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Black Hole Entropy in the O(N) Model

a r X i v :h e p -t h /9506182v 2 18 J u l 1995Black Hole Entropy in the O(N)Model

D.Kabat,S.H.Shenker,and M.J.Strassler

Department of Physics and Astronomy

Rutgers University

Piscataway,NJ 08855–0849

kabat,shenker,strasslr@https://www.wendangku.net/doc/0a14434842.html, We consider corrections to the entropy of a black hole from an O (N )invariant linear σ-model.We obtain the entropy from a 1/N expansion of the partition function on a cone.The entropy arises from diagrams which are analogous to those introduced by Susskind and Uglum to explain black hole entropy in string theory.The interpretation of the σ-model entropy depends on scale.At short distances,it has a state counting interpretation,as the entropy of entanglement of the N ?elds φa .In the infrared,the e?ective theory has a single composite ?eld σ～φa φa ,and the state counting interpretation of the entropy is lost.

hep-th/9506182

June 1995RU–95–34

1.Introduction

Within ordinary gravity,the dynamical origins of black hole entropy have always been obscure.But gravitation emerges as a low energy phenomenon from string theory, and Susskind and Uglum have proposed that,in terms of underlying string degrees of freedom,a state counting interpretation of black hole entropy is possible[1].

We will discuss black hole entropy within the following framework.The temperature of a black hole is?xed by requiring that the Schwarzschild metric,continued to imaginary time,provide a smooth solution of the Einstein equations.This forces the periodicity of the Euclidean time coordinate to be the inverse Hawking temperature.Entropy is obtained from a partition function by varying with respect to temperature;in the case of a black hole,this variation introduces a conical singularity on the horizon.The classical action for gravity evaluated on a space with a conical singularity gives rise to the classical Hawking–Beckenstein entropy[2,3],while quantum corrections to the classical entropy result from ?uctuations of the metric or matter?elds in the background with the conical singularity.

Through genus one,the string diagrams Susskind and Uglum claim are responsible for black hole entropy are shown in Fig.1.The genus zero diagram gives rise to the classical O(1/ˉh)Hawking–Beckenstein entropy of a black hole.It can be understood as counting the states of a half-string with its ends stranded on the horizon.Two diagrams arise at genus one.The?rst represents a closed string encircling the horizon,and can be understood as counting the states of a closed string.The second represents represents an interaction between a closed string and an open string stranded on the horizon.Such interactions can be thought of as making corrections to the state counting entropy which is present at genus zero.

This state counting interpretation of black hole entropy is lost at low energies,when string degrees of freedom are not visible.At low energies,the string diagrams reduce to the particle diagrams of Fig.2.Diagram(i)represents a term in the e?ective action that is localized on the horizon.It is responsible for the classical Hawking–Beckenstein entropy,but has no state counting interpretation.Diagram(ii)does have a state counting interpretation,as counting the states of a particle encircling the horizon.In diagrams(iii) and(iv)contact interactions with the horizon appear.These contact interactions do not admit state counting interpretations.Diagram(iii)can be represented as a path integral over particle paths that begin and end on the horizon.In diagram(iv)a?eld acquires an expectation value at one loop,which then couples to the horizon.In a supersymmetric

horizon

(i)(ii)(iii)

Fig.1.String diagrams that generate black hole entropy.

horizon

(i)(ii)(iii)(iv)

Fig.2.Low energy limit of the string diagrams.

string theory,all one point functions on the torus vanish,so this last diagram will not be present.In fact one point functions vanish to all orders in the genus expansion as long as supersymmetry is unbroken,so contributions of this general form can arise only below the scale of supersymmetry breaking.

In this paper,we will study corrections to the entropy of a black hole from an O(N) invariant linearσ-model.For simplicity we work in two Euclidean dimensions,and consider only in?nitely massive black holes,for which curvature vanishes everywhere on-shell.The σ-model exhibits some analogous e?ects within?eld theory.In particular,diagrams appear at short distances in theσ-model which are analogous to the string diagrams in Fig.1.In the low energy e?ective theory for theσ-model,diagrams appear which are analogous to the particle diagrams in Fig.2.Moreover,at short distances,theσ-model corrections to

the entropy have a state counting interpretation as entropy of entanglement.This state counting interpretation is lost at low energies.

This paper is organized as follows.In section2,we discuss the relationship between black hole entropy and entropy of entanglement,and show that this relationship leads to an underlying state counting interpretation of theσ-model contribution to black hole entropy.We also discuss the way in which the state counting interpretation of black hole entropy breaks down for non-minimally coupled?elds,such as appear in the low energy description of theσ-model.In section3,we renormalize theσ-model to determine its low energy e?ective action.In section4we compute the entropy of theσ-model,and discuss how its interpretation changes at di?erent length scales.In section5we discuss the implications of our results for string theory.

2.Entropy of Entanglement and Conical Singularities

In this section we discuss the relationship between corrections to the entropy of a black hole,obtained from a partition function on a cone,and entropy of entanglement[4–13,1]. For theσ-model we will argue that these two entropies are identical at short distances. This leads to an underlying state counting interpretation of theσ-model contribution to black hole entropy.We will also show that their equivalence is broken for?elds with a non-minimal coupling to curvature,such as arise in the low energy description of the σ-model.

We begin by de?ning entropy of entanglement.Consider a quantum?eld in?at space.Suppose that it is in its ground state,described by the pure density matrix|0><0|. Introduce an imaginary boundary that divides space into two regions,and form the reduced density matrixρred from|0><0|by tracing over all degrees of freedom located in one of the regions.Entropy of entanglement is de?ned by S=?Trρred logρred.For simplicity we will only treat the case where space is divided in half by an imaginary planar boundary.

Entropy of entanglement is ultraviolet divergent in most quantum?eld theories,so we must introduce an ultraviolet cuto?Λ.The“full entropy of entanglement”of the theory is de?ned as above in terms of all the dynamical degrees of freedom present in the theory up to the cuto?Λ.A Wilsonian e?ective action at a scaleμ<Λis constructed by integrating out the degrees of freedom with momentum betweenμandΛ.We introduce the notion of “e?ective entropy of entanglement,”that is,the entropy of entanglement calculated in the

e?ective theory at the scaleμ.When we state that,for some theory,black hole entropy and entropy of entanglement are identical,we are referring to the full entropy of entanglement.

Note that the e?ective entropy of entanglement is less than the full entropy of entan-glement:as degrees of freedom are integrated out,the e?ective entropy of entanglement decreases.In contrast,black hole entropy is de?ned in terms of the partition function on a cone,a quantity which does not change as degrees of freedom are integrated out.In the infrared,the di?erence between the e?ective and full entanglement entropies is accounted for by the appearance of terms in the Wilsonian e?ective action involving the background curvature.This is a general phenomenon,which we will explicitly see occur in theσ-model.

We now review the formal argument that relates entropy of entanglement to black hole entropy.The argument uses the Rindler Hamiltonian(generator of Euclidean rotations) H R.One can show that the reduced density matrix takes a thermal form in terms of H R, in thatρred=e?2πH R[14,15,1,8,16].This is a statement of the Unruh e?ect[17],namely, that the Minkowski vacuum state of a?eld is seen as a thermal state by a Rindler observer. The entropy of entanglement of a?eld is therefore the same as the thermal entropy which the?eld carries in Rindler space,which can be obtained by standard thermodynamics from the Rindler thermal partition function Z(β)=Tr e?βH R.But Z(β)is the partition function for the theory on a cone with de?cit angle2π?β,so Rindler thermal entropy is the same as the contribution of the?eld to the entropy of a black hole.

This formal argument for equivalence has been tested by explicit calculation in several theories.For free scalar and spinor?elds,the full entropy of entanglement,Rindler thermal entropy,and entropy on a cone are indeed equal[15,16,18].The equivalence does not always hold,however.For free vector?elds,singular e?ects arise at the origin,which make the Rindler Hamiltonian ill-de?ned,and cause the equivalence to break down[13].

For an interacting minimally coupled scalar?eld theory the equivalence holds to all orders in perturbation theory:black hole entropy and full entropy of entanglement are identical.In particular,the equivalence holds for theσ-model,and leads to a state counting interpretation of theσ-model contribution to the entropy of a black hole.The proof of equivalence is straightforward.To be concrete,we consider a scalar?eld in1+1dimensions with a1

2π2+

m2φ2+

whereπis the momentum conjugate toφ.

We?rst show that Rindler thermal entropy is the same as black hole entropy.Rindler thermal entropy may be obtained from the partition function Z(β)=Tr e?βH R.This partition function can be calculated in perturbation theory,by separating H R into free and interacting pieces.The Feynman rules for computing Z(β)are identical to the Feynman rules used to evaluate the partition function on a cone–in particular,the two point function is simply the Green’s function on a cone[15].Rindler thermal entropy and black hole entropy therefore have identical perturbation series.The two entropies will be equal provided the two calculations are cut o?in the same way;this can be achieved by using Pauli-Villars regulator?elds as in[12].

To complete the proof we show that Rindler thermal entropy is the same as entropy of entanglement[14,15,1,8,16].Suppose we divide space at the origin(r=0),and compute the entropy of entanglement of one half of space(θ=0)with the other half(θ=π). This is de?ned in terms of the reduced density matrixρred,constructed from the vacuum density matrix|0><0|by tracing over all degrees of freedom located atθ=π.To all orders of perturbation theory,and for any operator A located atθ=0,the expectation values Tr(ρred A)and Tr(e?2πH R A)are equal,because(as above)the Feynman rules are identical.This shows thatρred=e?2πH R,which implies that entropy of entanglement is equal to Rindler thermal entropy.

We now discuss a theory for which black hole entropy and full entropy of entanglement are not equivalent.Consider a scalar?eld in two dimensions with a non-minimal coupling to curvature.

S= M d2x√2gμν?μφ?νφ+12ξRφ2

A non-minimal coupling of this form will arise in the low energy description of theσ-model.To see that the non-minimal coupling breaks the equivalence,note that entropy of entanglement is de?ned in terms of the vacuum state of the?eld in?at Minkowski space, and should therefore be insensitive to any terms in the action that vanish in?at space. The partition function on a cone,on the other hand,is a?ected by a non-minimal coupling to the curvature present at the tip of the cone[19].This di?erence breaks the connection between entropy of entanglement and black hole entropy.

To justify the claim that entropy of entanglement is not a?ected by the non-minimal coupling,we work in?at space,and consider the e?ect of the non-minimal coupling in greater detail.In?at space,the non-minimal coupling only enters via a boundary term,

involving the extrinsic curvature K,which we must add to the action to make the energy-momentum tensor well-de?ned.

S boundary= ?M dsξKφ2

The vacuum wavefunctional in in?nite?at space can be obtained from a path integral on a semi-disc of radius L,{(x,y):x2+y2≤L2,y≥0}.As L→∞,with the?eld taken to vanish on the semi-circular boundary at x2+y2=L2,this path integral produces the desired?at space vacuum wavefunctional on the boundary at y=0.The bulk curvature vanishes on the semi-disc,while the extrinsic curvature is1/L along the semi-circular boundary,withδ-functions of strengthπ/2at the two corners.This leads to the vacuum wavefunctional

e?πξ(φ2(L,0)+φ2(?L,0))/2ψ0[φ]

ψ[φ]=N lim

L→∞

whereψ0[φ]is the vacuum wavefunctional for a minimally coupled?eld,and N is a nor-malization constant.We see that in?at space the non-minimal coupling only a?ects the vacuum state at spatial in?nity.Because we consider a massive?eld,correlation functions calculated in this vacuum are likewise only a?ected at spatial in?nity.

What is the entropy of entanglement of this state?Suppose we divide space at x=0. For a massive minimally coupled?eld the degrees of freedom at x=±L are exponentially uncorrelated with the degrees of freedom located across the division.From the form of the wavefunctional we see that they remain uncorrelated in the presence of the non-minimal coupling;it follows that the non-minimal coupling does not a?ect the entropy of entanglement.

The fact that the correlation functions are only a?ected at spatial in?nity has an important consequence.It implies that the Rindler Hamiltonian which generates angular evolution in?at space is not a?ected by the non-minimal coupling,aside from a possible surface term at spatial in?nity.Even with a non-minimal coupling,one can regard entropy of entanglement as counting the degrees of freedom of a well-de?ned dynamical system governed by this?at space Rindler Hamiltonian.

Note that,for non-zeroξ,the partition function on a cone is not generated by the?at space Rindler Hamiltonian–in fact it seems unlikely that the partition function on a cone can be generated by a Hermitian angular Hamiltonian at all,for reasons given below.We can now state more precisely why the equivalence between entropy of entanglement and

black hole entropy breaks down for a non-minimally coupled?eld.The Rindler Hamilto-nian is perfectly well-de?ned,and the reduced density matrix is given byρred=e?2πH R, so entropy of entanglement is equal to Rindler thermal entropy.But it is not equal to black hole entropy,because H R does not generate the partition function on a cone.This provides an interesting contrast to the way in which the equivalence is broken for a vec-tor?eld[13].For a vector?eld,singular e?ects present at the origin make the Rindler Hamiltonian ill-de?ned,even in?at space.

We now study the behavior of the same non-minimally coupled?eld in curved space. To compute the partition function on a cone,and to gain more insight into the e?ect of the non-minimal coupling,we introduce a particle path integral representation for the partition function.We work in two Euclidean dimensions,with a proper time cuto??as a regulator.

βF=

2 ∞?2ds4gμν˙xμ˙xν+m2+ξR(x) (2.1)

A careful de?nition of the particle path integral would have additional terms present in the action,beyond those we have indicated here[20].The naive action we use here is su?cient for our purposes,however,as we only wish to show that the e?ect of the non-minimal coupling is to produce an additional contact term in the partition function.To see this,we expand the free energy in powers of the non-minimal coupling,following[19].The scalar

curvature on a cone is R=2(2π?β)δ2(x)/√

2 ∞?2ds4gμν˙xμ˙xν+m2

+ξ(2π?β) ∞?2ds x(0)=x(s)=0D x(τ)exp ? s0dτ 1

with the horizon;as they are also higher order in the de?cit angle,they are not needed to calculate the entropy at the on-shell temperatureβ=2π.The path integrals can be evaluated(see e.g.[13],equation(2.4)),and the entropy is[19]

S= β?

2 1s e?sm2.

The partition function of a non-minimally coupled?eld on a cone does not have a state counting interpretation;note that,as a consequence of the contact term,it can make a negative contribution to the entropy of a black hole for some values ofξ.This suggests that the partition function of a non-minimally coupled?eld on a cone does not have a Hamiltonian description.A non-minimal coupling may arise from integrating out short distance degrees of freedom,in which case there can be a hidden state counting interpreta-tion in terms of the underlying degrees of freedom.Theσ-model will provide an example of this phenomenon.

As an aside,we note that the contact term arose from the explicit appearance of the curvature in the?rst quantized particle Lagrangian in(2.1).This phenomenon seems to be general.Spinors and minimally coupled scalars have classical?rst quantized particle Lagrangians in which curvature does not explicitly appear,1and their entropy merely re?ects the density of states.In contrast,non-minimally coupled scalars have a termξR in their particle Lagrangian,and similarly a vector particle,which has negative black hole entropy,also has a term involving the curvature in its world-line action[21].In both cases these explicit curvature terms produce a contact interaction when the particle path crosses the tip of the cone,and this contact interaction shifts the coe?cient of the leading divergence of the entropy.It is interesting to note that the N=1superstring does not have any explicit R dependence in its world-sheet Lagrangian,though the signi?cance of this fact is not clear to us.

3.Renormalization of the Sigma Model

In this section we renormalize theσ-model in curved space.The casual reader may wish to skip to the next section,where the results are discussed in a way that does not depend on the details of the calculations.

Theσ-model in two dimensions in a background metric gμνis de?ned by the Euclidean action for N real scalar?eldsφa

I[φa]= d2x√2gμν?μφa?νφa+18N(φaφa)2 .

As usual we introduce an auxiliary?eldσ=λ

g 12m2φφaφa?N2σφaφa

Note that1/N plays the role of aσloop counting parameter.The large N limit is taken withλheld?xed.This model has a double scaling limit,discussed in[22];by tuningλclose to the critical coupling we will makeσa propagating?eld at low energies,with a mass mσ?mφ.

To regulate the model we introduce a momentum cuto?Λ.In the remainder of this section we will calculate the e?ective action in curved space at next-to-leading order(N0) by renormalizing from the scaleΛdown to the scale mσ.We will keep only the leading logarithms that arise in perturbation theory.To calculate the entropy,we only need to work to?rst order in curvature.

3.1.Ultraviolet renormalization

In this section we renormalize from the ultraviolet cuto?Λdown to the scale of theφmass.Ultraviolet logarithms～logΛ

2 d2xσ Λμd2k k2+m2φ

μ2 d2xσ+···

This linear term in the e?ective action givesσa non-zero expectation value,which we absorb by replacingσ→σ+Σ(μ2)and renormalizing theφmass:m2φ(μ2)=m2φ(Λ2)+Σ(μ2).The constantΣ(μ2)is chosen to make the expectation value ofσvanish at the

scaleμ2.A logarithm also appears

=λ

φaφa Λμd2k k2+m2φ

=λ

μ2 d2x1

16πG(μ2) d2x√

log det(?+m2+ξR)

=?1

g ∞?2ds s+ 1

8π 1m2?2

d2x√

16πG(μ2)

μ2

(3.2)

These ultraviolet renormalizations of theφmass and the gravitational coupling absorb all logarithms which arise in running fromΛdown to the scale of the physical(pole)φmass, i.e.the scale at whichμ2=m2φ(μ2).We denote this pole mass by mφfrom now on.

3.2.E?ectiveσtheory

Below the scale mφthe?eldsφa become non-propagating auxiliary?elds.In this section we integrate them out in order to construct an e?ective theory involving only the light?eldσ.We also discuss the?ne tuning necessary to make mσ?mφ.

We must determine the gravitational coupling in the e?ective theory at next-to-leading order(N0),but we only need terms involvingσat leading order(N).The leading order(N) e?ective action is given by a one-loop determinant,which we now compute.By combining this determinant with our previous result(3.2)for the gravitational coupling,we will be able to determine the e?ective theory at the scale mφ.

Because we did not keep track of the?nite terms that involveσin the previous section, we must start the computation of the determinant from the ultraviolet cuto?Λ.

I[σ]=?N

gσ2+

2 Λμd2k k2+m2φ(Λ2)+Σ(μ2).

This is the same shift ofσwhich we performed previously in the ultraviolet renormalization, so the quantity m2φ(μ2)is the same at leading order(N0)as the renormalized mass m2φ(μ2) that was introduced in the last section.The renormalized e?ective action forσis given at leading order(N)in terms of m2φ≡ m2φ(0),

I[σ]=?N

gσ2+

2λ d2x√

48π m4φ d2x √

gμν?μσ?νσ?3 m2φσ2+σ3+···

? m4φlogΛ22Rσ2+···

(3.3)

Note that one can obtain the non-minimal curvature couplings ofσwithout evaluating any diagrams.A constant value ofσproduces a shift in theφmass,m2φ(Λ2)→m2φ(Λ2)+σ,so the coe?cients of the Rσand Rσ2terms may be found by di?erentiating the Schwinger–De Witt representation(3.1)with respect to m2.

The determinant(3.3)is the leading order(N)e?ective action forσ.It is a“classical”action,valid at any length scale,although the derivative expansion only makes sense below the scale mφ.This classical action is not su?cient for our purposes,however,as we must determine the e?ective gravitational coupling at next-to-leading order(N0).In the previous section we integrated outφandσdegrees of freedom at next-to-leading order to ?nd the running gravitational coupling(3.2),which is valid down to the scale mφ.Recall that mφis the pole mass incorporating ultraviolet e?ects to next-to-leading order(1/N). Integrating out the remaining auxiliaryφdegrees of freedom,from mφdown to zero, does not change the gravitational coupling at leading log order,3so(3.2)is the correct gravitational coupling to use in the e?ective action at the scale mφ.

I[σ]=?

gR+···

g 12m2σσ2+ m2φσ3+Rσ?1

m2φintoσ,which makesσdimensionless.The renormalized

parameters in this e?ective action are

3log

Λ2

λ?

m2σ.Note that this tuning makes the quartic coupling negative,so the theory is non-perturbatively unstable.Perturbation theory in1/N is well-de?ned,however,which is all that we require.By making this?ne tuning,we are driving the model close to its double scaling limit[22].We do not actually take the double scaling limit,because we will wish to preserve terms in our?nal result for the entropy that drop out in the double scaling limit mσ

3We ignore the large,but non-logarithmic,shift which occurs in the gravitational coupling as theφthreshold is crossed;this is responsible for the di?erence,at leading order(N),between the gravitational coupling in(3.3)and(3.4).This shift will be discussed brie?y in section5.

3.3.Infrared renormalization

We now run the e?ective theory (3.4)from m φdown to m σin order to ?nd the renormalized gravitational coupling at scales below m σ.We only keep the leading infrared logarithms ～log m φ

(2π)2

4π

log m 2φ

N m 2φμ2.This produces a shift of the gravitational

coupling,from the Rσterm in the action (3.4).A further logarithmic renormalization of the gravitational coupling arises from the

graphs

++ . . . =?7

μ2 d 2x √

G (μ2)=

m 2σ+7μ2=N m 2φ+ 12 m 2φ3

log m 2φ

4.Discussion of the Sigma Model

We will now use the results of the previous sections to calculate the black hole entropy of theσ-model,and discuss how its interpretation changes at di?erent length scales.We will see that it exhibits behavior analogous to the behavior of string theory proposed by Susskind and Uglum[1].

At the scale of the ultraviolet cuto?Λ,theσ-model is de?ned by the Euclidean action I[φa,σ]= d2x√2gμν?μφa?νφa+12λσ2+1

16πG(m2

) d2x√

48π d2x√2gμν?μσ?νσ+12Rσ2+···

The diagrams in this e?ectiveσtheory are shown in Fig.4.They are obtained from Fig.3by shrinking allφloops down to points.These graphs are analogous to the particle diagrams which arose as the low energy limit of string diagrams in Fig.2.At this scale the state counting entropy is only order N0.It measures the e?ective entropy of entanglement of the?eldσ,and is given by diagram(ii),which is generated by theσkinetic term in the e?ective action.The remainder of the partition function on a cone does not have a state counting interpretation in the e?ective theory.There is an order N contribution from the Einstein–Hilbert term,which generates diagram(i).This re?ects the underlying correlations between short distance degrees of freedom located on opposite sides of the horizon.There are also order N0contributions from the non-minimal curvature couplings ofσwhich generate diagrams(iii)and(iv).These re?ect the underlying correlations between short distance degrees of freedom localized on the horizon and long wavelength degrees of freedom.

(i)(ii)(iii)(iv)

Fig.4.Diagrams in the e?ectiveσtheory.

The e?ects of the non-minimal couplings in the low energy theory can be understood from these diagrams.The Rσ2coupling in diagram(iii)produces a contact interaction, which can be expressed as a path integral over particle paths which begin and end on the horizon,as was shown in detail in section2.In diagram(iv),we see thatσacquires an expectation value at order1/N in the infrared,which then couples to the curvature singularity via the Rσinteraction.Note that this diagram has a1/m2σpole,which will show up in the entropy.

Finally,we discuss the interpretation of the entropy below the scale mσ.There are no dynamical degrees of freedom left,but the e?ective action contains an Einstein-Hilbert term.

I[gμν]=?1

At leading log order the renormalized gravitational coupling is given by

3log

Λ2

m2σ

At this scale allσloops shrink down to points as well.The entire partition function is given by a contact term localized on the horizon,illustrated in Fig.5.No states are present at this scale;the state counting entropy vanishes below the scale mσ.All the underlying correlations responsible for the black hole entropy have been hidden in the low energy coe?cient of the Einstein–Hilbert term.

Fig.5.Diagram below the scale mσ.

Below the scale mσ,we can calculate the partition function,simply by evaluating the e?ective action on a cone with de?cit angle2π?β.Only the Einstein–Hilbert term contributes at?rst order in the de?cit angle.The integral of the scalar curvature is d2x√

?β?1

β=2πI[gμν]

12log

Λ2

m2σ

Note that the black hole entropy is determined by the low energy Newton’s constant.The quantum e?ects that correct the entropy also renormalize the gravitational coupling[24], in such a way that the entropy of a black hole is always given by Area

5.Additional Comments and Conclusions

What conclusions should we draw from these observations?First,we should stress that the qualitative resemblance of the behavior of theσ-model to that of string theory (as proposed by Susskind and Uglum[1])is no accident;the diagrams which contribute to the entropy would be present in a large class of models of extended objects.For example, as pointed out by Susskind[1],below the con?nement scale of QCD analogous diagrams generate a“classical”entropy of order N2c due to the gluons of the theory,and“quantum”corrections of order N0c from the glueballs of the low-energy theory.Similar behavior would be expected in matrix models of string theory.The universality of these phenomena justi?es our discussing them in some generality.

We now turn to the implications for string theory.Our study of theσ-model lends support to the ideas of Susskind and Uglum concerning string theory,in that it provides a toy model which displays analogous behavior.The analogy is based on a comparison of the string diagrams of Fig.1to theσ-model diagrams of Fig.3.According to the proposal of Susskind and Uglum,the string diagram(i)counts the number of states of a half-string stuck to the horizon.In theσ-model,diagram(i)counts states of the?eldφ,which may be regarded as“half-σ”states attached to the horizon.Diagram(ii)in the two theories counts states of the closed string and of theσ?eld,respectively.String diagram(iii)is the most interesting.It represents interactions between the half strings on the horizon and the closed strings outside the horizon.It has an analog in theσ-model,where both diagrams (iii)and(iv)explicitly arise from the interaction between theφ?elds on the horizon and theσ?elds.All this behavior is consistent with the proposals of[1].

As an aside,we note a subtlety that is avoided in our analysis of theσ-model,but deserves mention.In section3.2we constructed a low energy e?ective theory forσby integrating out theφ?elds.The lowest energyφmodes,with k2

A few additional comments are in order regarding the string diagram(iii)in Fig.1. According to[1]this diagram destroys the state-counting interpretation of the one-loop correction to the entropy(while maintaining it for the full theory)and is responsible for the vanishing of the one-loop correction to the entropy in the superstring.At low energies this term reduces to the particle diagrams(iii)and(iv)of Fig.2.We now discuss the lessons that have been learned from studying these?eld theory diagrams.

In the e?ective theory for theσ-model,the quantum contribution to the entropy is dominated by diagram(iv)of Fig.4.In this diagram,theφdegrees of freedom on the horizon couple to the vacuum loop of theσ?eld by exchanging aσparticle;theσis light and the diagram is enhanced quadratically in the infrared.Does this have an analog in string theory?In string theory,the analogous diagram(iv)of Fig.2can be realized as the coupling of half-strings to a vacuum loop via exchange of a light particle with vacuum quantum numbers–a dilaton or one of the moduli.However,in an exactly supersymmetric theory this diagram will vanish,since all scalar?eld tadpoles vanish on general grounds in superstring theory.We do expect such a process to contribute in a theory in which supersymmetry is broken non-perturbatively;however it will be suppressed by a model-dependent factor.It would be of interest to estimate this contribution in speci?c cases.

In a low energy description of a string model,even one with supersymmetry,we expect diagram(iii)in Fig.2to contribute to the entropy.This particle diagram occurs in fundamental theories of vector?elds[13]and of non-minimally coupled scalar?elds (as we discussed in section2,see also[19]),and causes the equivalence of full entropy of entanglement and black hole entropy to break down in those theories.Thus,even in exactly supersymmetric string theories,the one-loop entropy due to light?elds does not have a state-counting interpretation solely in terms of the light degrees of freedom.This is compatible with the suggestion of Susskind and Uglum regarding the interpretation of the one-loop entropy in string theory.However,we stress again that the state-counting interpretation of the black hole entropy is preserved if we consider the theory as a whole.

We note that,in theσ-model,it is natural to introduce an alternate renormalization scheme,in which diagrams(iii)and(iv)of Fig.3are viewed as corrections to theφpropagator and are absorbed into the de?nition of theφmass(see the discussion at the end of section3.3).Diagram(ii)is not naturally absorbed in this way.In this alternate scheme diagrams(i)and(ii)correctly count the states of the interactingφandσ?elds; thus,through rede?nitions of the parameters of the short distance theory,the part of the quantum entropy which does not have a state counting interpretation in the low energy

theory is absorbed into the classical entropy.We do not know if this type of scheme is generally available,or if it is speci?c to theσ-model.

In conclusion,our results provide an explicit?eld theoretic realization of the ideas of Susskind and Uglum[1].One must know the detailed physics of the short distance degrees of freedom in order to give a state counting interpretation of black hole entropy.Short distance degrees of freedom localized on the horizon give rise to the“classical”Hawking–Beckenstein entropy.The“quantum”corrections to this entropy due to light?elds do not have a state counting interpretation in terms of the low energy degrees of freedom, because interactions of the light?elds with the short distance degrees of freedom produce substantial shifts in the quantum entropy.These qualitative features should be expected in any model with extended objects,including QCD,matrix models and string theory.

Note Added

As this work was nearing completion we received a paper by https://www.wendangku.net/doc/0a14434842.html,rsen and F.Wilczek, hep-th/9506066,which has some overlap with this one.

Acknowledgments

This project was inspired by the ideas of Leonard Susskind.We are grateful to him and to John Uglum for numerous illuminating conversations.We also thank Sasha Zamolodchikov for a valuable discussion,and Finn Larsen and Frank Wilczek for discussing their work with us.Part of our work was completed at the Aspen Center for Physics.This research was supported in part by DOE grant DE-FG05-90ER40559.

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III. Complete the following sentences with the proper form of the given phrases. used to keep it up apply for it seemed that agree with be over give it a try try to 1. Do you ________________________me? 2. I think _______________________________ and I will find it a lot of fun. 3. When class _____________________, we talked about where to go for dinner. 4. __________________they couldn’t understand each. 5. I ____________________ a job in a big company. 6. He found it very difficult to _____________ every day. 7. My English ________ be poor but now I am good at it. 8. In English class, everyone _____________speak in English. IV. Choose the best answers. ( ) 1. In the USA, most people speak _________. A. America B. English C. England D. American ( ) 2. _______ has the largest number of speakers. A. China B. Chinese C. French D. English ( ) 3. Is English _______ for the job? A. must B. need C. a must D. a need ( ) 4. English keeps _________ all the time. A. change B. changing C. to change D. changed ( ) 5. He spent a lot of time ________ English these days. A. study B. to study C. is studying D. studying ( ) 6. In some Western countries, people try ______a black cat. A. to avoid B. avoid C. avoiding D. avoided Part II Oral practice I. Match the questions with the answers. 1. What is he? A. He is a tour guide. 2. Where is he from? B. He speaks English.

The Black Cat读后感

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The Black Cat 原典阅读

The Black Cat Edgar Allan Poe 1FOR the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence. Yet, mad am I not -- and very surely do I not dream. But to-morrow I die, and to-day I would unburthen my soul. My immediate purpose is to place before the world, plainly, succinctly, and without comment, a series of mere household events. In their consequences, these events have terrified -- have tortured -- have destroyed me. Yet I will not attempt to expound them. To me, they have presented little but Horror -- to many they will seem less terrible than barroques. Hereafter, perhaps, some intellect may be found which will reduce my phantasm to the common-place -- some intellect more calm, more logical, and far less excitable than my own, which will perceive, in the circumstances I detail with awe, nothing more than an ordinary succession of very natural causes and effects. 2From my infancy I was noted for the docility and humanity of my disposition. My tenderness of heart was even so conspicuous as to make me the jest of my companions. I was especially fond of animals, and was indulged by my parents with a great variety of pets. With these I spent most of my time, and never was so happy as when feeding and caressing them. This peculiarity of character grew with my growth, and, in my manhood, I derived from it one of my principal sources of pleasure. To those who have cherished an affection for a faithful and sagacious dog, I need hardly be at the trouble of explaining the nature or the intensity of the gratification thus derivable. There is something in the unselfish and self- 黑猫埃德加·爱伦·坡homely adj.平凡的；朴素的 unburthen vt. 使……安生succinctly adv.简洁地 expound vt.解释；阐述baroque n.巴洛克作品，这里指「奇谈」phantasm n.幻象；幻觉 succession n.继承顺序，自然演替 docility n.温驯 humanity n.仁爱；人道disposition n.性情；性格；气质 conspicuous adj.显眼的jest n.笑话；笑柄 caress vt.爱抚；抚摸sagacious adj.聪明的；精明的 gratification n.满足；满意derivable adj.可引出的；可诱导的 - 1 -

中英-the black cat

The Black Cat ----Edgar Allan Poe Array FOR the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence. Yet, mad am I not -- and very surely do I not dream. But to-morrow I die, and to-day I would unbur d en my soul. My immediate purpose is to place before the world, plainly, succinctly, and without comment, a series of mere household events. In their consequences, these events have terrified -- have tortured -- have destroyed me. Yet I will not attempt to expound them. To me, they have presented little but Horror -- to many they will seem less terrible than baroques. Hereafter, perhaps, some intellect may be found which will reduce my phantasm to the common-place -- some intellect more calm, more logical, and far less excitable than my own, which will perceive, in the circumstances I detail with awe, nothing more than an ordinary succession of very natural causes and effects. From my infancy I was noted for the docility and humanity of my disposition. My tenderness of heart was even so conspicuous as to make me the jest of my companions. I was especially fond of animals, and was indulged by my parents with a great variety of pets. With these I spent most of my time, and never was so happy as when feeding and caress ing them. This peculiarity of character grew with my growth, and, in my manhood, I derived from it one of my principal sources of pleasure. To those who have cherished an affection for a faithful and sagacious dog, I need hardly be at the trouble of explaining the nature or the intensity of the gratification thus derivable. There is something in the unselfish and self-sacrificing love of a brute, which goes directly to the heart of him who has had frequent occasion to test the paltry friendship and gossamer fidelity of mere Man. I married early, and was happy to find in my wife a disposition not uncongenial with my own. Observing my partiality for domestic pets, she lost no opportunity of procuring those of the most agreeable kind. We had birds, gold-fish, a fine dog, rabbits, a small monkey, and a cat.

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斐斐课件园 https://www.wendangku.net/doc/0a14434842.html,]乐于与他人合作的精神与竞争意识。教学重点： 1.能够用英文同时叙述出物品的颜色和名称。 2.能听懂、会说、会读以下句子 Whats this? Is it an orange? It is a black cat.. 并能进行扩展，在实际生活中灵活运用。教学难点： 1.熟练掌握本单元的单词和句型，能在实际生活中灵活运用。 2.创设英语情景，使学生正确使用英文同时叙述出物品的颜色和名称。教学意图： 1.培养学生[此文转于斐斐课件园 https://www.wendangku.net/doc/0a14434842.html,]对重点词汇和句子的认读能力。 2.培养学生[此文转于斐斐课件园 https://www.wendangku.net/doc/0a14434842.html,]的听说能力。 3.激发学生的求知欲，创设各种真实或接近真实的语言环境，让学生积极参与体验。教具准备：

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H a v e a P o s i t i v e A t t i t u d e i n L i f e After reading this story, I really shocked by what the man had done. He is a murderer who cruelly killed a black cat loved by his wife, then brutishly murdered his wife instead of his second cat, and hid her carcass in the wall. I was confused with the behavior of the man who did all that damn things. I think this man is abnormal, and it is terrible to live with such a person who has something wrong with his mind. I just wonder how can once a righteous and kind man turned out to be so merciless and why the author wanted to show the ugly aspect of human being. After doing some reading about the author, I found the main character in the story is almost the duplicate of author himself; he expressed his own feelings and thoughts through writing. The author lost his parents when he was very young, and he lacked of warm and parental care and sense of security during childhood. What’s more, there’s nobody for him to confess his sufferings and unhappiness, what he could do is to immerse him in alcohol to escape from the anguish. So the reasons were clear for the conversion of the man’s personality. Firstly, is the outside environment: He was always haunting in the bar and was alcoholic, which shows that his life is aimless and disordered. As a result, he was under control of alcohol. Though the surroundings are vital to one’s character development, but it’s not all. In the story, every time when the man did evil things, he blamed them on the alcohol and others. In my perspective, it was just his excuse to avoid the responsibility. As a human, you can’t give way to the madness; instead, you can have strong will to overcome it. Secondly, plenty of negative emotions such as jealousy, violence, irritation make the man envies to the black cat that was loved by his wife and he is violent and angry to the cat in that the cat was so close to the man that the man felt troublesome. When you have such complicated psychology, the most effective and useful way is to talk with your friends or relatives, but not bury deeply in heart, which will later result in vicious cycle. In a word, both inside and outside reasons lead the man to the depth of sin. He seemed accustomed to perform this crime that he had not felt shameful. So after killing the first cat, he acquired some kinds of satisfaction, which later made him tend to abuse things that he didn’t like and eventually killed his wife. So I draw a conclusion from this story: you should find ways to release your dissatisfactions encountered in lif e; don’t find any excuse s to get you away from punishing or it will lead you to crime abyss. Though environment is influential to one’s mood, you should keep a positive and opportunistic attitude toward life. Only in this way, you can drive all the devils away and lead a cozy and comfortable life.

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