they are motivated by the mathematical juxtaposition mentioned
above; on the other, they have, I believe, a certain physical
plausibility.

Let us reiterate that all the above definitions should be extended by
specifying the labeling set with involution.

Observation: {\bf Any modular tensor category gives rise to a state for
quantum gravity.}

The mathematical motivation for the above definitions should be clear;
it is just what we know how to produce.

In our language, a State for quantum gravity is the same thing as a
$3-D$  TQFT.  We should note some facts about the development of the
Ashtekar variables for general relativity, which make this physically
plausible.  Although a state is given by any invariant of {\bf links},
since the Hamiltonian constraint only gives a condition at the vertices
of a {\bf graph}, the states which are supported only on links seem to
have unsatisfactory properties.  In searching for solutions of the
Hamiltonian constraint which were nontrivial on {\bf graphs} Gambini et
al (5) found themselves reproducing the perturbative terms of the CSW
invariant.  Thus, it could well be that CSW gives the only physically
interesting states.  Should the connection with TQFT then be taken as
pure coincidence?

Let us now consider the physical ideas which underlie these definitions.
The first departure is attaching vector (``Hilbert'') spaces to skins of
observation, rather than observers.  This means that if we divide the
universe into ``system'' and ``observer'' the observer no longer
measures the state of the system, but only of that part of it which
impinges instantaneously on the observer.  This is commonly neglected in
ordinary quantum mechanics, but it seems reasonable that in quantizing
general relativity we would need to consider it.  I refer to this as
abandoning ``measurements at a distance''.

We are left with a picture in which states for a whole universe are
fixed things constructed out of special algebraic structures, while the
relational states which we measure live in spaces constructed by a
different recipe from the same structures.

Here we see a second physical departure.  The {\bf state of the universe
as a whole} becomes part of the framework of laws by which the parts
interact.

It has been noted for some time that when considering the entire
universe, the relationship between laws and initial conditions could be
different from what physicists commonly expect.  Hawking has suggested
determining initial conditions from laws.  I am proposing, rather that
the ``initial'' state of the universe (which does not change in time)
acts as a law.

It is not hard to create a quantum mechanical system in which the
initial state of a combined system acts as a law of motion for the two
parts [10], so perhaps this notion is not so implausible.

Before discussing how this theory might meet some of the titanic
challenges which await it, let me allow myself a few philosophic
observations.

Category theory was invented largely in order to rewrite mathematics in
the most coordinate invariant way possible.  The notion that it was
desirable to write things invariantly is something which 20th century
mathematics inherited from relativity.  (The first abstract definition
of a vector space is in Weyl's exposition of relativity {\bf Zeit}, {\bf
Raum}, {\bf Materie}.)

Thus, if the fully adequate formulation of relativity uses categories,
we would be coming full circle.

Beside the functorial nature of the {\bf result}, i.e.\ of a state of
quantum gravity; there is also a categorical structure, namely a modular
tensor category, at the heart of the construction [3] of a  $3-D$  TQFT,
which we are reinterpreting here.

I want to argue that the entire CSW invariant is more interesting than
its perturbative terms.  This is because modular tensor categories are
only slight modifications of Tannakian categories.  [6] Tannakian
categories are simply the categories of representations of groups.
Thus, tensor product categories are expressions of symmetries.  Modular
tensor categories are ``purely quantum'' versions of the same thing.
Thus, we can think of them as ``quantum symmetries''.  The fact that we
can construct our invariant states from MTC's can be expressed (somewhat
lavishly) by saying that in the quantum domain invariant states are
determined by symmetry.

Enough philosophy.  How can this theoretical model be compared with
experiment?  I believe that there are three main problems which must be
confronted: (A) making some physical interpretation of the states which
we attach to skins of observation (B) the problem of introducing time
and (C) how to include matter.  In the next 3 paragraphs, I shall say
what I know about these problems.

\vskip .2in

\subheading{III. States as Quantum Geometries.  Deformed Spin Networks}

The idea I discuss here was already proposed in [7].  I shall outline it
here for completeness, and to set it in a theoretical context.

The invariants which CSW theory assigns to embedded knotted labeled
graphs are deformed generalizations of the number called the
``evaluation'', which Penrose [8] assigned to a labeled graph called a
spin network.

The derivation of formula (1) uses only basic properties of the category
of representations of  $SU(2)$.  Thus, we can repeat the derivation of
(1) for CSW invariants of trivalent graphs, provided that we are careful
not to embed the graph in such a way that we have to unbraid it in order
to put it on a boundary.  This is not an issue for spin networks, since
over and under crossings are equivalent (i.e.\ the tensor category is
symmetric).

There is a natural generalization of spin networks to larger groups,
called {\bf fabrics} by Moussouris [11].  A formula like (1) can be
similarly deduced, except that the ``labelings'' include assigning
intertwining operators to internal faces.  (These are unique for
$SU(2)$.)

Thus, a formula like (1) but with more general labelings, including
labels on faces, can be derived for CSW invariants of graphs.

Formula (1) has the same form as the invariant of  $3$-manifolds due to
Viro and Turaev, but the geometric setting is different, and we are
obtaining the CSW invariant, instead of its absolute square.

Regge and Ponzano reinterpreted (1) as a discrete path integral for
$3-d$  quantum gravity.

The essential step for them was to reinterpret the spin label on each
edge as a length.  Assigning lengths to the edges of a triangulation is
an approximation to selecting a metric; summing over all labelings then
approximates a path integral over metrics.

Thus a labeling of the edges of a graph gives a sort of discretized
probability density of metrics or ``quantum geometry'', to the space it
surrounds.  For ordinary spin networks, these sums peak around flat
metrics, so the classical ``equations of motion'' match those for  $3-d$
euclidean general relativity.

Applying the same approach to states on the vector space attached to a
boundary surface yields a ``quantum geometry'' on {\bf part} of the
interior of a  $3$-manifold with boundary.  A small improvement of the
form of (1) would suffice to give a quantum geometry on the entire
interior.  Studying the recursion relations for quantum  $6J$  symbols
leads me to believe the improvement exists, but I have not constructed
it, and do no pursue it here.  At any rate, being able to interpret a
state on the boundary as probabilistic information about the geometry on
the interior brings us much closer to relating our formalism to
experiment.  If we could relate states on a boundary for CSW theory with
a larger group to a more complex geometry in the interior, mixing
intrinsic with extrinsic geometry (i.e.\ coupling gravity to Yang-Mills)
it would be even a larger step.

This brings us to the question of introducing matter into our picture.

\vskip .2in

\subheading{IV. Matter and Larger Groups}

The  $2 + 1 - D$  TQFT constructed by means of a modular tensor category
(or quantum group, etc.), can equally well be defined for some larger
semisimple compact lie group as for  $SU(2)$.  [Noncompact forms are
much thornier.]  It is tempting (to anyone who finds this paper
tempting), to try to interpret them as ``states of gravity plus
matter.''  This raises a key issue.

Issue: {\bf Are TQFT's associated to Larger Groups States for some
Quantum Theory?}

If the answer to this question is yes, then it is not necessary to worry
too much about finding the Yang-Mills Lagrangian in the theory, since
the action of the renormalization group will bring us to it if our
theory has the right symmetries.

As it turns out, it is possible to approach this problem, although not
in a mathematically rigorous way.  Peldan [12] has written down a
Hamiltonian constraint for an arbitrary lie group, which reduces to the
Ashtekar form for  $SU(2)$, and produces no second class constraints.
We can then translate our question into the simpler question of whether
the Chern-Simons State  $\ell^{2\pi ik cs(A)}$  solves Peldan's
generalized
Ashtekar Hamiltonian constraint (with cosmological constant).  This
reduces to some algebraic relations on the group, currently under study.
This would not, of course, provide a rigorous argument that the TQFT
associated to some special group is a quantum state for
gravity-plus-matter, but it would be very suggestive.

\vskip .2in

\subheading{V. The Problem of Time}

The description of the world which the picture we are discussing
provides would not, (even given the most optimistic outcome of the
unsettled questions) at this point, resemble the world we see.  This is
because the relative states which live on surfaces have no time
evolution.  Past, present and future are all one in them.

This is not a new problem in attempts to quantize gravity.  It is a
consequence of the vanishing of the Hamiltonian on physical states,
which is ultimately an expression of  $4-D$  diffeomorphism invariance.

There is a standard resolution of this question among relativists.  This
is to treat one part of the variables as a clock.  The relative states
of the rest of the variables then turn out to solve a Schr\"odinger
equation [13].

Thus, if we are to interpret the states in our model as giving an
evolution in  $4-D$  space time, we have to solve the problem of
representing clocks within the states of a TQFT.

There is another fundamental question which must be faced here: Can one
have a clock in a purely gravitational universe, or are clocks
necessarily material?  As Einstein once phrased a similar question: If
matter disappeared from the universe, would time as well?

Moncrief [14] has taken rather heroic measures, to construct a clock from
a gravitational universe.  However, it is hard to imagine doing general
relativity with only one clock, and measuring the volume of the universe
has little to do with what an observer timing some event normally does.
I therefore believe that the most likely route to an interpretation of
the picture we are developing must go through interpreting a larger
group as gravity plus matter, and constructing clocks from matter.

Although it is hard to construct a state involving time explicitly, it
is easy to see what mathematical form such a thing should take.  We
formulate such a picture, then make some speculations as to how it might
be constructed.

Definition: {\bf A \underbar{process} is an oriented $4$-manifold with
boundary with an embedded labeled branched surface which meets the
boundary transversely in a labeled knotted graph.}

Definition: {\bf An observer with no boundary is a \underbar{moment}.}

Definition: {\bf If  $A$  and  $B$  are moments and  $P$  is a process
with boundary  $\overline{A} \cup B$, then we say  $P$  is an
\underbar{intervening process} between  $A$  and  $B$.}

If  $P$  is an intervening process between  $A$  and  $B$, then
$\overline{P}$  is an intervening process between  $\overline{B}$  and
$\overline{A}$  (``time reversal'').

Now let us define the structure which seems natural to express time
evolution.  Suppose given a vector field  $\phi$  which is the gradient
of a morse function on  ${\Cal P}$  which intersects  $\overline{A}$
and  ${\Cal B}$  transversely, pointing ``inward'' on  $\overline{A}$
and ``outward'' on  ${\Cal B}$.  Dragging along  $\phi$  then gives a
functor  $F_\phi$  from the relative category of observation  ${\Cal
O}_A$  to  ${\Cal O}_B$.

Definition: {\bf We say that a state of gravity  $S$  is
\underbar{augmented} if for every two moments  $A$, $B$, process  $P$
intervening between them, and vector field  $\phi$  as above, we are
given a natural transformation  ${\Cal N}_{P,\phi} : S_A \to S_B \circ
F_\phi$.  We require the assignment to be natural, in the sense that the
composition of processes and vector fields (obvious definition) is
assigned the composition of natural transformations.}

Let us translate into more prosaic language.  A natural transformation
is given by maps between the images of objects, which make certain
squares commute.  These maps are between the Hilbert spaces of skins of
observation and their ``time evolutions'', i.e. draggings by  $\phi$,
and should be thought of as giving time evolution for relative states.
The commuting squares give a consistency relation, in which the state
$M$  observes in  $N$  changes to the same state in  $F_\phi (M)$  as
the state the later observer  $F_\phi (M)$  would observe from the
changed state in  $F_\phi (N)$, i.e. the following diagram commutes:
\pagebreak

\midspace{1.5in}

        (figure 1)

Conjecture: {\bf The natural world is described by an augmented state of
quantum gravity.}

We should note that even if we could construct an augmented state, the
physical interpretation of it would still involve all the subtleties of
clocks, lapses, shifts, etc. with which general relativity is so richly
endowed.

\vskip .2in

\subheading{VI. Some Observations on Augmented States}

The suggestion here is that quantum gravity is to be understood, {\bf
not} as a  $3 + 1 - D$  TQFT, but as a  $(2 + 1) - D$ TQFT with some
extension into four dimensions.  The extension is very ``categorical'',
a  $2 + 1 - D$  TQFT is a set of functors, we now need functors plus
natural transformations.

This is reminiscent of the conjecture, [3], since demonstrated by K.\
Walker [15], that the  $3 - D$  TQFT's constructed from modular tensor
categories related to WZW models can be interpreted as containing
information about a bounding  $4$-manifold, (thus removing the phase
ambiguity).

More recently, I.\ Frenkel and the present author [16] have been able to
associate to a quantum group a tensor  $2$-category, which is a sort of
``recategorification'' of a modular tensor category.  The relationship
between augmented states and  $2$-categories is very suggestive.  For
those not versed in category theory, let me remark that both (skins of
observation, inspections, processes with corners) and (categories,
functors, natural transformations) are  $2$-categories [17].

\vskip .2in

\subheading{VII. Possible Approximations}

How could we compute anything in this picture?  One suggestion, due to
L.\ Smolin, is to try to model an asymptotically flat situation.

We could imagine picking a particular weave to model the state in the
flat region, and couple it to our deformed spin networks in the
interior.

In general, it is possible that experiments could be modeled after some
hybrid of the loop states [18] in one half of the universe with the TQFT
state on the other.  This would correspond to breaking gauge invariance
in the ``apparatus''.

It may be possible to formulate a simplified model of matter by using
the representations of an inhomogeneous quantum group instead of a
larger simple group.

\vskip .2in

\subheading{VIII. Perspective and Apologia}

One way of looking at the proposal in this paper is that it is an
attempt to fuse the many worlds interpretation of quantum mechanics with
general relativity.  States become correlated only locally, i.e. only on
skins.  Somehow, the classical equation of motion must emerge in a
situation where many bodies measure one another's state very often.

The great hope of the subject would be that once we introduce clocks
Einstein's equations in  $4 - D$  emerge by a process analogous to the
emergence of flat  $3$-geometries in the work of Regge and Ponzano.

Finally, an apologia.  This is a very sweeping proposal, motivated
partly by work in quantum gravity, but also largely by mathematical
elegance, and the existence of certain elegant algebraic structures.  It
has no doubt escaped no ones attention that much of the necessary
delicate analysis is not done.  Nevertheless, my intuitive feeling is
that enough of the ingredients of nature have mathematical analogs here
to make the ideas worth pursuing.  Mathematical elegance has served
physics well enough in the past.



\bye



\documentstyle{report}

\begin{document}

{\LARGE  FOUR DIMENSIONAL TQFT; a Triptych}

\bigskip



by Louis Crane\footnote{ supported by NSF grant DMS-9106476}

Mathematics Department

Kansas State University

Manhattan KS, 66506

\bigskip

\begin{quotation}
{\small ABSTRACT: We discuss three possibly interrelated ideas.
One idea is a suggestion for a categorical approach to quantizing
gravity, one tells us how to lift a TQFT from D=3 to D=4, and one
tells us how to refine some modular tensor categories into
2-categories. Relations between the ideas are conjectured.}

\end{quotation}

Since it is impossible to state one idea adequately in 20 minutes, I
shall outline three ideas sketchily instead. I am submitting to this
conference, as joint
work with David Yetter, a preliminary version of a paper which details
one of the three ideas [1]. A second is written out in [2], and a third in
[3].

The three ideas I describe begin in different places, but ought to
connect with one another. They all revolve around the problem of
constructing Topological Quantum Field Theory [TQFT] in dimension 4.

\smallskip

{\em 1 Quantum Gravity and augmented TQFT}

\smallskip

There seems to be a connection between 3D TQFTs and the quantum theory
of gravity, as formulated in the Ashtekar variables. There are several
indications of this. One is that the Chern-Simons functional, which
Witten [4] used to formally construct 3D TQFT, is (also formally),
a solution of the constraints for general relativity in the Ashtekar
variables [5]. Another indication is the deformed spin network
interpretation
of the CSW TQFT, [6]  which shows that it is closely related to the
spin network formulation of 3D quantum gravity due to Regge and
Ponzano [7].

The deformed spin network picture gives us a picture of a 3D TQFT as
a rule which assigns to a 3-manifold with boundary a number, which
can be interpreted as an average over discretized 3D Riemannian metrics, when
we specify  a state in the space of conformal blocks of
the boundary surface. This goes a long
way towards giving an interpretation of the CSW TQFT as a state for
the quantum theory of gravity.

In order to create an interpretation for the CSW state which would be
related to physical experiments, it is necessary to reintroduce time
into
our picture, i.e. to make a 4-dimensional structure which is some sort
of extension of a 3d TQFT. The word I coin for the new structure is
``augmented.''

An analysis of what would be physically useful for a 4D interpretation
suggests a mathematically elegant form for what an augmentation should
be. A TQFT can be described as a functor from a cobordism category to
Vect (the category of finite-dimensional vector spaces)
It is easy to restrict a TQFT to subsets of some 3-manifold M and get a
functor on a relative cobordism category, of surfaces and cobordisms
embedded in M.

An augmentation of a TQFT then amounts to a rule, which assigns, to
any 4D cobordism between two 3-manifolds a natural transformation
between the two functors associated to the relative categories of the
two 3-manifolds. This translates physically into time evolution
operators on relative states, together with appropriate consistency
relations when observers observe one another evolving.

This leads us to the problem of trying to augment a 3D TQFT. Since
CSW theory can be constructed from a modular tensor category (MTC) [8],
we are led to consider two separate mathematical problems :

\smallskip

1) augmenting a 3D TQFT directly up to D=4, and

\smallskip


2)augmenting a modular tensor category into a 2-category.

\smallskip

I do not, at this point know how to use either of these ideas to
construct an augmentation in the sense described above, but there is
considerable progress to report on both, purely as mathematical problems.

\smallskip

{\em 2. Categorical construction of 4D TQFT.}

\smallskip

This construction is discussed in the short paper submitted to the
conference
jointly with D. Yetter.Consequently, I confine myself here to a few
simple remarks.

One important point is that we have a construction of a 4D TQFT from a type
of tensor category we are already familiar with, rather than a
hypothetical one which solves some list of axioms.

It is rather surprising that we get a 4D theory from a modular tensor
category, since the general intuition in the field is that 4D
invariants should come from 2-categories [9, 10]. The construction has
another rather surprising feature, which is that we put labels of
spins (or, more generally, of representations) on the 2-simplices
of the triangulation rather than on the edges.

We believe that these two facts suggest that our invariant is a
special case of one constructed from 2-categories. The MTC is playing
the role of a 2-category with one object, so that the edges are
labeled trivially, and the representations on the surfaces are really
to be thought of as morphisms.

As I shall mention below, this suggests that the 2-category I describe
in section 3 may have some relationship to 4D topology.

\smallskip

{\em 3. The Canonical Basis and 2-Categories}

\smallskip

This part of my talk represents joint work with Igor Frenkel [3].

\smallskip

There are several ways to construct the MTCs which are related to CSW
theory. One is to look at the representation theory of quantum groups
when q is a root of unity. Quantum groups admit a rigid and elegant
layer of structure related to the canonical basis of Lusztig [11], in
which the structure constants and tensor operators between
representations have purely integral coefficients, and with slight
modification of the algebra, even purely positive integral
coefficients.

At first glance, the structure related to the canonical basis seems
unrelated to the topological applications of quantum groups. We do
not know if we can use the canonical basis to produce 4D invariants,
but we do have a result which strongly suggests that possibility.

Our result is that the quantum groups can be ``categorified'',
ie. represented as the Grothendieck rings of tensor categories.
The categories of representations of the quantum groups can then
be recategorified into 2-categories.

We believe that this ``categorification'' is indicative of possible
4D topological significance because of an analogous phenomenon
which relates 2D and 3D TQFTs via categorification.

(A detailed discussion of the notion of categorification can be found
in [3]. Let me just explain, briefly, that if we have a category with
direct sum and tensor product, we can make a ring of the formal linear
combinations of objects in the obvious way. This we call the
Grothendieck ring. The reverse operation, i.e. from ring to category,
is called ``categorification''. It can be nonexistent or many-valued.
Clearly, a ring needs a basis in which its operations have positive
integral structure constants in order to be categorified. That is the
point of departure for the connection between the canonical basis and
categorification.)

 2D TQFTs are constructible from commutative
associative algebras with an inner product and suitable compatibility
conditions. On the other hand, 3D TQFTs are constructed from modular
tensor categories.

It turns out that the most physically interesting 2D TQFTs, namely the
G/G models, are constructed from the algebras constructed from the
chiral vertex operators of WZW models, called Verlinde algebras.
The Verlinde algebras are precisely the Grothendieck algebras of
the modular tensor categories which are related to CSW theories.
Thus the 3D CSW theories are categorifications of the G/G models.

As yet, it is only a matter of conjecture that the 2-category we
construct has any relationship with the 4D construction in part 2.

\smallskip

{\em 4. Connections}

\smallskip
It is not yet clear if the three ideas outlined above connect.
Recently, L. Smolin [12] has suggested that in order to make a 4-
dimensional interpretation of states for general relativity in the
Ashtekar variables, it is necessary to fill up space with a family of
surfaces on which we could locate clocks.

Since the 4D construction described in 2 above is really built up out
of 3D TQFT on a family of 2-surfaces in the 3-skeleton, it may be
possible to interpret our 4D construction as implementing Smolin's
suggestion under the assumption that the universe is in the CSW state.
If so, and if a steepest descent approximation exists for our 4D
theory analogous to Regge-Ponzano theory in D=3, then the relationship
between ideas 1 and 2 in this paper may be close.

\bigskip

\newpage

Acknowledgements:

Much of this work is joint work with D. Yetter and I Frenkel.
the ideas about quantum gravity grow out of years of work with L.
Smolin and C. Rovelli. I wish to thank J. Baez, J. Pullin, and
G. Zuckerman for helpful conversations.

\bigskip


References:
\smallskip

1.L. Crane and D. Yetter,  A Categorical Construction of 4D
Topological Quantum Field Theories, this volume

\smallskip

2. L. Crane, Categorical Physics, preprint

\smallskip

3. L. Crane and I. Frenkel, Categorification and The Construction of
Topological Quantum Field Theory, to appear in Conference Proceedings
AMS conference on Geometry, Symmetry and Topology, Amherst MA, 1992

\smallskip


4. E. Witten, Quantum Field Theory and The Jones Polynomial, IAS
preprint
HEP 88/33

\smallskip

5. B. Brugman, R Gambini and J. Pullin, Gen. Rel. Grav. in press

\smallskip

6. L. Crane, Conformal Field Theory, Spin Geometry, and Quantum Gravity,
Phys.  Lett. B v259 #3 (1991)

\smallskip

7. G. Ponzano and T. Regge, Semiclassical Limits of Racah Coefficients
in Spectroscopic and Group theoretical methods in Physics. ed F. Bloch
(North Holland, Amsterdam)

\smallskip


8. L. Crane, 2-d Physics and 3-d Topology, Commun. Math. Phys.
135 615-640 (1991)

\smallskip

9. D. Kazhdan, personal communication

\smallskip

10. M. Kapranov and V. Voevodsky, Braided Monoidal 2-Categories,
2-Vector Spaces and Zamolodchikov's Tetrahedra Equation,
preprint

\smallskip

11. G. Lusztig, Canonical Bases in Tensor Products, MIT preprint
1992, and references therein

\smallskip


12. L. Smolin, Time, Measurement, and Information Loss in Quantum
Cosmology, SUHEP preprint, 1992

\end{document}

