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Spacetime thermodynamics without hidden degrees of freedom Goffredo Chirco, Hal M. Haggard, Aldo Riello, Carlo Rovelli Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France. Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. (Dated: January 22, 2014) AcelebratedresultbyJacobsonisthederivationofEinstein’sequationsfromUnruh’stemperature, the Bekenstein-Hawking entropy and the Clausius relation. This has been repeatedly taken as evidence for an interpretation of Einstein’s equations as equations of state for unknown degrees of freedom underlying the metric. We show that a different interpretation of Jacobson result is possible,whichdoesnotimplytheexistenceofadditionaldegreesoffreedom,andfollowsonlyfrom the quantum properties of gravity. We introduce the notion of quantum gravitational Hadamard states, which give rise to the full local thermodynamics of gravity. I. INTRODUCTION measures the entanglement between adjacent spacetime 4 regions. Its finiteness is evidence for the quantization of 1 0 In a celebrated paper [1], Ted Jacobson presented the gravitational field: this is analogous to the fact that 2 a surprising thermodynamical derivation of Einstein’s the finiteness of the black-body electromagnetic entropy n equations, based on three inputs: (i) the vacuum of a is evidence for the quantization of the electromagnetic a quantum field theory in Minkowski space behaves as a field. J thermal state at the Unruh temperature We show that the Jacobson result is consistent with 1 this simpler and tighter scenario. The finiteness and the 2 T = (cid:126)a (1) universality of the entanglement entropy across space- 2π time regions indicates ultraviolet quantum discreteness, ] c foranobserverwithaccelerationa; (ii)thereisauniver- as it did for Planck and Einstein at the beginning of the q XX century. sal entropy density α=1/4(cid:126)G per unit area, associated - r to any causal horizon in a locally Minkowski patch of Section II summarizes basic facts about entropy in g spacetime, giving an entropy quantum theory. Section III discusses the relation be- [ tween entanglement entropy and temperature. In Sec- 1 A tion IV we interpret Jacobson’s result in these terms. In S = (2) v 4(cid:126)G Section V we discuss the dependence of the entropy on 2 the number of fields. 6 for a horizon region of area A; and (iii) a local entropy In Section VI we outline how the scenario we consider 2 balance relation 5 can be concretely implemented in loop quantum grav- . δE ity. In this context we introduce the Hadamard quan- 1 δS = (3) T tum gravitational states. In Section VII we discuss the 0 meaning of these states, which give rise to the correct 4 holds, where δE is an energy exchange. By interpret- 1 entropy and provide a foundation for the semiclassical ing δE as the energy of matter flowing across the local : statesofthegeometry,inthesenseof[5]. Wesummarise v Rindler horizon of the accelerated observer, and match- our results in the last Section. In the three appendices, i ing the variation of the area with the focusing effect of X we discuss the relation between thermometers and KMS spacetime curvature, Jacobson was able to show that if r states (Appendix A), as well as the relation between the a the three equations above hold for any local frame, then simplicity conditions and the area-energy relation (Ap- Einstein’s equations follow. pendix B); finally, in Appendix C, we recall the main This is a beautiful piece of theoretical physics. But lines of Jacobson’s original argument. its interpretation is not clear. A common understanding [1–3] takes the result as evidence that Einstein’s equations have a statistical origin and should be interpreted as equations of state for unknown underlying degrees of II. STATISTICAL AND ENTANGLEMENT freedom, with the metric being a macroscopic “coarse- ENTROPY grained” variable. In this paper we show that a different interpretation is possible. The alternative interpretation, which we develop We start by recalling some basic facts about entropy mostlyfollowing[4,5],isbasedonthefactthatthegrav- in quantum systems. Consider a quantum system whose itational field has quantum properties. The microscopic pure states are described by a normalised vector ψ in a degreesoffreedomarethoseofthequantumgravitational Hilbert space . A generic, not necessarily pure, state H field and the Einstein equations express only the classi- is described by a density matrix ρ : satisfying H → H callimitofthedynamics. Theentropyacrossthehorizon tr[ρ]=1 and ρ =ρ. To any such density matrix we can † 2 associate the von Neumann entropy Therefore we see that the von Neumann entropy can describetwo(relatedbut)distinctphysicalquantities: in SvN = tr[ρlnρ]. (4) the first example, it describes our ignorance of the spe- − cific microstate; in the second, it is a measure of entan- This entropy measures the lack of information about glement. The first describes simply a translation of the measurement outcomes that is there in addition to that standardclassicalentropyofGibbsandBoltzmann. The implied by the Heisenberg uncertainty relations. It second is a genuine quantum phenomenon, which does is important to distinguish two extreme examples of not exist in classical physics. sources for such uncertainty. The difference is at the core of the argument of this paper: III. ENTANGLEMENT THERMODYNAMICS a. Statisticalentropy. Considerasystemwithmany degrees of freedom and a small number of macroscopic When the von Neumann entropy describes statistical observablesdescribingit. Fixavaluea fortheseobserv- i ignorance, its relation with thermodynamical entropy is ables. The density matrix ρ representing this informa- ai manifest. In fact, the thermodynamics of quantum sys- tionaboutthesystemistheprojectorontotheeigenspace tems is described in all textbooks in terms of von Neu- determined by these values. The dimension n of this mannentropy. Istherearelationbetween,thermalquan- Hilbert space counts the number of (orthogonal) states titiesandvonNeumannentropywhenitdescribesentan- compatible with these values. The microcanonical von glement? Neuman entropy of ρ ai Entanglemententropyisdifferentfromstandardstatis- tical entropy in some respects, but several characteristic S = p lnp =lnn (5) vN − i i thermodynamical relations remain valid [6]. Let us illus- (cid:88)i trate this fact in a simple case. Consider a system with two degrees of freedom, respectively described by the measures our ignorance of the microstate, where the Hilbert spaces and ˜. In the tensor product Hilbert probability of every microstate is p = 1/n. This is the i H H space consider the pure state standard entropy of statistical mechanics. (In a similar manner one can construct the canonical version of the density matrix.) Ψβ = e−β2En n n˜ . (10) | (cid:105) | (cid:105)⊗| (cid:105) n (cid:88) b. Entanglement entropy. Consider now one of the where β is a parameter that we keep free, H n =E n subsystems of a larger system formed by two parts, de- | (cid:105) n| (cid:105) in , where H is the hamiltonian, and the basis n˜ in scribed by the Hilbert spaces H and H(cid:48), respectively. A ˜ His an arbitrary orthonormal basis. This is a h|ig(cid:105)hly genericpurestateofthecompositesystemcanbewritten H entangled pure state. The corresponding density matrix in the form obtained tracing over ˜ is H ρ =N e βH (11) Ψ = c n m (6) β β − | (cid:105) nm(cid:48)| (cid:105)⊗| (cid:48)(cid:105) n(cid:88),m(cid:48) where N is the normalization factor needed to have β tr[ρ] = 1. The expectation value of the energy H in where n and m are bases in the respective spaces. (cid:48) | (cid:105) | (cid:105) the reduced state is If we consider only measurements performed on the first system, these are fully characterised by the density ma- E(β)= Ψ H Ψ =tr[ρ H]= H (12) trix (cid:104) β| | β(cid:105) β (cid:104) (cid:105)ρβ and the von Neumann entropy is given by ρ=tr Ψ Ψ = p n n (7) H(cid:48)| (cid:105)(cid:104) | n| (cid:105)(cid:104) | n (cid:88) S(β)= tr[ρ lnρ ]. (13) β β − where Restricted to , the state looks “thermal” even if the H p = c 2. (8) systems as a whole is in a pure state and does not have n | nm(cid:48)| manydegreesoffreedom. Now,considerasmallvariation (cid:88)m(cid:48) δ Ψ of the state Ψ . The change in entanglement en- β β The von Neumann entropy is tr|opy(cid:105)S(β) is prop|orti(cid:105)onal to the change in the averaged energy, namely S = p lnp (9) vN n n − δS = tr[δρ lnρ ]=βtr[δρ H]=βδ H . (14) n β β β (cid:88) − (cid:104) (cid:105) and measures the amount of entanglement between the wherethefirstequalityfollowsfromtheidentitytr[δρ ]= β two systems. 0 and the second from the Gibbs form of (11). 3 By using (12) and defining spacetime region bounded by the horizon of an accelerated observer in the pure state 0 can be rewritten as M 1 | (cid:105) T , (15) ≡ β ρ=tr 0 0 =Ne βH, (20) ˜ M M − equation (14) takes the form of a thermodynamical rela- H| (cid:105)(cid:104) | tion where δS =δE/T. (16) 1 a(cid:126) T = = , (21) β 2π Equation(16)isarelationofequilibrium. However,there is no thermodynamical limit, no large number of degrees whichistheUnruhtemperatureassociatedtoanobserver of freedom in the argument above. In the simple context accelerating in the Minkowski quantum-field vacuum [9]. of a two-degrees of freedom system, T does not have an The Unruh temperature is a physical temperature: it interpretation as temperature. determines the transition probabilities of a system in- Ontheotherhand,ifwehavemanycopiesofthiscou- teracting with the quantum field and moving along the pled system, and a thermometer coupled with them all, accelerated trajectory. then T can be interpreted as a temperature. Indeed it This shows that under appropriate conditions the determines the raising and lowering transition probabil- quantum mechanical entanglement can define a temper- ities between thermometer eigenstates that thermalizes ature which is the temperature measured by a physical to the probability distribution thermometer. In particular, this happens when the density matrix obtained by reducing the state to a subalge- p e βE (17) − bra has the form (11) for a hamiltonian H generating a ∼ flowthatcanbeidentifiedwiththetimeevolutionofthe where E is the energy difference between the thermome- thermometer. ter states. This result follows from the fact that (11) is Parenthetically we note that the density matrix asso- a KMS state for the time flow, and its proof is recalled ciated to a subsystem can generically be written in the in Appendix A. form The example above might sound artificial, since the one-parameter family of states (10) was introduced ad ρ e H (22) − hoc. But it is not so artificial: in fact, its structure ∼ exemplifies that of the Minkowski vacuum 0M under forsome hermitianoperatorH. ItsufficestodefineH as | (cid:105) the split of the Fock space into the tensor product of a minus the log of ρ. The quantity H defined in this man- Hilbertspacecapturingthedegreesoffreedomaccessible ner is called the “entanglement hamiltonian”, “modular from within the horizon of a uniformly accelerating ob- hamiltonian”[7] or “thermal time” [10, 11] hamiltonian server,andthoseoutside(Rindlerwedgelocalization)[7]. in various contexts. It generates an “evolution” in the The Minkowski state is a pure state and when restricted Hilbert space , and the state ρ is a KMS state with totheRindlerwedgeobservablesitturnsouttobegiven respect to thisHflow. In general this flow does not corre- by the density matrix spondtoaflowinspacetime. Inthecaseoftherestriction of vacuum to a Rindler wedge, it does. ρ=tr˜ 0M 0M =Ne−2(cid:126)πK (18) The thermodynamical aspects of the setting consid- | (cid:105)(cid:104) | H ered by Jacobson can be understood in this framework. where K is the boost generator. This is the content of There is a quantum field in its vacuum seen by an accel- the Bisognano-Wichmann theorem [8], which can be de- erated observer. The observer interacts only with a sub rived rigorously from quantum field theory axioms, or algebra of the field observable algebra, and therefore de- formally with a variety of simple manipulations. It is a scribes the state of the field in terms of a density matrix consequence of Lorentz invariance and positivity of the (20). The entropy that quantifies the incertitude, due to energy. entanglement,ofanobserver’smeasurement,andtheex- The motion of a uniformly accelerated observer is pre- pectation value of the generator (19) of his proper-time cisely generated by K. The proper time τ of an observer translations are related by (14), namely moving at constant acceleration a is given by τ = η/a, whereηistheboostparameterofthetransformationgen- δS =δE/T. (23) erated by K. Therefore once rescaled by a, K generates propertime translationson theworldline ofa uniformly The Unruh temperature and the entropy accelerated observer, that is 2π δS = δE (24) H =aK (19) a(cid:126) is the generator of the evolution in the proper time in canbecomputeddirectlyfromtheformoftheentangled the Rindler wedge. As a consequence, the density ma- state ρ. These are relations that pertain to standard trixthatrepresentstheoutcomeofmeasurementsonthe quantum field theory and do not require any additional 4 hypothesis about underlying degrees of freedom to be andasaconsequenceiftheconstantαistobeuniversal, true. TheydependonthelocalLorentzinvarianceofthe area and energy variations must be related. Inserting vacuumandinparticularonitslocal(shortscale)conse- the value of T from equation (24) into (29) leads to the quences. General Relativity plays no role so far. Equa- fundamental relation tion (24) has nothing to do with the Einstein equations. a(cid:126) In particular, the relation δS = δE/T knows nothing δE = αδA. (30) 2π about the Newton constant or the Einstein equations. How do then Einstein’s equations emerge in Jacobson’s Notice that this relation has nothing thermodynamical argument? in it. It is a relation between energy and geometry. This relationwasderivedbyFrodden,GoshandPerez[19]ina different context as a consequence of the Einstein equa- IV. THE JACOBSON RESULT tions (see also [20]). The Frodden-Gosh-Perez relation reads The key ingredient of Jacobson’s derivation is the as- a δE = δA. (31) sumption that the entropy density per unit horizon area 8πG isnotonlyfinite,but,mostimportantly,universal. That andagreeswith(30)identifyingtheuniversalconstantα is, in the relation as α=1/4(cid:126)G. Thus, what Jacobson has shown in his derivation S =αA, (25) (which we recall in Appendix C) is that equation (31) is itisassumedthatαisfiniteanddoesnotdependonany notonlyaconsequenceofEinstein’sequations,butisalso detailofthephysicalsituation. Here,Aistheareaofthe a sufficient condition for the Einstein equations to hold. 2-dimensional spacelike bifurcation surface of the causal If we assume the validity of (31) in any frame, then the horizon comprising the wedge. Einstein equations follow. In this sense, Einstein’s equa- This universality is a strong assumption that falsely tionsareencodedintheproportionalitybetweenclassical may appear to be innocent. The fact that the entropy variation of energy and horizon area, as measured by a across a region is proportional to the area is common uniformly accelerating observer. when correlations are sufficiently short scale compared By itself, the fact that the Einstein equations can be to the geometry of the surface. However, in general the encoded in a single simple equation relating an energy proportionality constant will depend on the system and and a geometrical quantity is well known. Of course the on the state. Jacobson assumes it does not. trick is requiring that this holds everywhere and in any The entropy density of the density matrix (18) is in- frame, thus adding general covariance as an independent finite, because of the contribution of arbitrarily short input. Forinstance,seethenicepaperbyJohnBaezand wavelength correlations across the Rindler horizon. If Emory Bunn [21] where it is pointed out that Einstein’s we cut off the theory at some length (cid:96) , we obtain an equationsfollowentirelyfromtherequestthatthesecond p entropy that scales as 1/(cid:96) 2 [12–15], that is variation of the volume of a small sphere of particles is p proportional to the sum of the energy and momentum α S = cA (26) flow components in the centre of the sphere. Jacobson’s (cid:96)2 p isanelegantandnullversionofBaezandBunn’stimelike or α = αc. If we change the state, the area does not result. (cid:96)2p The important point of this discussion is that Jacob- change(weareonafixedgeometry)buttheentropydoes son’s result can be split into two parts. The second part change, therefore being the step from (30) to the Einstein equations. This is a nice piece of differential geometry, but has nothing δS =δ(αA)=(δα)A. (27) to do with thermodynamics. The first step is to get to If one instead assumes, as Jacobson does, that the pro- equation(30). Aswehaveseen,thisequationcomesfrom portionality constant between entropy and area is uni- considering the entanglement entropy across the horizon versal, then one either gets a contradiction (because one plus a universality assumption. Again, here statistical can change the entropy by changing the state), or one considerations play no role, and nothing points to un- hastorelaxthefactthatthegeometryisfixedandallow derlying degrees of freedom, or to a reading of the Ein- ittochangewithachangeinthestate,thusallowingthe stein equations as equations of state. The key is simply area to vary theuniversalityoftheentanglemententropy-density. We comment on this below. δS =αδA. (28) As we have seen above the entanglement entropy across V. UNIVERSALITY: THE SPECIES PROBLEM asurfacesatisfiestherelationδS =δE/T,whereE isan energy. Therefore we obtain TheinterpretationoftheBekensteinentropyasentan- δE =TαδA, (29) glement entropy has been called into question precisely 5 due to the unclear dependence of the entanglement en- gain an intuitive understanding of the fact that gravity tropy on the number of species at the cutoff scale. How cuts the degrees of freedom off in a universal manner as canentanglemententropynotbedependentonthenum- follows. The quantum gravitational regime starts when ber of species? thetotalenergydensityreachesthePlanckscale. Matter A recent result [4] has shown that for a low energy fluctuationsinteractgravitationally,sointhepresenceof perturbation of the Minkowski vacuum, the variation of more than one field, say n fields, this regime is reached entanglement entropy δS across the Rindler horizon is at a larger length scale than in the presence of a sin- independent from the number of matter species and in- gle field. More precisely, if we introduce a cutoff at the dependent from the ultraviolet cut-off, provided one in- scale (cid:96), on dimensional grounds the vacuum energy den- cludes gravitons together with matter fields, in a per- sity of the fluctuations of a single field is proportional to turbative approach. What happens is that the change (cid:96) 4. Within an (cid:96)3 volume, the (negative) gravitational − in the energy of the vacuum δE, expressed in terms of potentialenergyofthesefluctuationsgoesasG(cid:96) 3 andis − theenergy-momentumofmatterand gravitonsT ,isre- proportional to n2 because each field interacts with any µν latedtothed’Alambertianofthegravitonfieldbymeans otherfield. Weenterinastronggravitationalfieldregime of the perturbative Einstein equations, at the scale where these two energies balance, namely at thelengthscale√n(cid:126)G. Thus,forasinglefieldweexpect (cid:3)h = √8πG(T 1η Tσ), (32) a gravitational cutoff at the Planck scale √(cid:126)G, but for µν − µν − 2 µν σ n fields the cutoff scale is reached at a larger scale by a factor √n, that is, the cutoff scale (cid:96) depends on the from which number of fields as a δE = δA (33) 8πG (cid:96) √n. (35) ∼ follows. Thechangeintheareaofthecausalhorizonhas The entanglement entropy of n fields is nowasemiclassical interpretation. Classicalbackground A geometryisnotaffected,andtheperturbationofthearea S n (36) isdeterminedbythequantumgravitationalfieldh and ∼ (cid:96)2 µν given by the expectation value of the operator Aˆ associ- where (cid:96) is the cutoff scale. The last two equations to- ated to the first order variation of the area with respect getherindicatethatS doesnotdependonthenumberof to its background value, as a function of hµν. That is fields [22]. It is the universality of gravity, not statistics, δA=δ Aˆ tr[δρAˆ]. thatdeterminestheentanglemententropyuniversalityof (cid:104) (cid:105)≡ This perturbation controls the deflection of light rays the entropy density. relatedtotheexpansionoftheareaofthehorizonsurface, So far, we have stayed away from the Planck scale. It so that equations (16), (21) and (33) together provide a is time to ask what may happen there and complete the dynamical Bekenstein-Hawking area law relation, picture. δA δS = , (34) 4G(cid:126) VI. FINITENESS: THE JACOBSON RESULT FROM QUANTUM GRAVITY which relies on the entanglement of the gravitationally dressed vacuum fields. At the beginning of the XX century, Planck and Ein- Importantly, equation (34) does not depend on the steinwereledtotheexistenceofafundamentaldiscrete- physics at the UV scale, as δS = δE/T involves only ness in the structure of the electromagnetic field, and in differences in the low energy modes, while its universal facttothediscoveryoftheexistenceofthephoton, from character has a dynamical origin: it derives from the theabsenceofthe“ultravioletcatastrophe,”namelyfrom universal coupling of gravitons to the energy momentum the finiteness of the entropy of the electromagnetic field. tensor. The entropy of black body radiation, indeed, is given by At the perturbative level, this result provides a con- nectionbetweenBekenstein-Hawkingentropyandentan- glement entropy and indicates that the relation between 4 π2VU3 area and energy variations in (33) can only have a dy- S = 4 , (37) namical origin: it requires the Einstein equations. 3 (cid:114)15c3(cid:126)3 This connection between Bekenstein-Hawking entropy where V is the volume of the box and U the energy, and entanglement entropy is not in terms of statistical and this diverges if we take (cid:126) 0. Quantum theory → ignorance about mysterious underlying microscopic de- is the source of the Planck-size granularity that renders grees of freedom, it does not imply any “thermal” char- phase space volumes finite, and therefore avoids the di- acterization of the Einstein equations. vergence of the entropy. In the entropy (34), associated The universality of α is the effect of an active role of to a gravitational field, (cid:126) is in the denominator as well. gravitational dynamics at the quantum level. One can Thisindicatesthatitisquantummechanicsthatcutsoff 6 6 of a given area and with normal in direction ~n has the the divergence of the entropy. In fact, the formula in- form s = s ,where s isthestatewithmaximum dicates that there should be a physical (real) quantum f f f | i ⌦ | i | i magnetic number in direction ~n mechanical cutoff at the Planck scale. This is precisely the conclusion that one obtains from j,~n �j,j;j,+j . (35) a full quantum gravitational treatment of the problem | i⌘| i in the context of loop quantum gravity, where the dis- Now, the single puncture state s is an eigenstate of creteness of the geometry at the Planck scale appears | fi the area operator of the corresponding facet as a conventional quantization effect: the gravitational field determines lengths, areas and volumes; since the Af =8⇡G~� Lf , (36) gravitational field is a quantum operator, these quanti- | | ties are given by quantum operators. Planck scale dis- so that the facet has area Af = 8⇡G~�jf. The states cerreatteonrse,sswfhoilclhowinsdfircoamtesthmeinsipmeuctmranloann-avlaynsiisshoinfgsuscizhesopat- FIG. 1. IndFicIaGt.iv1e.pSiicntgulree,litnokbaendchiatsngdeuda.lbfaacseict.system: sin- |rsefpireasreenptaotsitounlawtehderteotlhiveerienlautniointa(r3y0i)rrheodludcsi.bFleorSaL(s2in,gCle) gle link with its dual facet. the Planck scale. This is the key result of the loop the- puncture, the relation K =� L then gives f f oryandisresponsiblefortheUVfinitenessoftheentropy The two spin j representations of Eq.(39) are each | | | | density. mtoappoploegdicianltfiheilsdwthaeyo,rtyh,earnefdordeefitnheessqpuinanjtucmomgpeonnereanltroefl- Kz = Af (37) Let us see how Jacobson’s result emerges from this tahteivsittyatienstphaeceSpliivnefsoainmaaspupbrsopaacche[o]f3. f 8⇡G~ framework. For this, we recall some basic notions from Condition (30) defines a subspace of , spanned by theInfullolotphegorrayv,irteyl,evqaunatnttoumthestparteessenotfdtihsceugsseioomne[2tr3y,2a4r]e. the subspaces Hpj,k w(Hhejγr)e∗T ⊗(Hjγ)S Hp,k (42) wThheisrereKlafztio=nKb~e·tw~neiesntthheebaoroesatodfuaaslptaocethliekefascuertfascuerf(ascaey. described by SU(2) spin networks. Let us consider here where we have indicated by and the source and tar- in the x y plane) and the boost (in the t z plane) is for simplicity a single link of the spin network. The cor- get ends of the link,pr=esp�ejc,tSivkely=. TjO;n the image of (t3h1e) at the ro�ot of the classical equation in (27).�Why so? responding quantum state is a function ψ(U) on SU(2), map Y , the boost generator K(cid:126) and the rotation genera- Letsfocusonthesinglelinkandthecorrespondingdual that isγ, the space and U is classically interpreted as the open path holon- tor L(cid:126) satisfy facet. Givenafunction (hl)ofSU(2)groupelementon omy of the gravitational Ashtekar connection along the the link, the maps Y sends (h ) to a generalised func- = ( j ) j . (32) � l link. AbasisofstatesisprovidedbytheWignermatrices H� �jK(cid:126) H=�jγ,j L(cid:126)⇤⌦H�j,j (43) tion � of Y� (gl) of SL(2,C) elements. The associated (Peter-Weyl theorem) (cid:104)⇣ (cid:105) (cid:104) (cid:105) ⌘ states � defined on the single-link Hilbert space of SL(2,C) functions of the form | i as matrix elements. (cid:104)U|j,m,n(cid:105)=Dm(jn)(U). (38) Inthec las=sicaltheocry,equ�ajti,ojn;((j4,3m))i,s(sja,tmisfi)e.dby(t3h3e) H�j =(H�jj,j)⇤⌦H�jj,j, (38) Equivalently, the state space associated to the link conjugat|e mi omentumj,omf,mth0e| Lorentz connect0ioin, which decomposes as H becomes the SjL,Xm(2,m,C0 ) generator in the quantum theory satisfy (30). (see Appendix B). This relation directly parallels (31) The tensor product decomposition of the single-link Among the general quantum geometrical states, the = ( ) (39) because (from the canonical analysis of the action) the Hilbert space is related to a local “partition of space” H ⊕j Hj∗⊗Hj astraeate-vseicnto(r33is) raerleattehdetooneL(cid:126)s dbyynAa(cid:126)m=ica8llπyγGre(cid:126)leL(cid:126)vaanntdfoK(cid:126)r tihse in two chunks, associated to the source and target nodes quantum description of general relativity. The space where j is the spin j irreducible representation of related to the energy by (19). H� comprising the link. In this sense, we wish to redefine H is not a subspace of the Hilbert space of the square inte- SU(2). The two factors are associated to the two ends the single-link Hilbert space as The SU(2) spin-network states represent the geome- grablefunctionsL [SL(2,C)], but it isnaturally isomor- of the links, respectively, as each transforms under the try of a spatial sec2tion of spacetime with normal tµ. If gauge transformation at one end. The dynamics of the wpheicmteoasLu2r[eStUh(e2)g]e.oTmheetrmyaopf a spacelike surface with a H�j =(H�jj,j)S ⌦(H�jj,j)T =HS ⌦HT (39) theoryisobtainedmappingthesestatestounitaryrepre- sentations of SL(2,C). A unitary representation (in the csuerrteaminenptrsebciysitoensY,s�ewl:leaVtcia(njng) !dweistVchr(i(Nb�je)f,tjah)ceetrsesfu.ltEoafchoufracme(te3ai4s-) wherewethinkabouttheindividualsubspacesasdefined apnridncaipcaolnsteirniueso)u[s25p]airsalmabeteellredpbyaR+disacnredtethsepinrekpr∈ese12nN- gcaoningvitsouabliesedeasscrpibu|nejdc,tmubyriin!ags|pt(h�inej,nsjue)tr;wfja,ocmreki[2li6n]k,,sweehiFchig.w1e. bbyy tthheeppaurntcittiuorneoofnthtehelindkuainl ftawcoethsaulfrflaincek.s lISn,lfTac,tg,iovnene tinattoionirrsepdaucceibilsedreenporteesdenHta(tpi,okn).s∈oTfhtishespSaUce(2d)ecsoumbgproosueps Csreeapnlrdles|sseLfn2(cid:105)t[StthhUee(s2gt)ea]ottmeoedHters�yc.roiNbfioanwgs,paSatUfiaa(cl2e)stesocptfiinoan-nqoeutfawnsopturakmces-tstauimtre-es cbaonuntdhainryksoufrftahcee bfaectewtee4natshseoctiwaotesdpatcoetchheunliknsk. as the as follows fshfaataaccveteienoShgfaawgsgiivmlilveanebnxenimaotrreuemsmasaeallmnlatdaµtge.wndieItnbthiycpnnaoNrrutmmifcaabucleleiarntrsi,ndfaird,esicerpteaaiccocthneio-o(cid:126)lnnink.(cid:126)neeT.schuIoinsrr-- linNk.owF,olretea|c�hih2aHlf�jofbtehae pliunrkelstattheeoref SisLa(n2,aCs)soocniatthede writing it, for notational simplicity, one can restrict to H(p,k) =⊕∞j=kHj (40) (cid:126)nrthe=sepsozûn,rdtfhaincegen[t?o a].lTinhkeosftaatespdines-ncreitbwinogrkagqraupahntpuumncstuurrfaincge mfreixededomstaatsesoicniaHteTd,tgoivlen, by traTcing over the degrees of where isthe(finitedimensional)SU(2)representation S j H s =Y j,j = (γj,j);j,+j . (44) of spin j. Therefore (p,k) admits a basis (p,k);j,m | f(cid:105) γ| (cid:105) | (cid:105) ⇢ =tr [� �]. (40) obtained diagonalizingHthe total angular mo|mentum L2(cid:105) T HS | ih | The state s is an eigenstate of the area operator and aTnhdetmhaepLtzha=t L(cid:126)giv·e(cid:126)zsctohmispionnjeecnttioonf,tahnedSdUefi(2n)essutbhgerolouopp. o3fGdtyhenneaemrtahilcesr,e|nlpoafltru(cid:105)imvsitaaylltcinoaenatrhbee“swcimorriptrlteiecsniptyao”sncdaoinnBdgFitfiaothnceeootrnyntowhreimthmalotirmsiveeidna-l Ttahinisedstaabteouetnc�odebsyapllertfhoerminifnogrmmaetaiosunretmhaetntcsanlocbaelisoebd- to the area | i quantum gravity covariant dynamics is given by tum conjugate to the gravitational connection [? ]. This, in in the “target” chunk of space associated to l . turn, reduces to a proportionality between the space-space and T Yγ :Hj →Hjγ (41) tidheenttiimfieeAd-szfwp|aistcfhe(cid:105)Kc=omth8peπobGnoe(cid:126)onsγttsLgozfefn|stehfria(cid:105)sto=mroa8mbπoeGvnet(cid:126)u,γmajn;d|tshtfhe(cid:105)e,fisresctocna(d4nc5ab)ne |j,m(cid:105)(cid:55)→|(γj,j);j,m(cid:105). whbiseerasenhaLolowzfgno=utsoL(cid:126)tbofet·hpzêro.fpaToctrhtteihonastatal(2tt7eo)sAc|a/sp8f⇡t(cid:105)uGraer(sestehemeAagppenppeeerndadlibxreylAaYt)i.γvTisthtoiics 4 This surface is defined by ~n, once the embedding SU(2)tµ Here γ R+ is the Immirzi parameter. thdeyunnaimtaicrsy. irreducibleSL(2,C)representationwherethe SL(2,C)isfixedbythechoiceoftµ. ⇢ ∈ 7 relation (43) holds. For a single puncture, the relation Therefore, given K =γ L , one finds f f (cid:104) (cid:105) (cid:104) (cid:105) K(cid:126) =γ L(cid:126) then gives f f (cid:104) (cid:105) (cid:104) (cid:105) δA f δS = (cid:104) (cid:105), (51) Kz = (cid:104)Azf(cid:105) (46) 4G(cid:126) (cid:104) f(cid:105) 8πG(cid:126) that is the area-entropy relation (34). We see here that where Kz is the boost dual to the facet surface. This thefinitenessoftheentropydensityreliesonthediscrete- f ness of the geometry at that scale and, in particular, on relation gives immediately equation (31), which is the the finiteness of the spectrum of the area operator. basis of the second part of Jacobson’s argument. From Equation (31) is a classical relation while (46) is as- thisrelationwecanobtaintheEinsteinequations. Notice sociated to a quantum mechanical system. To clarify that this is a relation between the area of a space like the relation between the two, we need to recall some ba- surface (say in the x-y plane) and the boost hamiltonian sic structure in quantum theory. This always refers to (in the dual t-z plane). As first observed by Smolin [27], results of a measurement performed by an external sys- this is a direct way of deriving the Einstein equations tem. Here the quantum system is a portion of quantum from the covariant loop quantum dynamics. spacetime. The external observer is an accelerated ob- Next,observethatthetensorproductstructurein(42) server which can measure local aspects of the geometry. is related to a local “partition of space” into two chunks, The states s describe a portion of a quantum surface, associated to the source and target nodes at the two f | (cid:105) a small facet with given area A and normal in direction ends of the link. In fact, the gravitational field opera- f (cid:126)n. The evolution of the facet state in spacetime as seen tors, which are given by the left invariant and right in- by an accelerated observer is generated by the unitary variant vector fields act respectively on the source and operator representing a Lorentz boost target end, and correspond to measurements performed on one side or the other of the facet [23]. It therefore sτ =U(τ)s (52) makessensetodefineahalf-linkHilbertspaceassociated | f(cid:105) | f(cid:105) to each end of the link. The short-scale correlations of where U(τ)=e(cid:126)iHτ, where the operator H is thetheoryarethencapturedbythecorrelationsbetween these two Hilbert spaces, namely by the failure of the H =(cid:126)Kfa (53) link states to be diagonal in this tensor decomposition. If |Φ(cid:105)∈(Hjγ)∗⊗(Hjγ) is a pure state of the link, then its with a being the acceleration of the observer. This op- restriction to one half-link is the density matrix erator generates the evolution of the state along a spe- cific isometry trajectory of the boost labelled by a. This ρ =tr [Φ Φ], (47) trajectory represents the world line of an observer who T HS | (cid:105)(cid:104) | moves in Minkowski space with uniform acceleration a. that encodes the results of the measurements performed In this sense, a classical measurement of the facet state on the target side only of the facet (with an analogous dynamics is associated to the covariant dynamics of the relation holding for the other half of the link). The von trajectory of the accelerated observer. This can be seen Neumann entropy of this density matrix measures the either passively as a motion of an observer in spacetime entanglement across the facet. or actively as the evolution of spacetime seen by an ob- Let us now consider a particular family of states Φ | 0(cid:105) server. suchthattheassociatedreduceddensitymatrixtakesthe Written in terms of the accelerated observer’s quanti- the form ties, the density matrix in (48) reads ρf =e−2πKf. (48) ρf =Ne−(cid:126)2πaH (54) whereK =K(cid:126) (cid:126)nistheboostgeneratorinthedirection f f· and the expectation value of the Hamiltonian operator normaltothefacet(thenotationρ =ρ =ρ indicates the symmetry of the reduced denfsity mSatrixT). We call H gives these states “Hadamard states” for a reason that will be A a f clear below. E = (55) 8πG These states have interesting properties. They have von Neumann entropy where E = H has now the dimensions of an energy. (cid:104) (cid:105) Thisisexactlyrelation(31),thecorephysicalinputfrom S = tr[ρflnρf]=2πtr[ρfKf]=2π Kf , (49) whichtheEinsteinequations, asshownbyJacobson, fol- − (cid:104) (cid:105) low in the classical limit. where K = Φ K Φ istheexpectationvalueofK f 0 f 0 f To the ensemble of single facet states given by (54), (cid:104) (cid:105) (cid:104) | | (cid:105) on Φ . Asshownin(24),avariationδρrelatesachange 0 the observer can effectively associate an absolute tem- | (cid:105) intheentropytoachangeintheexpectationvalueofK , f perature, via the general definition namely δE a(cid:126) δS =2πδ K . (50) T = = . (56) f δS 2π (cid:104) (cid:105) 8 becauseifitinteractswithalargenumberofthese,thisis What is the physical meaning of the Hadamard-like the temperature determining the transition probabilities states introduced in Sec. VI? These quantum states de- betweenitseigenstates. ThisistheUnruhtemperature.1 termine all the relations necessary for Jacobson’s calcu- Therefore all the ingredients for Jacobson’s derivation lation. They are analogous to those obtained restricting follow (see also [27]). theMinkowskyvacuumtotheRindlerwedge. Likethose Wehaveillustratedhowgeometricaldiscretenessarises states,theypreciselydescribetheshortscalecorrelations in theformalism of loop gravity and shownthat this dis- andtheeffectoflocalLorentzinvarianceandthepositiv- cretenessgivesrisetoallofJacobson’sinputsmicroscop- ity of the energy. Formally, a pure state on the link that ically. This line of thinking has exposed the possibility gives rise to the density matrix (48) is simply built by that local Lorentz invariance is encoded directly in the having the source component of the state related to the quantum correlations at the Planck scale and led us to target one by a Lorentz transformation with imaginary introduce quantum gravitational Hadamard states. We parameter iπ further develop this idea below. Φj,(cid:126)n =e πγj j,(cid:126)n . (57) − (cid:104) | (cid:105) (cid:104) | S T VII. THE ARCHITECTURE OF In turn, this form of the state is formally determined SEMICLASSICAL SPACETIME by requiring a global Lorentz invariance and positivity of energy that allows for an analytic continuation [8]. In covariant loop quantum gravity [23] the dynamics Thus, these states can be viewed as implementing the is captured by a relation between the area of a spacelike local properties that render a semiclassical geometry lo- surface (say in the x-y plane) and the boost (in the t-z cally Lorentz invariant. If every link of a spin network plane). This is beautifully made explicit by the result carries the correlations of these states, then the corre- of Jacobson, which shows that (31) yields the Einstein sponding quantum state has everywhere the short scale equations. correlationsofaconventionalquantumfieldtheorystate. ThedynamicalinputthatgivestheEinsteinequations These are therefore the states that may build up the in Jacobson’s derivation is not the thermodynamics. It architecture of a classical spacetime geometry [5]. In istherelationbetweentheboostgeneratorandthearea. quantum field theory on curved spacetime attention is This is the input that contains the Newton constant restricted to states having an appropriate short scale be- G and gives the dynamics. Then, the presence of the havior, which is determined by it being the same as that horizon leads to the entanglement that gives rise to the of a flat Lorentz invariant quantum field theory. States entropy, and the relation between this entropy and the withthislocalpropertyarecalledHadamardstates. This boost generator determines the KMS temperature. iswhy,byanalogy,wecallthestatesofSec.VIHadamard In the case of a black hole, as shown in [19], the en- states: theybehavelikequantumfieldstatesinthesmall. tropy at infinity, the energy at infinity (ADM mass) and Notice that the relation here is not at the infinitesimal the temperature at infinity (the Hawking temperature) level, but at the finite Planck-scale, namely across each are simply obtained by red shifting the local quantities singlelinkofaspinnetwork. Quantumgravityisatheory to infinity. Their relation is precisely the local Clau- without infinities. siusrelationredshiftedtoinfinity. Thisisunproblematic standard physics (much like the Tolman law) and is well understood. VIII. CONCLUSION Theseconsiderationsdonotputintoquestionthetight constraints of reciprocal consistency that gravitational Jacobson’sderivationoftheEinsteinequationsadmits dynamics and thermodynamics of the vacuum state im- aninterpretationdifferentfromthecommononewhereit poseoneachother;aconsistencythatJacobson’sderiva- isassumedtoindicatetheexistenceofmicroscopicstates tion sheds light on. It is quite remarkable, indeed, that forwhichtheEinsteinequationswouldbeanequationof so much of the gravitational dynamics is constrained by state. Thealternativeinterpretationisbasedontheiden- consistency with the thermodynamics of quantum fields. tificationoftherelevantdynamicssimplyasthequantum But thermodynamics does not and cannot give the full version of Einstein’s dynamics. gravitational equations, as is clear for instance from the The relevant microscopic degrees of freedom are the factthatgravitationaltheoriesdifferentfromgeneralrel- quanta of the gravitational field. These are discrete at ativity are also compatible with the field thermodynam- the Planck scale, and this is the source of the finiteness ics [28]. of the entropy. The interaction of any system with the gravitational field in a region of spacetime is described by the field operators of the quantum theory. As for the 1 Notice also that for a black hole the Hawking temperature at electromagnetic field, some of the quantum states of the infinity is just the red shifted version of the near horizon temperature,andtheasymptoticADMenergyisjusttheredshifted gravitational field have a semiclassical behavior. These localenergy. Thereforethestandardblackholethermodynamics appear as continuous geometries when probed at large follows. scales: the mean value of the field operators determines 9 the value of the geometrical observables. The approxi- ratio, but is also implied by it. mation holds only in the semiclassical limit, namely at This does not diminish the relevance of Jacobson’s re- scales larger than the Planck scale. These states present sult. To the contrary, it puts it in an even more inter- short scale correlations, like all the finite energy states esting light. Jacobson’s remarkable discovery is not that of conventional quantum field theory. But these correla- there are mysterious microstates beyond the quantized tions do not involve modes of arbitrarily high frequency, gravitational field. It is that the dynamics of gravity because of the Planck scale quantum discreteness. and the Einstein equations can be recovered from a sim- Because of the universality of the gravitational cou- ple relation between boost generator and area of its dual pling,andthankstoamechanismthatcanbeintuitively surface, and these quantities are related to the entangle- understood as explained in Section V, the entropy den- mententropyinsemiclassicalstates. Thisisastepahead sity measuring these correlations is universal. An exter- in unraveling the spectacular beauty and the simplicity nal observer accelerating in the semiclassical geometry of general relativity, not an indication of its limits. interacts only with observables on one side of its causal ——— horizonandcanthereforeassociateanentropydensityto Thanks to Eugenio Bianchi for crucial inputs in the the horizon, which is finite and universal. The dynamics species problem and for numerous discussions. HMH ac- ties any change in the area with an energy flow, there- knowledges support from the National Science Founda- fore,becauseoftheuniversality,thechangeintheareais tion (NSF) International Research Fellowship Program tied by the dynamics to a change in entropy, with a uni- (IRFP) under Grant No. OISE-1159218. versal temperature T. This temperature is determined by the entropy-energy relation and therefore, ultimately, simply by the local Lorentz invariance and the positivity of the energy of the local states showing short scale correlations. In spite of the fact that the source of the entropy is entanglement, this temperature behaves as a Appendix A: KMS and thermometers true temperature because the density matrix is a KMS state with respect to the flow that describes the evolu- Consideraquantumsystemdescribedbyanobservable tion of an observer accelerating in the mean geometry, algebra , with observables A,B,..., which is in a state and because this observer interacts with a large number A of degrees of freedom, all in this KMS state. It follows σ :A→C. Let αt :A→A with t∈R be a flow on the algebra. We say that the state σ is an equilibrium state from this that T is detected by a thermometer moving with respect to α if for all t, with the observer. t The relation (31) between area and energy is deter- σ(α A)=σ(A), (A1) t mined by the quantum dynamics of general relativity: it is true both in the quantum and the classical theory. and that its temperature is T =1/β if The universal relation between area and entropy is de- f (t)=f ( t+iβ) (A2) terminedbytheshortscalestructureofthesemiclassical AB BA − states, their local Lorentz invariance and the positivity where of the energy, as in the Bisognano-Wichmann theorem. In the deep ultraviolet this picture is confirmed by loop f (t)=σ(α AB). (A3) AB t quantum gravity, where all the input formulas of Jacob- Anequivalentformof (A2)isgivenintermoftheFourier son’s derivation can be recovered starting from the ele- mentary quantum dynamics of the theory, for appropri- transform fÃB of fAB as ate states that have semiclassical properties, which we f˜ (ω)=e βωf˜ ( ω). (A4) have called Hadamard states. AB − AB − The entropy in Jacobson’s calculation is therefore en- The canonical example is provided by the case where tanglement entropy across the horizon. It is a property is realized by operators on a Hilbert space where a A H of a pure state, when restricted to the sub algebra of ob- Hamiltonian operator H with eigenstates n and eigen- | (cid:105) servableslyingononesideofthecausalhorizon. Itisnot values E is defined, the flow and the state are defined n relatedtoanymysteriousunderlyingdegreesoffreedom. in terms the evolution operator U(t) = e iHt and the − The entropy balance relation does not imply statistical density matrix ρ=e−βH = ne−βEn|n(cid:105)(cid:104)n|, by uncertainty and in general holds also for entanglement entropy. αt(A)=U−(cid:80)1(t)AU(t) (A5) What Jacobson has shown is (i) that the area-energy and relation (31), combined with general covariance, determines the Einstein equations, a result in line with simi- σ(A)=tr[ρA] (A6) lar previous derivations of the Einstein equations from a single local equation, e.g. [21]; and (ii) that this dynam- respectively. In this case, it is straightforward to verify ics not only implies the universality of the entropy/area that equation (A1) and (A2) follow. 10 However,thescopeofequations(A1)and(A2)iswider Theseobservationsshowthatequations(A1)and(A2) than this canonical case. For instance, these equations define a generalization of the standard quantum statis- permit the treatment of thermal quantum field theory, tical mechanics, which fully captures thermal properties where the Hamiltonian is ill-defined, because of the in- of a quantum system. The structure defined by these finite energy of a thermal state in an infinite space. It equations is called a modular flow in the mathematical is indeed important to remark that these two equations literature: α is a modular flow, or a Tomita flow, for t capture immediately the physical notions of equilibrium the state ω, and is the basic tool for the classification of and temperature. For equation (A1), this is pretty ob- theC algebras. Inthephysicalliterature, thestateω is ∗ vious: the state is in equilibrium with respect to a flow called a KMS state (for the time flow). In order to show of time if the expectation value of any observable is time that a system is in equilibrium and behaves thermally independent. in a certain state, it is sufficient to show that the state The direct physical interpretation of equation (A2) is satisfies these two equations. In order for a KMS state less evident: it describes the coupling of the system with to behave as an equilibrium state at some temperature, a thermometer. however, the corresponding flow must be the evolution To see this, consider a simple thermometer formed by flow of the thermometer. Therefore the claim here is not a two-state system with an energy gap (cid:15), coupled to a that any KMS state, with respect to any flow, behaves quantum system S by the interaction term as a thermal state. Notice that in the context of Hilbert space quantum V =g(0 1 + 1 0)A. (A7) mechanics, a state can satisfy these two equations also if | (cid:105)(cid:104) | | (cid:105)(cid:104) | it is a pure state. The standard example is provided by where g is a small coupling constant. The amplitude for the vacuum state of a Poincaré invariant quantum field the thermometer to jump up from the initial state 0 theory, which is KMS with respect to the modular flow | (cid:105) to the final state 1 , while the system moves from an definedbytheboostinagivendirection. BeingPoincaré | (cid:105) initialstate i toafinalstate f canbecomputedusing invariant, the vacuum is invariant under this flow, and a | (cid:105) | (cid:105) Fermi’s golden rule to first order in g: celebrated calculation by Unruhshows that it is KMS at inverse temperature 2π. t Inthiscase,thephysicalinterpretationissimplygiven W (t)=g dt ( 1 f ) α (V) (0 + i ) + t (cid:104) |⊗(cid:104) | | (cid:105) | (cid:105) by the fact that an observer stationary with respect this (cid:90)−∞ t flow, namely an accelerated observer at unit accelera- =g dt eit(cid:15) f αt(A)i . (A8) tion, will measure a temperature 1/2π. Unruh’s original (cid:104) | | (cid:105) (cid:90)−∞ calculation, indeed, follows precisely the steps above in The probability for the thermometer to jump up is the equations (A7-A10). modulus square of the amplitude, summed over the final state. This is Appendix B: The proportionality between area and t t boost generator P+(t)=g2 dt1 dt2 ei(cid:15)(t1−t2)σ(αt2(A†)αt1(A)) (cid:90)−∞ (cid:90)−∞ The Einstein equations can be derived from the Holst where we have used the algebraic notation ω(A) = action [29, 30] iAi . If the initial state is an equilibrium state (cid:104) | | (cid:105) 1 1 t t S[e,ω]= ((cid:63)e e) F + e e F . P+(t)=g2 dt1 dt2 ei(cid:15)(t1−t2)fA†A(t1−t2). 8πG(cid:18)(cid:90) ∧ ∧ γ (cid:90) ∧ ∧ (cid:19)(B1) (cid:90)−∞ (cid:90)−∞ where e is the (co)tetrad one-form, F the curvature of and the integrand depends only on the difference of the the Lorentz connection ω, the star indicates the hodge times. The transition probability per unit time is then dual on internal indices and a contraction on the internal indices is implicit. On a spacelike boundary Σ, the dP p+ = dt+ =g2 fÃ†A((cid:15)) (A9) quantity 1 1 which shows that (A3) is precisely the quantity giving Π= ((cid:63)e e+ e e) (B2) thetransitionrateforathermometercoupledtothesys- 8πG ∧ γ ∧ tem. It is immediate to repeat the calculation for the isthederivativeoftheactionwithrespectto∂ω/∂t,and probability to jump down, which gives therefore is the momentum conjugate to the connection. p =g2 f ( (cid:15)). (A10) The (co)tetrad e can be chosen to (locally) map Σ into − AA† − a spacelike 3d linear subspace of Minkowski space. The And therefore (A4) expresses precisely the fact that subgroup of the Lorentz group that leaves this subspace the thermometer thermalizes at temperature β, that is invariant is the SO(3) rotations subgroup, and its exis- p /p =e β(cid:15) tencebreaksthelocalSO(3,1)invariancedowntoSO(3) + − −