PHYSICAL REVIEW D VOLUME 37, NUMBER 10 15 MAY 1988

Gravitational 8 states and the wave function of the universe

J. B. Hartle and D. M. Witt Department of Physics, University of California, Santa Barbara, California 93106

(Received 1 February 1988)

The naive no-boundary wave function of the universe is shown to be invariant under diffeomorphisms only for the simplest spacetime topologies. A more general construction which does give an invariant wave function of the universe is exhibited. Similar problems, some familiar, some not, are encountered in a wide range of theories whose physical configuration space is topolog- ically nontrivial. These include the theory of identical particles, Yang-Mills theory, higher- dimensional gravity, and membrane theories. The sum-over-histories formulation of quantum mechanics provides a unified approach to these problems and their resolution.

The ground states of quantum theories whose physical must have configuration space is topologically nontrivial can display a richer structure than those of theories with topological- WgQ)=a(g )Y(Q) , g € 9 , (1)

ly trivial configuration spaces. ' Identical particles pro- where typically a(g) is a phase, simple examples, the vide the most familiar example: The multiply connected ground-state wave function can be constructed by a E ~ - configuration space2 allows for ground states with various clidean sum over histories of the form statistics and parastatistics. The 0 states of the Yang- Mills field provide another example. In quantum gravi- ty, the 0 states have been studied in the canonical frame-

@ ( Q ) = d p ( H ) e x p [ - I ( H ) ] , P, c

(2) - .

work by several author^.^ However, because of the ab- sence of obvious candidates for the ground state no expli- cit examples of such states were found. Analogs of the ground state have been proposed in the sum-over- histories framework. In quantum cosmology the analog of the ground state for closed universes is the "no- boundary" Euclidean sum-over-geometries construction of Hawking and his co- worker^.^ This state is a candi- date for the wave function of the universe. Since it is an explicit state, one can ask whether or not it changes un- der the action of diffeomorphisms or, in other words, how is it related to the 0 states. In this paper we shall show that if any but the most trivial topologies are admitted to the sum over geometries which defines the no-boundary wave function then it is not invariant under all diffeomor~hisms. We shall show how this construction can be generalized to give a wave function of the universe which is invariant under all diffeomorphisms. Although we work in the sum-over-histories framework our con- siderations are essentially topological in nature so that it should be possible to formulate them in any approach to quantum gravity.

We begin by reviewing the common situation using the sum-over-histories formulation of quantum mechanics. Four ingredients are required for a sum-over-histories formulation: the histories { H J , the action Z (H ), the mea- sure p ( H ) , and the physical configuration space of ob- servable~ 0. For many cases of interest the physical configuration space O is an identification by a symmetry group 9 of an extended configuration space @ whose indi- vidual members are subsets of histories. Thus, 0= e / 9 . States can be represented as wave functions on @, that is as Y(Q), Q E @, but physically allowed states must reflect the symmetry which defines the observables. That is, one

where the sum is over histories which match the condi- tions determined by the subset Q and satisfy further con- ditions C, typically asymptotic ones, designed to single out the ground state. I t is, therefore, natural to extend this construction to more complicated theories and the no-boundary wave function for cosmology is constructed in this way. If (2) is, by itself, to yield a physically sensi- ble wave function it must satisfy (1). It will if t_he symme- try group 9 is extendible to a symmetry group 9 of the ac- tion and measure which leaves the class of histories C in- variant. Then, if g E 9 and g E are the corresponding extensions,

@ ( g ~ ) = J dp(H)exp[ -Z iH) ] gQz c

However, there are interesting cases when 9 is not exten- dible in this way.

Even if @ does not satisfy ( I ) , there is a simple con- struction to produce from it a wave function which does. Let .no9 be the group of disconnected components of 9. A ground-state wave function which satisfies (1) may be constructed as

The sum is over the equivalence classes f which are ele- ments of r09, gf is any representative o f f in 9, and X( f ) is a one-dimensional representation of ao9. For then, if h denotes the equivalence class of g,

2833 @ 1988 The American Physical Society

2834 J. B. HARTLE AND D. M. WITT - 37

Y(gQ)= 2 X( f )@(gfgQ) In this framework, the observables O are the three- f geometries of spacelike surfaces 2 embedded in M. The

= 2 X( f )@(gfhQ) extended configuration space @ is the space of three-

f metrics h on Z denoted by Riem(Z). The symmetry group 9 is the group of diffeomorphisms of S , Diff(Z).

= XX( f h ' ) @ ( g f ~ ) = x ( h p l ) ~ ( ~ ) , ( 5 ) Thus, O = Riem( Z ) /Diff( Z ). The diffeomorphism groups f of three-dimensional manifolds have been extensively an-

where we have used the fact which follows from (3) that @ is constant for each representative of an element of no9. Further considerations will restrict the possible one-dimensional representations. Thus, for example, in theories with an external time (e.g., familiar quantum theories) symmetries should be represented by unitary transformations and X( f ) must define a unitary represen- tation. We shall illustrate other restrictions below.

Before discussing the gravitational case it may be use- ful to recall some more familiar examples of the above procedure. The simplest is quantum mechanics on a mul- tiply connected space, for example, the circle. Then, with the definition of a suitable angular coordinate 4, @=R and 9 is the group of translations by 27. No element of 9 is extendible because a history might be any path in R with a fixed origin so that C is not translationally invari- ant. Since n o 9 = Z the ground-state wave function is

This is the familiar Bloch result for periodic potentials. Yang-Mills theory in the temporal gauge provides

another examplee3 @ is the space of all SU(2)-valued three-vector potentials A. 9 is the group of SU(2bvalued gauge transformations U ( x ) . The histories are vector po- tentials on spacetime and the class C defining the naive ground-state wave function through (2) consists of Eu- clidean vector potentials which vanish sufficiently fast at Euclidean infinity. The gauge transformations which are homotopic to the identity are called "small gauge trans- formations." Their defining property is that for any U ( x ) there is a homotopy U ( x , r ) such that U ( x , O ) = U ( x ) a n d U ( x , l ) = I . Thesecanbeextended to histories in the class C by using the homotopy to define spacetime gauge transformations. If ?=0 is the time coordinate of the boundary, define the extended gauge transformation to be U ( X , T ) for 0 < T < 1 and the identity otherwise. Gauge transformations not in the identity component -"big gauge transformations" -cannot be so extended for the extension would define a homotopy to the identity. Thus, no9=r0 (gauge transformations)=Z and

where gn is the representative of an element of ~ T ~ Q = Z . These are the familiar Yang-Mills 8 states.

We now turn to the gravitational problem. For simpli- city of exposition we shall restrict attention to purely gravitational quantum cosmology in which histories are metrics on a fixed four-manifold M. The generalizations to include matter, to other dimensions, and to scattering problems should be both straightforward and interesting.

alyzed in several papers.637 The basic building block for constructing the no-boundary wave function is the func- tional @( h ) on Riem(Z) defined by choosing M to be a four-manifold with a single boundary, a M = Z , and then forming the sum

Here, I ( g ) is the Euclidean gravitational action, a func- tional of metrics on M. The integral is over a contour of metrics, in general complex, such that each point along the contour is a unique nonsingular metric on M. This defines the class C. It is assumed invariant under the diffeomorphisms of M as is the action I and the measure p. As a consequence of diffeomorphism invariance the formal expression (8) will diverge being a finite factor times the infinite volume ?/ of the diffeomorphism group of M. For this reason we have included a factor of ?/-I

to cancel this divergence. Of course, in practice one works in some discrete approximation to this expressions in which physical quantities are finite. For our present concerns, however, (8) will be sufficient.

All the quantities I, p, C, and ?/ in (8) depend on M and so does the result @. We have suppressed this depen- dence for notational compactness. What manifolds con- tribute to the wave function of our universe is part of the specification of the law of initial conditions. It could be that M is fixed in which case the simplest choice would be part of s4 with a single s3 boundary. It is more natural, however, to expect that the wave function of the Universe is a sum of terms such as (8) over different mani- folds M, a sum over topologies. There are, however, different views on the appropriateness of such a sum,9 and certainly it presents issues of computational complex- ity. lo If any but the simplest choice of manifold contrib- utes to the sum defining the wave function of the Universe then there is the gravitational analog of the phe- nomena discussed generally above as we shall now show.

The symmetry group 9 of the extended gravitational configuration space is Diff(aM). The issue is whether the elements of this group can be extended to elements of Diff(M) which is the symmetry of the action and mea- sure. Elements in the connected component of the identi- ty are always extendible because we have if both f and g are connected and orle of them is extendible, then both are extendible. In particular, every diffeomorphism connected to the identity is extendible.

ProoJ: Suppose g :aM-aM extends to g : M + M and that g is connected to f. Let F ( x , t ) be a family of diffeomorphisms connecting f and g, i.e., F ( x , O ) = f ( x ) and F ( x , 1 ) = g ( x ) . Near a M use the coordinates x = ( x , t ) so that x is parallel to a M and t is normal to a M with (x,O) E a M . Define F:M-M by

GRAVITATIONAL 6 STATES AND THE WAVE FUNCTION O F . . . 2835

F ( x , t ) , O < t 5 1 ; F(x ).= g ( ~ ) otherwise .

F is a smooth diffeomorphism and an extension of f . Q.E.D.

Thus, the diffeomorphisms connected to the identity al- ways contain Diffid(aM) and we only need investigate the extension of Diff( aM)/Diffid( a M ) = roDiff( aM).

Next, we consider some examples. The three-sphere s3= ((w,x,y ,z) I w ~ + x ~ + ~ ~ + z ~ = 1 ] is the simplest boundary which can occur and the part of the four-

vanishing homotopy group all maps are determined up to homotopy by their action on that homotopy group. l 2 In particular, diffeomorphisms are determined up to homo- topy by the isomorphism they determine. The group of isomorphisms of Z2 is GL(2,Z), and for Z3 the group is GL(3,Z). This means that up homotopy T 3 has more diffeomorphisms than D' x T2. Therefore, there are diffeomorphisms of T 3 which do not extend to D X T2.

Thus, in general, @( h ) given by (8) will not be invariant under all classes of 7roDiff(aM). An invariant candidate for the wave function of the universe is [cf. (411

sphere bounded by an s3 is the simplest model for M. This M is topologically equivalent to the four-ball Y ( h ) = 2 X ( f ) @ ( g f h ) ,

f € 7ioDiff(dMl B 4 = f (w,x ,y ,z) 1 w 2 + x 2 + y 2 + z 2 < 11. NOW, r 0 ~ i f f ( s 3 ) =Z2. The group is generated by the orientation- reversing diffeomorphism r ( w,x,y,z) = ( - w,x,y,z). This is extendible to E~ in the natural way. Therefore, for the case when a M =s3 and M is a part of an s4 all elements of iff(^^) are extendible to symmetries of the action and measure, and @ is invariant under iff(^^). No further construction of the form (4) is needed.

It is not difficult, however, to find examples where not every class in roDiff(aM) is extendible to a Diff(M). Consider, for example, a manifold M with a M =s3 for which the orientation-reversing diffeomorphism r is ex- tendible. Such manifolds have the property that there is an orientation-reversing diffeomorphism on the closed manifold a = M u B 4. The orientation-reversing diffeo- morphism r extends to all of 2 because it extends to both B4, and M by hypothesis. The extension of r to 2 is orientation reversing because it reverses orientation on B4. Thus in order to find a manifold for which r is nonextendible, one only needs to find a closed four- manifold with no orientation-reversing diffeo- morphisms and remove the interior of a ball from it. A simple example of such a closed four-manifold is c P 2 . To see this, let f :cP2-cP2 be any smooth mapping. The second de Rham cohomology vector space H 2 ( c p 2 ) is W, and is generated by a single two-form a. The mapping f induces a linear map f * : H ~ ( c P ~ ) - H ~ ( c P ~ ) which is just the pullback of forms. The fact that f * is linear means that f * a = h a where h is a real number. Taking the wedge product of a with itself one obtains the four-form a A a. The pullback of this four-form is

By definition of degree, " it follows that deg f = h2 2 0. An orientation reversing map has degf < 0 ; therefore, C P ~ has no orientation reversing mappings. In particu- lar, c P 2 has no orientation reversing diffeomorphisms. Therefore, r is nonextendible to CP' minus the interior of a four-ball.

Another example is the four-manifold D 2 x T~ which has boundary T3. The fundamental groups of D ~ X T 2 and T 3 are Z 2 = Z x Z and Z ~ = Z X Z X Z , respectively. Moreover, both of these manifolds have contractible cov- ering spaces so all of the other homotopy groups vanish. Now, in general, given a diffeomorphism of an arbitrary manifold to itself it naturally induces an isomorphism of the homotopy groups. For manifolds with only bne non-

where gf is a representation o f f in Diff(aM) and X ( f ) is a one-dimensional representation of roDiff( aM). We now turn to a discussion of what representations are possible.

We first note that, if the action is fixed, then a straight- forward application of sum-over-histories principles fixes X( f ) to be 1. Y is a functional of the observables, the three geometries [ h ] on aM. By [h] we mean the class of metrics equivalent to h under Diff(aM). According to the sum-over-histories prescription T ( [ h ] ) is, up to nor- malization, the sum exp( - I ) over all histories in the defining class C which are consistent with the three- geometry h. The histories are four-metrics on M. The sum over metrics therefore includes a sum over all three- metrics which represent [h]. Noting (8) we can write the sum over all metrics as

where the integral is over Dif laM), m is an invariant measure, and V is for the moment an arbitrary normaliz- ing factor. Equation (1 1) can be rewritten

since every g E f can be represented as ggf for some par- ticular gf E f and g €Diffid(aM). But since all elements of Diffid(aM) are extendible to elements of Diff(M), we have @(ggfh ) = @(gf h ). The integration therefore simply produces a factor of the volume of Diff,,(dM) and if V is chosen to be that volume one has

which is (10) with X( f ) = 1. This result follows because the restriction that Y be a function only of three- geometries implies that Y is strictly invariant under Diff(aM) not just invariant up to a factor as in (1). It then follows from ( 5 ) that X( f ) = 1. Of course, to a con- siderable extent the content of such invariance depends on the form of the action. For example, the addition of a Chern-Simons term to the Yang-Mills action leads to a theory whose ground state is equivalent to the 0 states be- cause the action depends on a 0 parameter but in which the ground-state wave functionals are invariant under gauge transformations. Analogous terms for gravitation

2836 J. B. HARTLE AND D. M. WITT 37 -

have been discussed. l 3

The straightforward application of the sum-over- histories prescription described above may be the sim- plest way of constructing a candidate for the wave func- tion of the universe but it may not be the most general. Such a prescription applied to identical particles, for ex- ample, would lead one to Bose statistics. It is therefore of interest to ask whether there are independent restrictions on the one-dimensional representations X(f) of ~ ,Di f f ( aM) . There are at least two.

The assumption that Diff(aM) is a symmetry group of the observables is the assumption that physical predica- tions of Y ( h ) should be the same as those of Y(gh), g EDiff(aM). Since the absolute square of V, specifies the joint probabilities for observations on a spacelike surface, and since physical predictions are conditional probabili- ties constructed from ratios of joint probabilities, l 4 Y must form a representation of Diff( a M ) up to normaliza- tion. It is possible that higher-dimensional representa- tions could be of interest but let us, for simplicity, restrict attention to the one-dimensional cases. This is (1) but un- like familiar quantum mechanics, where the square of the wave function is a directly interpretable physical proba- bility, the factors a ( g ) = X ( f ( g ) ) are not restricted to be phases. Were there but a single manifold M which en- tered into the sum over histories defining the no- boundary wave function there would be a variety of choices for X( f 1. Consider the example discussed above where a M =s3 and M = C P ~ - B ~ so that T , D ~ ~ ~ ( s ~ ) = z ~ . All one-dimensional representations of finite groups are of the form exp(in8) where the integer labels the repre- sentation and 8 is real.15 The reason is that fk=l for some finite k so the representation functions must be periodic. The possibilities for 8, however, are consider- ably restricted if one assumes time reversal invariance. Then the wave function is real16 so that X( f 1 = i 1. An alternating representation with some X( f) = - 1 can be certainly realized on Z2 so that there is this possibility in addition to the straightforward X( f ) = 1.

However, we do not expect only a single manifold to contribute to the no-boundarv wave function but rather a sum of them. Each manifold must support the same rep-

resentation of ~ ,Di f f ( aM) . If there is but one nonzero contributor for which the @( h ) is invariant, then we are restricted to the simplest representation with X( f )= 1. This is the case with a M =s3 if both s4 and Cp2 contrib- ute. As .n,DiflaM) is extendible for the simplest M s we expect this generically to be the case.

We conclude that in the presence of a sum over topolo- gies the no-boundary wave functions for individual mani- folds display a richness of transformation properties un- der aoDiff(aM) but that the straightforward construction of an invariant wave function (1 3) is the simplest most natural candidate for the generalization of the wave func- tion of the universe.

One natural generalization of this work is to consider the wave function for gravity coupled to other fields. In this case, the wave function must be invariant under both diffeomorphisms and the symmetries of the matter fields. Hence, if there are inextendible symmetries of the matter fields the correct wave function includes a sum over both T, of diffeomorphisms and the symmetries of the matter fields. In particular for gauge fields, there are gauge transformations which do not extend to the four- manifold. This is the case even in flat space as was dis- cussed earlier. It was first discussed in the context of quantum cosmology by ~ k r t c h ~ a n " although it seems that he did not realize how to restore the invariance of the wave function.

The nonextendibility of diffeomorphisms did not de- pend on the action used in describing gravity. Therefore, any geometric theory with general coordinate invariance and nontrivial topology will have the same problems of the wave function or transition amplitudes being not in- variant under diffeomorphisms. Examples of such theories are higher-dimensional theories of gravity and membrane theories. In the theory of oriented strings, one does not have this problem because only orientation- preserving diffeomorphisms are allowed and the only orientation-preserving diffeomorphisms of the circle or interval are connected to the identity and thus extendible.

This work was supported in part by NSF Grant No. PHY85-06686.

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