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A spin-foam vertex amplitude with the correct semiclassical limit Jonathan Engle Department of Physics, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33433, USA Spin-foam models are hoped to provide a dynamics for loop quantum gravity. All 4-d spin- foam models of gravity start from the Plebanski formulation, in which gravity is recovered from a topological field theory, BF theory, by the imposition of constraints, which, however, select not only the gravitational sector, but also unphysical sectors. We show that this is the root cause for termsbeyondtherequiredFeynman-prescribedexponentialofitimestheactioninthesemiclassical limitoftheEPRLspin-foamvertex. Byquantizingaconditionisolatingthegravitationalsector,we modify the EPRL vertex, yielding what we call the proper EPRL vertex amplitude. This provides at last a vertex amplitudefor loop quantum gravity with thecorrect semiclassical limit. 2 Introduction Loop quantum gravity (LQG) [1–4] of- wellasthe persistence ofdegenerateconfigurations. The 1 0 fers a compelling kinematical framework in which dis- other 4-d spin-foammodels of gravityhave similar prob- 2 creteness of geometry is derived from a quantization of lems with a similar source [15–17]. In the present work, general relativity (GR) rather than postulated. This weshowhow,byformulatingtherestrictiontothegravi- n discreteness has enabled well-defined proposals for the tationalsectorclassicallyfirst,quantizingit,andincorpo- a J Hamiltonian constraint — defining the dynamics of the rating it into the EPRL vertex definition, one can define 0 theory — in which one sees how diffeomorphism invari- a modified vertex for which the extra terms are elimi- 1 ance eliminates normally problematic ultraviolet diver- nated, degenerate configurations are exponentially sup- gences. However, the lack of manifest space-time covari- pressed,andonefinallyachievesavertexwiththecorrect ] ance, inherent in any canonical approach to gravity, is semiclassical limit. This new modified vertex, which we c q often suspected as a reason for the presence of ambigu- call the proper EPRL vertex, additionally continues to - ities in the Hamiltonian constraint. This has motivated be compatible with loop quantum gravity, linear in the r g the spin-foam program [1, 5–7], which aims to provide boundary state, and SU(2) invariant. The condition of [ a space-time covariant, path integral version of the dy- linearityintheboundarystateisnecessarytoensurethat namics of LQG. The histories summed over in the path the final transition amplitude defined by the spin-foam 1 integral are those arising from loop quantization meth- model is linear in the initial state and anti-linear in the v ods. final state. 7 8 Attheendoftheletter,wealsoremarkuponapossible 1 The individual amplitudes in a spin-foam sum are de- relationtoOriti’sFeynmanpropagator[18],Teitelboim’s 2 rivedusingthePlebanski formulationofgravity,orvaria- causal propagator [19], as well as the positive frequency . tionsthereof. Inthisformulationofgravity,onetakesad- 1 condition used in loop quantum cosmology [20]. 0 vantageofthe factthatGRcanbe formulatedasatopo- 2 logical field theory whosequantizationiswell-understood To begin, we review the classical discrete framework, 1 —BFtheory [8]— supplementedby so-calledsimplicity reviewtheEPRLvertex,pointoutitsproblems,andthen : constraints. Withinthelastfewyears,aspin-foammodel derivethesolution,leadingtothedefinitionoftheproper v EPRL vertex. The present letter gives only a summary i of quantum gravity was, for the first time, introduced X of this work, further details being left to the two papers whose kinematics match those of LQG and therefore re- r alize the original goal of the spin-foam program: to pro- [21, 22]. a videapathintegraldyanamicsforLQG.Thisisknownin Review of EPRL and the problem with its asymp- theliteratureasEPRL[9,10];whentheBarbero-Immirzi totics. The quantum histories used in spin-foam sums parameter [11, 12] β, a certain quantization ambiguity, are usually based on a triangulation of space-time into is less than 1, this model is equal to the Freidel-Krasnov 4-simplices. The probability amplitude for a given spin- model [13]. Despite its success, the EPRL vertex still foam history breaks up into a product of amplitudes as- has difficulty in obtaining the correctsemiclassicallimit: sociated to each component of the triangulation [1, 7]. (non-geometric) degenerate configurations are not sup- The most important of these amplitudes is the vertex pressed [14], and even if one restricts to non-degenerate amplitude, which provides the probability amplitude for configurations, the semiclassical limit of the vertex am- dataassociatedtoasingle4-simplex. Inthefollowing,as plitude has four terms instead of the required one term we are concerned specifically with the vertex amplitude, of the form exponential of i times the action [14]. As we for conceptual clarity, we focus on a single 4-simplex σ. shall show in this letter, this difficulty is directly due to (Though the EPRL vertex has been generalized to ar- a problem with the way gravity is recovered from BF bitrary cells [23], we restrict ourselves to the simplicial theory: When one imposes the simplicity constraints, case,ascertainkeyargumentsweusewilldependonthe one isolates not only the gravitational sector, but three combinatorics of this case. See remarks at the end of sectors, and these sectors are directly responsible both thisletter.) Lettrianglesandtetrahedraofσ bedenoted for the extra terms present in the semiclassical limit, as respectively by f and t and decorations thereof. Fix a 2 transverse orientation of each f within the boundary of clarifies the interpretation of the LQG boundary states σ. For improved clarity and brevity we review here the in terms of the discrete classical Spin(4) BF theory re- definition of the basic variables in a way different from, viewed above. but equivalent to, [21, 22]. Let us fix an affine struc- The LQG Hilbert space associated to ∂σ is ture onthe 4-simplex σ; this is equivalentto fixing a flat L2(× SU(2)). A(generalized)spin-network Ψ in f {kf,ψft} connection ∂a on σ. The basic variables for a single 4- thisspaceislabeledbyonespinkf andtwostatesψft′ ∈ f assuinmbdpjel2ce0txtcaoolgnaesbisnrtauimneble5emrgeronoftuscpo{neBlsettImrfJae}inntt⊂ss:{Gs(o1t(.})4t)∈.σ‘or⊂TiehnSetpasietnio(a4nr)’e,, Vkgifv∗r,eenpψrfeetxsfepn∈litcaVittfliyopnberyoftrSiaUn(g2le).fΨ, w{khfe,ψreft}Vk∈dLen2(o×tefsStUhe(2s)p)inis- Gt⊲Bft =−Gt′⊲Bft′,(2.) ‘closure’, f∈tBft =0,and (3.) ‘linear simplicity’, (⋆Bft)0i = 0.PLet (1.)’ denote Ψ{kf,ψft}({gf}):= hψft′fρkf(gf)ψftfi (3) the part of constraint (1.) that is independent of the Yf Gt variables (BfIJtBftIJ=BfIJt′Bft′IJ, ǫIJKLBfIJtBfKtL= where ρ (g) denotes the representation matrix for g ∈ ǫ BIJBKL). (1.)’ and (3.) are imposed as a re- k IJKL ft′ ft′ SU(2) on V . The embedding ι : L2(× SU(2)) → striction on the allowed boundary data and histories in- k f L2(× Spin(4)) from LQG states to Spin(4) BF theory tegrated over in the vertex amplitude (see below). The f boundarystatesisdefinedintermsofthe(overcomplete) rest of (1.) and (2.) are imposed in the sense that viola- basis (3) by tions are exponentially suppressed by the vertex amplitude. Constraints(1.) and(2.) imply thatthere exists a uthnaiqt,uefotrwaol-lfto,rfmwBitµIhJν,fc∈onts,t[a2n1t, w24it]h respect to ∂a, such (cid:0)ιΨ{kf,ψft}(cid:1)({Gf}):=Yf hψft′f|ιkfρs−f,s+f(Gf)ιkf|ψftfi (4) where here and throughout this letter we set Gt⊲BfIJt =Z BIJ. (1) s± := 21|1±γ|k, ρs−,s+(G) denotes the spin (s−,s+) f representation of G ∈ Spin(4), ιk : Vk → Vs− ⊗ Vs+ denotes the indicated intertwiner scaled such that it Whentheconstraint(3.),linearsimplicity,isadditionally is isometric in the Hilbert space inner products, and imposed, this two-form field BIJ furthermore takes one of the three forms [21] µν ιk :Vj− ⊗Vj+ →Vk is its hermitian conjugate. In terms of the family of states (3), the operator Lî , ft (II±)BIJ =±12ǫIJKLeK ∧eL for some constant eIµ tahcetsqduiarencttizlyatoionnψof LasiftcainntbheecShpecinke(4d): quantum theory, ft (deg)ǫ ǫµνρσBIJBKL =0 (degenerate case) IJKL µν ρσ Lî ιΨ =ιΨ (5) ft {kf′,ψf′t′} {kf′,ψ˜f′t′} where the names for these sectors have been taken from [21,25]. Only(II+)yieldsatheorywithboththecorrect where ψ˜ft := Lîψft, and ψ˜f′t′ := ψf′t′ for f′ 6= f or gravitational degrees of freedom and the usual gravita- t6=t′,andwhereLî denotestheSU(2)generatorsacting tional action with the usual sign [21, 25]. intheappropriateirreduciblerepresentation,normalized The boundary phase space giving rise to the Spin(4) such that [Lî,Lˆj]=iǫijLˆk. k BF Hilbert space associated to the boundary of σ is pa- TheEPRLvertexamplitude AEPRL :L2(× SU(2))→ v f rameterized by the five group elements {Gt′ftf}f∈σ ⊂ C, in terms of the above is Spin(4) and the algebra elements {BIJ}, where t ,t′ ft f f wariethriensptheectbivoeulnydtahreyoteftσra,haenddroGnt′t‘a:b=ovGe−t’′1aGntd. T‘bheeloPwo’isf- AEvPRL(Ψ{kf,ψft}):=SpiZn(4)(cid:16)5Yt dGt(cid:17)(cid:0)ιΨ{kf,ψft}(cid:1)(Gt′ftf) son bracket structure is such that the combination JIJ := 1 B + 1 ⋆B IJ (2) =ZSpin(4)5(cid:16)Yt dGt(cid:17)Yf hψft′f|ιkfρ(Gt′ftf)ιkf|ψftfi. (6) ft 8πG(cid:18) ft γ ft(cid:19) The vertex amplitude can be rewritten using what we tgoenwerhaettehserleftteoqruarlisghtt′f torrantsfl,atrioesnpsecotniveGlyt′f.tfTahcecocrodrirneg- 3cΨa-{vlklefcL,tniovfritn}nei-sSlpapebezriealefled,tcbowyhietohrneenfts∈pstinat,tekasfn[d2p6ea]r.refE,oabachntadsinuoecndhesfurtaontmiet sponding generators of (internal) spatial rotations are ft then Lift := 12ǫijkJfjkt. |tkhe,sntateis,(w3h)ebrye|ske,ttniing∈ψVft′fde:=nohtkefs,t−henfint′fd|icaantdedψSftUf(:2=) In quantum theory, the simplicity constraint reduces f ftf k Perelomov coherent state [27]. One then has the boundary Hilbert space of the quantum BF theory topreciselythatofLQG,yielding anembeddingofLQG ALQG(Ψ )= (7) boundarystatesintoSpin(4)BFtheoryboundarystates v {kf,nft} [t1h0e].heLaerttuosf trheecaEllPtRhiLs veemrbteexddaimngplbitoutdhe,baencadubseeciatuisseaitt ZSpin(4)5(cid:16)Yt dGt(cid:17)Yf hkf,nft′f|ιkfρ(Gt′ftf)ιkf|kf,nftfi 3 The Spin(4) boundary state ιΨ represents a co- then use it to modify the EPRL vertex (6, 7). For each {kf,nft} herent state peaked on the classical configuration of two tetrahedra t,t′ define the sign βtt′({Gt˜t˜′}) by BIJ’s taking the values ft βtt′({Gt˜t˜′}):=−sgn ǫijk(Gtt1)i0(Gtt2)j0(Gtt3)k0· BfIJt =16πGkfδ0[InJft] (8) ·ǫlm(cid:2)n(Gt′t1)l0(Gt′t2)m0(Gt′t3)n0 (10) (cid:3) where n0ft := 0. The BfIJt values of the form (8) are where GIJ denotes the SO(4) matrix canonically asso- the most generalsatisfying the constraints (1.)’ and (3.) ciated to a given Spin(4) element G (see, e.g., [14, 22]), (linear simplicity) as enumerated earlier. Furthermore, t1,t2,t3 are the tetrahedra other than t,t′, in any order, the integral over the G ’s can be interpreted as a path and sgn is defined to be zero when its argument is zero. t integral over possible Gt’s, which one identifies with the We then have the result that the constant 2-form BµIJν paralleltransportsintroducedaspartofthediscreteclas- determined by equation (1) is in sector (II+) iff sical framework reviewed above. This identification and interpretation of the group integration variables in (7) βtt′({Gt˜t˜′})·(Gt)i0·(Lft)i >0 (11) is justified by the asymptotic analysis of (7) carried out for some pair of tetrahedra t,t′, where f = t∩t′ [22]. by Barrett et al. [14], in which one finds that the criti- Thisistheconditionwhichweseektoquantizeanduseto cal point equations for the group elements are precisely modifythevertexintegral(6,7). Normallythiswouldbe those satisfied by the parallel transports in the discrete donebyinsertingintothepathintegral(7)theHeaviside classical framework. function If the data {k ,n } satisfies k n = 0 (clo- f ft f∈t f ft issurteh)raeenddiismseuncshionthaal,t,tfhoerne,acfohrtP,eatchhe stp,aonnoefc{annft}cof∈nt- Θ βtt′({Gt˜t˜′})·(Gt)i0·(Lft)i , (12) (cid:0) (cid:1) struct a unique geometrical tetrahedron with face areas where Θ(·) is zero when its argument is zero. However, A(kf):=8πℓ2Pγkf, and such that ntf ·ntf′ =−cosθtff′ if one inserts this into (7), one obtains a vertex ampli- forallf,f′ ∈t,whereθtff′ denotestheinterioranglebe- tude which is non-linear in the boundary state, spoiling tween triangles f and f′ within the geometrical tetrahe- the property of the final spin-foam sum that it be lin- dronfort. Ifthesegeometricaltetrahedrafurthermorefit ear in the initial state and anti-linear in the final state together to form a consistent geometrical 4-simplex, the (a property necessary for the final spin-foam sum to be data {kf,nft} is called Regge-like. and the overallphase interpreted as a transition amplitude). Instead, we par- ofthecoherentstateΨ{kf,nft} canbefixeduniquely,giv- tially quantize theexpression(12)beforeinsertingitinto ing rise to what is called the Regge state ΨR{kf,nft} [14]. (6),byreplacingLift withSU(2)generatorsLî actingon For such states, the asymptotics of the EPRL vertex are the coherent state |k ,n i. Because the generators Lî f ft given by are peaked on Li = j ni when acting on the Perelo- ft f ft Av(ΨR{λkf,nft})∼λ−12 C1eiSR +C1e−iSR pmoosvectohheedreesnitresdtactoen|dkift,ionnft(i1,1s)ucinhtinhseesretmionicslawssililcastlillilmimit-, +(cid:0)C2eγiSR +C3e−γiSR , (9) and so remove the unwanted ‘non-gravitational’ sectors, (cid:1) while at the same time preserving the necessary linear- where ∼ indicates that the error term is bounded by a constanttimesthepowerofλ∈R+ indicated,andwhere ity in the boundary state. Thus we insert the following group-variable dependent operator on V : SR denotes the Regge action determined by the data kf {λk ,n }. To be noted is the presence in the asymp- f ft totics of not just the Feynman-prescribed exponential of Pˆt′t({Gt˜′t˜}):=P(0,∞) βt′t({Gt˜′t˜})·(Gt)i0·Lî (13) (cid:16) (cid:17) i times the action, but of 3 extra terms which spoil the usual proof of the correct classical limit. This is in ad- where P (Oˆ) denotes the spectral projector for the op- S dition to the problem of the existence of a class of de- erator Oˆ onto the portion S of its spectrum. Inserting generateconfigurationsyieldingasymptoticsthatpersist this into the vertex path integral (6), one obtains what as strongly as those of the Regge-like configurations in we callthe proper EPRL vertex amplitude. For ageneral (9). The asymptotics for this class of degenerate con- SU(2) spin-network state (3), it is explicitly given by figurations have not been shown here, but can be found in [14]. It turns out that the existence of these non- A(II+)(Ψ )= (14) v {kf,ψft} suppressed degenerate configurations, as well as of the ethxetrafacttertmhsatinlintehaerassimympplitcoittyicsall(o9w),sifsorprmecoirseelythdanuethtoe SpiZn(4(cid:16))5Yt dGt(cid:17)Yf hψft′f|ιkfρ(Gt′ftf)ιkfPˆt′ftf({Gt′ftf})|ψftfi. usual gravitationalsector (II+) as was shown in [21]. A condition selecting the gravitational sector and its Onecanequivalentlywritethevertexamplitudewiththe quantization. To solve the above problem, our strategy projector on the left side of each integrand factor [22]. is first to find a classicalcondition on the basic variables This vertex amplitude is manifestly linear in the bound- thatselectsthe(II+)sector,quantizethiscondition,and ary state (3), and one can furthermore show that it is 4 SU(2) invariant [22]. For the coherent state Ψ , removal is achieved in such a way that linearity in the {λkf,nft} for large λ, the proper EPRL vertex is exponentially boundary state is preserved, sothatthevertexamplitude suppressed unless {k ,n } describes a non-degenerate can continue to be used to define transition amplitudes f ft Regge geometry, in which case it now has precisely the between canonical states in the usual sense. required asymptotics [22] Beyond the removal of the degenerate sector, the propervertexalsoachievesthe removalofPlebanskisec- A(II+)(ΨR )∼λ−12eiSR. (15) v {λkf,nft} tor (II-). The importance of this lay first and foremost in the need to include only one copy of the gravitational Discussion. By implementing quantum mechanically sector, with the usual sign in the action. However, the a restriction to Plebanskisector (II+), the EPRL vertex removal of sector (II-) also has a tantalizing similarity amplitude has been modified, yielding what we call the to a selection of a single orientation of the space-time proper EPRL vertex. The resulting vertex is linear in manifold. If one could make this precise, one could po- the boundary state, SU(2) invariant, and possesses the tentially show that the restriction to (II+) achieved in correct semiclassical limit. the proper vertex is in fact conceptually the same as the Letusremarkfirstonthenon-trivialityoftheremoval restriction advocated by Oriti [18], and hence closely re- of the degenerate sector that has been achieved. In the lated to the causal propagator of Teitelboim [19], which work [17], the degenerate sector of the Freidel-Krasnov from the symmetry reduced model analysis [28, 29], has model (equal to EPRL for γ < 1) is removed by using been argued to be necessary for spin-foam models to be a path integral representation based on coherent states, consistent with loop quantum cosmology [20, 30]. similar to the path integral representation of the EPRL Two final open issues that remain are the general- vertexgivenin(7)above. However,inthe conclusionsof ization to the Lorentzian signature and the generaliza- the work [17], the authors state that they do not know tion to an arbitrary cell [23]. The generalization to the how rewrite the resulting restricted path integral as a Lorentzian signature is expected to be straightforward. spin-foamsum—thatis,asasumoverhistoriesofspin- Thegeneralizationtoanarbitrarycell,ontheotherhand, foamslabeledwithspins andintertwiners,similarto(6). willlikelyrequireonetounderstandthestrategyfroman The reason for this difficulty arguably can be traced to entirely new perspective, as the combinatorics of the 4- thesamereasonforourrejectionofthe “naive”prescrip- simplex are used in a key way not only in the derivation tion of inserting the non-quantized Heaviside function of the proper vertex, but in its very definition. (12) into (7): because the resulting transition amplitude is non-linear in the boundary state. Thus, as far as re- Acknowledgements. 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