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The basis of the Ponzano-Regge-Turaev-Viro-Ooguri

arXiv:hep-th/9304164v1 1 May 1993The basis of the Ponzano-Regge-Turaev-Viro-Ooguriquantum-gravity model is the Loop Representation basisCarlo RovelliPhysics Dept. University of Pittsburgh, Pittsburgh PA 15260, USAand Dipart.

di Fisica Universita’ di Trento and INFN sez. Padova, Italiarovelli@pittvms.bitnetJuly 25, 2018AbstractWe show that the Hilbert space basis that defines the Ponzano- Regge-Turaev-Viro- Ooguri combinatorial definition of 3-d Quantum Gravity is thesame as the one that defines the Loop Representation.We show how tocompute lengths in Witten’s 3-d gravity and how to reconstruct the 2-d ge-ometry from a state of Witten’s theory.

We show that the non-degenerategeometries are contained in the Witten’s Hilbert space. We sketch an exten-sion of the combinatorial construction to the physical 4-d case, by defining amodification of Regge calculus in which areas, rather than lengths, are takenas independent variables.

We provide an expression for the scalar productin the Loop representation in 4-d. We discuss the general form of a nonper-turbative quantum theory of gravity, and argue that it should be given by ageneralization of Atiyah’s topological quantum field theories axioms.1

The problem of describing physics at the Planck scale and the quantumproperties of gravity, is the problem of understanding what is a non- trivialgenerally covariant quantum field theory. The last years have seen manydevelopments in our understanding of these theories: Witten’s introductionof topological field theories, in their two versions, a’ la Chern- Simon [1]and a’ la Donaldson [2]; Atiyah’s axiomatization of these [3]; dynamical tri-angulations techniques [4]; Ashtekar reformulation of general relativity [5],which opened the way to the Loop Representation [6], which lead to discoversolutions of the Wheeler-DeWitt equation, the relevance of Knot Theory inquantum gravity and a discrete structure of space at the Planck scale [7];Turaev and Viro’s [8] reformulation of the Ponzano-Regge model [9] in termsof quantum groups, which provides a combinatorial definition of 3- d topo-logical field theories; Crane and Vetter’s [10] extension of this constructionto 4-d; the pioneering work of Ooguri, which in 3-d has tied the Euclidean,combinatorial, canonical and topological definitions of quantum gravity [11],and in 4-d has opened the path to the Crane-Vetter work.

These resultsshare a remarkable common flavor, besides, of course, the common long termaim of quantizing gravity. In this paper, we find the bridge between the 3-dPonzano-Regge-Turaev-Viro-Ooguri (PRTVO) model and the Loop Repre-sentation, we discuss the physical interpretation of 3-d quantum gravity, andwe sketch a general theory of physical 4-d gravity in which all these linesmay converge.Ponzano and Regge [9] considered Regge-calculus [13] in 3-d, but madethe ansatz that the lengths li of the Regge-calculus links be constrained tobe half integers: li = ji = 1/2ni (integer ni).

Half integers ji can be in-terpreted as labels of SU(2) representations. The Regge calculus action canthen be written as a very simple expression, which is essentially a sum overthe tetrahedra of the triangulation of the 6-j symbols of the 6 (half- integer)lengths li = ji of the links of each tetrahedron.

The partition function ofQuantum Gravity can then be constructed by fixing a sufficiently thin trian-gulation ∆, and summing over its colorings c (assignments of half integersto every link). The reason for taking half-integer lengths, as well as the rela-tion between lengths of links and SU(2) representations appeared to be quitemysterious at the time.

In this paper we throw some light on this relation.Turaev and Viro [8] were able to show that the Ponzano–Regge partitionfunction is independent from the triangulation chosen (and transformed it ina finite sum by replacing SU(2), with a corresponding quantum group with2

a finite number of representations). Ooguri [11] has related the quantizationof 3-d quantum gravity based on this model to the Witten quantization ofthe same theory.

Ooguri construction can be summarized in short as follows.The quantum states of the Ponzano-Regge theory have to be taken, followingAtiyah’s general formulations of topological quantum field theories, as quan-tum combinations Φ∆(c) of the colored triangulations (∆, c) induced on the2-d boundary ∂M of the 3-d manifold M. In Witten theory, quantum statesare wave functions Ψ(ω) over the moduli space of the flat SU(2) connectionsAIa(x) on the 2-d boundary (ω being equivalent classes of AIa(x)’s). Oogurirelates the two representations of the theory byΨ(ω) =XcΦ∆(c) Ψ∆,c(ω),(1)where we have absorbed in Φ∆(c) a normalization factor appearing in eq.

(16)of Ref. [11].

The ”matrix elements of the change of basis”⟨ω|∆, c⟩= Ψ∆,c(ω)will described in a moment. Louis Crane and Lee Smolin have suggestedthat there may be a direct relation between the Ooguri construction and theLoop Representation [14]; in this letter, we show, indeed, that (1) is nothingbut the Loop Transform [6], which relates the connection representation ofQuantum Gravity with the Loop Representation.

The Ooguri representationΦ∆(c) is therefore essentially equivalent to the Loop Representation. In themaking this relation explicit, we will provide a physical interpretation of thePonzano-Regge ansatz according to which the Regge-calculus links have half-integer length and are related to SU(2) representations, and we will find thephysical justification of the Ooguri construction.The functions Ψ∆,c(ω) introduced by Ooguri, are constructed as follows.Given the triangulation ∆, we construct the trivalent graph dual to ∆.

Thearc Ci of this graph crosses the i-th link of the triangulation ∆; we associateto Ci the SU(2) representation ji, where ji is the half-integer that colorsthe i-th link of ∆.We then associate to Ci the function over flat SU(2)connections AIa given by the Wilson line Uji[A, Ci] = P exp{RCi AIatIjidxa},where the su(2) generators tIji are taken in the ji representation. Next, weconsider the product of all these Wilson lines, where 6-j symbols are used tocontract the indices at the trivalent intersections.

The resulting object is a3

function of the triangulation ∆, the coloring c and the (flat) connection A.It is gauge invariant, and thus it defines a function over the moduli space ofthe flat SU(2) connections for every (∆, c); this function is Ψ∆,c(ω) .In order to relate this construction with the Loop Representation, thefirst observation is that, since any representation of SU(2) is obtained bytensor multiplication of the j = 1/2 representation with itself, a Wilson lineUj[A, Ci] in the j representation can be expressed by means of 2j Wilsonlines U1/2[A, Ci] in the 1/2 representation. We exploit this fact by replac-ing each arc Ci of the trivalent graph with precisely 2ji lines, each carryinga U1/2[A, Ci] parallel transport matrix.

Accordingly, the sum at the triva-lent intersections obtained with the 6-j symbols, can simply be replaced bythe sum over all the possible rootings of these lines at the intersection. Fi-nally, Ψ∆,c(ω) can be reexpressed as a combination of products of tracesof holonomies of A along the resulting closed loops, all taken in the 1/2representation.

This follows from elementary properties of SU(2) representa-tion theory. In other words, the colored triangulation (∆, c) uniquely deter-mines an ensemble E∆,c = {α1, α2, ...} of multiple loops (sets of closed loops)αi = (αi1, αi2, ...αiN), where αij are (single) loops; each multiple loop αi hav-ing the property that precisely 2j single loops cross a link of the triangulationwith color j.

The ensemble E∆,c is defined as the set of all the homotopicallyinequivalent multiple loops with this property. By construction we have themain relation:Ψ∆,c(ω) =Xαi∈E∆,cYiTrU1/2[A, αij]Now, given a multiple loop αi, the productQi TrU1/2[A, αij] is nothing butthe loop state |αi⟩, written in the connection representation, namely⟨A|αi⟩= Ψαi =YiTrU1/2[A, αij](2)This relation is at the roots of the Loop Representation.

Using this relation,and its gauge invariance, we have⟨ω|∆, c⟩= Ψ∆,c(ω) =Xαi∈E∆,c⟨A|αi⟩=Xαi∈E∆,c⟨ω|αi⟩,or|∆, c⟩=Xαi∈E∆,c|αi⟩. (3)4

Eq. (3) provides an identification between the Loop representation basisstates |α⟩and the Ooguri states |∆, c⟩.

The relation is many-to-one becausethe loop states are not independent (they form an overcomplete basis). Thisrelation is our first result.In Ooguri’s work, the relation (1) is postulated, and the equivalence of thecombinatorial theory with Witten’s quantization is derived a posteriori byshowing the isomorphism of the two structures.

Still, the half-integer lengthsremain as mysterious as they were in the original Ponzano-Regge paper. Toprovide an interpretation of this fact, let us calculatethe lengths of thelinks of a triangulation in a fixed quantum state of the gravitational field.

Arecent calculation in (3+1)-d gravity, shows that the area of any surface isquantized in the Loop Representation in multiples of 1/2 (in Planck units);the area being precisely given by the number of intersections of the surfacewith the loops of the quantum state. It is natural to suspect that a similarrelation may work in one dimension less.

In fact, let us show it does. Thelength l of a curve C in 3-d gravity is given byl[C] =ZC dtsdCadtdCbdt gab =ZC dtsdCadtdCbdt EIcEIcǫacǫbd,where EIa is the variable conjugate to the connection and ǫac is the antisym-metric two dimensional pseudotensor.

We refer to [6] for the notation. Inorder to promote l[C] to an operator, we have to deal with the product ofthe two E’s.

Following Ref. [7], we point split the product EE, by means ofthe gauge invariant two-point objectTrhU[A, γ′ǫx]Ea(γǫx(0))U[A, γ′′ǫx]Eb(γǫx(π))i== T ab[γǫx](0, π).Here γǫx is a loop with radius ǫ, and center in x, γ′ǫx and γ′′ǫx are its twocomponents from the value 0 to π and from π to 0 of the loop parameter, andin the second equality we have introduced the standard Loop Representationnotation [6] for this operator.

Note that classically we havelimǫ→0 T ab[γǫx](0, π) = det ggab(x)The operator corresponding to the observable T ab[γǫx](0, π) is well defined;using the Loop Representation, it is given [6] by⟨α|T ab[γǫx](0, π) ==Zα dsdαads δ2(α(s), γǫx(0))Zα dudαbdt δ2(α(u), γǫx(π))Xi⟨α#su,iγǫx|5

Following ref. [7], we may regularize the delta functions by a further re-placement of γǫx by means of a one parameter family of loops.

We can thencompute the action of the operator l[C] on a loop state ⟨α|. The square rootof the square of the (regularized) delta function gives an absolute value; inthe limit the intersection rearrangement gives just a multiplicative factor,and taking the limit we obtain, in Planck units,⟨α|l[C] = 12Zα dsZC dtdαadsdCbdt ǫab ⟨α|The double integral is precisely the (positive) intersection number betweenC and α.

Thus we arrive at the following results: i. the length of every curveC is quantized in units of 1/2; ii. if the gravitational field is in the state ⟨α|,the length of a curve C is given (reinserting conventional units) byl[C] = n[C, α]LP lanck2,where n[C, α] is the number of times α crosses C.Now we can return to the PRTVO model.Result i.above impliesthat summing over independent states in quantum gravity means to sum justover quantized half-integer lengths, precisely as in the Ponzano Regge ansatz.Moreover, the relation between the 2-dimensional colored triangulation onthe boundaries of the manifold M and the Witten theory is now physicallytransparent: Recall that the i-th link Ci of the triangulation, with has colorji is crossed precisely by 2ji loops, thus n[Ci, α] = 2ji; therefore, using resultii., the physical length in Planck units in the quantum state defined by theseloops isli = 1/2n[C, α] = 122ji = jiThus, the physical length of the i-th link in Ooguri state is precisely equalto its coloring.

Therefore, the state Ψ∆,c(w), which Ooguri associates to thecolored triangulation (∆, c) is a quantum state in which the physical lengthof the links of the triangulation is precisely equal to their coloring, in Planckunits. This is our main result.An important consequence of this result is the fact that it throws lighton the confused issue of the existence of an ”unbroken phase” and a ”brokenphase” in quantum gravity.

This issue concerns the existence of states cor-responding to non-degenerate metrics in a quantization in which fields are6

excitation around the zero metric configuration (as opposed to flat). The con-siderations above show that as far as 2+1 gravity is concerned, the problemdoes not sussist.

Indeed, in an generic state |α⟩, the diffeomorphism-invariantfunctions of the metric are computable, and, in general, they take values cor-responding to non-degenerate metric configurations. The core of the subteltyis the diffeomorphism invariance of the theory.

The number of independentinvariant observables in the theory is finite in the quantum theory as well asin the classical theory . To clarify this point, let us consider a 2-d space man-ifold with the topology of a torus.

The ”no-loop” state |0⟩gives a vanishingmetric tensor. However, let us consider the state |α, β⟩, where α and β arethe two independent non contactible loops of the torus (the correspondingstate in the Witten representation is easily obtained from Eq.

(2)). In thisstate, the minimal length of any curve wrapping once around the torus is12LP lanck.

Now let us consider the 2-d space manifolds of the classical theory.Since spacetime if flat, we can always go to a flat euclidean 2-d manifoldby means of a gauge transformation (a 3-d diffeomorphism). The physicallymeanigful information about space is contained in two numbers that charac-terise a flat torus.

Thus, in this theory an arbitrary non-degenerate flat spacemetric is characterized by these two variables. Say, for definitness, two radii.Now let us return to the quantum theory.

In the state |α, β⟩, the two vari-ables have a well-defined non-vanishing value. Thus the quantum state |α, β⟩corresponds to the classical non-degenerate configuration in which space isformed by a flat torus in which the two radii are long 12LP lanck each.

Big-ger spaces are obtained by wrapping the loops that define the state moretimes around the torus. The extrinsic curvature of these spacelike surfaces,then, will of course be completely indetermined, due to Heisenberg principle.This example shows that, at least as far as the 2+1 theory is concerned, thenon-degenerate geometries live in the same Hilbert space as the completelydegenerate state.Before getting to the second part of this work, where we discuss the 3+1theory, let us note that the relation between the Loop Representation theoryand the PRTVO model allows us to write the scalar product of two loopstates |γ⟩and |γ′⟩by means of a sum over colorings: we put γ and γ′ on theboundaries of a (topologically trivial) three manifold M, we fix a triangulation∆, and we have⟨γ|γ′⟩=Xc(γ,γ′)YPR(c),7

where the sum is over all the triangulations c(γ, γ′) such that the coloring ofthe links of the boundaries is determined by the number of times the loopsγ and γ′ cross the link, and byQ PR(c) we indicate the Ponzano-Reggeproduct of the coloring.The above result indicates a direction for constructing the physical 4-dtheory. Let us consider the 4-d manifold M, with, say, two boundaries ∂M1and ∂M2.

We begin by fixing a 4-d triangulation ∆of spacetime, whichinduces 3-d triangulations of the two boundaries. In 4-d, they are the areasof surfaces, not the lengths, that are naturally quantized in 1/2 the Planckunit.

Thus, it is natural to use as independent variables for Regge calculusthe areas of the faces, rather than the lengths of the links. A key observationis that a 4-d simplex has the same number (10) of faces (2-d symplices)and links (1-d simplicies).

Therefore we can generically invert the relationbetween lengths and areas, and express the lengths of the 10 links of each4-simplex as functions of the areas of the 10 faces. Let a1 ... a10 be the areasof the 10 faces of a fixed 4-simplex s. The Regge action of the simplex canbe expressed as a function of these areas: SRegge(s) = SRegge(a1 ... a10).

Notethat SRegge must be a function of 10 variables with the full symmetry of the4-simplex.We suspect that the corresponding quantity SRegge(a1 ... a10),seen as function of half-integer variables, has an interpretation in terms ofgroup representation theory (at least in the large ai limit) analogous to the6-j symbols interpretation of its 3-d analog, but we have not found it. 1 Theabove construction defines a combinatorial quantum theory in 4-d (for a fixedtriangulation).

In the absence of boundaries, we haveZ(∆) =XcZRegge(∆, c) =Xc of ∆Ysexp{SRegge(c)}where the sum is over the colorings, the product over the 4- simplices. Thestates of this theory are given by the induced colorings of the induced tri-angulation on the 3-d boundary of the 4-d triangulated spacetime.

Thesestates are physically determined by the fact that there is a three dimensionaltriangulation of space such that the (2-d) surfaces of the triangulation have1The reason the group SU(2) is still the relevant group, in spite of the fact we are onedimension above, is in the very roots of Ashtekar’s magic construction: the (complexified)Lorentz group splits naturally in two (complexified) SO(3) groups, its self-dual and anti-selfdual parts, and, as shown by Ashtekar, the full theory can be constructed using onlyone of the two SO(3) components.8

assigned areas; these areas being half-integers in Planck units. These statesare therefore in correspondence with the states of the Loop Representation.The correspondence being given by the result of ref.

[7], that the physicalarea of any surface is 1/2 the number of loops that cross the surface. Sincethe states can be written in terms of loop states, and viceversa, this theorydefines a scalar product in loop space by:⟨γ|γ′⟩∆=Xc(γ,γ′)of ∆ZRegge(∆, c),where γ is on ∂M1 and γ′ is on ∂M2, and the sum is over all the colorings of theinterior areas, the colorings of the areas on the boundaries being determinedas 1/2 the number of times the loops cross the surface.

The idea that statesof the Loop Representation can be better understood in terms of the areathey induce on a 3-d triangulation was proposed by Lee Smolin [15].We still do not have a well defined diffeomorphism invariant theory, sincethe above construction depends on the triangulation ∆.2We eliminatethis dependence by summing over all the triangulations of M. The key pointin this construction is that this is possible to sum over the triangulations,because we have a way of naming the quantum states in the boundaries,which is independent from the triangulation of the boundary itself, namelythe loop basis. Thus, we define a theory asZ =X∆Xc of ∆ZRegge(∆, c)(4)and the scalar product between two loop states by⟨γ|γ′⟩=X∆⟨γ|γ′⟩∆(5)(If in Eq.

(4) we perform the sum over the colorings first, the theory takes aform strictly related to the dynamical triangulations approaches to quantumgravity [4]; this relation, we believe, deserves to be better explored.) Theimportant point is that scalar product defined in this way is invariant under2An optimistic hope would be that the scalar product above does not depend on thetriangulation.

While we do not hope for so much, still we do not think this be totallyimpossible, as some considerations below may suggest.9

independent diffeomorphisms on each of the two loops, because so is the sum.Therefore it defines a scalar product ⟨K|K′⟩between knot states by⟨K(γ)|K(γ′)⟩= ⟨γ|γ′⟩These equations provide a formal expression for the Hilbert structure of quan-tum gravity. We expect that Eq.

(5) defines a projector on the knot states,which projects on the solutions of the Wheeler DeWitt equation, as it hap-pens in 3-d [11]. This is our main proposal for a 4-d theory.To clarify the meaning of Eqs.

(4) and (5), let us rewrite them in termsof the original continuous Ashtekar’s variables A and E. We have 3⟨γ|γ′⟩=Z[d4A][d4E] exp{−1/¯hSE[4A,4 E]}Ψγ(A)Ψ∗γ′(A)whereΨγ(A) = Ψγ(3A) = ⟨3A|γ⟩= trP expIγ Aand each loop is in one of the two boundaries of spacetime. It is easy to con-vince oneself that this is the correct formal expression for the scalar productin the Loop Representation, up the problem of definition and finiteness of thefunctional integration.

To our knowledge, this expression was first suggestedby Maurizio Martellini [16]. This is the connection representation analog ofHawking expression [17]⟨Ψ|Φ⟩=Z[d4g] exp{−1/¯hSE[4g]}Ψ∗(3g)Φ(3g′)where the integration is over all the 4-d metrics 4g on M, and 3g and 3g′are the restrictions of 4g to the two components of the boundary of M. Ofcourse these functional integrals do not mean anything until a definitionis provided; in particular, if we want to compute them in a perturbation3 Ooguri suggests [12] that in a one may have two conjugate variables in the continuumversion of the theory, which correspond respectively to the choice of the triangulation andthe choice of the coloring in the combinatorial version of the theory.

Note the similar-ity between the Ashtekar’s action S[A, E] =RF ∧E ∧E, with the (topological !) BFtheory action S[A, B] =RF ∧B which seems to underlie the Ooguri-Crane-Vetter invari-ants [10,12].

The relation between the construction proposed here and the (triangulationindependent) Ooguri- Crane-Vetter construction deserves to be studied in detail .10

expansion around flat space, we encounter the weel known gravitational di-vergences. The combinatorial expression given above is a proposal for thisdefinition.4 In this continuum case, Hawkings does indeed give a formalderivation of the fact that the functional integral defines a projection onthe solutions of the Wheeler DeWitt equation.

This is an indirect supportof our expectation that Eq. (5) projects on the solutions of the hamiltonianconstraint.We conclude with a consideration on the formal structure of 4-d quantumgravity, which is important to understand the above construction.

Standardquantum field theories, as QED and QCD, as well as their generalizations likequantum field theories on curved spaces and perturbative string theory, aredefined on metric spaces. Witten’s introduction of the topological quantumfield theories has shown that one can construct quantum field theories definedon a manifold which has only its differential structure, and no fixed metricstructure.

The theories introduced by Witten and axiomatized by Atiyahhave the following peculiar feature: they have a finite number of degreesof freedom, or, equivalently, their quantum mechanical Hilbert spaces arefinite dimensional; classically this follows from the fact that the number offields is equal to the number of gauge transformations. However, not anydiff-invariant field theory on a manifold has a finite number of degrees offreedom.

Witten’s gravity in 3-d is given by the action S[A, E] =R F ∧E,which has finite number of degrees of freedom. Consider the action S[A, E] =R F ∧E ∧E, in 3+1 dimensions, for a (self dual) SO(3,1) connection A anda (real) one form E with values in the vector representation of SO(3,1).This theory has a strong resemblance with its 2+1 dimensional analog: itis still defined on a differential manifold without any fixed metric structure,and is diffeomorphism invariant.

We expect that a consistent quantizationof such a theory should be found along lines which are more similar to thequantization of theR F ∧E, theory than to the quantization of theories onflat space, based on the Wigthman axioms namely on n- points functionsand related objects. Still, the theoryR F ∧E ∧E has genuine field degreesof freedom: its physical phase space is infinite dimensional, and we expectthat its Hilbert state space will also be infinite dimensional.There is a4 Of course, each of the above equations can be immediately generalized to non-trivialtopologies, Hartle-Hawking states, disconnected universes, and so on, if one is interestedin this kind of physics directions.11

popular belief that a theory defined on a differential manifold without metricand diffeomorphism invariant has necessarily a finite number of degrees offreedom (”because thanks to general covariance we can gauge away any localexcitation”). This belief is of course wrong.

A theory as the one definedby the actionR F ∧E ∧E is a theory that shares many features with thetopological theories, in particular, no quantity defined ”in a specific point”is gauge invariant; but at the same time it has genuinely infinite degrees offreedom. Indeed, this theory is of course nothing but (Ashtekar’s form of)standard general relativity.The fact that ”local” quantities like the n-point functions are not appro-priate to describe quantum gravity non-perturbatively has been repeatedlynoted in the literature.

As a consequence, the issue of what are the quantitiesin terms of which a quantum theory of gravity can be constructed is a muchdebated issue. The above discussion indicates a way to face the problem: Thetopological quantum field theories studied by Witten and Atiyah provide aframework in terms of which quantum gravity itself may be framed, in spiteof the infinite degrees of freedom.

In particular, Atiyah’s axiomatization ofthe topological field theories provides us with a clean way of formulating theproblem. Of course, we have to relax the requirement that the theory has afinite number of degrees of freedom.

These considerations leads us to proposethat the correct general axiomatic scheme for a physical quantum theory ofgravity is simply Atiyah’s set of axioms [3] up to finite dimensionality of theHilbert state space. We denote a structure that satisfies all Atiyah’s axioms,except the finite dimensionality of the state space, as a generalized topologicaltheory .The theory we have sketched is an example of such a generalized topolog-ical theory.

We associate to the connected components ∂Mi of the boundaryof M the infinite dimensional state space of the Loop Representation of quan-tum gravity. Eq.

(5), then, provides a map, in Atiyah’s sense, between thestate spaces constructe on two of these boundary components. Equivalently,it provides the definition of the Hilbert product in the state space.One could argue that the framework we have described cannot be consis-tent, because it cannot allow us to recover the ”broken phase of gravity” inwhich we have a nondegenerate background metric: in the proposed frame-work one has only non-local observables on the boundaries, while in thebroken phase a local field in M has non-vanishing vacuum expectation value.We think that this argument is weak because it disregards the diffeomor-12

phism invariance of the theory: in classical general relativity no experimentcan distinguish a Minkowskian spacetime metric from a non-Minkowkianflat metric. The two are physically equivalent, as two gauge-related Maxwellpotentials.

For the same reason, no experiment could detect the absolute po-sition of, say, a gravitational wave, (while of course the position of an e.m.wave is observable in Maxwell theory). Physical locality in general relativ-ity is only defined as coincidence of some physical variable with some otherphysical variable, while in non general relativistic physics locality is definedwith respect to a fixed metric structure.

In classical general relativity, thereis no gauge-invariant obervable which is local in the coordinates. Thus, anyobservation can be described by means of the value of the fields on arbitraryboundaries of spacetime.

This is the correct consequence of the fact that”thanks to general covariance we can gauge away any local excitation”, andthis is the reason for which one can have the ADM ”frozen time” formalism.The spacetime manifold of general relativity is, in a sense, a much weakerphysical object than the spacetime metric manifold of ordinary theories. Allthe general relativistic physics can be read from the boundaries of this man-ifold.

At the same time, however, these boundaries still carry an infinitedimensional number of degrees of freedom.Finally, we must recall that the computation of the evolution of expec-tation values in physical time (as opposed to coordinate time, which has nodiffeomorphism invariant meaning) requires the use a physical clock coupledto the theory (in principle this could also be a component of the gravi-tational field itself) [18].In this sense the integration (or the sum) overthe M is physically very analogous to the derivation of the propagator ofa relativistic particle by means of an integral over the paths xµ(τ), whereµ = 0, 1, 2, 3; in the particle case too, indeed, the scalar product betweentwo wave functions on Minkowski space can be obtained by integrating overa five dimensional manifold that interpolates two Minkowski spaces (see forinstance [19]). Physical evolution, of course, is in xo, not in τ, namely in4-d, not in 5-d.

In addition, we should also recall that the quantization ofthe physical area is a non-gauge invariant result, unless reinterpreted in somesuitably gauge-invariant context [20].Summarizing, we have have shown the following:i. The ”colored triangulation” basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantization of 3-d gravity is precisely the Loop Representation basis.ii.

In Witten’s theory, we can compute the lengths of the arcs of a Regge13

traingulation, (up to diffeomorphisms).iii. We can interpret the quantization of the length in half integer units inphysical terms: the spectrum of the length operator has discrete half-integereigenvalues.iv.

These lengths are related to the SU(2) representations because thequantum states that diagonalize the lengths are given (in the connectionrepresentation) by Wilson lines that cross the curve 2j times, or, equivalently,by one Wilson line in the j representation that crosses the curve.Motivated by these results, we have sketched a 4-d combinatorial theory,based on a modification of Regge calculus. Many questions remain open as faras this theory is concerned, the most relevant ones being the relation betweenthe Euclidean and the Lorentzian theory, and the convergence propertiesof the sum (3).Finally, we have proposed that Atiyah’s axiomatizationof topological field theories can be extended also to theories with infinitedegrees of freedom, and that this extension can be takes as the general formalstructure of a quantum theory of gravity.References[1] E Witten, Comm Math Phys 121 (1989) 351; Nucl Phys B322 (1989)622; B330 (1990) 285; B311 (1988/89) 46; Nucl Phys B340 (1990) 281[2] E Witten, Comm Math Phys 117 (1988) 353[3] MF Atiyah, The Geometry and Physics of Knots , Accademia Nazionaledei Lincei, Cambridge University Press 1990; Publ Math Inst hautes EtudesSci Paris 68 (1989) 175[4] J Ambjorn, B Durhuus, T Jonsson, Mod Phys Lett A6 (1991) 1133;J Ambjorn, Nucl Phys B25A (1992) 8; J Ambjorn, J Jurkiewicz Phys LettB278 (1992) 42.

ME Agishtein, AA Migdal, Mod Phys Lett A6 (1991) 1863.Godfrey, M Gross Phys Rev D43 (1991) R1749; M Gross, Nucl Phys B20(1991) 724[5] A Ashtekar, Phys Rev Lett 57 (1986) 2244; Non-perturbative canonical14

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Forthe 2+1 dimensional theory see: A Ashtekar, V Husain, C Rovelli, J Samuel,L Smolin, Class and Quant Grav 6 (1989) L185[7] A. Ashtekar, C. Rovelli, L. Smolin, Phys Rev Lett 69 (1992) 237[8] V Turaev, O Viro, Topology 31 (1992) 865[9] G Ponzano, T Regge, ”Semiclassical limits of Racah Coefficients” inSpectroscopy and Group theoretical methods in Physics. Ed F Bloch, NorthHolland, Amsterdam 1968[10] L Crane, D Yetter, ”A categorical construction of 4D topologicalquantum field theories”, Kansas State University preprint 1992[11] H Ooguri, Nucl Phys B382 (1992) 276[12] H Ooguri, Mod Phys Lett A7 (1992) 2799[13] T Regge, Nuovo Cimento 19 (1961) 551[14] L Smolin, personal communication; L Crane, ”Categorical Physics”,Kansas State prep 1992[15] L Smolin, ”Time, measurement and information loss in quantumcosmology”, Syracuse preprint 1993[16] M Martellini, personal communication[17] SW Hawking, in General Relativity, an Einstein centenary Survey ,eds.

SW Hawking, W Israel, Cambridge University Press 1979[18] C Rovelli, Class and Quant Grav 8 (1991) 297; 8 (1991) 317; PhysRev D42 (1991) 2638; D43 442 (1991)[19] PD Mannheim, Phys Lett 166B (1986) 191[20] C. Rovelli, ”A Generally covariant quantum theory and a predictionon quantum measurement of geometry”, sumbitted to Nucl Phys (1992).15

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출처: arXiv:9304.164 • 원문 보기