---

Finite diffeomorphism invariant observables

arXiv:gr-qc/9302011v1 10 Feb 1993Finite diffeomorphism invariant observablesin quantum gravityLee Smolin∗Department of Physics, Syracuse UniversitySyracuse, New York 13244 U.S.A.ABSTRACTTwo sets of spatially diffeomorphism invariant operators are constructedin the loop representation formulation of quantum gravity.This is doneby coupling general relativity to an anti- symmetric tensor gauge field andusing that field to pick out sets of surfaces, with boundaries, in the spa-tial three manifold. The two sets of observables then measure the areas ofthese surfaces and the Wilson loops for the self-dual connection around theirboundaries.

The operators that represent these observables are finite andbackground independent when constructed through a proper regularizationprocedure. Furthermore, the spectra of the area operators are discrete sothat the possible values that one can obtain by a measurement of the areaof a physical surface in quantum gravity are valued in a discrete set thatincludes integral multiples of half the Planck area.These results make possible the construction of a correspondence be-tween any three geometry whose curvature is small in Planck units and adiffeomorphism invariant state of the gravitational and matter fields.

Thiscorrespondence relies on the approximation of the classical geometry by apiecewise flat Regge manifold, which is then put in correspondence witha diffeomorphism invariant state of the gravity-matter system in which thematter fields specify the faces of the triangulation and the gravitational fieldis in an eigenstate of the operators that measure their areas.∗smolin@suhep.phy.syr.edu1

1IntroductionThe problem of observables is probably the key problem presently con-fronting non-perturbative approaches to the quantization of the gravita-tional field.The problem is difficult because it reflects, in two differentways, the problem of making sense of a diffeomorphism invariant quantumfield theory.Already at the classical level, it is non-trivial to constructand interpret functions of the dynamical variables that are invariant underspacetime diffeomorphisms. While, it is true, one can describe in words acertain limited number of them, what is needed is much more than this.First of all, we must have an infinite number of observables, correspondingto the number of physical degrees of freedom of the theory.

Second, to makethe transition to the quantum theory one must know their Poisson algebra.When we come to the quantum theory, new problems emerge becauseessentially all diffeomorphism invariant observables involve products of localfield observables, and so are not directly defined in the representation spaceof any quantum field theory. They must be defined through a limiting pro-cedure which is analogous to the regularization procedures of conventionalquantum field theories.

Furthermore, it is necessary to confront the factthat none of the conventional regularization or renormalization procedurescan be applied to this case. This is because they all depend on the presenceof a background metric.

It is then necessary to define new regularizationprocedures which may be applied to field theories constructed without abackground metric. Additional background structure does come in the defi-nitions of the regulated operators; what must be shown is that in the limitsthat the regulators are removed the resulting action is finite and backgroundindependent.In the last year and a half, some progress[1, 2] has been made on the prob-lem of observables in the context of a nonperturbative approach to quantumgravity based on the Ashtekar variables[3] and the loop representation[6, 7]1.This approach is based on taking as a starting point a quantum kinemati-cal framework which is based on non-Fock representations[8, 9, 2] of certainnon-local observable algebras.

It seems necessary to pick non-Fock represen-tations as the starting point for the construction of diffeomorphism invariantquantum field theories to avoid the dependence of the Fock structure on afixed background metric.What has been learned using this approach may be summarized as1Reviews of previous work in this direction are found in [4, 5, 2].2

follows[1, 2]:a) It seems impossible to construct background independent renormal-ization procedures for products of local fields. This is because any localrenormalization procedure is ambiguous up to a local density.

As a con-sequence, because we are working in a formalism in which the basic localobservable is a frame field, this theory has no operator which can representthe measurement of the metric at a point. This further means that diffeo-morphism invariant operators cannot normally be constructed from integralsover products of local fields.b) Despite this, there are several non-local observables that can be con-structed as operators acting on kinematical states by a regularization pro-cedure appropriate to the non-perturbative theory.

In all of the cases inwhich a well defined operator exists in the limit that the regularization isremoved, that operator is finite and background independent. Among theseoperators are those that measure the area of any given surface, the volumeof any given region and the spatial integral of the norm of any given oneform.

By measurements of these observables the metric can be determined,in spite of the fact that there is no local operator which can represent themetric.c) The connection between background independence and finiteness ismost likely general, as there are arguments that for any operator that canbe constructed through a point splitting regularization procedure of the typeused in [1, 2], background indepdence implies finiteness[10].d) The spectra of the operators that measure areas and volumes arediscrete series of rational numbers times the appropriate Planck units.e) Using these results, the semiclassical limit of the theory can be under-stood. Given any classical three metric, slowly varying at the Planck scale, itis possible to construct a kinematical quantum state which has the propertythat it is an eigenstate of the above mentioned operators and the eigenvaluesagree with the corresponding classical values in terms of that metric, up toterms of order of the inverse of the measured quantity in Planck units.These results are very encouraging, but they are subject to an importantlimitation.

They concern the kinematical level of the theory, which is theoriginal state space on which the unconstrained quantum theory is defined.The physical states, which are those states that satisfy the Hamiltonian anddiffeomorphism constraints, live in a subspace of this space.It would then be very desirable to find results analogous to these holdingfor physical operators. In this paper I report results which bring us signif-icantly closer to that goal.

These include the construction of a number of3

operators which are invariant under spatial diffeomorphisms. For example,as I will show below, it is possible to construct a diffeomorphism invariantoperator that measures the area of surfaces which are picked out by the val-ues of some dynamical fields.

Just as in the kinematical case, the spectrumof this operator is discrete and includes the integral multiples of half thePlanck area.The basic idea on which these results are based is to use matter fieldsto define physical reference frames and then use these to construct diffeo-morphism invariant observables. Of course, the idea of using matter fieldsto specify dynamically a coordinate system is very old.It goes back toEinstein[12], who pointed out that in order to realise the operational defini-tion of lengths and times in terms of rulers and clocks in general relativity,it was necessary to consider the whole dynamical system of the rulers andclocks, together with the gravitational field.

The application of this idea tothe quantum theory was first discussed by DeWitt[13], and has been recentlyrevived by Rovelli[14, 16] and by Kuchar and Torre[15].In a paper closely related to this one, Rovelli has used a scalar field topick out a set of surfaces whose areas are then measured. In this paper I takefor the matter field an antisymmetric tensor gauge field, with dynamics asfirst written down by Kalb and Ramond[17].

There are two reasons for this.First, as we will see below, the coupling of the antisymmetric tensor gaugefield to gravity is particularly simple in the Ashtekar formalism, which allowsus to hope that it will be possible to get results about physical observables,which must commute also with the Hamiltonian constraint.The secondreason is that, as I will describe, the configurations of the Kalb-Ramondfield can be associated with open surfaces, which has certain advantages.Now, it is clear on the kinematical level that if one measures the area ofevery two dimensional surface one determines the spatial metric completely.One can then imagine that if one has a finite, but arbitrarily large, set ofsurfaces, one can use measurements of their areas to make a partial mea-surement of the metric. Such an arrangement can serve as a model of anapparatus that might be used to measure the gravitational field, becauseindeed, any real physical measuring device returns a finite amount of infor-mation and thus makes only a partial measurement of a quantum field.

As Idiscuss below, there are a number of results and lessons that can be learnedabout measurement theory for quantum gravity by using a finite collectionof surfaces as a model of a measuring apparatus.Once we have a set of finite spatially diffeomorphism invariant operators,defined by using matter fields as a quantum reference frame, it is interest-4

ing to try to employ the same strategy to construct physical observables2.One can add to the theory additional matter degrees of freedom which canrepresent physical clocks and use these to construct operators which com-mute with the Hamiltonian constraint but describe measurements made atparticular times as measured by the physical clock. This idea is developedin [24].

Furthermore, once one has physical operators that correspond tomeasurements localized in space and time by the use of matter fields toform a spacetime reference system, it is possible to give a formulation of ameasurement theory which may be applied to quantum cosmology. A sketchof such a measurement theory is also developed there.This paper is organized as follows.

In the next section I show how tocouple an antisymmetric tensor gauge field to gravity. This is followed bysection 3 in which I show how to quantize the tensor gauge field in termsof a surface representation that is closely analogous to the abelian looprepresentation for Maxwell theories3.

I then show how to combine theseresults with the loop representaion of quantum gravity and how to constructdiffeomorphism invariant states of the coupled system. Here it is also shownhow to construct the diffeomorphism invariant operator that measures thearea of the surface picked out by the quantum state of the antisymmetrictensor gauge field.

In section 4 I consider a straightforward extension ofthese results in which certain degrees of freedom are added which result inquantum states which are labled by open rather than closed surfaces. In thenext section, which is section 5, I show how the matter field may also beused to construct a diffeomorphism invariant loop operator.

This has theeffect of adding Wilson loops of the left-handed spacetime connection aroundthe edges of the surfaces picked out by the quantum states of the matter.These results are then used in section 6, where I show how to constructquantum reference systems by combining the surfaces from a number ofindependent matter degrees of freedom to construct simplicial complexes.2Note that, as pointed out by Rovelli[16], there is a model diffeomorphism invarianttheory in 3 + 1 dimensions whose Hamiltonian constraints is proportional to a linearcombination of the gauge and diffeomorphism constraints.This is the Husain-Kucharmodel[18], which corresponds, at the classical level, to a limit of general relativity inwhich the speed of light has been taken to zero. The physical state space of this model isjust the space of diffeomorphism invariant states of quantum gravity and the physical innerproduct is known and takes a simple form in the loop representation[18].

The observables Iconstruct here are then examples of physical observable if we adjoin to the Husain-Kucharmodel the antisymmetric gauge field.3Rodolfo Gambini has kindly informed me that many of the results of this section werefound previously in [19].5

We find here a very interesting correspondence between certain peicewise-flat Regge manifolds and the elements of a basis of diffeomorphism invariantstates of the coupled matter- gravity system. I also sketch an approach toa measurement theory for quantum gravity which is based on the resultsdescribed here[24].

Finally, the implications of these results are the subjectof the conclusion.2Coupling an antisymmetric tensor gauge theoryto gravityAn antisymmetric tensor gauge field[17] is a two form, Cab = −Cba subjectto a gauge transformation generated by a one form Λa by,δCab = dΛab. (1)It’s field strength is a three form which will be denoted Wabc = dCabc.

Thecontribution to the action for these fields coupled to gravity is, in analogywith electromagnetism,SC = k4Zd4x√ggadgbegcfWabcWdef(2)where k is a coupling constant with dimensions of inverse action and all thequantities are, just for moment, four dimensional.In the Hamiltonian theory4 its conjugate momenta is given by ˜πab =−˜πba so that{Cab(x), ˜πcd(y)} = δ[ca δd]b δ3(y, x)(3)where the delta function is understood to be a density with the weight onthe first entry and ˜πcd is also a density. We will also find it convenient towork with the dualized fields˜W ∗= 13!ǫabcWabc(4)andπ∗a = 12ǫabc˜πbc.

(5)4From now on we consider that the indices a, b, c, ... are spatial indices, while indicesi, j, k, ... will be internal SO(3) indices. Densities, as usual, are sometimes, but not always,indicated by a tilde.6

The gauge transform (1) is then generated by the constraintG = ∂c˜πcd = dπ∗cd = 0(6)This field can be coupled to gravity in the Ashtekar formalism by addingto the Hamiltonian constraint the termCmatter = k2( ˜W ∗)2 + 12kπ∗aπ∗b˜˜qab(7)and adding to the diffeomorphism constraint the termDmattera= π∗a ˜W ∗. (8)We may note that the term added to the Hamiltonian constraint is nat-urally a density of weight two, so that it is polynomial without the necessityof changing the weight of the constraint by multiplying by a power of the de-terminant of the metric, as is necessary in Maxwell or Yang-Mills theory[20].The antisymmetric tensor gauge field can be understood to be a theoryof surfaces in three dimensions in the same sense that Maxwell theory is atheory of the Faraday flux lines[21].

By (6) there is a scalar field φ such thatlocally5π∗a = dφa(9)The equipotential surfaces of φ define a set of surfaces which are the ana-logues of the Faraday lines of electromagnetism. Further, any two dimen-sional surface S defines a distributional configuration of the πab by,πabS (x) =Zd2Sab(σ)δ3(x, S(σ))(10)here σ are coordinates on the surface.

Note that πabS is automatically di-vergence free.These are completely analogous to the distributional con-figurations of the electric field[2, 22] that may be associated to a curve γby˜Eaγ(x) =Zdγa(s)δ3(x, γ(s))(11)We can define a diffeomorphism invariant observable which depends on boththe metric and ˜πcd which has the property that when ˜πcd has such a dis-tributional configuration it measures the area of that surface. This can be5Global considerations play no role in this paper.7

defined either by generalizing the definition of the area observable[1, 2] ormore directly byA(π, ˜E) ≡Q(π, ˜E) =Z q˜˜qabπ∗aπ∗b(12)An equivalent expression for this is given byA(π, ˜E) = limN→∞NXN=1qA2approx[Ri](13)where space has been partitioned into N regions Ri such that in the limitN →∞the regions all shrink to points. Here, the observable that is mea-sured on each region is defined by6,A2approx[R] ≡ZRd3xZRd3y T ab(x, y)π∗a(x)π∗b(y)(14)To show the equivalence between these two expressions, we may startwith (12) and regulate it the way it is done in the quantum theory byintroducing a background euclidean coordinate system and a set of testfields fǫ(x, y) byfǫ(x, y) ≡pq(x)ǫ3θ[ ǫ2 −|x1 −y1|]θ[ ǫ2 −|x2 −y2|]θ[ ǫ2 −|x3 −y3|](15)In these coordinateslimǫ→0 fǫ(x, y) = δ3(x, y)(16)We can then writeA(π, ˜E) = Q(π, ˜E) = limǫ→0Zd3xsZd3yZd3zT ab(y, z)π∗a(y)π∗b(z)fǫ(x, y)fǫ(x, z)(17)When the expression inside the square root is slowly varying in x we canreexpress it in the following way.

We divide space into regions Ri which6Here T ab(x, y) is defined as follows[2]. Let there be a procedure, based on a backgroundflat metric, to associate a circle γx,y to every two points in the three manifold Σ. Thendefine T ab(x, y) ≡T rUγxy(y, x) ˜Ea(x)Uγxy(x, y) ˜Eb(y), where Uγ(x, y) ≡PexpGRγ A isparallel transport along the curve γ from x to y.8

are cubes of volume ǫ3 centered on the points xi = (nǫ, mǫ, pǫ) for n, m, pintegers. We then write,A(π, ˜E)=limǫ→0Xiǫ3sZd3yZd3zT ab(y, z)π∗a(y)π∗b(z)fǫ(y, xi)fǫ(z, xi)=limN→∞NXN=1qA2approx[Ri](18)If we now plug into these expressions the distributional form (10) it isstraightforward to show thatA(πS, ˜E) =ZS√h(19)where h is the determinant of the metric of the two surface, which is givenby h = ˜˜qabnanb where na is the unit normal of the surface.3QuantizationIt is straightforward to construct an algebra of loops and closed surfaces tocoordinatize the gauge constraint surface, corresponding to the impositionof both G = 0 and the SU(2) Gauss’s law of the gravitaional fields.

We mayassociate to every closed surface S a gauge invariant observable,T[S] ≡eık RS C.(20)Conjugate to T[S] we have the observables ˜πab(x) which satisfy the algebra,{T[S], ˜πab(x)} = ıkZd2Sab(σ)δ3(x, S(σ))T[S](21)We would like now to construct a representation of this algebra as analgebra of operators. We can construct a surface representation in whichthe states are functions of a set of closed surfaces Ψ[{S}] 7.

In order toimplement the abelian gauge invariance we require that these states areinvariant under reparametrization invariance and satisfy two relations. First,we require thatΨ[S ◦S′] = Ψ[S ∪S′](22)7Such a representation was first constructed in [19].9

where S and S′ are any two surfaces that touch at one point and S ◦S′is the surface made by combining them. Second, we require that Ψ[S] =Ψ[S′] whenever eRS F = eRS′ F for every two form F. We then define therepresentation by,ˆT[S′]Ψ[S] = Ψ[S′ ∪S](23)andˆπab(x)Ψ[S] = ¯hkZd2Sab(σ)δ3(x, S(σ))Ψ[S].

(24)It then follows that the operators satisfy[ ˆT[S], ˆπab(x)] = −¯hkZd2Sab(σ)δ3(x, S(σ)) ˆT [S](25)We should now say a word about dimensions. In order that the interpre-tation of A[π, ˜E] as an area work, it is necessary that ˜πbc have dimensionsof inverse length, from which it follows from (7) that k have dimensionsinverse to ¯h and that the dimensions of Cab are mass/length.

This choiceis consistent with both the Poisson bracket and the requirement that theexponenent in (20) be dimensionless.We now bring gravity in via the standard loop representation[6]. Thestates are then functions, Ψ[α, S], of loops and surfaces.

We may introducea set of bra’s < α, S| labled by loops and surfaces so that,Ψ[α, S] =< α, S|Ψ >(26)We then want to express the area observable (13) as a diffeomorphism in-variant operator and show that it does indeed measure areas. It is straight-forward to show that the bras< α, S| are, for nonintersecting loops α, eigenstates of the operator ˆA.

Thisoperator may be constructed by using the expression (13) as a regulariza-tion, in the way described in detail in [2]. A straightforward calculationshows that for the case of nonintersecting loops8< α, S| ˆA2approx[R] = (¯hkl2P lanck2)2I[α, S ∩R]2 < α, S|(27)8The appearance of the Planck area is due to the presence of G in the definition of theparallel propogators for the Ashtekar connection, A, as in the kinematical case [1, 2].

Thisis due to the fact that it is GAa that has dimensions of inverse length. The dimensionalityof the gravitational constant thus manifests itself in the appearance of the Planck area inthe operator algebra for quantum gravity.10

where I[γ, S] is the intersection number given by,I[γ, S] ≡Zdγa(s)Zd2Sbc(σ)δ3(S(σ), γ(s))ǫabc(28)and where S ∩R means the part of the surface that lies inside the region.It then follows from (13) that< α, S| ˆA = ¯hkl2P lanck2I+[α, S] < α, S|(29)where I+[α, S] represents the positive definite unoriented intersection num-ber which simply counts all intersections positively. Thus, we see that theoperator assigns to the surface an area which is given by ¯hkl2P lanck/2 timesthe number of intersections of the loop with the surface9.The action (29) of the area operator is diffeomorphism invariant, becausethe surface is picked out by the configuration of the field.

(One may checkthat this is also the case when the loop has an intersection at the surface. )The operator is then well defined acting on states of the formΦ[{α, S}] =< {α, S}|Φ >(30)where {...} denotes equivalence classes under diffeomorphisms.

On the spaceof diffeomorphism invariant states we can impose the natural inner product.Again, restricted to the case of nonintersecting loops this must have theform,< {α, S}|{β, S′} >= δ{α,S}{β,S′}(31)where the delta function is a kronocker delta of knot classes. The definitionof the inner product on intersecting loops may be obtained by imposingreality conditions.

The complete set of reality conditions at the diffeomor-phism invariant level is not known, but it is known that an inner productthat satisfies (31) is consistent with the requirment that ˆA be a hermitianoperator.We may then conclude that the spectrum of ˆA is discrete. It consists firstof the series integer multiples of ¯hkl2P lanck/2, together with a discrete seriesof other eigenvalues that come from eigenstates similar to those discussedin [2] in which the loops have intersections at the surfaces.FInally, may then note that if we require that the diffeomorphism invari-ant operator yield, when acting on kinematical states of the kind described in9When the loop α has intersections at the surface S there are additional terms in theaction of the area operator[2].11

[1, 2], the same areas as the kinematical area operator, we get the conditionthat,k = 1¯h(32)With its coupling thus set by ¯h, the antisymmetric tensor gauge field is thenin a sense purely a quantum phenomena.4Adding a boundaryIn the next section I am going to make use of the quantum antisymmetrictensor gauge field to construct a quantum reference system for measuring thediffeomorphism invariant states of the gravitational field. For this and otherpurposes, it is convenient to have states which are labled by open surfaces inaddition to those described in the previous section in which gauge invariancerestricts the surfaces to be closed.

As I will now describe, there is a verysimple way to do this, which is analogous to the Abelian Higgs model andwas described first by Kalb and Ramond[17]. We will see that by couplingthe Cab field to a vector field in a way that preserves the gauge invariance(1) we open up the possibility for our surfaces to have boundaries.Let us consider then adding to the system described by (2) an ordinaryabelian gauge field, ba, with an Abelian gauge group given byδba = ∂aφ(33)where φ is a scalar field.

We may couple this field to the Abelian tensorgauge field by supplementing the gauge transformations (1) byδba = Λa(34)Thus, we see that this vector field can be set to zero by a gauge transform.A field strength for ba that is invariant under both abelian gauge invariancesmay be defined byFab = dbab −Cab(35)To define the dynamics of this coupled system we add to the action thetermSb = k4Zd4x√ggabgcdFacFbd(36)We can define a constained Hamiltonian system by adding (2) and (36) tothe gravitational action. If the conjugate momenta to the ba are labled as12

˜pa the diffeomorphism and gauge constraints (8) and (6) are nowDa = ˜W ∗π∗a + ˜pcFac(37)andGa = ∂b˜πab + ˜pa(38)The Hamiltonian constraint has additional terms, which are given by12k ˜pa˜pbqab + k2det(q)FacFbdqabqcd(39)Note that the new terms are non-polynomial, when expressed in terms ofthe canonical variables ˜Eai , as in the case of the Maxwell and Yang-Millstheories[20]. (As in that case this can be remedied by multiplying throughby det(qab).) Finally, there is a new constraint,g = ∂c˜pc(40)which generates (33).

This, however, is not independent of (38) as∂aGa = g(41)As a result, there are now three independent gauge constraints and six eachof canonical coordinates and momenta.Thus, the theory now has threedegrees of freedom per point. The two additional degrees of freedom arereflected in the fact that in addition to the one gauge invariant field ˜W ∗,we now have the gauge invariant two form Fab.

Three of these four gaugeinvariant degrees of freedom are independent, because we havedFabc = −Wabc(42)As a result, we can associate gauge invariant observables to each open surfaceS. This is given byT[S] = e1kRS F(43)The poisson brackets of this with the canonical momenta ˜πab and pa aregiven by{˜πab(x), T[S]} = −ıkZd2Sab(σ)δx(x, S(σ))T[S](44){pa(x), T[S]} = ıkZdsδx(x, ∂S(s)) ˙∂S(s)T[S](45)13

The surface representation defined by (23) and (24) can be extended inthe obvious way. The arguements of the states are now open surfaces andthe obvious combination laws of surfaces hold.

In addition to (24), whichstill holds, there is the operatorˆpa(x)Ψ[S] = ¯hkZdsδx(x, ∂S(s)) ˙∂S(s)Ψ[S]. (46)Finally, one may check that the gravitational degrees of freedom may beadded and the area operator defined, so that all the results of the previoussection extend naturally to the surface representation with boundaries.5A diffeomorphism invariant loop operatorGiven that the matter fields specify a set of surfaces with boundaries, wemay imagine constructing a diffeomorphism invariant holonomy operator,analogous to the T 0[α] operators of the kinematical theory, in which theloop α is given by the boundary ∂SI of the surface determined by the I’thmatter field.To do this we first need to construct an appropriate diffeomorphisminvariant classical observable that will measure the holonomy of Aia on suchloops.

This can be done by using the fact that ˜pa, by virtue of its beinga divergence free vector density, defines a congruence of flows. These flowsmay be labeled by a two dimensional coordinate σα, with α = 1, 2, whichmay be considered to be scalar fields on Σ that are constant along the flows.The idea is to define a generalization of the trace of the holonomy from acurve to a congruence by taking the infinite product of the traces of theholonomies over each curve in the congruence.

This may be done in thefollowing way.Each divergence free vector density may be written as a two form interms of the two functions σα as [21],p∗ab = (dσ1 ∧dσ2)ab(47)where the σα are two scalar functions that are constant along the curvesof the congruences and so may be taken to label them. The curves of thecongruences may be written as γap(σ, s) and satisfy,˙γap(σ, s) ≡dγap(σ, s)ds= ˜pa(48)14

We may note that because each ˜pa is divergence free it is the case thatthrough every point x of Σ there passes at most one curve of the congruence.We will denote this curve by γp(x). We may take as a convention that if nocurve of the congruence passses through x we have γp(x) = x, which is justthe degenerate curve whose image is just the point x.

Further, note thatwe assume that either appropriate boundary conditions have been imposedwhich fix the gauge at the boundary or we are working in the context of aclosed manifold Σ, for which the curves γp(x) are closed.We may then define a classical observable which is the trace of the holon-omy of the connection around the curve γp(x).W[p, A](x) = TrUγp(x)(49)where Uγ is the usual path ordered holonomy of A on the curve γ. We maynote that the observable W[p, A](x) transforms as a scalar field.We may now write a diffeomorphism and gauge invariant observablewhich isT[p, A] ≡eRdσ1dσ2LNTrUγp,σ(50)To show that this is indeed diffeomorphism invariant, as well as to facilitateexpressing it as a quantum operator, it is useful to rewrite it in the followingway.

Let S˜p be an arbitrary two surface subject only to the condition thatit intersects each curve in the congruence determined by ˜pa exactly once sothat I[γp,σ, S˜p] = 1. Then we may writeT[p, A] = eRd2Sab˜p p∗abLNW [p,A](51)The diffeomorphism invariance of this observable is now manifest.

To seewhy this form may be translated to a diffeomorphism invariant quantumoperator, we may note that it reduces to a simple form if we plug in for ˜pathe distributional divergence free vector density˜paα ≡Zdsδ3(x, α(s)) ˙αa(s). (52)It is then not hard to show thatT[pα, A] = TrPeRα A = T[α](53)We may now define a quantum operator ˆT corresponding to (51) byreplacing ˜pa with the corresponding operator (46),ˆT = T[ˆp, ˆA].

(54)15

As all the operators in its definition commute, there is no ordering issue. Itis then straightforward to show that< {S, γ}| ˆT =< {S, γ ∪∂S}|(55)That is, the action of the ˆp operators in (51) is by (46) to turn theoperator into a loop operator for the holonomy around the surface.

Theresult is that what the operator ˆT does is to add a loop to the diffeomorphismequivalence class {S, γ} which is exactly the boundary of the surface. Thus,we have succeeding in constructing a diffeomorphism invariant loop operator.I close this section by noting two extensions of this result.

First, if oneconsiders the case of Maxwell-Einstein theory, where both fields are treatedin the loop representation10, one has an analogous operator, where ˜pa shouldbe taken to be just the electric field.In this case, if a diffeomorphisminvariant quantum state is given by Ψ[{α, γ}] where α are the abelian loopsthat represent the electromagnetic field and γ are the loops that representthe gravitational field and ˆT is the operator just described we haveˆTΨ[{α, γ}] = Ψ[{α, γ ∪α}]. (56)That is, the operator puts a loop of the self-dual graviational connectionover each loop of the electromagnetic potential.Second, all of the considerations of this paper apply to the system whichis gotten by taking the G →0 limit of general relativity in the Ashtekarformalism[30].

This limit yields a chirally asymmetric theory whose phasespace consists of all self-dual configurations together with their linearizedanti-self-dual perturbations.In this case there are operators ˜eai and Aiawhich are cannonically conjugate, but the internal gauge symmetry is theabelian U(1)3 reduction of the internal SU(2) gauge symmetry. One thenhas an operator analogous to (51), which is justT[e, A]G→0 ≡eRσ ˜eai Aia(57)The corresponding quantum operator has the effect of increasing the windingnumbers of loops that are already present.

It is also interesting to note thatin this case, T[e, A]G→0 commutes with the Hamiltonian constraint, so thatit is actually a constant of the motion [30].10The loop representation for Maxwell fields is described in[28, 29] and the coupling ofMaxwell to gravity in the Ashtekar formalism is described in [20].16

6A quantum reference systemI would now like to describe how the preceding results can be used to con-struct a physical interpretation of a very large class of diffeomorphism in-variant states. As I mentioned in the introduction, the idea of using matterfields to provide a dynamically defined coordinate system with respect towhich a diffeomorphism invariant interpretation of the gravitatational fieldscan be defined was introduced into quantum gravity by De Witt’s paper [13]in which he applied the Bohr-Rosenfeld analysis to the problem of the mea-surability of the quantum gravitational field.

It is interesting to note that inthis paper DeWitt concluded that it was impossible to make measurementsin quantum gravity that resolved distances shorter than the Planck scale.The results of the present paper reinforce this result and add to it two im-portant dimensions: first that, at least in one approach, it is impossible tomeasure things smaller than Planck scales because the fundamental geomet-rical quantities are quantized in Planck units and second, that it is areasand volumes11, and not lengths, whose measurements are so quantized.Let us consider, for simplicity, that there are many species of antisymmetric-tensor gauge fields, (CIab, bIa), labled by the index I = 1, ..., N, where N canbe taken arbitrarily large. This is a harmless assumption as long as we areconcerned only with spatially diffeomorphism invariant states.

I will comeback to this point in the conclusion.By the straightforward extension of all the results of the previous section,quantum states are now functions of N surfaces, SI, so thatΨ[{γ, SI}] =< {γ, SI}|Ψ >(58)We may note that the space of diffeomorphism equivalence classes, {γ, SI}of loops and N labeled open surfaces is countable12. The diffeomorphisminvariant state space of quantum gravity coupled to the N antisymmetrictensor gauge fields then has a countable basis given byΨ{α,S′I}[{γ, SI}] = δ{α,S′I}{γ,SI}(59)in the case that the loop γ is not self-intersecting.

In the intersecting case,the form of the basis elements is more complicated because of the presenceof the non-trivial relations among intersecting loops which result from theidentities satisfied by SU(2) holonomies. For the kinematical case, these11For the volume operator see [2].12Note that each surface may be disconnected.17

relations, and the effect on the characteristic inner product are describedin [2]. For the present, diffeomorphism invariant, case they have not yetbeen completely worked out.

However, for the results I will describe belowit is sufficient to restrict attention to diffeomorphism equivalence classesinvolving only non-intersecting loops.Let us now consider a particular subspace of states of this form which aredefined in the following way. Let us consider a particular triangulation of thethree manifold, Σ, labled T .

It consists of some number, M, of tetrahedra,labled Tα, where α = 1, ..., M, that have been joined by identifying faces.Let us call the faces FI and let us consider only T that contain exactlyN faces so that I = 1, ..., N. The idea is then to use this triangulation toconstruct a quantum coordinate system by identifying each face FI with thesurface SI which is an excitation of the I’th matter field.We do this in the following way. For each such triangulation of Σ we canconsider a subspace of states, which I will denote ST , which consists of allstates that have the formΨ[{γ, SI}] = δ{FI}{SI}ψ[{γ, SI}].

(60)where the δ{FI}{SI} is, again, a topological Kronocker delta that is equalto one if and only if each surface SI can be put in correspondence withthe face FI such that all the topological relations among the surfaces arepreserved. Such an arrangement of surfaces can be taken to constitute aquantum reference frame.

The states in ST can then take any value as wevary over the countable set of diffeomorphism equivalence classes in whichthe loops are knotted and linked with the surfaces in T and with each otherin all possible diffeomorphically inequivalent ways.If we impose an additional restriction, we can make a correspondencebetween a basis for ST and a countable set of peicewise flat three dimen-sional manifolds based on the simplicial complex T . This restriction is thefollowing: in any three dimensional simplical complex the number of faces,F(T ) is greater than or equal to the number of links, L(T ) [23].

For a rea-son that will be clear in a moment, let us restrict attention to T such thatF(T ) = L(T ).Let us then consider the characteristic basis for ST given by (59) with{SI} = T . In any such state we may then associate a definite value for thearea of each face in T , which is given by the eigenvalue of ˆAI.We may then associate to each set of areas AI a piecewise flat manifold,which I will call M{AI,SI}, which is composed of flat tetrahedra glued to-gether with the topology of T such that the areas of the faces are given by18

the AI. We know that generically this can be done, because such piecewisegeometries are determined by the edge lengths of the triangulation, and wehave assumed that the number of edges in T is equal to the number of faces.Thus, we may in general invert the N relations between the edge lengthsand the areas of the faces to find the edge lengths.

However, when doingthis, we need to be careful of one point, which the following.Note that we have chosen the signs while taking the square root in (13)so that all areas are positive. However, if we consider a tetrahedron in T ,there is no reason for the areas of the four sides to satisfy the tetrahedralidentities, which imply that the sum of the areas of any three sides is greaterthan the area of the fourth side.

This means that we cannot associate toeach tetradhedra of T a metrically flat tetrahedra, if we require that thesignature of its metric be positive definite. Instead, we must associate a flatmetric of either positive or negative signature, depending on whether or notthe classical tetrahedral identities are satisfied.

Thus, whether a particularsurface of a particular tetrahedra is spacelike, timelike or null depends onhow the identities are satisfied in that tetrahedra.However, each surface bounds two tetrahedra and there is no reason thatthe signiture of the metric may not change as the surface is crossed. Thus,a surface may be, for example, timelike with respect to its imbedding inone of the tetrahedra it bounds, and spacelike in another, as long as theabsolute values of the areas are the same.

Similarly, when the edge lengthsare determined from the areas it is necessary to use the appropriate formulafor each tetrahedra, which depends on the signature of the metric in thattetrahedra.Thus, the result is that the piecewise flat manifold M{AI,SI} that isdetermined from the N areas AI in general contains flat tetrahedra withdifferent signatures, patched together so that the absolute values of the areasmatch. Additional conditions, which are precisely the tetrahedral identities,must be satisfied if the geometry of M{AI,SI} is to correspond to a positivedefinite metric on Σ.We may note also that the correspondence between the piecewise flatthree geometry, M{AI,SI}, and the diffeomorphism equivalence classes {γ, SI}is not one to one.

Given M{AI,SI} we have fixed only the topology of thesurfaces and their intersection numbers with the loops.There remain acountable set of diffeomorphism equivalence classes with these specifications;they are distinghished by the knotting of the loops and their linking witheach other.19

Of this remaining information, a certain amount may be said to corre-spond to information about the spacial geometry that cannot be resolvedby measurements made using the quantum coordinate system T . We mayimagine further refining the quantum reference system by introducing newsurfaces by subdividing the tetrahedra in T .

If we consider how this maybe done while keeping the toplogical relations of the loops with themselvesand with the original set of surfaces fixed, we may see that there is a sensein which we can obtain a more precise measurement of the spatial quantumgeometry associated with the topology of the loops, γ. Of course, there isalso a danger that by subdividing too much we may reach a point whereadditional surfaces tell us nothing more about the quantum geometry; theinformation about the matter state and the quantum geometry is entangledand cannot be easily separated.

In a further work, I hope to return to theproblem of how to disentangle the geometrical from the matter informationin such measurements.At the same time, it is clear that there is information in the topology ofthe loops that is not about the spatial geometry and so cannot be resolved byfurther refinement of the simplex based on the matter state. This includesinformation about the routings through intersections.

It is clear from thisand earlier considerations that the routings through the intersections carryinformation about the degrees of freedom conjugate to the three geometry.One can obtain this conjugate information by measuring the operatorsT I ≡T[∂SI defined in the previous section. If one measures all N of theseoperators, rather than the N areas, one determines the parallel transportof the Ashtekar connection around the edges of each of the faces of thesimplex T .

Essentially, this means that one determines, instead of the areasof the faces, the left handed curvatures evaluated at the faces. There is alsoa classical description that can be associated with this measurement, it isdescribed in [24].I would like to close this section by describing the sense in which theresults just described suggest an approach to a measurement theory forquantum gravity13.

The idea is to extend the principle enunciated by Bohrthat what is observed in quantum mechanics must be described in terms ofthe whole system which includes a specification of both the atomic systemand the measuring apparatus. In the case of quantum gravity, the quantumsystem is no longer an atom, it is the whole spacetime geometry.

As thequantum system is no longer microscopic, but in fact encompasses the whole13These remarks are enlarged in [24].20

universe, we can no longer treat the measuring instrument classically whilewe treat the spacetime geometry quantum mechanically. Thus, it is nec-essary that a measuring system that is to be used to determine somethingabout the spacetime geometry must be prepared for the measurement byputting it in some definite quantum state.In this paper I have described two conjugate sets of measurements, whichdetermine either the areas of or the left handed parallel transport aroundareas a set of N surfaces.

However, the basic features of the measurementprocess and how we describe it should extend to more general measurements.Any measurement theory must have two components: preparation andmeasurement.If we are to use this measuring instrument to probe thequantum geometry, we must prepare the quantum state of the measuringinstrument appropriately. As we are interested in describing the theory ata diffeomorphism invariant level, we must give a diffeomorphism invariantspecification of the quantum state of the measuring instrument such that,when we act on the combined gravity matter state with the area operatorswe measure a set of areas which are meaningful.Now, the requirement of diffeomorphism invariance forbids us from prepar-ing the measuring system in some state and then taking the direct productof that ”appratus state” with a state of the system.

Instead, the preparationof the measuring system must be described by restricting the system to anappropriate diffeomorphism invariant subspace of the combined apparatus-gravity system. Thus, what I have done above is to prepare the quantumstate of the whole system in a way appropriate to the specification of themeasuring instrument by restricting the topological relations among the Nsurfaces so that they are faces of a given simplex T .

This is done by re-stricting the quantum state of the system, prior to the measurement, to beof the form (60). After we have made this restriction, we can be sure thatthe results of the N measurements will be a set of N areas that can beascribed to the faces of the simplex T .

Thus, the result of the measurementof the area operators on prepared states of the form (60) is to produce apartial description of the spatial geometry which is given by the piecewiseflat manifold Mγ,SI.Now, the N area operators ˆAI commute with each other, but they donot make a complete set of commuting observables. This is because to eachsuch peicewise flat manifold, which encodes the results of the N observables,there are a countably infinite number of diffeomorphism invariant states inthe subspace ST which are degenerate as far as the values of the ˆAI are21

concerned.We would then like to ask whether we can add operators to the ˆAI tomake a complete set of commuting operators. We certainly can extend theset, by subdividing some or all of the tetrahedra in T to produce a simplicialcomplex with more surfaces.

This would correspond to introducing morematter fields, which would make it possible to specify more surfaces whosearea is to be measured and by so doing make a more refined measurementof the quantum geometry. But, notice that there is a natural limit to howmuch one can refine one’s observations of the quantum geometry becauseone can never measure the area of any surface to be less than one-half Planckarea.Further, note that no matter how large N is, and no matter how the Nsurfaces are arranged topologically, there are always a countably infinite setof states associated to each measurement of the ˆAI’s.

In this sense, it seemsthat one can never construct a physical measuring system that suffices toextract all the information out of the quantum gravitational field. This is,of course, just a reflection of the fact that the quantum gravitational fieldhas an infinite number of degrees of freedom, while any physical measuringinstrument can only record a finite amount of information.

However, notethat we have come to an expression of this fact in a way that is completelydiffeomorphism invariant. In particular, we have a characterization of a fieldwith an infinite number of degrees of freedom in which we do not say howmany degrees of freedom are associated to each ”point.” This is very good,as we know that no diffeomorphism invariant meaning can be given to apoint of space or spacetime.Of course, there remains one difficulty with carrying out this type ofinterpretation, which is that the problem of time in quantum gravity mustbe resolved so that we know how to speak of the time of the observation.

In[24] I show that the resolution of the problem of time may be carried out us-ing the ideas of spacetime diffeomorphism invariant observables of Rovelli14[11]. To do this one constructs the physical time dependent operators thatcorrespond to the ˆAI and T I.

These depend on a time parameter τ which isthe reading of a physical clock built into the measuring instrument. The Noperators ˆAI(τ) will then commute with the Hamiltonian constraint, and soact on physical states and their eigenvalues return the values of the areas ofthe N surfaces when the physical clock reads τ.

Given this, there seems to14Note that such observables have been described in the 2 + 1 case by Carlip[25], in theGowdy model by Husain[26] and in the Bianchi-I model by Tate[27].22

be no obstacle to the observer employing the projection postulate and say-ing that the quantum state of the matter plus gravity system is projectedinto a subspace of the physical Hilbert space spanned by the appropriateeigenstates of the ˆAI(τ) and that this is something that occurs just afterthe measurement is made, in spite of the fact that she and her apparatusare living inside the quantum system under study.Thus, despite various assertations to the contrary, there seems to be nodifficulty in applying a Copenhagen-like description of the measuring processto the case of quantum cosmology in spite of the fact that the measuringinstrument is inside the universe. As long as we can prepare the measuringinstrument in such a way that the quantum state of the whole matter-gravitystate space is inside a subspace of the state space associated with a particularspecification of the measuring instrument one can assign meaning to a setof commuting observations.

The implications of this are discussed furtherin [24].Finally, we may note that one finds that the ˆAI(τ = 0) are equal tothe area operators constructed in this paper[24], so that the quantization ofareas becomes a physical prediction based on the spectra of a set of physicaloperators of the theory.7ConclusionsI would like to close by making a number of comments about the implicationsof the results obtained here.1) We see that in each case when we have succeded in constructing thedefinition of an operator in such a way that when it is diffeomorphism invari-ant it is automatically finite. This is in accord with the general argumentsthat all spatially invariant diffeomorphism invariant operators must be finitethat were given previously in [2, 10].

This suggests strongly that the problemof constructing a finite theory of quantum gravity can be to a great extentresolved at the diffeomorphism invariant level. The reason is that once oneimposes spatial diffeomorphism invariance there is no longer any physicalmeaning that can be given to a point in space.

As a result, although thetheory still has an infinite number of degrees of freedom, in the sense dis-cussed in section 6, it is no longer meaningful to speak of the field as havinga certain number of degrees of freedom per point. Instead, there seeems tobe a natural limitation to how many degrees of freedom there can be insideof a Planck volume due to the discreteness of the spectra of the geometrical23

operators that measure area and volume[2]. This in turn suggests that theproblem of finiteness has little to do with the dynamics of the theory or thechoice of matter couplings, which are coded in the Hamiltonian constraint.2) We also see that the conclusions of previous analyses of the measure-ment problem in quantum gravity by DeWitt and others are confirmed.

Thekey conclusion of these works was that it should be impossible to meaning-fully resolve distance scales shorter than the Planck scale. We see that this isthe case here, because the possible values of physical areas that can be got-ten from a diffeomorphism invariant measurement procedure are quantizedin units of the Planck area.We also see that any particular configuration of the matter fields thatare used to define the reference system can only be used to resolve a certainfinite amount of information about the space time geometry.This is aconsequence of using the quantum theory to describe the reference systemas well as the gravitational field.

This is certainly consistent with the generalobservation that diffeomorphism invariant measurements are about relationsbetween the gravitational field and the measuring instruments. If we wanta measurement system which is able to resolve N different spatial distances,it had better come equiped with N distinguishable components.3) We see that a large class of doubts about the physical applicability ofthe description of quantum states of geometry by means of the loop repre-sentation can now be put to rest.

Note that any spacial geometry in whichthe components of the curvatures are small in Planck units can be approx-imated by a Regge manifold in which the areas of the faces are integralmultiples of half the Planck area. As a result we see from the correspon-dence between Regge manifolds and quantum states arrived at in section6 that any such spatial geometry can be associated with a diffeomorphisminvariant quantum state in the loop representation.

This allows us to extendthe discussion of the classical limit of quantum gravity developed in [1] tothe diffeomorphism invariant level.4) It would be very interesting to be able to characterize the quantumgeometry associated with diffeomorphism invariant states of the pure grav-itational field. The results obtained with matter fields as reference systemssuggest that there should be a basis of states which are diagonal in someset of diffeomorphism invariant operators which measure the three geome-try and that this basis contains the characteristic states of non-intersectingknots and links.The problem is to construct an appropriate set of dif-feomorphism invariant classical observables which are functions only of thegravitational field and translate them into quantum operators while preserv-24

ing the diffeomorphism invariance. We already know how to construct a fewsuch operators, which measure the areas of extremal surfaces and, in thecase that it is spatially compact, the volume of the universe[2].One approach to the construction of such observables could be by mim-icking the results of this paper by constructing observables that measurethe areas of surfaces on the faces of a given simplex, and asking that all theareas are extremized as the whole simplex is moved around in the geometry.Constructions along this line are presently under study.5) Given the present results, a new approach to the construction of thefull dynamical theory becomes possible.

This is to impose an inner productconsistent with the reality conditions at the diffeormorphism invariant leveland then project the Hamiltonian constraint into the resulting Hilbert spaceof diffeomorphism invariant states.The physical state space would thenbe found as a subspace of the space of diffeomorphism invariant states.constraint The main difficulties facing such an approach are the problemof expressing both the reality conditions and the Hamiltonian constraint indiffeomorphism invariant forms.6) As I commented above, the form of the Hamiltonian constraint (7)for gravity coupled to the simple, massless antisymmetric tensor gauge fieldis particularly simple.It would be very interesting to see if solutions tothe Hamiltonian constraint for the coupled gravity matter system couldbe obtained, if not exactly, in the context of some perturbative expansion.It is intersting to note that exact solutions can be obtained in the strongcoupling limit in which k is taken to zero (because k is inverse to what wouldusually be written as the coupling constant). In this limit, only the secondterm of (7) survives.

It is easy to show, using a regularization of the typeintroduced in [6] that one then has a class of solutions of the form Ψ[{S, γ}]in which the loops never intersect the surfaces or in which the loops alwayslie in the surfaces. It would be very interesting to then develop a strongcoupling expansion to construct approximate expressions for solutions forfinite k. It would also be very interesting to see if one could recover fromthe semiclassical limit of the gravity matter system the solutions to theSchroedinger equation described in [19].7) It is interesting to note that surfaces play an interesting role in twomathematical developments connected to the Ashtekar variables and theloop representation.

In [33] Baez extends the loop representation to the casein which the spatial manifold has a boundary, and shows that in this casethere is an interesting algebra of operators that acts on the diffeomorphisminvariant states. In [34], Crane proposes a new interpretative scheme for25

quantum gravity in which Hilbert spaces of states coming from conformalfield theories are defined on surfaces which are identified with observersand measuring instruments. Both proposals need to be completed by theconstruction of explicit diffeomorphism invariant observables associated tosurfaces, and it would be interesting to see if the operators described herecan thus play a role in these proposals.8) Finally, I would like to address the issue of the use of N separate mat-ter fields to label the operators that measure the areas of the N surfaces.This is clearly necessitated by the idealization in which I use the values ofa field to specify a set of physical surfaces in a very simple way.

The pointis that there must be a physical way to distinguish the N different surfacesin terms of the configurations of the matter fields. In real life, in whichmeasuring instruments of arbitrary complexity are constructed from a smallnumber of fields there is no difficulty with specifying quantum states associ-ated to some degree of precision with an arbitrary number and configurationof surfaces.

In a realistic situation the configuration is complex enough toallow an intrinsic labeling of the different surfaces15 Of course, another is-sue also arises when we construct the surfaces out of realistic physical fields,which is that there will be restrictions to the accuracy of the measurementsof thea areas due to the fact that matter is made out of atoms. It is not,however, impossible that such limitations can be overcome by a clever useof matter and other fields to specify very small surfaces.

What the presentresults suggest, however, is that no matter how clever we are with the designof our measuring instruments, it will be impossible to measure the area ofany physical surface to be less than half the Planck area.ACKNOWLEDGEMENTSI would like to thank Abhay Ashtekar, Rodolfo Gambini and Carlo Rovellifor critical readings of a draft of this manuscript.I am very grateful tothem and to Julian Barbour and Louis Crane for crucial conversations aboutthis work and the general problem of constructing diffeomorphism invariantobservables. I would also like to thank Alan Daughton and Rafael Sorkin forconversations about simplicial manifolds.

This work was supported by theNational Science Foundation under grants PHY90-16733 and INT88-1520915This accords with the observation stressed by Barbour that real physical observablesare well defined because the world is sufficiently complex to allow the events of spacetimeto be distinguished by the values of the physical fields [35].26

and by research funds provided by Syracuse University.References[1] A. Ashtekar, C. Rovelli and L. Smolin, Physical Review Letters 69(1992) 237-240. [2] L. Smolin, Recent developments in nonperturbative quantum gravity inthe Proceedings of the 1991 GIFT International Seminar on Theoreti-cal Physics: ”Quantum Gravity and Cosmology, held in Saint Feliu deGuixols, Catalonia, Spain (World Scientific, Singapore,in press); Whatcan we learn from the study of non-perturbative quantum general rela-tivity?

to appear in the procedings of GR13, IOP publishers. [3] A. V. Ashtekar, Physical Review Letters 57, 2244–2247 (1986) ; Phys.Rev.

D 36 (1987) 1587. [4] A. Ashtekar , Non-perturbative canonical gravity.

Lecture notes pre-pared in collaboration with Ranjeet S. Tate. (World Scientific Books,Singapore,1991); in the Procedings of the 1992 Les Houches lectures.

[5] C. Rovelli, Classical and Quantum Gravity, 8 (1991) 1613-1676. [6] C. Rovelli and L. Smolin, Knot theory and quantum gravity Phys.

Rev.Lett. 61, 1155 (1988); Loop representation for quantum General Rela-tivity, Nucl.

Phys. B133 (1990) 80.

[7] R. Gambini and A. Trias, Phys. Rev.

D23 (1981) 553, Lett. al NuovoCimento 38 (1983) 497; Phys.

Rev. Lett.

53 (1984) 2359; Nucl. Phys.B278 (1986) 436; R. Gambini, L. Leal and A. Trias, Phys.

Rev. D39(1989) 3127.

[8] D. Rayner, Class. and Quant.

Grav. 7 (1990) 651.

[9] A. Ashtekar and C. J. Isham, Inequivalent observer algebras: A newambiguity in field quantization and Representations of the holonomyalgegbra of gravity and non-abelian gauge theories, Syracuse Universityand Imperial College preprints (1991). [10] C. Rovelli, unpublished.27

[11] For general discussions of the problem of diffeomorphism invariant ob-servables and the related problem of time see recent reviews by K.Kuchar,Time and interpretations of quantum gravity in the Proceed-ings of the 4th Canadian Conference on General Relativity and Rela-tivistic Astrophysics, eds. G. Kunstatter, D. Vincent and J. Williams(World Scientific, Singapore,1992); C. Rovelli, Phys.

Rev. D 42 (1991)2638; 43 (1991) 442; in Conceptual Problems of Quantum Gravity ed.A.

Ashtekar and J. Stachel, (Birkhauser,Boston,1991); Class. QuantumGrav.

8 (1991) 317-331 and L. Smolin, Time, structure and evolutionSyracuse Preprint (1992); to appear (in Italian translation) in the Pro-cedings of a conference on Time in Science and Philosophy held at theIstituto Suor Orsola Benincasa, in Naples (1992) ed. E. Agazzi.

Syra-cuse preprint, October 1992. [12] A. Einstein, in Relativity, the special and the general theory (Dover,NewYork).

[13] B. S. DeWitt, in Gravitation, An Introduction to Current Research ed.L. Witten (Wiley, New York,1962).

[14] C. Rovelli, Class. and Quant.

Grav. 8 (1991) 297,317.

[15] K. Kuchar and C. Torre, Utah preprint (1991). [16] C. Rovelli, A generally covariant quantum field theory Trento and Pitts-burgh preprint (1992).

[17] M. Kalb and P. Ramond, Phys. Rev.

D9 (1974) 2273. [18] V. Husain and K. Kuchar, Phys.

Rev. D 42 (1990) 4070.

[19] R. Gambini, Phys. Lett.

B 171 (1986) 251; P. J. Arias, C. Di Bartolo,X. Fustero, R. Gambini and A. Trias, Int.

J. Mod. Phys.

A 7 (1991)737. [20] A. Ashtekar, J. Romano and R. S. Tate, Physical Review D40 (1989)2572-2587.

[21] E. T. Newman and C. Rovelli, Phyys. Rev.

Lett. 69 (1992) 1300-1303.

[22] L. Smolin, Modern Physics Letters A 4 (1989) 1091- 1112. [23] I would like to thank Alan Daughton for a helpful conversation aboutthis.28

[24] L. Smolin, Time, measurement and loss of information in quantumcosmology Syracuse preprint 12/92, to appear in the Brill Feschrift, ed.B.-L. Hu and T. Jacobson (Cambridge University Press,1993). [25] S. Carlip, Phys.

Rev. D 42 (1990) 2647; D 45 (1992) 3584; UC Davispreprint UCD-92-23.

[26] V. Husain, Class. Quant.

Grav. 4 (1987) 1587.

[27] R. S. Tate, Constrained systems and quantization, Lectures at the Ad-vanced Institute for Gravitation Theory, December 1991, Cochin Uni-versity, Syracuse Univesity Preprint SU-GP-92/1-4; An algebraic ap-proach to the quantization of constrained systems: finite dimensionalexamples Ph.D. Dissertation (1992) Syracuse University preprint SU-GP-92/8-1. [28] A. Ashtekar and C. Rovelli, Quantum Faraday lines: Loop representa-tion of the Maxwell theory,Syracuse preprint 1991.

[29] R. Gambini and A. Trias, Phys. Rev.

D23 (1981) 553, Lett. al NuovoCimento 38 (1983) 497; Phys.

Rev. Lett.

53 (1984) 2359; Nucl. Phys.B278 (1986) 436; R. Gambini, L. Leal and A. Trias, Phys.

Rev. D39(1989) 3127.

[30] L. Smolin, The GNewton →0 limit of Euclidean quantum gravity, Clas-sical and Quantum Gravity, 9 (1992) 883-893. [31] R. Gambini, Loop space representation of quantum general relativityand the group of loops, preprint University of Montevideo 1990, PhysicsLetters B 255 (1991) 180.

R. Gambini and L. Leal, Loop space coor-dinates, linear representations of the diffeomorphism group and knotinvariants preprint, University of Montevideo, 1991; M. Blencowe, Nu-clear Physics B 341 (1990) 213.; B. Bruegmann, R. Gambini and J.Pullin, Phys. Rev.

Lett. 68 (1992) 431-434; Knot invariants as nonde-generate staes of four dimensional quantum gravity Syracuse UniversityPreprint (1991), to appear in the proceedings of the XXth InternationalConference on Differential Geometric Methods in Physics, ed.

by S.Catto and A. Rocha (World Scientific,Singapore,in press). [32] B. Bruegmann and J. Pullin, On the Constraints of Quantum Gravity inthe Loop Representation Syracuse preprint (1992), to appear in NuclearPhysics B.29

[33] J. Baez, University of California, Riverside preprint (1992), to appearin Class. and Quant.

Grav. [34] L. Crane, Kansas State Univesity preprint, (1992).

[35] J. B. Barbour,to appear in the Procedings of the NATO Meeting onthe Physical Origins of Time Asymmetry eds.

J. J. Halliwell, J. Perez-Mercader and W. H. Zurek (Cambridge University Press, Cambridge,1992); On the origin of structure in the universe in Proc. of the ThirdWorkshop on Physical and Philosophical Aspects of our Understandingof Space and Time ed.

I. O. Stamatescu (Klett Cotta); Time and theinterpretation of quantum gravity Syracuse University Preprint, April1992.30

---

출처: arXiv:9302.011 • 원문 보기