Entropy, Contact Interaction with Horizon and Dark Energy
EEntropy, Contact Interaction with Horizon and Dark Energy
Ariel R. Zhitnitsky
Department of Physics & Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada
We present some arguments suggesting that the mismatch between Bekenstein- Hawking entropyand the entropy of entanglement for vector fields is due to the same gauge configurations whichsaturate the contact term in topological susceptibility in QCD. In both cases the extra term with a“wrong sign” is due to distinct topological sectors in gauge theories. This extra term has non-dispersivenature, can not be restored from conventional spectral function through dispersion relations, and cannot be associated with any physical propagating degrees of freedom. We make few comments on someprofound consequences of our findings. In particular, we speculate that the source of the observeddark energy may also be related to the same type of gauge configurations which are responsible forthe mismatch between black hole entropy and the entropy of entanglement in the presence of causalhorizon.
I. INTRODUCTION
The relation between black hole entropy and entropyof entanglement for matter fields has been a subject of in-tense discussions for the last couple of years, see reviews[1–4] and original references therein. There are many sub-tleties in relating these two things. The present work isconcentrated just on one specific subtlety first discussedin [5]. Namely, it has been claimed [5] that for spinszero and one-half fields the one loop correction to theblack hole entropy is equal to the entropy of entanglementwhile for spin one field the black hole entropy has an extraterm describing the contact interaction with the horizon.Precisely this contact interaction with the horizon is themain topic of the present work. Before we elaborate onthis subject we want to make one preliminary remarkregarding the term “black hole entropy”. As it has beenargued in a number of papers, see e.g.[6–8] and also reviewpapers [1–4] the notion of black hole entropy should applynot just to black holes but to any causal horizon (“blackhole entropy without black holes”). We adopt this view-point, and in fact we shall not discuss black hole physicsin this paper at all. Rather, the main application of ourstudies will be cosmology of the expanding universe andits causal horizon. For short, we shall use term “entropy”through the paper.The unique features of the contact term related to thevector gauge field in the entropy computations can besummarized as follows [5] (see also follow up paper [9]) :a) the contact term being a total derivative can berepresented as a surface term determined by thebehaviour at the horizon;b) this term makes a negative contribution to the blackhole entropy.c) therefore, it can not be identified with entropy ofentanglement which is intrinsically positive quantity.d) this contribution does not vanish even in two dimen-sions when the entropy of entanglement is identicallyzero as no physical propagating degrees of freedomare present in the system; e) this contribution is gauge invariant in two dimen-sions and gauge dependent in four dimensional case[9];f) the technical reason for this phenomenon to happenis as follows. One can not use the physical Coulombgauge (when only physical degrees of freedom arepresent in the system) as it breaks down at theorigin, where A θ is ill defined. Therefore, an al-ternative description in terms of a covariant gauge(when unphysical degrees of freedom inevitably ap-pear in the system) should be used instead.g) in this covariant description the entropy is obtainedby varying the path integral with respect to thedeficit angle of the cone as explained in [5]. Sucha procedure can (in principle) lead to a negativevalue for the entropy. In fact, it does come outnegative [5].The main goal of the present work is to argue that thepresence of this “weird” term is intimately related to thewell known property of gauge theories when a summationover all topological sectors must be performed for the pathintegral to be properly defined. We explain how all fea-tures a)-g) from the list above can be natural understoodwithin our framework when summation over topologicalsectors is properly taken into account. In Minkowskispace the corresponding procedure is known to produce anon-dispersive contact term with a “wrong sign” whichplays a crucial role in resolution of the so-called U (1) A problem in QCD. Exactly this non-dispersive contact termeventually becomes the “weird” term with properties a)-g) when we go from conventional Minkowski space intocurved/time-dependent background with a causal horizon.Our consideration in this paper will be based on analysisof the local characteristics (such as topological suscepti-bility, free energy density, etc) computed deep inside thehorizon region. It is very different from the computationof a global characteristic such as the total entropy for ablack hole when the closest vicinity of the horizon (justoutside of it) plays the crucial role in the computations.Nevertheless, we remain sensitive to the existence of hori-zon because our analysis is based on consideration of some a r X i v : . [ h e p - t h ] N ov specific topologically- protected quantities. Essentially,by analyzing very unusual features a)-g) listed above welearn some important lessons on behaviour of the groundstate resulting from merely existence of a causal horizonin the presence of the gauge degrees of freedom in thesystem.The workflow is as follows. In section II we presentour arguments for two dimensional case when all compu-tations can be explicitly performed. We generalize ourarguments for four dimensional case in section III. Weargue that this term is indeed gauge dependent in fourdimensions in abelian case as explicit computations ofref. [9] suggest. However, we shall argue that this termbecomes a gauge independent in non-abelian case. Wemake few comments on some profound consequences ofour findings in section IV, where we speculate that thesource of the observed dark energy might be related tothe same gauge configurations which are responsible forthe mismatch between black hole entropy and the entropyof entanglement. II. TOPOLOGICAL SECTORS, CONTACTTERM WITH “WRONG SIGN”, AND ALL THATFOR 2D QED IN RINDLER SPACE
First of all, we shall demonstrate below the presenceof a nonconventional contribution into the energy witha “wrong sign” in Minkowski space. This contributionis gauge invariant, it exists even in pure photo-dynamicswhen no propagating degrees of freedom are present inthe system. It can not be removed by any means (suchas redefinition of the energy) as it is a real physical con-tribution. In particular, the anomalous Ward Identities(which emerge when the massless fermions are added intothe system) can not be satisfied without this term. Weshall argue that this term can be treated as a contactterm, and in fact is related to the existence of differenttopological sectors in this (naively trivial) two dimen-sional photo-dynamics. In different words, the presence ofdifferent topological sectors in the system, which we callthe “degeneracy” for short , is the source for this contactterm which is not related to any physical propagatingdegrees of freedom.As the next step, we shall discuss the same system inthe presence of the horizon in the Rindler space. We shall Not to be confused with conventional term “degeneracy” when twoor more physically distinct states are present in the system. In thecontext of this paper the “degeneracy” implies there existence ofwinding states | n (cid:105) constructed as follows: T | n (cid:105) = | n + 1 (cid:105) . In thisformula the operator T is the large gauge transformation operatorwhich commutes with the Hamiltonian [ T , H ] = 0. The physicalvacuum state is unique and constructed as a superposition of | n (cid:105) states. In path integral approach the presence of n differentsectors in the system is reflected by summation over k ∈ Z in eq.(10, 11,12). argue that the contact term (which emerges itself as a re-sult of topological features of gauge theory) demonstratesthe “weird” and strange properties listed above in thepresence of the horizon.In what follows it is convenient to study the topologicalsusceptibility χ (rather than free energy itself) which isrelated to the θ dependent portion of the free energydensity as follows χ ( β, θ = 0) = − ∂ F vac ( β, θ ) ∂θ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 , (1)where θ is the conventional θ parameter which entersthe Lagrangian along with topological density operator,see precise definition below. We always assume that θ = 0, however χ ( θ = 0) (cid:54) = 0 does not vanish, and infact is the main ingredient of the resolution of the U (1) A problem in QCD [10–12], see also [13–15]. Free energyitself F vac ( θ ) can be always restored from χ as dependenceon θ is known to be F vac ∼ cos θ . As we show below, thetopological susceptibility χ (and therefore F vac ), beingthe local characteristics of the system, nevertheless arequite sensitive to merely existence of horizon, even whencomputed far away from it. As we shall see this sensitivityis related to the degeneracy of the system and topologicalnature of χ . A. Topological susceptibility and contact term
The simplest (and physically attractive) choice isCoulomb gauge when no physical propagating degreesof freedom are present in the system, and therefore thedynamics must be trivial. It is well known, why this naiveargument fails: the vacuum in this system is degener-ate, and one should consider an infinite superposition ofof the winding states | n (cid:105) as originally discussed in [16].Such a construction in Coulomb gauge restores the clusterand other important properties of quantum field theory.The vacuum in this gauge is characterized by long rangeforces (if charged physical fermions are introduced intothe system). This long range force prevents distant re-gions from acting independently. We believe that preciselythis feature leads to the difficulties mentioned in [5] incomputations of the entropy in physical Coulomb gaugein two dimensions, where a covariant gauge has been usedinstead.As our goal is to make a connection with computationsof ref. [5] we shall not elaborate on Coulomb gauge in thepresent paper any further, but rather consider a covariantgauge to study this system. In the covariant Lorentzgauge there are no long range forces. Instead, a new in case of infinite manifold (rather than finite size β = T − ) thefree energy from relation (1) becomes the conventional vacuumenergy as employed in study of the U (1) A problem in QCD in[10–12]. (unphysical) degrees of freedom emerge in the system, seeprecise definition below.We want to study the topological susceptibility χ inthe Lorentz gauge defined as follows , χ ≡ e π lim k → (cid:90) d xe ikx (cid:104) T E ( x ) E (0) (cid:105) , (2)where Q = e π E is the topological charge density and (cid:90) d x Q ( x ) = e π (cid:90) d x E ( x ) = k (3)is the integer valued topological charge in the 2d U (1)gauge theory, E ( x ) = ∂ A − ∂ A is the field strength.The expression for the topological susceptibility in 2dSchwinger QED model is known exactly [17] χ QED = e π (cid:90) d x (cid:20) δ ( x ) − e π K ( µ | x | ) (cid:21) , (4)where µ = e /π is the mass of the single physical state inthis model, and K ( µ | x | ) is the modified Bessel functionof order 0, which is the Green’s function of this massiveparticle. The expression for χ for pure photo-dynamics isgiven by (4) with coupling e = 0 in the brackets whichcorresponds to the de-coupling from matter field ψ , i.e. χ E & M = e π (cid:90) d x (cid:2) δ ( x ) (cid:3) . (5)The crucial observation here is as follows: any physicalstate contributes to χ with negative sign χ dispersive ∼ lim k → (cid:88) n (cid:104) | e π E | n (cid:105)(cid:104) n | e π E | (cid:105)− k − m n < . (6)In particular, the term proportional − K ( µ | x | ) with neg-ative sign in eq. (4) is resulted from the only physicalfield of mass µ . However, there is also a contact term (cid:82) d x (cid:2) δ ( x ) (cid:3) in eqs. (4), (5) which contributes to thetopological susceptibility χ with the opposite sign , andwhich can not be identified according to (6) with anycontribution from any physical asymptotic state.This term has fundamentally different, non-dispersivenature. In fact it is ultimately related to different topo-logical sectors of the theory and the degeneracy of theground state [18] as we shortly review below. Withoutthis contribution it would be impossible to satisfy theWard Identity (WI) because the physical propagating de-grees of freedom can only contribute with sign ( − ) to thecorrelation function as eq. (6) suggests, while WI requires χ = 0 in the chiral limit m = 0. One can explicitly check Here we use Euclidean metric where path integral computations(4) have been performed. factor e π in front of (4) does not vanish in this limit as it is dueto our definition (2) rather than result of dynamics that WI is indeed automatically satisfied only as a re-sult of exact cancellation between conventional dispersiveterm with sign ( − ) and non-dispersive term (5) with sign(+), χ = e π (cid:90) d x (cid:20) δ ( x ) − e π K ( µ | x | ) (cid:21) (7)= e π (cid:20) − e π µ (cid:21) = e π [1 −
1] = 0 . B. The origin of the contact term– summation overtopological sectors.
The goal here is to demonstrate that the contact termin exact formulae (4), (5) is a result of the summation overdifferent topological sectors in the 2d pure U (1) gaugetheory as we now show. We follow [17] and introducethe classical “instanton potential” in order to describethe different topological sectors of the theory which areclassified by integer number k defined in eq. (3). Thecorresponding configurations in the Lorentz gauge on twodimensional Euclidean torus with total area V can bedescribed as follows [17]: A ( k ) µ = − πkeV (cid:15) µν x ν , eE ( k ) = 2 πkV , (8)such that the action of this classical configuration is12 (cid:90) d xE = 2 π k e V . (9)This configuration corresponds to the topological charge k as defined by (3). The next step is to compute thetopological susceptibility for the theory defined by thefollowing partition function Z = (cid:88) k ∈ Z (cid:90) D Ae − (cid:82) d xE . (10)All integrals in this partition function are gaussian andcan be easily evaluated using the technique developed in[17]. The result is determined essentially by the classicalconfigurations (8), (9) as real propagating degrees offreedom are not present in the system of pure U (1) gaugefield theory in two dimensions. We are interested incomputing χ defined by eq. (2). In path integral approachit can be represented as follows, χ = e π Z (cid:88) k ∈ Z (cid:90) D A (cid:90) d xE ( x ) E (0) e − (cid:82) d xE . (11) When m (cid:54) = 0 the WI takes the form χ ∼ m |(cid:104) ¯ ψψ (cid:105)| . It is alsoautomatically satisfied because µ = e π + O ( m ), and cancellationin eq. (7) is not exact resulting in behaviour χ ∼ m in completeaccord with WI [19]. This gaussian integral can be easily evaluated and theresult can be represented as follows [18], χ = e π · V · (cid:80) k ∈ Z π k e V exp( − π k e V ) (cid:80) k ∈ Z exp( − π k e V ) . (12)In the large volume limit V → ∞ one can evaluate thesums entering (12) by replacing (cid:80) k ∈ Z → (cid:82) dk such thatthe leading term in eq. (12) takes the form χ = e π · V · π e V · e V π = e π . (13)Few comments are in order. First, the obtained expressionfor the topological susceptibility (13) is finite in the limit V → ∞ , coincides with the contact term from exactcomputations (4), (5) performed for 2d Schwinger modelin ref. [17] and has “wrong” sign in comparison withany physical contributions (6). Second, the topologicalsectors with very large k ∼ √ e V saturate the series(12). As one can see from the computations presentedabove, the final result (13) is sensitive to the boundaries,infrared regularization, and many other aspects which arenormally ignored when a theory from the very beginning isformulated in infinite space with conventional assumptionabout trivial behaviour at infinity. Last, but not least: thecontribution (13) does not vanish in a trivial model whenno any propagating degrees of freedom are present in thesystem! This term is entirely determined by the behaviourat the boundary, which is conveniently represented by theclassical topological configurations (8) describing differenttopological sectors (3), and accounts for the degeneracyof the ground state . We know that this term must bepresent in the theory when the dynamical quarks areintroduced into the system. Indeed, it plays a crucial rolein this case as it saturates the WI as formula (7) shows. C. The ghost as a tool to describe the contact term
The goal here is to describe exactly the same contactterm (5, 13) without explicit summation over differenttopological sectors, but rather, using the auxiliary ghostfields as it has been originally done in ref. [16] (usingthe so-called the Kogut-Susskind dipole). This auxiliaryghost field effectively accounts for the degeneracy of theground state as discussed above. The computations inboth refs. [5] and [16] are performed precisely in termsof the same auxiliary scalar field defined as follows A µ = (cid:15) µν ∂ ν Φ . (14) One can check that the contribution resulting from the quantumfluctuations about the classical background (8) does not changethe result (13). Indeed, the corresponding extra “quantum” con-tribution e π · (cid:82) d x [ δ ( x ) − V ] = 0 vanishes as expected. See footnote 1 for clarification of the term “degeneracy”. In thegiven context the degeneracy implies the summation over k ∈ Z in eq. (10, 11,12). This formal connection allows us to make a link betweenexpressions (5, 13) for the contact term with “wrong” signcomputed in our framework in terms of the auxiliary scalarfield as described below and the entropy computationsperformed in ref. [5] featuring the “weird properties” e)-g)as listed in Introduction.Our starting point is the effective Lagrangian describ-ing the same two dimensional gauge system. However,now the theory is formulated in covariant Lorentz gaugein terms of the scalar fields [16]. The crucial elementaccounting for different topological sectors of the underly-ing theory, and corresponding degeneracy of the groundstate, does not go away in this description. Rather, thisinformation is now coded in terms of unphysical ghostscalar field which provides the required “wrong” sign forcontact term (5, 13).Precise construction goes as follows. The effective La-grangian describing the low energy physics (in Minkowskimetric) is given by [16]: L = 12 ∂ µ ˆ φ∂ µ ˆ φ + 12 ∂ µ φ ∂ µ φ − ∂ µ φ ∂ µ φ (15) − µ ˆ φ + m |(cid:104) ¯ qq (cid:105)| cos 2 √ π (cid:104) ˆ φ + φ − φ (cid:105) . The fields appearing in this Lagrangian are φ = the ghost , φ = its partner , (16)while ˆ φ is the only physical massive degree of freedom. Itis important to realize that the ghost field φ is alwayspaired up with φ in each and every gauge invariant matrixelement, as explained in [16]. The condition that enforcesthis statement is the Gupta-Bleuler-like condition on thephysical Hilbert space H phys which reads like( φ − φ ) (+) |H phys (cid:105) = 0 , (17)where the (+) stands for the positive frequency Fouriercomponents of the quantized fields. One can easily un-derstand the origin for a wrong sign for the kinetic termfor φ field. It occurs as a result of (cid:3) operator when theMaxwell term E ∼ (cid:3) is expressed in terms of the scalarfield (14). As usual, the presence of 4-th order operator isa signal that the ghost is present in the system. Indeed,the relevant operator (cid:2) (cid:3)(cid:3) + µ (cid:3) (cid:3) which emerges for thissystem can be represented as the combination of the ghost φ and a massive physical ˆ φ using the standard trick bywriting the inverse operator as follows1 (cid:3)(cid:3) + µ (cid:3) = 1 µ (cid:18) − (cid:3) − µ − − (cid:3) (cid:19) . (18)This is a simplified explanation on how sign ( − ) emerges inLagrangian (15) describing auxiliary φ field, see originalpaper[16] for details.The contact term in this framework is precisely rep-resented by the ghost contribution[18, 19] replacing thestandard procedure of summation over different topologi-cal sectors as discussed above II B. Indeed, the topolog-ical density Q = e π E in 2d QED is given by e π E =( e π ) √ πe (cid:16) (cid:3) ˆ φ − (cid:3) φ (cid:17) [16]. The relevant correlation func-tion in coordinate space which enters the expression forthe topological susceptibility (2) can be explicitly com-puted using the ghost as follows χ ( x ) ≡ (cid:68) T e π E ( x ) , e π E (0) (cid:69) = (cid:16) e π (cid:17) πe (cid:90) d p (2 π ) p e − ipx (cid:20) − p + µ + 1 p (cid:21) = (cid:16) e π (cid:17) (cid:20) δ ( x ) − e π K ( µ | x | ) (cid:21) (19)where we used the known expressions for the Green’sfunctions (the physical massive field ˆ φ as well as the ghost φ field) determined by Lagrangian (15) and switchedback to Euclidean metric for comparison with previousresults from sections II A, II B.The obtained expression precisely reproduces exactresult (4) as claimed. In the limit e → − ) in the commutation relationsN = (cid:88) k (cid:16) b † k b k − a † k a k (cid:17) (20) (cid:104) b k , b † k (cid:48) (cid:105) = δ kk (cid:48) , (cid:104) a k , a † k (cid:48) (cid:105) = − δ kk (cid:48) one can in fact check that the expectation value for anyphysical state vanishes as a result of the subsidiary condi-tion [16, 18]: (cid:104)H phys | N |H phys (cid:105) = 0 , ( a k − b k ) |H phys (cid:105) = 0 . (21)This vanishing result (21) obviously implies that no en-tropy may be produced in Minkowski space. In differentwords, the fluctuations of unphysical fields described byoperator (20) do not lead to any physical consequences(except for merely existence of the contact term (5) asalready discussed).We shall see in next subsection how this simple picturedrastically changes when we consider the very same sys-tem but in presence of the horizon. We shall argue thatthe number density N of “fictitious particles” with wrongcommutation relations starts to fluctuate in the presenceof the horizon, in contrast with eq. (21). Therefore, weformulate a conjecture that precisely these fluctuationsare responsible for term with a “wrong sign” in entropycomputations [5, 9]. The corresponding contribution, as we already mentioned, is not related to any physical prop-agating degrees of freedom but rather, is due to presenceof topological sectors in gauge theories (and the degener-acy of the ground state as its consequence, see footnote1 for clarification on terminology) which eventually leadto a non-dispersive contribution in topological suscepti-bility. To simplify things in what follows we consider asimple Rindler space when the Bogolubov’s coefficientsare exactly known. However, we argue that a genericcase (when horizon is present in the system) leads to verysimilar conclusion. D. Rindler space
The total entropy with “weird” properties listed inIntroduction was computed a while ago [5], and thereis no reason to review these results in the present paper.These original results have been reproduced in [9] byusing another technique. Furthermore, in the same paper[9] it has been demonstrated that in two dimensionalcase the final result is gauge invariant, and therefore, itobviously represents a physically observable quantity. Aswe mentioned earlier, we are not interested in computingthe global characteristics such as total entropy. Rather,we are interested in computing some local properties, suchas topological susceptibility, or θ − dependent portion ofthe energy density (1) in the presence of the horizon.However, we shall argue below, the source for “weird”features in both cases is the same, and, in fact, related tofundamental properties of gauge theories as discussed insection II B.As we explained above, the presence of different topo-logical sectors in gauge theory (and the degeneracy of theground state as its consequence) leads to contact term (5)even when no physical propagating degrees of freedom arepresent in the system. In physical Coulomb gauge thisterm manifests itself as the presence of long range forcewhich prevents distant regions from acting independently.The same feature but in covariant Lorentz gauge is ex-pressed in terms of new (unphysical) degrees of freedom(16) which emerge in the system and effectively repro-duce the contact term as explicit computations show (19).While these unphysical degrees of freedom do fluctuate,these fluctuations do not lead to any physical observableexpectation values in Minkowski space (21) as a resultof cancellation between two unphysical fields similar toconventional Gupta-Bleuler condition in QED when twounphysical photon’s polarizations cancel each other. Wewant to see how this conclusion changes when a horizonis present in the system.One can repeat the construction of previous sectionII C to describe (unphysical) degrees of freedom but inRindler space [18]. A Rindler observer in (R,L) wedge willmeasure the number density of unphysical states usingdensity operator N ( R,L ) which is given byN ( R,L ) = (cid:88) k (cid:16) b ( R,L ) † k b ( R,L ) k − a ( R,L ) † k a ( R,L ) k (cid:17) . (22)The subsidiary condition (17) defines the physical sub-space for accelerating Rindler observer (cid:16) a ( R,L ) k − b ( R,L ) k (cid:17) (cid:12)(cid:12)(cid:12) H ( R,L )phys (cid:69) = 0 , (23)such that the exact cancellation between φ and φ fieldsholds for any physical state defined by eq. (23), i.e. (cid:68) H ( R,L )phys | N ( R,L ) |H ( R,L )phys (cid:69) = 0 (24)as it should. However, if the system is prepared as theMinkowski vacuum state | (cid:105) then a Rindler observer us-ing the same operator for N ( R,L ) (22) will observe thefollowing number density in mode k , (cid:104) | N ( R,L ) | (cid:105) = (cid:104) | (cid:16) b ( R,L ) † k b ( R,L ) k − a ( R,L ) † k a ( R,L ) k (cid:17) | (cid:105) = 2 e − πω/a ( e πω/a − e − πω/a ) = 2( e πω/a − , (25)where we used known Bogolubov’s coefficients mixingthe positive and negative frequency modes for operators b ( R,L ) k , a ( R,L ) k describing unphysical fluctuations [18].One can explicitly see why the cancellation (21) of un-physical degrees of freedom in Minkowski space fail to holdfor the accelerating Rindler observer (25). The technicalreason for this effect to occur is the property of Bogol-ubov’s coefficients which mix the positive and negativefrequencies modes. The corresponding mixture can notbe avoided because the projections to positive -frequencymodes with respect to Minkowski time t and positive-frequency modes with respect to the Rindler observer’sproper time η are not equivalent. The exact cancellationof unphysical degrees of freedom which is maintainedin Minkowski space can not hold in the Rindler spacebecause it would be not possible to separate positive fre-quency modes from negative frequency ones in the entirespacetime, in contrast with what happens in Minkowskispace where the vector ∂/∂t is a constant Killing vector,orthogonal to the t = const hypersurface. The Minkowskiseparation is maintained throughout the whole space asa consequence of Poincar´e invariance. It is in a drasticcontrast with the accelerating Rindler space [18].The nature of the effect is the same as the conven-tional Unruh effect[20] when the Minkowski vacuum | (cid:105) is restricted to the Rindler wedge with no access to theentire space time. An appropriate description in this case,as is known, should be formulated (for R observer) interms of the density matrix by “tracing out” over thedegrees of freedom associated with L -region. In this casethe Minkowski vacuum | (cid:105) is obviously not a pure statebut a mixed state with a horizon separating two wages,which is the source of the entropy. In contrast with Unruheffect[20], however, one can not speak about real radiationof real particles as the ghost φ and its partner φ arenot the asymptotic states and the corresponding positivefrequency Wightman Green function describing the dy-namics of these fields vanishes [18]. In different words, these auxiliary fields contribute to the non-dispersive por-tion of the correlation function in eqs. (4,5,7), but notto conventional dispersive part which is unambiguouslydetermined by the absorptive function as conventionaldispersion relation dictates.Few more comments on (25) are in order. The effectis obviously sensitive to the presence of the horizon, and,therefore is infrared (IR) in nature. The IR nature of theeffect was anticipated from the very beginning as formu-lation of the problem in terms of auxiliary fields (16) issimply a convenient way to deal with different topologi-cal sectors of the gauge theory in covariant gauge (andthe degeneracy of the ground state as their consequence)instead of dealing with the long range forces in the uni-tary Coulomb gauge as discussed in sections II B andII C. Also: the contribution of higher frequency modesare exponentially suppressed ∼ exp( − ω/a ) as expected.The interpretation of eq. (25) in terms of particles is veryproblematic (as usual for such kind of problems) as typicalfrequencies when the effect (25) is not exponentially small,are of order ω ∼ a , and notion of “particle” for such ω isnot well defined.We do not attempt to reproduce known results onentropy from ref. [5] based on non-vanishing expectationvalue for number density operator (25). First of all, it isnot obvious what would be the physical meaning of sucha computation based on expectation value (25) for theoperator which satisfies “wrong” commutation relation(20). Furthermore, it is not obvious how to interpret Nparticles from eq.(25) when entire notion of particles isnot even defined for relevant parameters. Indeed, as weargued above the effect is large (cid:104) | N | (cid:105) ∼ λ ≥ a − which is the size of the horizonscale.Our goal here is in fact quite different. We want toargue that the source for the “wrong sign” in entropycomputations [5] (featuring the “weird properties” aslisted in Introduction) and the source for the “wrong sign”for the contact non-dispersive term (discussed in presentpaper) are in fact have the same origin. In addition to thearguments presented above, we note that the technicalcomputations of the entropy performed in [5] are actuallybased precisely on the same representation for A µ field (14)describing fluctuations of unphysical auxiliary degrees offreedom. This representation for A µ field in our formalismeventually leads to the expression for the contact non-dispersive contribution ∼ δ ( x ) with “wrong” sign (19)and non vanishing number density (25) while in ref.[5] thevery same representation for A µ field (14) leads to the“wrong” sign for the entropy. Furthermore, the contactterm (5) can be represented as a surface term, χ E & M ∼ (cid:90) d x (cid:2) δ ( x ) (cid:3) = (cid:90) d x ∂ µ (cid:18) x µ πx (cid:19) , (26)analogous to the “weird” contribution in the entropy com-putations [5, 9]. It is important to realize that the contactterm (5, 13, 26) is a result of summation over all topo-logical sectors with inclusion of all quantum fluctuationswhich account for the degeneracy of the ground state asdiscussed in section II B. At the same time, quite mirac-ulously, the final result (5, 13, 26) can be interpreted asa surface integral of a single classical configuration of apure gauge field A clµ ∼ ∂ µ φ cl defined on a distance surface S and characterized by unit winding number χ E & M ∼ (cid:73) S A clµ dl µ π = (cid:73) S rA clθ dθ π = (cid:90) dθ π ∂φ cl ∂θ . (27)These observations strongly suggest that the term witha “wrong” sign in the expression for entropy derived in[5] has exactly the same origin as the “wrong” sign forthe contact term (5) as in both cases the relevant physicsis determined by the surface integrals, not related to anyphysical propagating dynamical degrees of freedom. Fur-thermore, in both cases the sign of the effect is oppositeto what one should expect from physical degrees of free-dom, and, finally, in both cases the starting point (formalrepresentation for A µ field (14)) is the same. • Therefore, we conjecture that: the surface term witha “wrong sign” in entropy computations [5, 9] and the“wrong sign” in topological susceptibility (2,5,13) are bothoriginated form the same physics, and both related tothe same (topologically nontrivial) gauge configurations,and must be present (or absent) in both computationssimultaneously. In both cases a “wrong sign” emerges dueto unphysical degrees of freedom fluctuating in far infrared(IR) region. The technical treatments of these terms in ourframework and in ref. [5] of course is very different: we useconventional Hamiltonian approach supplemented by thecondition (17) while in ref. [5] the Rindler Hamiltonian isill defined on the cone, and computations are performedusing some alternative methods. Nevertheless, in ourframework, we interpret the fluctuations (25) of “fictitiousparticles” with “wrong” commutation relations (20) as adifferent manifestation of the same physics which led to awrong sign in entropy computations [5, 9]. An additionalargument supporting our conjecture will be presented inthe next section where we show that these very differentquantities nevertheless behave very similarly when thesystem is generalized from two to four dimensions, andtherefore, they must be originated from the same physics.Our final comment here is this: the IR physics pene-trates into the physical gauge invariant correlation func-tion (2) not due to the massless degrees of freedom in thephysical spectrum (there are none in fact), but rather, asa result of degeneracy of the ground state and summationover all topological sectors in gauge theory as discussedin sections II A, II B. The ghost (16) in this frameworkis simply a convenient tool to account for this far IRphysics as it effectively accounts for the non-dispersivecontact term with a “wrong sign” (19). It fluctuates inthe presence of the horizon (25), and responsible for a“wrong” sign in entropy computations, according to ourconjecture. However, it remains unphysical auxiliary fieldas it does not belong to the physical Hilbert space (and itnever becomes an asymptotic state capable to propagateto infinity) [18]. It is interesting to notice that there are other known examples when the degeneracy of the groundstate in the presence of the horizon leads to mismatchbetween black hole entropy and entropy of entanglement,see Appendix A for references and details.
III. GENERALIZATION TO 4-D CASE
The goal of this section is twofold. First, in next sub-section III A we make few comments on generalization oftwo dimensional results discussed above to four dimen-sional QED. In this case the corresponding calculationsof the entropy are known [5, 9]. Analysis of these resultsfurther support our conjecture on common nature of thesurface term with a “wrong sign” in entropy computations[5, 9] and “wrong sign” in topological susceptibility asthe behaviour of the system follows precisely the patterndictated by the conjecture . Secondly, in section III B wediscuss four dimensional non-abelian gauge theories whencorresponding computations of the entropy are not yetknown. Nevertheless, based on our conjecture on com-mon origin of these two different phenomena, we predicta possible outcome if the corresponding computations areperformed.
A. Four dimensional abelian QED
We start by reviewing the basic results of refs. [5, 9]on entropy computations in four dimensional case. Inoriginal paper [5] the gauge invariance of the “surfaceterm” has not been tested. This question has been specif-ically discussed in followup paper [9] where it has beendemonstrated that in two dimensions the result is indeedgauge invariant and coincides with the original expressionfound in ref. [5]. However, a similar analysis in four di-mensions turned out to be much more subtle, see detailsin [9]. In particular, it has been found that this term isgauge dependent in four dimensional abelian case, andtherefore, it was discarded [9].How one can understand such puzzling behaviour ofthe system when one jumps from two to four dimen-sions? If one accepts our conjecture formulated abovethen this puzzle has a very natural explanation. Indeed,the photon field in two dimensions has nontrivial topolog-ical properties formally expressed by the first homotopygroup π [ U (1)] ∼ Z . It implies the degeneracy of theground state when each topological sector | n (cid:105) is classifiedby integer number. Precisely this feature leads to nonvanishing topological susceptibility with a “wrong” signin two dimensions (5). The same degeneracy leads tonontrivial instanton solutions (8) interpolating betweendifferent topological sectors which saturate the topologicalsusceptibility (13) with “wrong sign”.In contrast to two dimensional case, in four dimensionsone should not expect any contact term with a “wrongsign” similar to (5) as the third homotopy group is trivial, π ( U (1)) ∼
1, there is no degeneracy of the ground stateas there is only a single trivial vacuum state. Therefore,one should not expect any non trivial surface terms in en-tropy computations in four dimensions. This expectationbased on our conjecture is supported by explicit compu-tations [9] where it was shown that in four dimensionalQED the surface term is gauge dependent and must beconsistently discarded. In fact, we consider these argu-ments as further support for our conjecture formulatedabove as the behaviour of the system follows precisely thepattern dictated by this conjecture .Our final comment is on interpretation of the surfaceterm with a “wrong” sign given in conclusion of ref. [9],where it has been suggested that, quote, “effective low-energy string theory which does not coincide with theordinary QFT” in principle may produce some surfaceterm with a “wrong” sign. We want to comment here,that in fact, very ordinary QFT may produce such kind ofterms, which however, are non-dispersive in nature, andnot related to any physical propagating degrees of freedomas explained in previous section II B for two dimensionalcase. As we argue in next section, such a behaviour is nota specific feature of two dimensional physics, but in factvery generic property in four dimensions as well. However,these nontrivial properties emerge in four dimensions onlyfor non abelian gauge fields when the third homotopygroup is non-trivial, π [ SU ( N )] ∼ Z , the ground state isdegenerate and each topological sector | n (cid:105) is classified byinteger number similar to two dimensional case consideredin section II. The contact term with a “wrong sign” similarto eq. (5) is expected to emerge in this case as a resultof nontrivial topological features of four dimensional non-abelian gauge theories. B. Four dimensional non-abelian QCD
The goal of this section is to argue that all key elementsfrom previous section II are also present in four dimen-sional QCD. In fact, the presence of the contact termwith “wrong sign” in topological susceptibility in QCDis a crucial element of resolution of the so-called U (1) A problem [10, 11]. The difference with two dimensionalcase is that in strongly coupled QCD we can not performexact analytical computations similar to (5,13). However,one can use an effective description in terms of auxiliaryghost field [11] to compute the non-dispersive contributionto topological susceptibility with “wrong sign”. This com- The argument presented above is based on observation that 4 D space time (where computations [9] have been performed) hastrivial topological properties. One can consider, instead, lesstrivial case when 4 D space time is represented, for example, by atorus, in which case the relevant homotopy group could be non-trivial, π ( U (1)) ∼ Z , and contact term with a “wrong sign” inentropy computations may occur. In principle, this is a testableproposal. Technically, though, it could be quite a challengingproblem. putation which employes the Veneziano ghost is directanalog of derivation of eq. (19) when the Kogut Susskindghost was used. Essentially, our goal here is to point outthat the relevant features in 2d QED (discussed in sectionII when all computations can be explicitly performed)and in 4d QCD (when the final word is expected to comefrom the lattice numerical computations) are almost iden-tical. To further support these similarities we presentsome QCD lattice numerical results explicitly measuringthe term with a “wrong sign” in topological susceptibilitysimilar to eqs. (4,19). Based on these observations, our conjecture essentially implies that the entropy computa-tions in four dimensional non-abelian gauge theories mustreveal a contribution with “wrong sign” as the crucialelement, the degeneracy of the ground state, is present inthe system. Moreover, it must be gauge invariant (andtherefore, physical) in contrast with 4D QED computa-tions where it was shown to be gauge variant [9], andtherefore, was discarded.Our starting remark is that the expression for topologi-cal density operator q = ∂ µ K µ = g π (cid:15) µνρσ G aµν G aρσ = (cid:3) Φ (28)being represented in terms of auxiliary scalar field Φ hasexactly the same form as in 2d Schwinger model, seesection II C. The Φ field in formula (28) is defined as K µ ≡ ∂ µ Φ and is the direct analog of representation (14)for 2d model. Our next remark is that four-derivative op-erator (cid:82) d x q ∼ (cid:82) d x ( (cid:3) Φ) is expected to be inducedin effective low energy Lagrangian as argued by Veneziano[11, 12] in his resolution of the U (1) A problem. As aresult of generating of q operator the relevant structurewhich emerges in the effective Lagrangian and describ-ing this system is identical to 2d QED case, i.e. it hasprecisely the same structure ∼ Φ (cid:2) (cid:3)(cid:3) + m η (cid:48) (cid:3) (cid:3) Φ. Thecorresponding path integral (cid:82) D Φ can be treated exactlyin the same way as it was treated in 2d QED, i.e. it canbe represented as the combination of the ghost φ and amassive physical ˆ φ field using the same trick by writingthe inverse operator as follows1 (cid:3)(cid:3) + m η (cid:48) (cid:3) = 1 m η (cid:48) (cid:32) − (cid:3) − m η (cid:48) − − (cid:3) (cid:33) , (29)in complete analogy with 2d case, see eq. (18). In fact,one can show that the relevant part of the low energy QCDLagrangian in large N c limit in the form suggested byVeneziano [11, 12] is identical to that proposed by Kogutand Susskind for the 2d Schwinger model (15), whereone should replace 2 √ π → f − η (cid:48) and µ → m η (cid:48) such thatthe scalar fields φ , φ , ˆ φ have appropriate (for 4D case)canonical dimension one, see [21] for details. This formalsimilarity leads to almost identical computations (in terms not to be confused with conventional Fadeev Popov ghosts. FIG. 1. The density of the topological susceptibility χ ( r ) ∼(cid:104) q ( r ) , q (0) (cid:105) as function of separation r such that χ ≡ (cid:82) drχ ( r ),adapted from [23]. Plot explicitly shows the presence of thecontact term with the “wrong sign” (narrow peak around r (cid:39)
0) represented by the Veneziano ghost in our framework. of the ghost) of the topological susceptibility in 4d QCDand in 2d QED. Indeed, by repeating all our previous stepsleading to eq. (19) with known Green’s functions whichfollow from (15) and with known expression for topologicaldensity operator q ( x ) ∼ ( (cid:3) ˆ φ − (cid:3) φ ) one arrives to thefollowing expression for topological susceptibility in 4dQCD in the chiral limit m = 0, χ QCD ≡ (cid:90) d x (cid:104) T { q ( x ) , q (0) }(cid:105) (30)= f η (cid:48) m η (cid:48) · (cid:90) d x (cid:2) δ ( x ) − m η (cid:48) D c ( m η (cid:48) x ) (cid:3) , where D c ( m η (cid:48) x ) is the Green’s function of afree massive particle with standard normalization (cid:82) d xm η (cid:48) D c ( m η (cid:48) x ) = 1. In this expression the δ ( x ) rep-resents the ghost contribution while the term proportionalto D c ( m η (cid:48) x ) represents the physical η (cid:48) contribution, see[21, 22] for details. The ghost’s contribution can be alsothought as the Witten’s contact term [10] with “wrongsign” which is not related to any propagating degrees offreedom. The topological susceptibility χ QCD ( m = 0) = 0vanishes in the chiral limit as a result of exact cancellationof two terms entering (30) in complete accordance withWI. When m (cid:54) = 0 the cancellation is not complete and χ QCD (cid:39) m (cid:104) ¯ qq (cid:105) as it should. Of course χ = 0 to any order in perturbation theory because q ( x ) is a total divergence q = ∂ µ K µ . However, as we learntfrom [10, 11], χ (cid:54) = 0 due to the non-perturbative infrared physics.One can interpret field K µ as a unique collective mode of theoriginal gluon fields. It describes the dynamics of the degeneratestates | n (cid:105) representing the topologically nontrivial sectors of theground state, it leads to a pole in unphysical subspace in theinfrared, and finally, it saturates the contact term with a “wrongsign” in topological susceptibility (30). Similar to eq. (26) for two dimensional system, thenon-dispersive term with “wrong sign” in topologicalsusceptibility (30) in four dimensional QCD can be alsorepresented as a surface integral χ ∼ (cid:90) d x (cid:2) δ ( x ) (cid:3) = (cid:90) d x ∂ µ (cid:18) x µ π x (cid:19) . (31)In case of 2d QED we could compare our ghost’s basedcomputations (19) with exact results (4,5) and with ex-plicit summation over different topological sectors in pure E & M when no propagating degrees of freedom are presentin the system (13). We do not have such a luxury in caseof 4d QCD. Nevertheless, we can compare the ghost’sbased computations in 4d QCD given by eq.(30) with thelattice results, see e.g. [23]. We reproduce Fig.1 fromref.[23] to illustrate few elements which are crucial for thiswork and which are explicitly present on the plot. Firstof all, there is a narrow peak around r (cid:39) r ∼ fm with the opposite sign. Boththese elements are present in the lattice computations asone can see from Fig.1. The same important elements arealso present in our ghost’s based computations given by eq.(30). In different words, the QCD ghost does model thecrucial property of the topological susceptibility relatedto summation over topological classes in gauge theories.This feature can not be accommodated by any physicalasymptotic states as it is related to non-dispersive con-tribution in the topological susceptibility as explainedabove in section II B, and elaborated further in AppendixA where this feature is explained as a result of differencesin definition between the Dyson’s T-product and Wick’sT- product.Our next step is to describe the behaviour of thesame system (more precisely, the behaviour of the non-dispersive term in eq. (30, 31) proportional to the δ ( x )function) in Rindler space in the presence of the horizon.We consider a simple case when the acceleration is suffi-ciently large Λ QCD (cid:29) a (cid:29) m |(cid:104) ¯ qq (cid:105)| such that interactionterm in (15) can be neglected and the Bogolubov’s coef-ficients are exactly known. In this limit one can repeatall previous steps to arrive to the same Planck spectrum(25) for number density fluctuations of “fictitious parti-cles” with “wrong” commutation relations [18, 24]. Thisformula (up to some irrelevant numerical coefficient) hasbeen reproduced in ref.[25] using a different technique.We interpret these fluctuations precisely in the same wayas we did in section II D in 2d case when we interpretedthese fluctuations of “fictitious particles” with “wrong”commutation relations as a different manifestation of thesame physics which led to a “wrong sign” in entropycomputations [5, 9]. As we emphasized before the cor-responding contribution is not related to any physicalpropagating degrees of freedom but rather, is due to topo-logical sectors in gauge theories which eventually lead toa non-dispersive contribution in topological susceptibility.Analysis of 2d case led us to the conjecture formulatedat the end of section II that both phenomena (“wrong sign”0in entropy computations and “wrong sign” in topologicalsusceptibility) are originated from the same physics de-termined by surface dynamics of the “fictitious particles”.In this section we demonstrated that all relevant featuresare also present in 4d non-abelian QCD. Therefore, basedon series of arguments presented above, it is naturallyto assume that there will be a mismatch between blackhole entropy and the entropy of entanglement in 4d QCDwith the same “weird” features as listed in Introduction.However, in contrast with 4d QED we expect that thesurface term with “wrong sign” in entropy will be a gaugeinvariant quantity similar to 2d QED case discussed in sec-tion II. This is essentially a prediction which follows fromthe conjecture . As we already mentioned there are otherknown examples when the degeneracy of the ground statein the presence of the horizon leads to mismatch betweenblack hole entropy and entropy of entanglement, see Ap-pendix A for the details and references. One could arguethat the dynamics of “fictitious particles” on a surfaceshould be governed by the corresponding Chern-Simonsaction. We leave this subject for a future study[26]. IV. CONTACT INTERACTION ANDPROFOUND CONSEQUENCES FOREXPANDING UNIVERSE
This portion of the paper is much more speculativein nature than the previous sections. However, thesespeculations may have some profound consequences onour understanding of expanding universe we live in whenthe horizon is inherent part of the system. Therefore, weopt to present these speculation in the present work.Non-dispersive contribution with a “wrong sign” intopological susceptibility (30) obviously implies, as eq.(1) states, that there is also some energy related to thiscontact term determined by the surface dynamics of “ficti-tious particles”. This θ − dependent portion of the energy,not related to any physical propagating degrees of free-dom, is well established phenomenon and tested on thelattice; it is not part of the debates. What is the partof the debates and speculations is the question on howthis energy changes when background varies. In differ-ent words, the question we address in this section canbe formulated as follows. How does the non-dispersivecontribution to the θ − dependent portion of the energyvary when conventional Minkowski background is replacedby expanding universe with the horizon size L ∼ H − determined by the Hubble constant H ? Here and in what follows we use parameter H ∼ L − as a typicaldimensional factor characterizing the visible size of our universe.We do not assume at this point that it is described by FRW metricwith a single parameter H . In fact, it could be much more genericconstructions when the spatial hypersurfaces are embedded in acompact 3d manifold such as, for example, Bianchi I geometrywith few additional parameters. We refer to Appendix B of ref.[21] for a short review on this subject in a given context. The motivation for this question is as follows. We adoptthe paradigm that the relevant definition of the energywhich enters the Einstein equations is the difference ∆ E ≡ ( E − E Mink ), similar to the well known Casimir effect whenthe observed energy is in fact a difference between theenergy computed for a system with conducting boundaries(positioned at finite distance d ) and infinite Minkowskispace. In this framework it is quite natural to define the“renormalized vacuum energy” to be zero in Minkowskivacuum wherein the Einstein equations are automaticallysatisfied as the Ricci tensor identically vanishes. Fromthis definition it is quite obvious that the “renormalizedenergy density” must be proportional to the deviationfrom Minkowski space-time geometry. This is in fact thestandard subtraction procedure which is normally usedfor description the horizon’s thermodynamics [27, 28] aswell as in a course of computations of different Green’sfunction in a curved background by subtracting infinitiesoriginated from the flat space [29]. In the present contextsuch a definition ∆ E ≡ ( E − E Mink ) for the vacuum energyfor the first time was advocated in 1967 by Zeldovich [30]who argued that ρ vac ∼ Gm p with m p being the proton’smass. Later on such a definition for the relevant energy∆ E ≡ ( E − E Mink ) which enters the Einstein equations hasbeen advocated from different perspectives in a numberof papers, see e.g. relatively recent works [31–38] andreferences therein.This is exactly the motivation for question formulatedin the previous paragraph: how does ∆ E scale with H ?The difference ∆ E must obviously vanish when H → E ∼ exp( − Λ QCD /H ) ∼ exp( − ) as QCD has a mass- gap ∼ Λ QCD , and there-fore, ∆ E must not be sensitive to size of our universe L ∼ H − . Such a naive expectation formally follows fromthe dispersion relations similar to (6) which dictate thata sensitivity to very large distances must be exponentiallysuppressed when the mass gap is present in the system.However, as we discussed at length in this paper, alongwith conventional dispersive contribution we also havethe non-dispersive contribution (30, 31) which emerges asa result of topologically nontrivial sectors in four dimen-sional QCD. This contact term may lead to a power likescaling ∆ E ∼ H + O ( H ) + ... rather than exponentiallike ∆ E ∼ exp( − Λ QCD /H ) because this term (in ourframework) is described by massless ghost field (29) asdiscussed in section III B. The position of this unphys-ical massless pole is topologically protected as eq. (28)states, which eventually may result in power like scaling∆ E ∼ H + O ( H ) + ... rather than exponential like . If a system is characterized by a single parameter, the curvature,then one should expect, on dimensional ground, that the first non-vanishing term in this expansion should be quadratic ∆ E ∼ H rather than linear. However, in a generic case one expects a linearnon-vanishing term ∆ E ∼ H + O ( H ) + ... , see Appendix B ofref. [21] for the details. E could be small but not exponentially small.Similar assumption based on very different arguments wasalso advocated in [31–34]. This postulate on Casimir-like scaling ∆ E ∼ H + O ( H ) has recently received a solidtheoretical support as reviewed below. It is importantto emphasize that this term with power like behaviouremerges as a result of non-dispersive nature of topologicalsusceptibility (30), such that no violation of unitarity,gauge invariance or causality occur when the theory isformulated in terms of the unphysical ghosts [18]. If true,the difference between two metrics (expanding universeand Minkowski space-time) would lead to an estimate∆ E ∼ H Λ ∼ (10 − e V ) , (32)which is amazingly close to the observed DE value today.It is interesting to note that expression (32) reduces toZeldovich’s formula ρ vac ∼ Gm p if one replaces Λ QCD → m p and H → G Λ QCD . The last step follows from thesolution of the Friedman equation H = 8 πG ρ DE + ρ M ) , ρ DE ∼ H Λ (33)when the DE component dominates the matter compo-nent, ρ DE (cid:29) ρ M . In this case the evolution of the uni-verse approaches a de-Sitter state with constant expansionrate H ∼ G Λ QCD as follows from (33).There are a number of arguments supporting the powerlike behaviour ∆ E ∼ H + O ( H ) in gauge theories. Firstof all, it is an explicit computation in exactly solvable two-dimensional QED discussed in section II and defined in abox size L . The model has all elements crucial for presentwork: non-dispersive contact term (5) which emerges dueto the topological sectors of the theory (13), and whichcan be described using auxiliary fictitious ghost fields(19). This model is known to be a theory of a singlephysical massive field. Still, one can explicitly compute∆ E ∼ L − which is in drastic contrast with naively ex-pected exponential suppression, ∆ E ∼ e − L [19]. It isimportant to emphasize that this correction ∆ E ∼ L − while computed in terms of the ghost’s (unphysical) de-grees of freedom in our framework, nevertheless representsa gauge invariant physical result. In different words, thefinal result ∆ E ∼ L − is not related to any violation ofgauge invariance though it is computed using auxiliaryfictitious ghost fields similar to computation of the contactterm (19).One more support in power like behaviour is an explicitcomputation in a simple case of Rindler space-time in fourdimensional QCD in the limit when a Rindler observeris moving with acceleration Λ QCD (cid:29) a (cid:29) m |(cid:104) ¯ qq (cid:105)| whenthe interaction term in eq. (15) can be neglected [18,24, 25]. These computations explicitly show that thepower like behaviour emerges in four dimensional gaugesystems in spite of the fact that the physical spectrumis gapped. In different words, a power like behaviour is not a specific feature of two dimensional physics as somepeople (wrongly) interpret the results of ref. [19].Another argument supporting the power like correctionsis the computation of the contact term in four dimensionalQCD defined in a box size L . The computations areperformed using the so-called instanton liquid model [39].While the motivation for analysis [39] was quite differentfrom our motivation, these model-based computationsnevertheless explicitly show the emerges of power likecorrections to non-dispersive portion of the topologicalsusceptibility.Power like behaviour ∆ E ∼ L − is also supported byrecent lattice results [40], see also earlier paper [41] withsome hints on power like scaling in drastic contrast withnaive expectations ∆ E ∼ exp − (Λ QCD L ). The approachadvocated in ref.[40] is based on physical Coulomb gaugewhen nontrivial topological structure of the gauge fields isrepresented by the so-called Gribov copies. It is very dif-ferent from our approach where we advocate the auxiliaryghost’s description to account for this physics. Eventu-ally, the physical results must not depend on the differenttechnical tools which are used in different frameworks.However, it is not a simple task to demonstrate an inde-pendence of the results from an employed technique in astrongly coupled gauge theory!Finally, Casimir like scaling ∆ E ∼ L − can be tested inthe so-called “deformed QCD” in weakly coupled regimewhen all computations are under complete theoreticalcontrol [42]. One can explicitly demonstrate that for thesystem defined on a manifold size L the θ - dependentportion of the energy shows the Casimir-like scaling E = − A · (cid:2) BL + O (cid:0) L (cid:1)(cid:3) despite the presence of a massgap in the system, in contrast with naive expectation E = − A · [1 + B exp( − L )] which would normally originatefrom any physical massive propagating degrees of freedomconsequent to conventional dispersion relations.Another remark worth to be mentioned is that the signof ∆ E ≡ ( E − E Mink ) is always expected to be negative in conventional quantum field theory computations. Thisis due to the fact that some modes can not be accommo-dated in a system with a nontrivial geometry/boundaries,and therefore the absolute value of E Mink > E which cor-responds to ∆
E <
0. The Casimir effect is the well knownexample when the sign ( − ) emerges as a result of thissubtraction procedure. The non-dispersive contributioninto the energy, on the other hand, being represented bythe ghost in our framework will lead to an opposite sign∆ E >
0. The positive sign for ∆
E >
E > E ∼ L − in drastic contrast withnaive expectations ∆ E ∼ exp − (Λ QCD L ) which shouldoccur from any conventional physical massive propagatingdegrees of freedom. If true, one can interpret the extracontribution to the energy (32), which we identify withDE, as a result of contact interaction with the horizon.This interpretation is consistent with interpretation of theterm with a “wrong sign” in entropy computation in twodimensions [5, 9], and it is also consistent with our inter-pretation presented in section II D, see comments afterformula (25). This interpretation is also consistent witharguments [5] suggesting that this term corresponds to acontact interaction with horizon in the description of theblack hole entropy within a string theory formulation [43]. CONCLUSION. FUTURE DIRECTIONS.
The main result of this paper is presented in formof a conjecture formulated at the end of section II andelaborated in section III. Essentially, the basic idea isthat the surface term with a “wrong sign” in entropycomputations [5, 9] and the contact term with “wrongsign” in topological susceptibility are both originatedform the same physics, and both related to the samegauge configurations related to the nontrivial topologicalstructure of the theory. If this conjecture turns out tobe correct, it would unambiguously identify the nature ofthe well known mismatch between computations of theblack hole entropy and entropy of entanglement for vectorgauge fields. Similar mismatch (but in quite differentcontext) was discussed also in [44–46]. In both cases themismatch is a result of degeneracy of the ground statein the presence of the horizon, see Appendix A for thedetails.Another, much more profound consequence is that thesame physics which is responsible for “wrong sign” contactterm in topological susceptibility, and “wrong sign” con-tribution in the entropy computations [5, 9], may in factlead to extra vacuum energy (32) (which is identified withobserved DE) in expanding universe in comparison withMinkowski space. This extra energy emerges in gaugetheories with multiple topological sectors as a result ofmerely existence of a causal horizon at distance L ∼ H − .Similar phenomenon as we already mentioned occurs indifferent systems [45, 46] when extra energy emerges as aresult of dynamics of the “soft modes” at the horizon. Thedegeneracy of the vacuum state in the system discussedin [45, 46] is achieved by non-minimally coupling with ascalar field, see Appendix A for the details, while in ourcase the presence of topologically distinct sectors in thesystem is an inherent feature of the QCD dynamics.Here are some features of these unique gauge config-urations which are responsible for “wrong sign” in theentropy computations and “wrong sign” in topologicalsusceptibility and which are characterized by very exoticproperties drastically different from everything previously known:a) a typical wavelength of fluctuations of the auxiliary“fictitious particles” is determined by the horizon scale, λ k ∼ /H ∼
10 Gyr, while smaller λ k (cid:28) /H are ex-ponentially suppressed (25). Therefore, these modes donot gravitationally clump on distances smaller than theHubble length, in contrast with all other types of matter,and can be identified with the observed properties of DE.Such very large wavelengths prevent us from adopting ameaningful scattering-based description, as the notion ofparticle is not even defined;b) the corresponding fluctuations are observer dependent,similar to the Unruh radiation, in contrast with any othertypes of radiation, see section II D and also [18] for detaildiscussions on problem of measurements in these circum-stances;c) the co-existence of the two drastically different scales(Λ QCD ∼
100 MeV and H ∼ − eV) is a direct conse-quence of the auxiliary conditions (17,23) on the physicalHilbert space rather than an ad hoc built-in feature such assmall coupling or/and extra symmetries in a Lagrangian.So, essentially our proposal for the DE can be formu-lated as follows. The source for both: DE and mismatch between the black hole entropy and entropy of entangle-ment is the same and related to the dynamics of topo-logical sectors of a gauge theory in the presence of thehorizon. In other words, the relevant gauge configura-tions which are responsible for DE are exactly the samewhich are responsible for the “wrong sign”-contributionin computations of refs [5, 9]. Precisely this contributionrepresents the mismatch between the black hole entropyand entropy of entanglement. In both cases the sourcefor the extra term is the degeneracy of the vacuum statewhich is represented by different topological sectors, andin our framework is described by the Veneziano ghost.One should also add that a phenomenological analysisof the DE model based on this idea and represented byeqs.(32, 33) has been recently performed in [47] withconclusion that this model is consistent with all presentlyavailable observational data.Another important result of this work can be formulatedas follows. When a system is promoted from Minkowskispace to a an expanding universe, we expect a power likecorrections ∆ E ∼ H + O ( H ) rather than exponentiallysuppressed corrections, ∆ E ∼ exp − (Λ QCD /H ). Thishappens in spite of the fact that the physical Hilbertsubspace contains only massive propagating degree offreedom, and naively the sensitivity to very distant re-gions should be exponentially suppressed. However, thepresence of the non-dispersive contributions (originatedfrom degenerate topological sectors of the theory), whichcan not be associated with any physical asymptotic states falsifies this naive argument . Explicit computations in 2dQED [19], in 4d weakly- coupled “deformed QCD” [42]and numerical studies in real four dimensional QCD [40]where power like behaviour ∼ L − is indeed observed,supports our claim.What is more remarkable is the fact that some of funda-3mental properties of gauge theories discussed in this papercan be, in principle, experimentally tested in relativisticheavy ion collider (RHIC) at Brookhaven and heavy ionprogram at LHC. In the “little bang” at RHIC the hori-zon appears as a result of induced acceleration a ∼ Λ QCD which itself emerges as a consequence of high energy col-lision. The acceleration a ∼ Λ QCD is a universal numberwhich is determined by strong QCD dynamics, does notdepend on energy or other properties of the colliding par-ticles, and plays the role of Hubble constant H ∼ − eVof expanding universe, see [24] for the details. ACKNOWLEDGEMENTS
I am thankful to participants of the workshop “largeN gauge theories” at the Galileo Galilei Institute, wherethis work has been presented, for useful and stimulatingdiscussions. I am also thankful to an anonymous refereefor few inspiring questions. This research was supportedin part by the Natural Sciences and Engineering ResearchCouncil of Canada.
Appendix A: Subtleties in definition of energy.
It is quite natural to expect that there should be someenergy associated with fluctuations of fictitious particles(25). As we argued above these fluctuations, which reflectthe nontrivial topological structure of the gauge theory, isthe source for missmatch between the black hole entropyand entropy of entanglement in the presence of the horizon.It is clear that there is some ambiguity in definition ofthis type of energy due to a number of reasons. Firstof all, as we discussed in the text the relevant physicsis determined by the dynamics on the surface ratherthan in the bulk of the space-time. Therefore, it is notobvious if a simple insertion ω k into definition (22) wouldproperly reflect this feature. Another comment is thatthe physics of fluctuations (25) is the observer dependentproperty similar to the Unruh effect as discussed in sectionII D. Therefore, all subtleties related to the Unruh effectare also present here. Also, very large wavelengths ofthese fluctuations prevent us from adopting a conventionaldescription in terms of particles, as the notion of particleis not even defined.Finally, and most importantly, the contact interactionwhich is the main subject of this paper, can not be ex-pressed in terms of any physical states, but rather isformulated in terms of fluctuations of fictitious particleswith “wrong” commutation relations. These unphysicalstates contribute to the non-dispersive portion of thecorrelation function, not to the dispersive part which isunambiguously determined by physical spectral functionthrough conventional dispersion relations. Our descrip-tion in terms of the ghost is simply a convenient way tostudy this IR physics in a covariant gauge. The samephysics in Coulomb (physical) gauge when the ghost de- grees of freedom are not present in the system leads to thelong range forces as discussed in simple two dimensionalmodel long ago [16]. These long range forces preventdistant regions from acting independently. The vacuumin this system is degenerate, and one should consideran infinite superposition of of the winding states | n (cid:105) asoriginally discussed in [16]. We think that precisely thisfeature prevented the author of ref. [5] to use the phys-ical Coulomb gauge in two dimensions in computationsof the entropy, where a covariant gauge has been usedinstead. The same physics in Coulomb gauge in 4d QCDis reflected by existence of the so-called Gribov copies,and one should use some numerical lattice methods tostudy the relevant physics [40].The reason why we pay so much attention to the topo-logical susceptibility χ and the corresponding contactterm ∼ δ ( x ) which enters the expressions for topologicalsusceptibility (4) and (30) is due to its relation to the θ dependent portion of the vacuum energy as eq. (1) states.Therefore, the presence of non-dispersive contributionsin χ automatically implies presence of the correspondingnon-dispersive contribution in the vacuum energy E vac .At the same time as we discussed in the text the non-dispersive contribution in χ (and in the vacuum energy)can be interpreted as a result of nontrivial topologicalstructure of the gauge theories. These discussions explic-itly demonstrate some subtleties on possible definition ofthe vacuum energy which should accommodate the physicsrelated to the contact term discussed in this paper.It is quite fortunate that in specific case with compu-tations of χ we can easily separate non-dispersive andphysical contributions. This is due to the fact, that thesetwo very different terms contribute to χ with oppositesigns as discussed in the text. Indeed, the topologicalsusceptibility in pure YM gauge theory (no quarks, andno η (cid:48) contribution) according to [10, 11] is given by χ Y M ≡ i lim k → (cid:90) d xe ikx (cid:104) T { q ( x ) , q (0) }(cid:105) W == − λ Y M < , λ Y M = f η (cid:48) m η (cid:48) , (A1)where (cid:104) ... (cid:105) W stands for Wick T- product, see below. Theexpression on r.h.s of eq. (A1) corresponds to the sub-traction constant ∼ δ ( x ) in eq. (30). This term hasa “wrong sign” (in comparison with contribution fromany real physical states), the property which motivatedthe term “Veneziano ghost”. Indeed, a physical state ofmass m G , momentum k → (cid:104) | q | G (cid:105) = c G contributes to the topological susceptibility with the sign All formulae in this Appendix A are written in Minkowski space incontrast with our discussions in the text, where a comparison withpath integral computations and lattice numerical computations(which are always performed in Euclidean space) was made. Inparticular, there is factor “i” in the definition of the correlationfunction (A1). i lim k → (cid:90) d xe ikx (cid:104) T { q ( x ) , q (0) }(cid:105) D ∼ (A2) i lim k → (cid:104) | q | G (cid:105) ik − m G (cid:104) G | q | (cid:105) (cid:39) | c G | m G ≥ , where (cid:104) ... (cid:105) D stands for Dyson T- product, see below. How-ever, the negative sign for the topological susceptibility(A1) is what is required to extract the physical mass forthe η (cid:48) meson, see the original reference [12] for a thor-ough discussion. The difference between behaviour (A1)and (A2) is related to inequivalent definitions of thesecorrelation functions. The behaviour (A2) corresponds tothe usual Dyson T-product when only physical states con-tribute, while eq. (A1) corresponds to Wick T-productobtained by variation of the partition function over θ parameter. The difference in the definitions constitutesprecisely the subtraction constant ∼ δ ( x ) in eq. (30).The WI expressed as χ QCD ( m q = 0) = 0 is satisfied forthe Wick T-product, not for the Dyson T-product.It is interesting to note that an analogous phenomenon(but in quite different context) was discussed in ref. [45,46], where it was observed that there are two definitionsof energy when the difference is saturated by the so-called“soft modes” fluctuating in far infrared at the horizon.First definition is the canonical energy determined bythe hamiltonian which is generator of translations of thesystem along the timelike Killing vector field ξ µ . Sincethe Killing vector ξ µ vanishes at the bifurcation surfaceof the Killing horizons, the corresponding Hamiltonianis degenerate. Therefore, one can add to the system anarbitrary number of “soft modes” without changing thecanonical energy. These “soft modes” contribute to thesurface integral which was interpreted as a Noether chargeof some non-minimally coupled scalar field φ s . Preciselythis contribution of the “soft modes” distinguishes two different definitions of the energy. As argued in ref. [45,46] precisely the dynamics of “soft modes” constitutes thedifference between Bekenstein -Hawking entropy and theentropy of entanglement.It is very similar to our case when we argued that a“wrong sign” in entropy computations of refs [5, 9] is aresult of degeneracy in the gauge theory represented in ourframework by the fluctuations of the fictitious particleswith a typical wavelength of order of the horizon scale, λ k ∼ H − . In different words, the “soft modes” from[45, 46] play the same role as pure gauge fields whichdescribe different topological sectors of the theory in ourcase. In both cases the effect emerges as a result of thedegeneracy of the ground state in the presence of thehorizon, and in both cases the difference is determinedby some surface integrals. Further to this analogy, twodifferent types of energies which can be reconstructedfrom two different definitions of topological susceptibility(A1) and (A2) precisely correspond to two different typesof energies discussed in refs. [45, 46].The final comment on the similarities between the twovery different systems is as follows. The relevant gaugeconfigurations which are responsible for mismatch be-tween the black hole entropy and entropy of entanglementfrom computations [5, 9] are exactly the same which areresponsible for the contact term in topological suscepti-bility (30). This mismatch is very similar to extra termdiscussed in [45, 46] which is resulted from the dynam-ics of the “soft modes” at the horizon. In both casesthe mismatch is a result of degeneracy present in bothsystems. In refs. [45, 46] this degeneracy is a result ofnon-minimally coupling with scalar field φ s . 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