Physics Letters B 536 (2002) 114–120

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Entropy of bosonic open string and boundary conditions

M.C.B. Abdallaa, E.L. Graçab,c, I.V. Vanceaa,b

a Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, SP, Brazilb Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, CEP 22290-180 Rio de Janeiro RJ, Brazil

c Departamento de Física, Universidade Federal Rural do Rio de Janeiro, BR 465-07-Seropédica CEP 23890-000 RJ, Brazil

Received 20 March 2002; accepted 18 April 2002

Editor: L. Alvarez-Gaumé

Abstract

The entropy of the states associated to the solutions of the equations of motion of the bosonic open string with combinationsof Neumann and Dirichlet boundary conditions is given. Also, the entropy of the string in the states|Ai 〉 = αi−1|0〉 and|φa〉 = αa−1|0〉 that describe the massless fields on the world-volume of the Dp-brane is computed. 2002 Elsevier ScienceB.V. All rights reserved.

1. Introduction

Recently, there has been some interest in formulat-ing the D-branes at finite temperature for several rea-sons. On one hand, the relation between the D-branesand the field theory at finite temperature represents aninteresting problem by itself which could help us toget a better understanding of the physical propertiesof D-branes. Some progress in this direction has beendone in the low energy limit of string theory wherethe D-branes are solitonic solutions of (super)gravity.In this limit, the thermodynamics of D-branes hasbeen formulated in the framework of (real time) path-integral formulation of field theory at finite temper-ature [1–9]. On the other hand, one would like tounderstand the statistical properties of some systemsthat can be described in terms of D-branes and anti-

E-mail addresses: mabdalla@ift.unesp.br (M.C.B. Abdalla),edison@cbpf.br (E.L. Graça), ivancea@ift.unesp.br (I.V. Vancea).

D-branes as, for example, the extreme, near-extremeand Schwarzschild black-holes [10–27].

In the other well understood limit of string the-ory, the perturbative limit, the geometric informationabout the D-branes is lost. In that case, the D-branesare more appropriately described by a superposition ofcoherent states in the Fock space of closed string sec-tor [28–34] that should satisfy a certain set of bound-ary conditions. They correspond to a combination ofDirichlet and Neumann boundary conditions whichshould be imposed at the endpoints of the open stringin the open string channel. The intuitive interpreta-tion of D-branes as coherent boundary states is main-tained at finite temperature if one formulates the prob-lem in the framework of the Thermo Field Dynamics(TFD). The reason for that is that the thermal depen-dence is implemented through the so-called thermalBogoliubov operators [35] which preserve the form ofthe relations at zero temperature. Working with TFDinstead of real-time path integral formalism representsjust a convenient way of formulating the problem since

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)01801-4

M.C.B. Abdalla et al. / Physics Letters B 536 (2002) 114–120 115

it is known that the two formalisms are equivalent atthe thermodynamical equilibrium. The main idea ofTFD is to interpret the statistical average of any quan-tity Q over a statistical ensemble as the expectationvalue ofQ in a thermal vacuum

(1)Z−1(βT ) Tr[Qe−βT H

]= ⟨⟨0(βT )

∣∣Q∣∣0(βT )⟩⟩,

where βT = (kBT )−1 and kB is the Boltzmann’sconstant [35]. The thermal vacuum is a vector in thedirect product space of the original Fock space by anidentical copy of it. The second copy of the Fock spaceprovide the necessary states to describe the thermaleffects.

The TFD was used to discuss the ideal gas ofstrings, to construct a bosonic open string field theoryat finite temperature [36–38] and to prove its renor-malizability [39–41,47,48]. These studies were mo-tivated by the need of understanding string cosmol-ogy and the ensembles of strings in general. However,when one applies the TFD to D-branes one has to takesome care [42–45]. Indeed, from the beginning, theD-branes are defined as states in the Fock space ofthe first quantized single string. The conformal fieldtheory that underlies the construction describes thebosonic vacuum of string theory. When passing to fi-nite temperature, one interprets the thermal string as amodel forthe thermal excitations of the bosonic vac-uum of string theory. Therefore, the TFD is applied tothe conformal field theory in two dimensions. Thus,the thermal D-branes should be interpreted as a coher-ent boundary states in the Fock space of the thermalexcitations. Another remark is that according to (1),the entropy of the system is defined as the thermal vac-uum expectation value of the entropy operator. How-ever, our basic system is the string and we can computethe entropy of the thermal string excitations by apply-ing (1). What about the entropy of the D-brane states?At the moment, there is no known theory in which theD-branes are described by vacuum-like states or bystates created from a vacuum. Thus, in order to com-pute the entropy of D-brane states, we have to calcu-late the average value of the entropy operator of thebosonic string in the thermal D-brane state. These tworemarks make our TFD approach to the thermal stringsdifferent from the previous ones.

The point of view adopted in [42–45] is that ofD-branes as states in the closed string sector. The TFDis applied to it. However, for the sake of completness,

one should construct the thermal theory for the openstring sector as well. In this case, one can wonder if theboundary conditions have any influence on the ther-mal properties of the system. The entropy of the openstring in its thermal vacuum is given by the sum overall space–time directions of the entropy of the scalarfields and thus it does not depend on the boundaryconditions. However, if the string is in a state that de-pends on the boundary conditions, the entropy com-puted as the average of the entropy operator in thatstate depends on them. Since the bosonic vacuum de-pends on what sort of boundary conditions we imposeto the equations of motion (for example, the Dirich-let boundary conditions will indicate the presence ofbranes), we would like to look to the entropy in thestates associated to the general solutions of the equa-tions of motion with a mixture of Dirichlet and Neu-mann boundary conditions. The aim of the present Let-ter is to present the TFD construction for the bosonicstring in the sense explained above, i.e., for bosonicstring vacuum, and to compute its entropy in stateswith explicit dependence on boundary conditions. Inparticular, we calculate the entropy of open strings inthe thermal massless states which can be interpreted asthermal massless fields living on the D-brane. Thereare at least two reasons for which such of computationshould be performed. Firstly, it represents a naturalstep in developing a theory of string vacua and ther-mal branes in TFD approach. In this context, the statesthat present an explicit dependence on the boundarystates play a fundamental role. Secondly, it might helpto understand the entropy of the D-branes in the per-turbative limit of string theory.

2. The bosonic string in TFD

Consider a bosonic open string in flat space–time. In order to avoid the introduction of ghosts,we choose to work in the light-cone gaugeX0 ±X9 so that the spacetime indices run over the setµ,ν = 1,2, . . . ,24. The string can be quantized in thecanonical formalism by interpreting the most generalsolution of the equations of motion as an operatoron the Fock space and by imposing the canonicalcommutation relations. The dependence on the world-sheet parameters(τ, σ ) is determined by the boundaryconditions which can be of the Neumann (N) or of

116 M.C.B. Abdalla et al. / Physics Letters B 536 (2002) 114–120

the Dirichlet type, independently at each end of thestring. Let us consider the string with NN boundaryconditions. The other cases are treated in exactly thesame manner. The general solutionsXµ(τ,σ ) of theequations of motion of the bosonic open string withthe Neumann boundary conditions at the two ends aregiving by the following relations

Xµ(τ,σ ) = xµ + 2α′τpµ

(2)+ i√

2α′∑n=0

αµn

ne−inτ cosnσ.

Here, xµ and pµ are the canonical coordinates andmomenta of the center of mass of the string. It is usefulto introduce the oscillator operators

Aµn = 1√

nαµ

n ,

(3)Aµ†n = 1√

nα

µ−n, n > 0,

that satisfy the canonical commutation relations amongthemselves and commute with the coordinates and mo-menta of the center of mass. The vacuum is defined tobe invariant under translations

(4)|Ω〉 = |0〉|p〉,where

(5)Aµn |0〉 = 0, ∀n,

(6)pµ|p〉 = pµ|p〉.Now let us construct the thermal bosonic string.

According to TFD, firstly one has to duplicate thesystem. The identical copy of it, denoted by˜, mustbe independent of the original string [35,42–44]. TheHilbert space of the total system is the tensor productof the two Hilbert spacesH andH. The operators forthe two strings commute among themselves. By du-plicating the system, new degrees of freedom are as-sociated to the string. They provide enough room toaccommodate the thermal properties of the bosonicvacuum of string. To this end, the new copy of thestring cannot represent a physical system. Otherwise,the tilde Hilbert space will describe just the dynamicaldegrees of freedom of the second string, and no de-grees of freedom will be left for the thermal effects.1

1 We would like to thank to A.L. Gadelha for illuminatingdiscussions on this point.

They can be implemented in the enlarged Hilbert spaceby constructing a map called the thermal Bogoliubovtransformation that maps the vacuum of the string tothe thermal vacuum defined by (1). It turns out thatthere are several possibilities to construct such of mapand that the corresponding operators generate a ther-malSU(1,1) group. But if we stick to unitary and tildeinvariant transformations we are left with just a singleoperator in the present case [45,46]. The Bogoliubovtransformation is defined for each oscillatorn in eachdirectionµ by the following relation

(7)Gµn = −iθn(βT )

(An · An − A†

n · A†n

).

Here, θn(βT ) is a parameter related to the Bose–Einstein distribution of the oscillatorn. The dot in (7)represents the Euclidean scalar product in the targetspace. The vacuum of the oscillators atT = 0 is givenby the following relation

(8)∣∣0(βT )

⟩⟩= ∏m>0

e−iGm∣∣0⟩⟩,

where|0〉〉 = |0〉|0〉 and

(9)Gn =24∑

µ=1

Gµn .

The total vacuum atT = 0 is constructed in the sameway as (4), i.e.,

(10)∣∣Ω(βT )

⟩⟩= ∣∣0(βT )⟩⟩|p〉|p〉.

The state (8) is called the thermal vacuum since it isannihilated by all thermal annihilation operators con-structed by acting with the Bogoliubov operators (9)onto the oscillator operators of string

Aµn (βT ) = e−iGnAµ

n eiGn,

(11)Aµn (βT ) = e−iGnAµ

n eiGn.

The same action of the Bogoliubov operators gives anyother thermal operator. In particular, one constructs thecreation operatorsAµ†

n (βT ) and Aµ†n (βT ) which, to-

gether with (11) define the set of thermal string oscilla-tors since they satisfy the same commutation relationsas the operators atT = 0. The coordinates and the mo-menta of the center of mass of string are invariant un-der the Bogoliubov map. Thus, one can construct thestring solutionXµ(βT ) atT = 0 by replacing the oper-ators in (2) with the corresponding operators atT = 0.

M.C.B. Abdalla et al. / Physics Letters B 536 (2002) 114–120 117

In exactly the same way one can construct the genera-tors of the Virasoro algebra in terms of thermal oscil-lator operators and show that the conformal symmetryof the solution is preserved. Consequently, the Bogoli-ubov transformation maps the two string copies intotwo thermal string copies [43,44]. However, the inter-pretation in terms of original and tilde string modesis lost at non-zero temperature since the Bogoliubovoperators (7) mix the two copies of strings. Also, thethermal vacuum should be invariant under the tildeoperation and therefore the tilde and non-tilde zero-temperature excitations are generated simultaneouslyat T = 0. This is an intrinsic fact of the TFD con-struction. Heuristically, one would say that the thermalstring excitations, being a mixture of zero-temperaturestring and tilde string excitations, carry thermal de-grees of freedom beside dynamical degrees of freedom(in zero-temperature language) which is in agreementwith our intuition.

3. The entropy in states with boundarydependence

Since we are treating the string as a set of bosonicoscillators, the entropy operators for the bosonic openstring follow directly from the definition [35]

(12)

K =24∑

µ=1

∞∑n=1

(Aµ†

n Aµn logsinh2 θn

− Aµn Aµ†

n logcosh2 θn

),

(13)

K =24∑

µ=1

∞∑n=1

(Aµ†

n Aµn logsinh2 θn

− Aµn Aµ†

n logcosh2 θn

).

The vacuum of the present theory is invariant underthe tilde operation and all the physical information iscontained in the operators without tilde. Therefore, weare going to calculate the matrix elements only of theentropy (12). The basic idea is to compute firstly thematrix elements to the contribution to the entropy ofthe excitations in each direction of space–time. Thus,it is useful to write (12) in the form

(14)K =24∑

µ=1

Kµ,

where the notation is obvious.The entropy of the string, as given by the for-

mula (1), represents the sum of the entropy of all oscil-lators and it is straight-forward to write it down. Next,we would like to know the entropy of the string instates associated to the most general solution of theequation of motion. This is a similar situation to com-puting the entropy of D-branes. In this case, the en-tropy is a world-sheet function and its dependence onthe world-sheet parameters is dictated by the bound-ary conditions imposed on the string equations of mo-tion. Upon quantization, the general solution of thestring equations of motion with given boundary con-ditions at the two ends becomes an operator acting onthe thermal Fock space. The state|Xµ(βT )〉〉 obtainedby applying (2) on the thermal vacuum (10) can beassociated to it. Let us work out the matrix elements〈〈Xµ(βT )|Kρ |Xν(βT )〉〉 for the NN boundary condi-tions in some detail. The computations involving theother boundary conditions follow the same line.

The matrix element〈〈Xµ(βT )|Kρ|Xν(βT )〉〉 canbe split in two parts: one part containing only theoscillator contribution and a part that contains thecoordinates and the momenta of the center of mass⟨⟨Xµ(βT )

∣∣Kρ∣∣Xν(βT )

⟩⟩= CM terms

(15)

− 2α′ ∑n,k,l>0

ei(l−n)τ

√ln

cosnσ coslσ

× [(T1)

µρνnkl + (T2)

µρνnkl

],

where

(T1)µρνnkl

= ⟨0∣∣⟨1µ

n

∣∣ ∏m>0

e−iGmAρ†k A

ρk logsinh2 θk

×∏s>0

eiGs∣∣1ν

l

⟩∣∣0⟩ ⟨p∣∣q⟩〈p|q〉,

(T2)µρνnkl

(16)

= −⟨0∣∣⟨1µn

∣∣ ∏m>0

e−iGmAρk A

ρ†k logcosh2 θk

×∏s>0

eiGs∣∣1ν

l

⟩∣∣0⟩ ⟨p∣∣q⟩〈p|q〉

and

(17)∣∣1µ

l

⟩= Aµ†l |0〉.

118 M.C.B. Abdalla et al. / Physics Letters B 536 (2002) 114–120

We consider the usual normalization of the momentumstates in a finite volumeV24 in the transverse space

(18)〈p|q〉 = 2πδ(24)(p − q),

(19)(2π)24δ(24)(0) = V24.

It follows from a simple algebra that the pure oscillatorcontribution to the matrix element of the entropyoperator has the following form⟨⟨Xµ(βT )

∣∣Kρ∣∣Xν(βT )

⟩⟩= CM terms

− 2α′(2π)(48)δµνδ(24)(p − q)δ(24)(p − q

)

(20)

×∑n>0

1

ncos2 nσ

[log(tanhθn)2δρν

− δρρ∑k>0

δkk

].

The terms containing the center of mass can be di-vided at their turn in to two sets: the ones contain-ing only the coordinates and momenta operators andthe ones containing the contributions from the oscilla-tors. The terms in coordinates and momenta can becalculated by using the completness relation of theeigenstates of momenta operators|x〉 together with thecoordinate-momenta matrix

(21)〈x|p〉 = (2πh)−12eip·x/h.

In order to compute the terms containing oscillatorcontribution one can either express the oscillatoroperators at zero temperature in terms of operatorsat T = 0 or write the vacuum atT = 0 in terms ofvacuum at zero temperature. The two ways lead tothe same result, however, in the first case one hasto deal only with polynomial relations in creationand annihilation operators at finite temperature. Usingthe properties of the Bogoliubov operators deducedin [43,44] and after a simple algebra one can showthat the contribution coming from terms that mixthe center of mass operators with the oscillator partcancels. The non-vanishing terms are all proportionalto 〈〈0(βT )|Kρ |0(βT )〉〉 which represents the entropyof an infinite number of bosonic oscillators in theρth directions of space–time. Putting all the resultstogether we obtain the following relation⟨⟨Xµ(βT )

∣∣Kρ∣∣Xν(βT )

⟩⟩

= −(2πh)−24

[(2πh)24(2α′τ

)2pµp′νδ(24)

(p − p′)

+ 2α′τ(I

µ2 p′ν + I ′ν

2 pµ)

+ Iµ2 Iν

2

∏j =µ,ν

Ij

1

]× δ(24)

(p − p′)

×∑m=1

[nρ

m lognρm + (

1− nρm

)log(1− nρ

m

)]− 2α′(2π)(48)δµνδ(24)

(p − p′)δ(24)

(p − p′)

(22)

×∑n>0

1

ncos2 nσ

[log(tanhθn)2δρν

− δρρ∑k>0

δkk

],

where the unidimensional integrals on the finite do-mainsx ∈ [x0, x1] are given by

(23)I1 = −ih(p′ − p

)−1[e

ih (p′−p)x1 − e

ih (p′−p)x0

],

(24)

I2 = −ih(p′ − p

)−1

×[−ihI1 + x1e

ih (p′−p)x1 − x0e

ih (p′−p)x0

].

Here, the momentum of the final and initial states aredenoted by|p〉 and|p′〉, respectively, and

nρm = ⟨⟨

(βT )∣∣Aρ†

m Aρm

∣∣0(βT )⟩⟩

(25)= sinh2 θm

represents the number of string excitations in thethermal vacuum.

Relation (22) represents the matrix element ofthe entropy operatorKρ between two general statesdescribed by solutions of the equations of motion ofthe open bosonic string with NN boundary conditions.The total energy is a sum over all directions in thetransverse space. Since the effect of the boundarycondition appears only in the dependence on theworld-sheet parameters, it is easy to write down thesimilar relations for the other boundary conditionsDD, DN and ND. In these cases, there are no operatorsassociated with the coordinate and momenta of thecenter of mass of the open string but rather constantposition vectors associated to the endpoints of string.Therefore, there is no contributions from these terms

M.C.B. Abdalla et al. / Physics Letters B 536 (2002) 114–120 119

to the entropy. The terms that mix the coordinates ofthe endpoints with the oscillators vanish for the samereason as in the NN case. The only non-vanishingterms in the matrix elements of the entropy are

DD: ⟨⟨Xµ(βT )

∣∣Kρ∣∣Xν(βT )

⟩⟩= 2α′(2π)(48)δµνδ(24)

(p − p′)δ(24)

(p − p′)

×∑n>0

1

nsin2 nσ

(26)×[

log(tanhθn)2δρν − δρρ∑k>0

δkk

],

DN: ⟨⟨Xµ(βT )

∣∣Kρ∣∣Xν(βT )

⟩⟩= 2α′(2π)(48)δµνδ(24)

(p − p′)δ(24)

(p − p′)

×∑

r=Z+1/2

1

rsin2 rσ

(27)×[

log(tanhθr)2δρν − δρρ

∑k>0

δkk

],

ND: ⟨⟨Xµ(βT )

∣∣Kρ∣∣Xν(βT )

⟩⟩= 2α′(2π)(48)δµνδ(24)

(p − p′)δ(24)

(p − p′)

×∑

r=Z+1/2

1

rcos2 rσ

(28)×[

log(tanhθr)2δρν − δρρ

∑k>0

δkk

].

Here, Z + 1/2 are the half-integer numbers. Thecontribution of just a single field is obtained by takingµ = ρ = ν. The relations (22), (26), (27) and (28)represent the entropy of the states associated to thegeneral solutions of the equations of motion. Theygive the entropy as a function on the world-sheet,therefore, it should not be taken for the entropy of thebosonic string vacuum which is simply the sum overall space–time directions of the entropy of masslessscalar bosons and it does not depend on the boundaryconditions as it should.

4. Discussions

The matrix elements (22), (26), (27) and (28) canbe used to calculate the entropy of various states ofopen strings with different boundary conditions. As an

example, consider the massless states of strings endingon one Dp-brane. It is known that these states describemassless fields on the world-volume of the brane.They form anU(1) multiplet Aj = α

j

−1|0〉, wherej = 1,2, . . . , p and a set of(24 − p) scalarsφa =αa

−1|0〉 wherea = p + 1, . . . ,24. The contribution tothe entropy of the corresponding string states with DD,DN and ND boundary condition can be computed bytruncating (26), (27) and (28) to the first oscillator termor computing the matrix elements ofKρ in the thermalstates

(29)∣∣Ψ λ(βT )

⟩⟩= αλ−1(βT )∣∣0(βT )

⟩⟩multiplied by the function onτ andσ that is obtainedfrom the corresponding boundary conditions. A sim-ple algebra gives the following expression for the en-tropies of theU(1) multiplet and the set of scalars,respectively,

(30)

EA = 2α′p(

−sin2 σ∑n=1

logcosh2 θn

+ 4 cos2 σ∑

r∈Z+1/2

logcosh2 θr

),

(31)

Eφ = 2α′(24− p)

(−sin2 σ

∑n=1

logcosh2 θn

+ 4 cos2 σ∑

r∈Z+1/2

logcosh2 θr

).

In (30) and (31) the first term represents the contribu-tion from the DD sector while the second one is ob-tained from DN and ND sectors.

In conclusion, we have calculated the entropy ofstring in states that depend explicitly on the bound-ary conditions. To this end, we have picked up themost general solution of the equations of motion withall possible combinations of Dirichlet and Neumannboundary conditions and we computed the matrix ele-ments of the entropy in the state associated to it. Therelevant relations are (22), (26), (27) and (28). It is in-teresting to observe that only the NN entropy dependson h. In the semiclassical limith → 0 the pure mo-mentum contribution becomes irrelevant. The domi-nating term is the same that dominates the infinite ten-sion limit α′ → 0. In this case the entropy of the DD,DN and ND strings vanishes.

120 M.C.B. Abdalla et al. / Physics Letters B 536 (2002) 114–120

In particular, we have applied the general results tothe computation of the entropy of the massless stringstates that generate the field theories on the world-volume of a Dp-brane. In the case of a stack ofN par-allel D-branes the gauge multiplet is ofU(N) and inorder to find the total entropy of it we just have to sumover the internal indices. However, the scalars becomenon-commutative and the present approach shouldbe extended to include non-commutative fields [49].Constructing the D-branes in the thermal string fieldtheory represents a different but interesting topic. Ifsuch of theory were given, one could get insights inthe problem of the entropy of the D-brane which, inthe present formulation, seems to suffer from certainanomalies [44].

Acknowledgements

We would like to thank N. Berkovits, A.L. Gadelha,P.K. Panda, B.M. Pimentel, H.Q. Placido and W.P. deSouza for useful discussions. I.V.V. also acknowledgeto S.A. Dias and J.A. Helayël-Neto for hospitality atDCP-CBPF and GFT-UCP where part of this workwas done. I.V.V. was supported by a FAPESP postdocfellowship.

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