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Quantum topology and quantisation on the lattice of topologies

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1989 Class. Quantum Grav. 6 1509

(http://iopscience.iop.org/0264-9381/6/11/007)

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Class. Quantum Crav. 6 (1989) 1509-1534. Printed in the UK

Quantum topology and quantisation on the lattice of topologies

C J Isham Blackett Laboratory, Imperial College of Science and Technology, South Kensington, London SW7 2BZ, U K

Received 16 June 1989

Abstract. The concept of ‘quantum topology’ is studied via a quantisation of the set . (X ) of all topologies on a given set X . A natural lattice structure exists on this set induced by the idea of one topology having more, or less, open sets than another. This is used to provide a basic set of functions on T ( X ) which generate a commutative algebra (the v operation on the lattice) whose spectral theory forms the basis for a general quantisation. It is shown that the analogue of a ’distributional’ topology is an ideal in the lattice . r ( X ) and the spectral theory is used to place a natural topology on the set of all such ideals. The next step is to discuss the existence of variables conjugate to the basic functions on T ( X ) . These are associated with the A operation of the lattice on itself and result in a complete algebra for the quantum theory on T ( X ) that is analogous in many respects to the canonical commutation relations in a conventional quantum theory. The paper con- cludes with discussions of the role of ‘time’ and of some model Hamiltonians that are suggested by the lattice structure on T ( X ) .

1. Introduction

Throughout most of the history of research into quantum gravity there has been a sustained interest in the idea that the space, or spacetime, on which the gravitational field is defined should itself be subject to the laws of quantisation. However, it is not at all clear a priori how the concept of ‘quantum topology’ is to be understood or, indeed, if it can be made compatible at all with the laws of quantum physics which are usually formulated within a framework which includes a fixed background space and time.

The first serious investigation into quantum topology was made by Wheeler (1964) working from within the canonical approach to quantum gravity. He argued that whereas a n ordinary field can become arbitrarily large (even distributional) with no implications for the manifold on which it is defined, this is not the case for the Riemannian metric in general relativity. In particular, the anticipated large, ‘non- smooth’ quantum fluctuations in the metric might cause space to ‘pinch off’ in some way and hence to induce a topology change.

Wheeler’s ideas were seminal in generating interest in the subject of quantum topology but, in practice, most work in this area did not progress much beyond artistic impressions of the ‘wormholes’ that were expected to become a prominent feature of physical space at scales of the Planck length or less. A more technical line of development began with the Hawking postulate that the basic quantum-mechnical transition amplitude K ( g , , E, ; g,, E*) for going from a Riemannian 3-metric g, on a 3-manifold I;, to a metric g, on a second 3-manifold E2, should be computed with a

0264-9381/89/ 11 1509 + 26$02.50 @ 1989 IOP Publishing Ltd 1509

1510 C J Isham

functional integral

where the sum includes any four-dimensional manifold M whose boundary is the disjoint union of the pair of 3-manifolds El and I;*, and the integral is over all Riemannian signature 4-metrics on M which induce the 3-metrics g , and g, on C1 and E2 respectively (Hawking 1979). Thus a topology change from one 3-manifold to another is mediated by the cobordisms provided by the interpolating 4-manifolds M. An extreme example is provided by the Hartle and Hawking (1983) ansatz for the creation of the universe in which the ‘initial’ 3-manifold is absent and (1.1) becomes the wave-function for the 3-metric on the single 3-manifold boundary.

This picture is pleasing aesthetically, but it is not clear how literal an interpretation it can sustain. The action S ( g ) which appears in the functional integral is usually taken to be the standard classical Einstein action, augmented perhaps by a cosmological term and/or contributions from various matter fields. But such a quantum theory is perturbatively non-renormalisable and therefore, barring the discovery of some miraculous non-perturbative technique, (1.1) could be expected to give reliable results only in the semiclassical regime and after the introduction of an effective cut-off at the Planck length. In particular, the pictures of topology change will only be meaningful if these effects manifest themselves at a scale that is significantly larger than the Planck length.

There may be some reason for such a scale to arise, but it is not what has been generally expected from quantum gravity. The dominant view has always been that it is at the Planck length that something ‘funny’ happens to the structure of spacetime, and the challenge is to discover what this might be. How one rises to this challenge depends very much on one’s general view of the quantum gravity programme. One long-held belief is that both the classical gravitational field and the spacetime on which it is defined will emerge as the low-energy limit of a quantum theory that is quite unlike any naive quantisation of the classical theory of general relativity. The concept of ‘quantising’ topology is as inappropriate in these circumstances as is the idea of quantising Riemannian geometry. Superstring theory is the current apotheosis of this school of thought, although it should be noted that the desire to reproduce the graviational field has been distinctly more successful (at least perturbatively) than the more ambitious scheme of recovering spacetime itself.

The approach to quantum topology adopted in this paper is quite different and is concerned with seeing how far conventional quantum-theoretical ideas can be taken by applying them to the class of all topological spaces. The operative word is ‘all’ since I wish to consider general topological spaces, not just differentiable manifolds. The motivation for such a Draconian step lies in the firm belief that the breakdown of the functional integral (1.1) at the Planck length signals not merely the failure of the classical field equations but also the whole edifice of differential geometry upon which they are built. A number of workers in the past have been unhappy about an unconditional application of the continuum picture of spacetime. This was certainly one of the motivations behind Penrose’s (1971) invention of spin networks (and their twistorial descendants), and also the more recent work by Finkelstein (1988) on a novel spacetime microstructure, and by Bombelli et a1 (1987, 1988) and others (for example, Szabo 1986) on the possibility of an underlying causal/partial-ordering for spacetime.

Quantum topology 1511

The intention of this present paper is to develop an implementation of the idea of ‘quantum topology’ which is based on a certain lattice structure (in the algebraic sense) possessed by the set T(X) of all topologies on a fixed set X. The aim is to see to what extent quantum states can be written as wavefunctions $ ( T ) of topology together with the associated concepts of eigenstates of topology, superposition of such states, and the like. Thus we are attempting a sort of ‘canonical’ quantisation of a system whose configuration space is T (X) , although it should be emphasised that it is by no means clear that the conventional ideas of ‘time’ or ‘dynamical evolution’ are applicable in this situation.

The paper starts with a general discussion of some of the issues that must be addressed in any attempt to construct a quantisation on the set of all topologies. The crucial step is the identification of a suitable set of functions on T ( X ) whose Abelian algebra can be used as the foundation for the quantum theory. The resolution of this problem with the aid of the lattice structure on T(X) is explained in $ 3. In 5 4 we will consider the question of conjugate variables (i t is not easy to differentiate with respect to a topology!) and illustrate the general theory with the example of a finite set X . Much of this discussion applies to a general lattice but the next section is peculiar to the topological case and concerns the important issue of the role of the group Perm(X) of all bijections of X onto itself-arguably the natural ‘gauge group’ of the theory on T ( X ) . In S: 6 we confront the difficult question of the status of the concept of ‘time’ and the associated notion of dynamical evolution. Some simple model Hamiltonians are proposed which are suggested rather naturally by the algebraic structure of the lattice. The paper concludes with a discussion of some of the outstand- ing problems and possible future lines for research.

2. The main problem

Our aim is to construct a quantum theory for a system whose ‘configuration space’ is the class of all topologies. We recall that the technical definition of a topology on a set X is a collection T of subsets of X (called ‘open’) such that

A, B E T implies A n B E T (2.1)

A, E T, i E I (an arbitrary index set) implies U A, E 7.

Thus the open subsets in a topology always include the empty subset 0 and the set X itself and are closed under the formation of finite intersections and arbitrary unions. The family of all topologies would consist of all such collections T on all possible sets X. However, this family is enormous-so big in fact that it is not a set. If it were, it could be equipped with a topology (in many different ways) and would therefore have to be member of itself, which is not permitted in set theory. We will avoid this logical problem by restricting our attention to the family T ( X ) of all topologies on a jixed set X , which is a proper set.

It is appropriate to recall that topologies T~ and T~ on a set X are said to homeomorphic if there is a bijection i: ( X , T ~ ) + (X, T ~ ) which is continuous and whose inverse is also continuous ((X, T ) denotes the set X equipped with the topology 7).

1512 C J Isham

This bicontinuity is equivalent to saying that the map i sets up a one-to-one equivalence between those subsets of X that are open in the r1 topology and those that are open in the topology. Thus the group Perm(X) of all bijections of X acts as a group of transformations of r ( X ) with orbits which are the homeomorphism classes of topology.

This action of Perm(X) is relevant to the important question of how the set X of ‘primordial’ points is to be selected. It can be argued that the only feature of X that should be of any real significance is its cardinality 1x1-that is, we d o not wish to distinguish between a pair of models for X if the points in the two spaces can be put in bijective correspondence. In particular, this would apply to the action of Perm(X) on a specific choice of X , which implies, for example, that any topology placed on T(X) as part of the quantisation programme should be invariant (up to homeomorph- isms) under this action.

However, we could go further and argue that the entire theory should be invariant under the action of Perm(X) , which thereby becomes a sort of ‘gauge group’ for the theory-an extension perhaps of the diffeomorphism-group invariance of conventional quantum gravity. Thus the ‘configuration’ space would now be the set r ( X ) / P e r m ( X ) of all homeomorphism classes of topology. This factoring out by the group Perm(X) deprives the individual points in X of any fundamental status. Such a position is attractive to those (including myself) who view the concept of a space or spacetime ‘point’ as a mathematical abstraction of dubious ontological validity.

We will return later to the specific implications for the quantisation programme of adopting Perm(X) as a gauge group, but we must consider first the quantisation of a system whose configuration space is the set T ( X ) . This is a non-trivial problem since there is no unique a priori way of quantising a system with a given configuration space Q, not even when it is a differentiable manifold. It is tempting perhaps to postulate that quantum states can be represented by complex-valued functions on Q (or cross sections of a vector bundle over Q) whose Hilbert space structure is defined by integrating with respect to some measure p on Q. However, one implication of such a scheme is that any bounded continuous function f : Q + C can be made into a self-adjoint operator by defining

(h) ( 4 1 := f ( 4 1 cL( 4 ) (2.2)

and even when Q is simply a vector space E we know that this does not always work. The difficulty arises when E is infinite dimensional since there are then many classical functions which cannot be admitted as quantum operators. Thus, in quantum field theory, it is usually impossible to define an operator version of ( q 5 ( ~ ) ) ~ : we must instead restrict our attention to polynomials (or suitable limits of polynomials) in the smeared fields q 5 ( f ) where the test function f belongs to E.

The crucial observation in quantum field the:ry is that the smeared operator fields generate a unitary representation U ( f ) := exp iq5(f) of the vector space E

U(f1) U ( f * ) = U ( f , +fi) (2.3)

and the appropriate spectral theory for the Abelian group E (typically a nuclear topological vector space) shows that if R is a cyclic vector in any cyclic representation then

Quantum topology 1513

for some probability measure p on the topological dual E’ of E. In addition, the Hilbert space is isomorphic to L 2 ( E ’ , d p ) with the operator fields $( f ) acting as

Thus the domain space of the quantum states is not the configuration space E itself but rather the distributional dual E‘ of E. So the representation theory of E would have forced the discovery of distributions in quantum field theory even if they had not already appeared in classical physics !

There is no reason to expect quantum theory on T ( X ) to be any less pathological than that on an infinite-dimensional vector space, but at the moment we cannot even get started since, so far, no structure at all has been placed on .(X). In any event, 7 ( X ) is most unlikely to be a manifold (infinite dimensional or otherwise) and so none of the traditional geometrical approaches to quantisation will be of any use. However, bearing in mind the analogue of conventional quantum field theory, it is clear that the following questions are of central interest.

(i) Is there a preferred minimal Abelian algebra of functions on T ( X ) ? Hopefully, this would play a role analogous to that of the smeared fields 4 ( f ) in quantum field theory.

(ii) Is it necessary to use a space T(X)’ of ‘distributional’ topologies? If so: what is a ‘distributional’ version of a topology? is there an appropriate topology on ~ ( x ) ’ ? is T ( X ) dense in such a space? This would be analogous to the way in which any

can this topology be chosen to be Perm(X) invariant? (iii) Is there a spectral theorem for the minimal algebra that justifies choosing the

Hilbert space to be L2(?(X), d p ) [or L*(T(X)’, d p ) ] ? Ideally such a theorem would: provide the appropriate definition of a distributional topology, provide a natural topology on the set of all of them, give some insight into how the measure p on T(X)’ is to be chosen and the role

played by equivalence classes of such measures. (iv) What are the variables conjugate to the functions on Q = T ( X ) selected in ( i )?

When Q is a manifold, the conjugate variables are associated with generators of diffeomorphisms of Q and are essentially vector fields which act on wavefunctions on Q as partial differential operators. With due care, this picture can be generalised to the infinite-dimensional case, but it does not provide much guidance in handling a space like 7(X) which is not a manifold. We certainly do not expect to be able to write literally a ‘momentum’ operator as -id+( ~ ) / a r !

(v) How (if at all) is the Perm(X) invariance to be implemented in the quantum theory?

It is clear that at the heart of the quantisation programme lies the problem of finding a suitable preferred algebra of observables on Q = 7(X). The q variables normally arise as generators of transformations along the p directions on the classical phase space T*Q. However, in the absence of any natural ‘cotangent’ structure associated with T ( X ) (and, a fortiori, no group of transformations), we will adopt instead the philosophy espoused by Mackey (1963) of basing the quantisation on collections of ‘yes-no’ questions about the system. These will be represented in the quantum theory by commuting Hermitian projection operators, and the goal is find a set which satisfy an algebra with a spectral theory that can be invoked to produce a framework within which arbitrary representations can be discussed.

distribution can be approximated arbitrarily closely by a smooth function;

1514 C J Isham

Bearing in mid the definition of a topology on X as a collection T of subsets of X satisfying the conditions in (2.1), one obvious family of questions is to ask for each subset A of X , ‘In the topology 7; is the subset A open?’ The associated set of functions on ~ ( x ) is

f A : r ( X ) .+ (0 , l } c R A c X (2.6) where

if A is open in T

otherwise. . f A ( r ) := (2.7)

This might suggest an ‘Ising model’ approach in which the ‘sites’ are the subsets of X and a ‘spin up’ (t) or ‘spin down’ (J) is assigned to each site/subset according to whether or not the set is open. The empty subset P, and X itself are always ‘spin up’ since they are open in every topology. But we must also take into account the fact that if A and B are open in some topology then so are A U B and A n B. Thus, unlike in the real Ising model, the 1 and t labels cannot be assigned independently to each site.

For example, when X has just three elements { a , b, c } , the set of all subsets can be represented as the lattice of eight elements shown below. An upward pointing line from a subset A to a subset B means that (i) A is a proper subset of B, and ( i i ) there is no third subset C, other than A or B, such that A c G c B. The meet and join of A and B are simply A n B and A u B respectively. Thus, for example, if T is assigned to the subsets {a , b } and { b, c}, it must also be assigned to the subset { b } = { a , b } r ) { b, c} .

At the level of the functions fA, A c X , this ‘non-local’ coupling is reflected in the relations

f A = l and f B = l implies fAnt3 = and f A u B = (2.9) plus a suitable extension for arbitrary unions. These relations are equivalent to the inequalities

(2.10)

It is difficult to know how to implement relations of this type since there is no obvious spectral theory associated with systems of inequalities. We will concentrate instead on a larger family of functions on r ( X ) that satisfies a set of algebraic equalities associated with a natural lattice structure possessed by T ( X ) .

3. Lattice structure

3.1. The lattice r(X)

Let us recall that a iattice (Birkhoff 1967, Gratzer, 1978) is a partially-ordered set in which each pair of elements { a , b } has a least upper bound, denoted a v b, and a

Quantum topology 1515

greatest lower bound, denoted a A b. These lattice operations are idempotent, commuta- tive and associative-that is, they satisfy the relations

Idempotency : a v a = a U A U = U

Commutativity: a v b = b v a a A b = b r, a

Associativity: ( a v b ) c = a v ( b v c)

( a A b ) A c = a A ( b A c).

Each of the operations A and v makes the lattice into a semigroup (there are no inverses) and our aim is to exploit this algebraic structure in the quantisation pro- gramme.

A natural partial ordering is associated with any lattice via the definition

a s b if and only if a A b = a (3.1)

and the first step in placing a lattice structure on T ( X ) is to note the well known partial ordering of topologies defined by

5-1 6 7 2 if r1 c r2 (3.2)

where the set inclusion sign c includes the possibility of equality; the notation r1 < r2 will be employed whenever it is important to emphasise that T~ # r 2 . If r, and T* are a pair of topologies satisfying (3.2) (so that every r ,-open set is also .r,-open) then r1 is said to be weakerlcoarser than T ? , and T~ is strongerljner than 7 , . The weakestlcoarsest topology is {a, X } and the strongestljnest is P(X)-the set of all subsets of X . These extreme topologies will sometimes be denoted by 0 and 1 respectively.

Lattice operations on T ( X ) can be defined by

T~ A r2:= T , n T~ = { A c X I A is open in both r1 and T ~ }

r1 v T*:= coarsest topology containing { A , n A21A, E r , , A2 E T ~ } . (3.3)

It is a standard exercise to show that these operations are compatible with the partial ordering in the sense that ( i ) T , A T- is the finest topology that is coarser than both T ,

and T ~ , and (ii) r , v T~ is the coarsest topology that is finer than both T , and T ~ .

It is instructive to study a few simple examples where X is a finite set. The simplest is when X has one element, { a } say, for which there is just the single topology {@, { a } } .

If X = { a , b } there are four topologies arranged in the following lattice:

0. I0, X l

where a line drawn upwards from T , to r2 means that (i) T , is strictly coarser than T ~ ,

and (ii) there is no intermediate topology which is strictly finer than T , and strictly coarser than r 2 .

1516 C J Isham

The first really interesting example is when X is a set {a , b, c} of cardinality three. The lattice diagram for this case is shown below using a notation which has been chosen for maximum typographical simplicity. For example, ab( ab ) ( ac) means the topology whose open sets other than (3 and X are the subsets { a } , { b } , { a , b } and {a , c}.

The lattice of all topologies on X possesses many interesting properties; a useful review article is Larson and Andima (1975). Some examples are given below.

(i) T ( X ) is complete. Hence the lattice contains the meets and joins of arbitrary sets of elements, not merely finite ones (which is all that is guaranteed by the algebraic axioms of a lattice).

(ii) . (X) is complemented (Steiner 1966). Thus, given any topology 7, there exists some other topology T’ such that

T A 7 ‘ = 0 and 7 V 7’” 1. (3.4)

However, unlike (for example) the Boolean algebra of all subsets of X , the complements in 7 ( X ) are not unique. In fact, when X is infinite, each topology 7 other than (0, X } and P ( X ) has at least 1x1 different complements 7’.

(iii) This is related to the fact that for 1x1 > 2, T ( X ) is non-distributive and non- modular. This reduces the number of po’tential operations that could be employed in a quantum scheme. In the concluding section I will comment on some additional possibilities that arise when constructing a quantum theory on a distributive or modular lattice.

(iv) The lattice T ( X ) is atomic. In fact, for each A c X , 7, = ((3, X , A } is an atom (that is, T~ > 0 with no intermediate topology) and every topology 7 is determined by these atoms in the sense that

(3 .5 )

Note that the relation T~ s 7 is precisely equivalent to the statement that the subset A is open in the topology T, that is, A E T. In the example X = { a , b, c}, the atoms are the six topologies b, ( a b ) , a, ( a c ) , (c) and (bc ) .

(v) The lattice 7 ( X ) is also anti-atomic. That is, there exist topologies T~ with the properties that (i) rd < 1 with no intermediate topologies, and ( i i ) every topology is uniquely determined by the anti-atoms that lie ‘above’ it. in the example X = { a , b, c}, the anti-atoms are the topologies bc( bc)( a c ) , bc(ab)( bc), ab( ab ) (bc ) , ab( a b ) ( a c ) , a c ( a b ) ( a c ) and ac(ac ) (bc ) . (The equality of the numbers of anti-atoms and atoms

7 = V{ T A / TA s T } .

Quantum topology 1517

stems from the self-duality of the lattice T ( X ) for 1x1 = 1-3. This is not true when 1x1 > 3.) In general, the anti-atoms are topologies of the form

T ( ~ , ~ ~ = { A c X l x A or A E %} (3.6)

where % is any ultra-filter not equal to the principal ultra-filter of all subsets of X containing the point x E X . For a finite set X, every ultra-filter is principal whereas, for non-finite X, the construction of a non-principal ultra-filter invariably involves the axiom of choice. Since no explicit use is made of this concept in this paper the reader is referred to the literature for details concerning these exotic objects (Bourbaki 1966).

(vi) The lattice structure also has important properties vis a vis the action of the group Perm(X). It is easy to see that these induced maps on T ( X ) preserve all the lattice operations and are therefore lattice automorphisms. Furthermore, Hartmanis (1958) and Frohlich (1964) showed that if IX/=cT, the group of automorphisms A u t ( r ( X ) ) is actually equal to Perm(X)-that is, the topological properties of an element of T ( X) are determined solely by its position in the lattice. (For finite X there is an additional automorphism induced by the transformation on the atoms T~ + T,,L

where A' is the set-theoretic complement of the subset A of X . ) This equality of Aut( ~ ( x ) ) with Perm(X) reinforces the idea that Perm( X ) is the natural 'gauge group' of the theory and therefore has important implications for the quantum theory.

It is evidently of some interest to know how many topologies can be placed on any given set X . Note first that

lP(X)l = 2IX and lP(P(X))l = 2 2 ' . (3.7)

l T ( x ) I I P ( P ( X ) ) l = Z 2 ' . (3.8)

Now, each topology T is a member of P ( X ) and hence T (X)C P ( P ( X ) ) . Thus

For infinite X , the number of atomic topologies is clearly only 2Ix (the cardinality of the set P ( X ) of all subsets of X ) but Frohlich (1964) showed that the cardinality of the set of anti-atoms is 2' '. Hence the cardinality of T ( X ) is equal to its set-theoretic maximum 2* '. Thus if our set X of primordial points is infinite the number of topologies on X is two orders of infinity higher-a rather large number by the standards of conventional theoretical physics!

The sizes of T ( X ) that have been computed for finite X are listed in table 1 taken from Larson and Andima (1975).

The value of ~ T ( X ) / for an arbitrary finite set X does not seem to have been worked out explicitly although it is known that if n := 1x1, 2" s IT(X)I s 2"("-".

1 1 2 4 3 29 4 355 5 6 942 6 209 527 7 9 535 241

1518 C J Isham

3.2. The conjguration variables

We come now to the critical question of selecting a set of functions on T ( X ) to use as a basis for a quantisation scheme. One natural family of ‘yes-no’ functions associated with the lattice structure is

if 7 6 T’

otherwise. R,( 7 ’ ) :=

In particular,

if A is open in T (i.e. ~~6 T)

otherwise. R,, ( 7 ) =

(3.9)

(3.10)

and hence the functions fA = R,, discussed earlier are included in this set. Since T ( X ) is atomic, this set of variables clearly distinguishes between the different topologies and should therefore be large enough to generate all other functions on T ( X ) .

We recall that we are looking for an algebra of functions on Q = T ( X ) , and therefore it is crucial to note that

R , , ( T ’ ) & ( T O = 1 if 7, G T‘ and ~~6 7‘

= O otherwise.

However, 7, , T~ 6 T’ if and only if 7, v T> < T’ and hence

R,, R T ? = R,, Y 7 2 .

(3.11)

(3.12)

Thus we d o indeed have closure with the set of functions generating a representation of the v-semigroup operation in the lattice. One consequence of (3.12) is

( & I 2 = R, for all T E T ( X ) (3.13)

and hence, in the quantum theory, ITS ~ ( x ) } can be expected to be a family of Hermitian projection operators satisfying (3.12).

When 1 T( X ) i is finite, it is easy to produce representations of this algebra by defining states to be vectors $(T) with

if T C r’ otherwise.

(l?,l,b)( 7‘) := {:(T’) (3.14)

However, when / T ( X ) I is infinite, more care is needed. In particular, account must be taken of the afore-mentioned possibility that the domain of the state vectors is a space of ‘distributional’ topologies in which T ( X ) is only a dense subset.

3.3. Spectral theory of {&/T E r (X) )

The key to finding such a space lies in the spectral theory of the Abelian algebra generated by the operators { l?,I T E T ( X ) } . The equation ( g7)2 = k7 implies that the eigenvalues of l?, are 0 or 1. Also

[E,, , &I = 0 (3.15)

and hence there should be a complete set of simultaneous eigenvectors lh) such that

g T l h ) = h (T) lh ) (3.16)

Quantum topology 1519

where h( T) = 0 or 1. The maps h : T ( X ) + (0 , l} c R are of considerable interest since they form the domain space of the state vectors-in Dirac notation, + ( h ) = (hi$). Note that wavefunctions with indices are n:t really excluded by this argument: they would correspond to a situation in which { R , ~ T E T(X)} is not a complete commuting set of operators. The extra quantum numbers might be expected to be associated with the action of the variables conjugate to the functions R, but we shall not consider this possibility here.

To find out more about the h maps, note that (3.12) implies h(.r,)h(.r,) = h ( T 1 v 7 2 ) V T l , T2E T ( X ) (3.17)

so that h is a v homeomorphism (or semicharacter) from T(X) into {O, l}c R, . The lattice T(X) is a semigroup with respect to the v operation and, as we shall see, the representations of such semigroups can be described in terms of the maps h as a precise analogue of the role played by the characters of an Abelian group.

Define I , := { T I ~ ( T ) = l}. Then (a) T, , r2 E I, implies 7, v T* E I , (b) If T’S T, then T ’ V T = T and hence (3.17) implies that h ( ~ ’ ) h ( ~ ) = h ( ~ ) . Then

These are precisely these properties for I , to be an ideal in the lattice T ( X ) . if T E I,, h ( ~ ’ ) = 1 so that T ’ E I , .

Conversely, if I is an ideal, define h, : T ( X ) R by if T E I it otherwise.

h l ( T ) : = (3.18)

However, the definition of an ideal is equivalent to the statement that T ~ , T~ E I if and only if T, v T~ E I. Hence h, satisfies (3.17). Furthermore, h , , > ( ~ ) = h ( ~ ) so that h + I,, is a bijection and therefore the domain space of the quantum states is the set I(T(X)) of all ideals in the lattice.

The important conclusion of this analysis is that the space of quantum states is expected to be equivalent to a Hilbert space of the form L 2 ( I ( ~ ( X ) ) , d p ) for some measure p on I ( T ( X ) ) . It is useful to emphasise this analogy of the space of ideals with the vector space of distributions by writing h, ( T) as ( I , 7). Thus the basic eigenvalue equation (3.16) becomes R,II) = (I, T ) ~ I ) which translates into the action on (I, E

( & N I ) = ( 4 T ) + ( I ) (3.19)

which should be compared with the quantum field theoretic analogue (2.5). Note that the set of ideals in any lattice is itself a lattice in which the meet operation is just set intersection n (Gratzer 1978). This feature will be exploited shortly in the context of the quantum variables complementary to R,.

One reason why distributions are so useful in physics is that they can be approxi- mated as closely as one likes by smooth functions-that is, the space of smooth functions is a dense subset of the space of distributions. In this sense, a distribution is merely some sort of ‘ideal’ limit of a sequence of classical functions. We would like something similar to apply in the case of T ( X ) and the first observation in this direction is that T ( X ) is indeed a subset of I(T(X)). More precisely, there is an injection

4x1 + I ( T ( X ) ) T H & ( T ) : = { X E 7(x)lXS T} (3.20) which maps each topology T into the principal ideal which it generates. This map is surjective if T ( X ) is finite since then every ideal is principal:

I . . .1 (V{TEI}) . (3.21)

L 2 ( 1 ( d X ) ) , d p )

1520 C J Isham

Thus, for finite X , our expectation that state vectors can be written as functions on T ( X ) is justified and (3.19) reproduces (3.14). However, in a general infinite lattice, ideals exist which are not principal and, in the present situation, the most that can be hoped for is that T ( X ) is a dense subset of 1 ( 7 ( X ) ) . But, of course, this concept is only meaningful if a topology has been placed on I ( T ( X ) ) .

There are many ways in which this space might be topologised and it is important to select one that is compatible with the use to which it will be put in the quantum theory. Fortunately, this is precisely what arises as part of the rigoroAus version of the discussion above on the simultaneous eigenvectors of the operators {RT/7 E ~ ( x ) } . The full mathematical analysis employs the Gel'fand spectral theorem (Rudin 1973) which, when applied to the commutative C" algebra generated by these operators, shows it is isomorphic to the algebra generated by the functions R: on I ( T ( X ) ) defined by

R i ( I ) : = ( I , T ) (3.22) which is the natural extension to the complete ideal space of the functions defined in (3.9) on the topologies/principal ideals.

An integral part of the Gel'fand theory is the construction of a natural topology on 1 ( 7 ( X ) ) , which is the weakest topology such that these functions are continuous. A basis for the open sets is all finite intersections of the subsets of I ( T ( X ) ) of the form

Q T . : = { I E I ( T ( X ) ) I T E I } (3.23) plus their set-theoretic complements.

From our point of view, this topology on T ( X ) has many desirable properties. (i) I ( T ( X ) ) is compact and Hausdorfl This is the nicest type of space on which

to d o measure theory. (ii) It is a straightforward application of the definition of the topology on the space

of ideals to show that the map in (3.20) embeds r ( X ) as a dense subspace of I ( T ( X ) ) . This is a precise analogue of the situation in conventional QFT in which the space of smooth functions is dense in the space of distributions. However, note that there is no particular reason for thinking that the topology on T ( X ) is first-countable (in particular, it may not be a metric space) and hence the limits may involve nets or filters rather than just sequences.

(iii) The automorphism of the lattice T ( X ) induced by a bijection of X passes to an automorphism of the lattice I ( T ( X ) ) which can be shown to be a homeomorphism with respect to the spectral topology on I ( T ( X ) ) . This is precisely what is required if it is only the cardinality 1x1 of X that is to be of any fundamental significance.

(iv) The operation m, : T ( X ) -+ T ( X ) , 7 ' - T v T' is continuous with respect to this topology, as is its extension to 1 ( 7 ( X ) ) defined by

m 7 ( I ) : = L ( 7 ) n I . (3.24) which is consistent with the operation on T ( X ) since the principal ideals obey the relation J ( T ) n & ( T ' ) = & ( T A 7'). In fact the semigroup map

(3.25)

(3.26)

Quuntum topology 1521

The spectral theory of the algebra generated by the operators {R,IT E T ( X ) } can be extended well beyond the Gel’fand results. In particular, there is a precise analogue of the well known theory which expresses all representations of an Abelian group in terms of its characters.

First, if p is any regular measure on the compact Hausdorff space I ( T( X ) ) then

K ( T ) : = 1 ( I ,T )dP( I ) (3.27) I ( T ( X ) )

is a positive semi-definite function on T ( X ) . That is

(3.28)

for all finite sets c,. . . c, E C. But the crucial result is the converse (Berg et a1 1984) which affirms that to each

such positive semi-definite function K , there exists a regular measure on I ( T ( X ) ) such that (3.27) is true. The analogous statement in conventional quantum field theory is that any such function on the topological vector space E of smooth test functions generates a unitary representation f - U ( f ) of E with ~ ( f ) = (a, U ( f ) a ) and with this expectation value being expressible as a ‘Fourier transform’ over E‘ in the form of (2.4). Thus, at least in principle, these spectral-theorem results on the algebra of operators {$?,I E T ( X ) } provide a way of placing the lattice-based theory of quantum topology on a respectable footing.

4. The complementary variables

4.1. The A operation

We must consider now the vital question of the construction of complementary ‘momentum’ variables. In conventional quantum theory, these variables usually come from a group acting on the configuration space Q. The most familiar situation is when Q is a manifold. Momentum observables are then defined to be the self-adjoint generators of representations of a Lie group which acts on Q. For example, if Q is a finite-dimensional vector space we define

where this family of unitary operators is related to its generators j3 by U, = exp(ij3 a ) for all vectors a E Q. These generators can be recovered from the unitary operators by an appropriate differentiation with respect to a.

A more appropriate situation for our present purposes is when Q is a discrete space. This is closer to the situation of interest in which Q = T ( X ) . A useful example is to take Q to be the integers. We define

( U a $ , ) ( x ) = $ ( x + a ) (4.1)

Thus T is unitary with respect to the usual 1’ scalar product, but there is no correspond- ing generator. Instead, there are the Hermitian operators

so that q := ( T + T’)/2 r :=(T-T. ’ ) /2 i (4.3)

( . r r $ ) ( n ) = ( $ ( n + 1) - $(n - 1))/2i (4.4)

1522 C J Isham

which is the nearest we get to ‘differentiation’ in this case. Note that q is not an independent observable since from T & T = TT’ = 1 there follows

q2+ %-2= 1. (4.5)

Clearly, T ( X ) is more like the second example than the first. Having already obtained a representation of the v operation (from the R, operators) it seems natural to complement this by choosing for the analogue of a group action on Q = T ( X ) , the action of the semigroup T ( X ) with respect to the A operation. Thus we define

(M&)(T’):= $ ( T A 7’) (4.6)

( M , $ ) ( I ) := $ ( & ( T I n 1 ) . (4.7)

and can extend this to an operation on L 2 ( 1 ( ~ ( X ) , d h ) by defining

This gives the following algebra which is in many respects the analogue of the canonical commutation relations in conventional quantum theory:

RTR7 = R T V T

M,MT = MTA, (4.8)

M& = (&(TI, T’)K MT. We will ‘axiomatise’ this as being the basic algebra for quantum topology on the lattice

Note that because T ( X ) is a semigroup under the A operation, rather than a group, M , cannot be expected to be unitary. Indeed, since ( MT)’ = M,, M , is a projection operator, albeit non-Hermitian. If it was also unitary we would have for all states 4, $

T ( W .

(49 $ ) = ( M T 4 9 MT$)=(M74> M T M T $ ) = ( 4 , M7$) (4.9) A

and hence M7 = 1.

representation and, correspondingly, the associated Hermitian operators are In fact, M , is more like a creation or annihilation operator than a unitary group

% - 7 : = ( ~ T - ~ ; ) / l q 7 : = ( ~ , + M : ) (4.10)

which, since M , is not unitary, do not satisfy the identity qs+ 7 ~ : = 1 (cf (4.5)). On the other hand, the relation ( M,)* = M , implies that

(qr- l )2-(7TJ2= 1 (4.11)

The basic algebra in (4.8) can be summarised by defining the operators T(,,,,2):=

T T l , T , ) T T i , T , ) = (&(‘2)> T : ) T T ~ Y 7 1 , T 2 A T 2 ) * (4.12)

But note that, by virtue of the discussion above, there is no uniform Hermiticity or unitarity property for these composite operators.

so it is still the case that qT and T , are dependent variables.

R,, M-, which satisfy the composite relations

4.2. The example of aJinite set X

When X is a finite set, the Hilbert space is L 2 ( 7 ( X ) , dp)-CI“X’I, with the inner product between a pair of vectors being the finite sum

( $ , 4 ) = c h ( T ) $ ( T ) * 4 ( T ) . (4.13)

Quantum topology 1523

The eigenstates of the operators R, can be written as I T ‘ ) , so that

if r 6 r’ otherwise.

(4.14)

The adjoint operator M i acts to weaken the topology:

M:/r‘)= ( p , ( r ’ ) / p ( r A ~ ‘ ) ) ‘ ’ ~ \ r A 7’). (4.15)

Thus the minimal operation by such an ‘annihilation’ operator is to weaken a topology by the action of an anti-atom. Note that if we had wished the minimal action to be ‘adding’ a set A to r (by forming r v rA), it would have been necessary to start with the basic functions

if r z r’ otherwise

S,( 7’) = (4.16)

which satisfy the relations

’Ti ’ 7 2 = A 7 2 * (4.17)

This gives an alternative, and complementary, quantisation of the lattice of topologies. When 1x1 is finite, these two different quantisations can be expected to be unitarily equivalent in pairs. But the situation for infinite X is less clear and depends on detailed properties of the measures on the space of ideals and dual ideals (which arise in the second scheme).

Returning again to the example of a finite set X , the most interesting of the operators is M,. This ‘creation’ operator acts to strengthen the topology as follows:

We see that the sum is over the set m i 1 ( r’) which will typically contain more than one element. This contrasts sharply with the more familiar examples of annihilation and creation operators. The origin of this characteristic feature in quantum topology is the basic fact that the map m,: r ( X ) + r ( X ) is many-to-one. Thus M , not only strengthens a topology, it also broadens the state in the sense that an eigenstate of topology becomes a linear superposition of eigenstates.

Note also that by using strings of M , and M i operators one can move ‘sideways’ along the lattice as well as up and down a chain. This is the positive side of M , being non-unitary. Note in this context that if r1 < r2 < r3 (strict inequalities) and if r2 is compact and Hausdorff, then (i) r1 cannot be Hausdorff, and (ii) r3 cannot be compact. This is because if i : X + Y is a continuous bijection between topological spaces X and Y, and if X is compact and Y is Hausdorff, then i is a necessarily a homeomorphism. Hence ‘strengthening’ or ‘weakening’ topologies can only produce one compact Haus- dorff space per linear chain. So a fair amount of side-hopping is necessary if we wish to keep to such ‘nice’ topologies!

5. The role of Perm(X)

We turn now to the important question of what role the permutation group Perm(X) is to play in the quantum theory. We have argued earlier that it may be reasonable to think of Perm(X) as the gauge-group of the theory and, correspondingly, to regard

1524 C J Isham

the quotient space “ i (X) /Perm(X) as the true ‘configuration space’. However, in this context it should be noted that the action of Perm(X) on T(X) is not free and many topologies have a non-vanishing ‘little group’. Indeed, the little group of a topology T is just the group of all homeomorphisms of (X, T ) . For example, when X = { a , b, c}, the action of Perm(X) on T(X) has nine orbits whose little groups at a representative element are given in table 2 .

Table 2.

As far as the quantum theory is concerned, there appear to be three different ways in which one might proceed.

( i ) Quantise on T ( X ) and then impose constraints on ‘physical’ state vectors and/or observables in the well known manner advocated by Dirac (or in one of the BRS

extensions of this method). (ii) Fix a gauge for Perm(X) and then quantise the remaining ‘physical’ degrees

of freedom. (iii) Try and quantise directly on the space T(X) /Pe rm(X) of homeomorphism

classes of topologies. For most gauge theories, the second ‘brute-force’ method is often employed as a

consistency check on the other, more elegant, approaches. However, in our case it is not at all clear how to set about specifying a gauge for the group Perm(X) and of necessity we are obliged to consider one of the other two schemes. In many ways the third is perhaps the most attractive but it is not that obvious how to proceed in this direction either since, although each 4 E Perm(X) induces an automorphism of the lattice structure on T ( X ) , it is not a lattice congruence relation and hence T(X) /Perm(X) does nor inherit any lattice structure from T(X)-that is, if T~ = T ; and T* = T ; it is not necessarily the case that T~ A T? (7 , v T ~ ) is homeomorphic to T I A 7;

( T : v 7;). For example, when X = { a , 6, c} the topology b ( a c ) is homeomorphic to c( a b ) via the permutation that exchanges the points 6 and c. However, b ( a c ) A a ( u b ) = 0 whereas c ( a 6 ) A a ( a b ) = ( a b ) .

On the other hand, . r (X) /PermiX) does possess a natural partial ordering which might be of some use. The ordering between a pair of homeomorphism classes of topology CY and /.3 is defined as

a“@ iff ~ T ~ E C Y and T ~ E / . ~ such that 7 , s ~ ~ . ( 5 . 1 )

For example, the ordering diagram for the case X = {a , 6, c} is given below, where [ 71

denotes the equivalence class (orbit of Perm(X)) containing the topology T.

Quantum topology 1525

Note that T~ =s r2 implies [ T ~ ] s [ 7J but the converse is not true in general and there may exist ‘extra’ links between a pair of equivalence classes [ 71] and [ r2]. This occurs when

(i) rl K T ~ ; and (ii) there exist topologies ri and r; such that T { < T ; with 71 = 7: and r2 = T ; . For

example, in the case above, we have ( a b ) < a ( a b ) and hence [ ( ab ) ]< [a (ab ) ] . However, we also have

( a b ) < c( ab ) ( 5 . 2 )

and c( ab) = b( ac)-that is [ c(ab)] = [ b( ac)]-via the permutation of X = {a , b, c} which exchanges the points b and c. This induces the additional ordering relation

[ ( a b ) ] < [ W a c ) ] (5 .3 )

as shown by the broken line in the diagram. Of course, which links are deemed to be ‘extra’ depends on the choices that are made for a representative topology in each equivalence class of topologies. In an intrinsic sense, all links have equal status and there is no real distinction between them.

The non-lattice nature of this partially-ordered set is illustrated by the sub-block

which shows, for example, that the ‘join’ of the pair of equivalence classes [(ab)] and [ a ] would have to be both [ a (ab ) ] and [b(ac)]. We shall see below how this partial ordering can be used in the quantum theory.

Note that this ordering on T(X) /Perm(X) can also be derived from the ordering on T ( X ) defined by

7 1 < 7 2 if 371 -- r, and 74 = r2 with r{ 7;. (5.4)

Equivalently, T~ < 72 if there exists a continuous bijection i : ( X , T ~ ) -+ (X, T ~ ) . This is a special case of the natural ordering on the class of all topological spaces ( Y , T )

defined by ( Y, , T ~ ) < ( Y 2 , T ~ ) if there exists an injection i : Yz -+ Y , which is continuous. This provides an elegant way of introducing sets X of different cardinalities. If we wish to remain within the category of sets we can d o so by requiring all spaces Y to have cardinal number less than or equal to some fixed one. It remains to be seen if

1526 C J Isham

this extension of the scheme being discussed in this paper can be applied to the quantum theory.

Let us return to the problem at hand and consider the imposition of Dirac-style constraints on the Hilbert space L’(Z(T(X)), d p ) associated with the quantum theory on T ( X ) . A representation of Perm(X) can be defined by

(u (A1)$ , ) ( f j = +(A4-1(f,)) 11 E Perm( X ) ( 5 . 5 )

which is unitary if y is Perm(X) invariant. If p is only quasi-invariant (that is, if the transformed measure y := A * p has the same sets o f measure zero as the original measure p j , a unitary representation can be obtained in the usual way by defining

( U ( N $ ) ( I ) = ( d p l ( I ) /dp j1’2$(A-1(I ) ) (5.6)

where d p , ( I ) / d p is a Jacobian (Radon-Nikodym derivative) of the transformed measure y , with respect to 3u.

A simple calculation shows that

U ( A ) R , U ( A - ’ = R , t 7 , ( 5 . 7 )

so that the R, variables transform covariantly with respect to Perm(X). The same is not automatically true of the h.3, operators unless the Jacobian a(h, I ) := d p ,( I ) / d y satisfies the condition

(5.8)

Of course, if y is Perm(X)-invariant (so that ct = 1) then this is necessarily true. In any event, we will assume from now on that such a measure has been found so that

l ) G k ’ ( , 1 - ’ , $ ( T ) f? 14 ’ ( I ) ) = 1.

U(h )M,U(A) - ’ = M,(,, 15.9)

in which case we will say that the quantum topology representation is Perm(X)- equivariant.

To illustrate the basic idea let us consider the case where 1x1 is finite. Then from (5.8)-(5.9) it follows that gauge-invariant operators can be constructed by defining

(5.10)

where 0 is an orbit of Perm(X) on ~ ( x ) . For example, when X ={a , b, c} we have

(5.11)

The crucial question for us is whether or not these new operators satisfy any sort of algebra. By explicit calculation we find that, whereas R ( o h ) & ( a c ) = Rha(ah),ac), the new operators satisfy the relation

R [ ~ o / , l ] R ; b ( a i ) ] = R r h a ( n h l j a i ) l + K [ b l o c ~ l (5.12)

where, it should be noted, the two entries on the right-hand side are precisely the two elements which appear in the partial ordering diagram for d X ) / P e r m ( X ) as the ‘least upper bounds’ of the equivalence classes [ ( a b ) ] and [ h( a c ) ] . Similarly

M[,ab)lM(h(a< 11 = 6Mo+ M [ W ) , (5.13)

and again we see the appearance of the appropriate elements from the partial ordering diagram. This is not a coincidence but follows from the general theorem (for finite 1x1).

Quuntum topology 1527

Theorem. If 0 and 0’ are two orbits of Perm(X) on T(X), then there exist positive integers n,(O, 0’) such that

RoRo = c n,(O, O’)Ro, (5.14)

where the sum is over all orbits 0, that pass through topologies T v T‘ for some T E 0 and T ’ E 0’.

1

ProoJ:

and so

RoRo = c c R,”,. 7t 0 T E 0’

Each term T v T’ on the right-hand side must lie on some orbit. But then, if 4 E Perm(X), ~ ( T V 7’) = &(T) v +(T‘) and 4 ( ~ ) (resp. $(T’)) lies on the same orbit 0 (resp. 0’) as T (resp. 7’) and is hence included in the sum over this orbit. I t follows that the right-hand side also includes a contribution from $(T v 7’) and, since this is true for all 4, we see that the T v T’ terms always appear in complete orbits of Perm(X). QED

Note that since the orbits which appear are precisely all those that pass through topologies of the form T v T’ for some T E 0 and T‘E Of, they are indeed given by the links in the partial ordering diagram for T(X)/Perm(X).

A similar result holds for products of MrT1. However, the situation is more complicated for mixed products. For example, when X = { U , b, c}:

M [ h ( u c ) ] K [ ( n h ) ] = R(uh)Mc(rrh,+ R(hclMa(bc) + R(<ulMbica) (5.15)

and once again we see the effects of the ‘extra’ links in the partial ordering diagram for T(X)/Perm(X). But the right-hand side is not of the form ROMo or sums of such. On the other hand, it is of the form C R,M, where ( T, 7’) belongs to an orbit of Perm(X) on the Cartesian product T(X) x r ( X ) .

This suggests strongly that gauge-invariant operators should be of the form

To= c R,M, ( T, 7’) E 0

where 0 is an orbit in T ( X ) x T ( X ) . Indeed,

(5.16)

Theorem.

TOTO, = c n, (0, 0’) To, (5.17)

for some integers n,(O, 0’) where i labels an orbit of Perm(X) in T(X) x T(X) .

ProoJ:

1528 6 J Isham

However, if ( a v c, b A d ) belongs to an orbit 0” in r ( X ) x ~ ( x ) , and if 4 E Perm(X) , then

4 ( a v c, b A d ) = ( 4 ( a ) v 4(cL 4 ( b j A 4 ( d ) ) Thus complete orbits appear on the right-hand side. Furthermore, for any 4 E Perm(X), ( $ ( b ) , c ) = 1 if and only if ( J 4 ( b ) , 4 ( c ) ) = 1. Q E D

By these means we have succeeded in finding a set of gauge-invariant operators which satisfy a well defined algebra. This is the outcome of the Dirac constraint method as applied to the observables in the original Q = T(X) theory. A similar analysis can be used to study ‘physical’ states, although there is a possibility here of the analogue of ‘ 6 factors’-that is, physical states might only be invariant up to an overall phase factor.

Note that although this approach has been phrased in the context of the first of our three a priori ways of handling the gauge group, it is clear that the procedure can be reversed and Perm(X )-invariant quantisation dejined to be the construction of representations of the algebra (5.17), where the numbers n,(O, 0‘) are determined from the partial-ordering diagram for r ( X ) / P e r m ( X ) . This could be regarded as a bonajide quantisation of the system whose configuration space is the space T ( X ) / P e r m ( X ) of homeomorphism classes of topology.

The technique discussed above is rather obvious when X is finite but it gets trickier in the infinite-dimensional case when it becomes necessary to integrate in some way over the orbits of Perm(X) on T(X) x r ( X ) . One possibility is to put a suitable topology on Perm(X) so that it becomes a topological group with something like a Haar measure. The topological structure should presumably be such that the map Perm(X) x r ( X ) + r (X) is continuous with respect to the topology on ~ ( x ) . Given such a situation we could define

(5.18)

The analogue in the finite case would involve replacing (5.10) with the slightly different definition

(5.19)

where Perm(X). denotes the little group of Perm(X) at the point T E r ( X ) . One way of equipping Perm(X) with a suitable topology might be to embed it as

a closed (and therefore compact) orbit in the compact Hausdorff space I ( r ( X ) ) . The minimum needed for this is a point in I ( T ( X ) ) whose isotropy group is trivial-that is, a topology on X with n o non-trivial homeomorphisms. When X = {1,2, . . . n } is finite, an example of such a topology is

T:={0,{1},{1,2},{1,2,3}.*.{1,2 ) . . . n - l } , X } (5.20)

since any non-trivial permutation of X must affect at least one of these subsets thus giving a trivial isotropy group.

This suggests that it might be necessary in the infinite case to invoke some sort of well ordering of X in order to construct an analogous topology. However, this remains a task for the future and, for the moment, the ideas outlined above for a finite X are only suggestive of what might be done when 1x1 = CD.

Quantum topology 1529

6. Dynamics of quantum topology

Arguing by analogy with conventional quantum gravity, there appear a priori to be two quite different ways of approaching the concepts of ‘time’ and ‘dynamics’ within a theory of quantum topology. The first is to employ a functional integral over all topologies whose boundary (in the topological sense) is the disjoint union of a pair of topologies T, and r2. This is a topological analogue of the cobordism picture of topology change in the context of the Euclidian quantum gravity programme. The major problems would be to construct an analogue of the j,w R ( g ) action of the conventional theory and to show that this classical action, and the spacetime manifold on which it is defined, can be recovered in some suitable semiclassical limit. A distance function defined on the set of all topologies of the given type might be useful in this context (assuming that the space of all such topologies is metrisable, which is doubtful). But of course there is a distinct possibility that the associated rigorous measure theory will require the introduction of an analogue of the distributional dual on which the measures in quantum field theory are defined.

A quite different approach is to introduce the concepts of ‘time’ and ‘dynamical evolution’ within the context of the ‘canonical picture’ developed in this paper in which state vectors are thought of as functions +(T) of topologies T (or, more generally, as functions on the space of ideals in T ( X ) ) . The applicability of the concept of time is as obscure within this framework as it is in the cobordism picture. The central problem is a suspicion that the continuum nature of the time line is an inconsistent concept at the scale of the Planck length/time. If this is true, it means that the conventional picture of an evolving wavefunction is only valid at scales well above this scale and hence it is incorrect simply to attach a time label to the quantum state $(T). Any proper discussion of this issue will require an understanding of how classical general relativity is to be rederived from, or inserted into, the purely topological ideas being discussed here.

However, these worries notwithstanding, it is interesting to see if there is any concept of ‘time’ evolution or whatever which is natural in some sense to the structure of the lattice T ( X ) itself. One possibility might be to adopt a discrete model for time and argue that the evolution of $(T) is governed by a difference equation with an appropriate concept of locality in T. Thus one might postulate that the difference +(T, t ) - +(T, t + 1) is determined by the equations (taking for simplicity the case of finite X ) :

$(T, t ) - $ ( T , t + l ) = $(TI, t )T(T’ , 7) (6.1) 7 E N ( 7 )

where T(T’, 7) is the evolution matrix and the sum ranges over topologies that are suitably ‘near’ to T.

In the context of the lattice structure of T ( X ) , a natural choice for N ( T ) might be all the topologies that ‘cover’ T or are covered by it. (A topology T’ covers a topology T if T’> T and there is no topology T” such that T’> T”> T.) There have been several studies of covering relations for general topologies (for example, Larson and Thron 1972) but it remains to be seen if any of this material can be used to construct a discretised type of dynamics.

Another possibility is to override our reservations concerning continuous time and see if there are natural models for ‘conventional’ Hamiltonians within the lattice structure of T ( X ) . The critical properties required of such operators are Hermiticity

1530 C J Isham

and positivity. In the case of finite X , the basic operators acting on eigenstates I T ’ ) were given in (4.14)-(4.15), (4.18) and, motivated by the analogy of the familiar simple harmonic oscillator, the two obvious families of positive, Hermitian operators are MZM, and M,M:. By a simple calculation we find

and

Thus we see that a natural candidate for a ‘free’ Hamiltonian is

(6.4)

since the basic states I T ) arc eigenstates of the operators M : M , with eigenvalues proportional to the p measure of the set m;’(T’). Then

Hob) = E,,(+) (6 .5 ) where

Note that every E,(T) contains a contribution from a = 1 = P(X) (the discrete

(6.7)

topology) and furthermore

m;’{T} = { 7‘11 A 7’ = T } = { T }

and hence

Thus the ground state of the ‘free’ Hamiltonian in this quantisation scheme is the eigenstate 11) corresponding to the discrete topology, and the ground-state energy E,, is h(1).

If 7 is an anti-atom a then only a = T contributes to the sum in (6.8). Also,

WZ,’{CX)={T’(T’A c U = @ } = { c Y , I} (6.9)

(6.10)

m;’{T} = { T ’ I T ’ z T } = ? ( T ) (6.11)

and hence

Eo(7) = h( l )+PL( , r (T ) )h (T )+ c h(a)/L.[m,’(.)I lP(7). (6.12)

Note that h can be any non-negative function. However, this arbitrariness is cut down to a considerable extent if the Hamiltonian Ho is required to be gauge-invariant under the action of Perm(X) in the sense that

U ( A ) H o U ( h ) - ’ = Ho. (6.13)

{ a l T < a < l )

Quantum topology 1531

For example, in a Perm(X)-equivariant quantisation, the same value of h must be assigned to any homeomorphic topologies.

The other natural family of Hamiltonians are functions of the quadratic operators M,MS. It follows from (6.3) that the states 17) are not eigenstates of these operators. Thus if some ‘perturbing’ function of them is added to H,, the resulting Hamiltonian will describe genuine quantum mechanical changes of topology whose transition probabilities can be computed using standard perturbation theory. However artificial this model may seem, it does show that it is possible in principle to construct bona jide quantum theories in which topology changes can occur.

7. Conclusions

We have seen that a quantisation of the set of all topologies on a given set X can be achieved with the aid of the preferred set (3.9) of functions R, which satisfy the v algebra of the lattice T(X). A key ingredient is the spectral theory of this algebra which shows that all cyclic representations can be realised on a Hilbert space L*(I ( T ( X ) ) , d p ) where is some measure on the compact space I ( T ( X ) ) of all ideals in T(X). The complementary variables are defined via the operators M , in (4.7) which close on the A algebra of the lattice and intertwine with the R, variables to give the basic algebraic structure in (4.8). The operation (3.9), ( & $ ) ( I ) = ( I , T ) $ ( Z ) , can be vieAwed as an analogue of the representation of smeared quantum fields as (( p ( f ) $ ) ( x ) = (,y,f)$(,y). Similarly, the definition (4.7) of the conjugate operation, ( M , $ ) ( I ) = $ ( J ( T ) n I ) is reminiscent of the quantum field theoretic operator V(g) = exp i.rr(g) defined as

( V ( g ) $ ) ( x ) = $(x + g ) (7.1) where the function g E E on the right-hand side is regarded as an element of E‘ via the embedding of E in E‘ as a dense subspace. This similarity between the canonical field operations of conventional quantum field theory and the operators of quantum topology is certainly suggestive and implies that it might be possible to regard T(X) x I ( . ( X ) ) as some sort of ‘phase space’ for the topological system. However, there are also important differences between these two cases. Firstly, in quantum topology it is the Hermitian operators R, which generate the fundamental Abelian algebra, whereas in the quantum field theory this role is played by the unitary operators U ( f ) . Secondly, and of considerably more significance, the operators in (7.1) are required to be unitary so that their generators, the conjugate fields n-(g), are self-adjoint. As it stands, (7.1) only yields a unitary operator if the measure p on E‘ is invariant under translations by elements of the dense subspace E. If, as is likely, the measure is only quasi-invariant, then (7.1) needs to be augmented by the addition of a suitable Jacobian factor on the right-hand side.

However, the situation for quantum topoLogy is rather different. We can certainly add a multiplier and redefine the operator M , as

(7.2) with appropriate ‘cocycle’ conditions on the multiplier p to preserve the basic algebraic relations (4.8). But the absence of the unitarity condition means there is considerably less restriction in the choice of the multiplier p than there is in quantum field theory. Similarly, it is not clear if there is anything in quantum topology that replaces the feature in conventional unitary representation theory whereby equivalent measures

(G7$)(I) = P(7 , W(i(7) n I )

1532 C J Isham

(that is, measures with the same sets of measure zero) yield representations that are unitarily equivalent. This potential proliferation of representations needs further study, especially when X is infinite. It would be of great help in this context if an analogue could be found of the Gaussian measure in conventional quantum field theory which provides the Hilbert space for the free field. It seems likely that the construction of any such measure would require the introduction of a background metric (or even manifold) topology on X, and this in turn raises the issue of how spaces of this type fit into the lattice scheme. This problem is of considerable importance since it relates to the whole question of how conventional general relativity and differential geometry are to be derived from, or inserted into, this topological discussion. Ideally, structures of this type would presumably ‘emerge’ from the theory in some sort of low-energy limit.

Another major problem requiring clarification when X is infinite is the possibility of turning Perm(X) into a topological group so that the sums used in 5 5 to construct gauge-invariant observables can be extended to integrals in an appropriate way.

At a somewhat more general level, one might wonder if there are other quantisation schemes related to the one presented here. One possibility is to adopt the same strategy as in this paper but with a different set of basic observables on T ( X ) . In this context it is worth noting that there are probability theorists who bring to the problem of making any and all mathematical structures ‘probabilistic’ the same degree of enthusiasm that some theoretical physicists devote to the task of ‘quantising’ everything in sight! In particular there is a well developed field of ‘probabilistic topology’ (Frank 1971, Schweizer and Sklar 1983) in which the analogue of our functions R, is played by the family of functions f y , A , where x E X and A c X, defined by

if x E cl,(A) otherwise (7.3)

where cl,(A) is the closure of the subset A = X in the topology T. The associated characteristic sets are

C,,,:={TE T ( X ) ~ X E ~ ~ , ( A ) } (7.4) which is a lower set in the lattice T(X) but not apparently an ideal. This, and the fact that the functions fr,A d o not seem to possess any useful algebraic properties, is basically because the ‘closure functions’ cl, : P ( X ) + P ( X ) only satisfy the string of inequalities:

clTV+(A) c cl,(A) n cl7.(A) c cl,(A) U cl,.(A) c ~ 1 , , ~ , ( ~ ) (7.5) whereas, for example, to show that CX,* is an ideal it would be necessary to prove that cl,(A) n cl,,(A) = cl,,,,(A).

There are many other interesting observables in quantum topology, even if they cannot readily be viewed as part of a set of basic functions in the sense above. A particularly interesting example are the cohomology groups H k ( X , T), where k (respec- tively T) is any positive integer (respectively Abelian group). These can be regarded as (Abelian group valued) functions on T(X) and can be converted into real-valued ‘observables’ by the family of yes-no questions which asks whether any particular cohomology group is equal to a particular Abelian group. An especially relevant formulation of cohomology in this context would that due to Cech who defined these groups in terms of coverings of X by collections of open sets.

More adventurous possibilities exist than merely changing the basic set of functions on T(X) . For example, there are several interesting sublattices of T ( X ) which might be used instead of T(X) itself; one much-studied example is the set of all T, topologies (that is, given any pair of points in X there always exists an open set which contains

Quantum topology 1533

one of them and excludes the other). The only a priori restriction on the ‘configuration space’ of our quantum theory is that it should include manifolds (and presumably metric spaces) in order that the classical limit of standard general relativity can be included at some time. This allows for the possibility of generalising smooth manifolds with structures that are similar to, but not the same as, conventional topologies. A good example is the set of Convergence structures on X . Roughly speaking, a conver- gence structure consists of a specification of which sequences of elements in X are deemed to ‘converge’, and to what. A metric topology is uniquely determined by its convergent sequences, but there exist convergence structures that are not the set of convergent sequences for any topology on X . The set of all convergence structures on X can be made into a lattice but, unlike the lattice of topologies, it is distributive. This allows several new and interesting operations in the quantum theory. For example, complements are unique in a distributive lattice and therefore one can define a n operator like

(7.6) where a’ denotes the complement of a. A possible application of this type of operation to Ashtekar’s new variables is contained in Rayner (1989).

There are a variety of other possible generalisations or modifications of the scheme discussed in this paper, but rather than muddy the water any further let me conclude instead by voicing a doubt about whether the concept of ‘quantum topology’ may not after all be intrinsically inconsistent. If conventional approaches to quantum gravity (including string theory) are to be attacked on the grounds that they incorporate smooth manifolds as a fundamental a priori ingredient, could it not be argued that the use of the real o r complex numbers in quantum theory is likewise inconsistent at the scale of the Planck length? The crucial point is the plausible argument that the origin of the real/ complex numbers in quantum theory is that measurements of any physical quantity ultimately involve measurements of distances in space and/or intervals of time, and these are modelled precisely by the real numbers. Thus the use of real/complex numbers in quantum theory presupposes the applicability of the differential-manifold picture of space or time. But it is the suggested inapplicability of such a picture which motivated the attempt to quantise all topologies in the first place! One possible implication of this situation is that the concept of quantum topology is only meaningful if attention is restricted to metric topologies since the real numbers are involved in their definition in an intrinsic way (Isham 1988).

The more iconoclastic idea that quantum theory itself is inconsistent at the Planck length has long haunted the dreams of people working in quantum gravity, and it may well be true. However, it is important to see how far conventional ideas of quantum theory can be made compatible with the more esoteric ends of quantum gravity, and I hope that in the particular case of ‘quantum topology’ the present paper has shown that it is possible to construct a plausible mathematical framework which incorporates this idea without doing too much violence to standard quantum theory. Of course, if someone asks me ‘how d o you actually observe one of your so-called ‘observables’ R, ?’ then, as is often the case in quantum gravity,. . . !

( 6aG)( 1 ) := $ ( U ’ A I )

Acknowledgments

I am most grateful to Paul Renteln, Yura Kybushin and especially Noah Linden for many discussions on the subject of quantum topology.

1534 C J Isham

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