Tachyon driven quantum cosmology in string theory

PHYSICAL REVIEW D 71, 063517 (2005)

Tachyon driven quantum cosmology in string theory

H. Garcıa-Compean,1,* G. Garcıa-Jimenez,2,† O. Obregon,2,‡ and C. Ramırez3,x

1Departamento de Fısica, Centro de Investigacion y de Estudios Avanzados del IPN, P.O.Box 14-740, 07000, Mexico D.F., Mexico2Instituto de Fısica de la Universidad de Guanajuato, P.O. Box E-143, 37150 Leon Gto., Mexico

3Facultad de Ciencias Fısico Matematicas, Universidad Autonoma de Puebla, P.O. Box 1364, 72000 Puebla, Mexico(Received 4 January 2005; published 18 March 2005)

*Electronic†Electronic‡ElectronicxElectronic

1550-7998=20

Recently an effective action of the SDp-brane decaying process in string theory has been proposed.This effective description involves the Tachyon driven matter coupled to bosonic ten-dimensional Type IIsupergravity. Here the Hamiltonian formulation of this system is given. Exact solutions for thecorresponding quantum theory by solving the Wheeler-deWitt equation in the late-time limit of therolling tachyon are found. The energy spectrum and the probability densities for several values of p areshown and their possible interpretation is discussed. In the process the effects of electromagnetic fields arealso incorporated and it is shown that in this case the interpretation of tachyon regarded as ‘‘matter clock’’is modified.

DOI: 10.1103/PhysRevD.71.063517 PACS numbers: 98.80.Qc, 04.20.Jb, 04.65.+e, 04.60.Ds

I. INTRODUCTION

In recent years a great deal of attention has been paid tothe classical cosmology of effective string theories (see forinstance, Ref. [1]). These models have become a suitabletest arena in regard to the difficulties of describing itsdynamics at string or M-theory level. One successful pro-posal connected with these ideas is the braneworld cos-mology that leads to interesting consequences at the levelof the classical equations of motion where the high energycorrections to the Friedman equation were found [2].

Recently an effective model of a coupled system of ten-dimensional Type II supergravity and tachyon driven mat-ter has been proposed [3–5]. This model was considered asa good candidate for describing the effective action of aspacelike Dp-brane, called SDp-brane, which describes(time-dependent) decaying processes in string theory. Inthis model our universe is contained in the bulk and thebrane is considered as a topological defect that is em-bedded also in the bulk. The spacelike brane solutions infield theory were first considered in [6] and can be under-stood as follows: in string theory the open string spectrumof the Dp-brane has tachyonic modes. In the effective fieldtheory the tachyon has a symmetric potential V�T� � 0,with a central maximum and two symmetric minima whichindicate that the system is unstable around the maximum ofthe tachyon potential (for a review, see [7]). If all theDirichlet boundary conditions on the brane are spacelike,then usual Dp-branes arise. However if one of theseDirichlet conditions is timelike, then the tachyon rollsdown the potential and a time-dependent process occurs[6]. The full picture of passing from one minimum to theother is called SDp-brane solution. As soon as the tachyonfield rolls down from the top of V�T� towards one of its

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05=71(6)=063517(8)$23.00 063517

minima, it starts to excite open and closed string modes insuch a way that the energy of the unstable Dp-brane isradiated away into the bulk. When the tachyon arrives to itsminimum, the radiation is in the form of only closed stringsbecause open strings cannot exist in the bulk. This has beencomputed explicitly, see [8,9] and references therein.

On the other hand, based in prior work [10], Sen pro-posed a field theory describing the dynamics of the rollingtachyon [11–13]. In this context, he found that the tachyonfield can be interpreted as the time in quantum cosmology[14]. This was done by coupling the ’’tachyon matter’’ to agravitational field and then performing its canonical quan-tization. From it, a Wheeler-deWitt equation turns out,which can be regarded as a time-dependent Schrodingerequation for this gravity-tachyon matter system. The cou-pling of the tachyon to gravity has been studied in connec-tion with classical cosmological evolution [15–17]. Inparticular its role related to inflation has been discussed,see [18] and references therein. The classical solutions tosupergravity including S-branes have been worked out insome cases, see, e.g., [19–21]. In Ref. [22] solutions of theEinstein-Maxwell effective description, in four dimen-sions, of the rolling tachyon of the S0-brane proposed inRef. [6], have been found.

Further, in [3–5] the bosonic sector of the effective ten-dimensional supergravity action, coupled to tachyonicmatter, under a maximal symmetric ansatz ISO�p� 1� �SO�8� p; 1�, has been considered. There, the time-dependent models were extensively studied and some clas-sical solutions to supergravity with SDp-branes have beenworked out. For recent developments in this direction seeRef. [23]. As we mentioned, in view of the lack of atachyonic quantum theory at the level of the whole stringtheory, it is of interest to consider the quantization of theeffective field theory. In order to describe the behavior ofthe SDp-brane in the effective theory, it is natural to foliatespace-time along the time coordinate, which leads us toconsider a minisuperspace approach. Although minisuper-

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GARCIA-COMPEAN, GARCIA-JIMENEZ, obregon, ramırez PHYSICAL REVIEW D 71, 063517 (2005)

space contains a further reduced sector of the degrees offreedom of the full string theory, at the level of the effectivetheory it becomes a suitable approach to have under controlsome features of the classical theory, as well as the lowestexcited states of the genuine quantum dynamics of theSDp-branes [17]. Thus, it is expected that the canonicalquantization description presented here corresponds to thes-wave approximation of the low energy quantum theoryof the effective action of Refs. [3–5]. Furthermore it couldgive some insight into the solution of the Wheeler-deWittequation for finite values of the tachyon T. The quantumproperties of the considered field theory seem to be aninteresting problem by itself, as already pointed out by Senin Ref. [14], where he considers a quantum cosmologymodel coupled to the tachyon matter. The effective model[3–5] we are going to consider can be understood as acosmological model with dilaton and RR fields, driven bythe tachyon matter. We show that the proposal by Sen,concerning the interpretation of the tachyon as time, in thelate ‘‘time’’ decoupling limit, is valid for the model underconsideration. In this case we find an exact wave function,finite and continuous everywhere for the correspondingSchrodinger equation. The associated ‘‘probability den-sity’’ shows an infinity of continuous degenerated maximadescribing a path in minisuperspace. Its behavior withrespect to some interesting values of p is also shown.Moreover, we will show that even for the next orderapproximation from the late-time decoupling limit (stillwith T large but with nonvanishing V�T� and F �T�, seeRef. [12,13] and Section 2 in Ref. [14]), the RR couplingallows an interpretation of the tachyon as time, in this casewith a Schrodinger equation with a time-dependentpotential.

It should be remarked however that in the presence of auniform electric field, the interpretation of the tachyon astime seems to be spoiled. In the late-time limit, the tachyondoes not decouple from the electric field. This electric fieldhas been considered, for example, in connection with whathas been called the carrollian confinement mechanism foropen string states [18,24].

This paper is organized as follows: in Sec. II we brieflydiscuss the model proposed in Refs. [3,4] for the effectiveaction of a SDp-brane. In Sec. III we find the Hamiltonianconstraint for the system. Section IV is devoted to the studyof quantum solutions with the rolling tachyon approxima-tion in the decoupling limit. We also comment about theinterpretation of tachyon as time to a first order approxi-mation around the limit at T ! 1. In Sec. V, we includeelectric and magnetic fields and exhibit the relevant part ofthe Hamiltonian in the late-time limit. Our conclusions arefinally presented in Sec. VI.

II. THE EFFECTIVE ACTION

The case we analyze here is that of the low energyeffective action of the closed string interaction with the

063517

rolling tachyon matter. This can be done by means of anaction Sbrane given by the Dirac-Born-Infeld action of thetachyon plus a Wess-Zumino term describing its couplingto the RR fields. To this, the action Sbulk of the backgroundten-dimensional type II supergravity is added, from whichwe will consider only the bosonic sector. The action pro-posed in [3,4] for this theory is:

S Sbulk � Sbrane; (1)

Sbulk 1

16G10

Zd10x

��g

p�R�

1

2�@��2

�ea�

2�p� 2�!F2p�2

�; (2)

Sbrane �

16G10

Zdp�2xk%?��V�T�e

��A

p�

��

16G10

Z%?F �T�dT ^ Cp�1; (3)

where G10 is the Newton’s constant in the ten-dimensionaltheory, � is the brane coupling, a � �3� p�=2 is thedilaton coupling, A detA��; A�� g��e

�=2 �

@�T@�T is the tachyon metric, F �T� is the factor ofcoupling between the tachyon and the RR fields Cp�1,and V�T� is the tachyon potential. %? is the ‘‘density ofbranes’’, which does not depend on the parallel coordinatesof the brane xk:Greek indices �;� 0; 1; . . . ; p� 1; labelthe time and parallel coordinates (denoted by k ) to theSDp-brane. Latin indices i; j 1; ::; 8� p label the per-pendicular coordinates of the brane (denoted by ? ), andcapital letters A;B; . . . ; etc. stand for space-time coordi-nates of the bulk.

Following Refs. [3,4], the simplest model that we canstudy is assuming the homogeneous (but nonisotropic)FRW metric. Making the space decomposition into maxi-mal symmetric direct product ISO�p� 1� � SO�8� p; 1�we have the metric

ds2 �N2�t�dt2 � a2k�t�dx2

k� a2?�t�dx

2?; (4)

where ak�t� and a?�t� are the parallel and perpendicularscaling factors of the brane and N�t� is the lapse function.In this homogeneous model the space parallel (perpendicu-lar) to the brane is characterized by the curvature constantkk (k?). In [4,22] it was noticed that the SDp-brane is notsuitable to be localized by means of a delta function,because it could break at short scales the R-symmetrypresent in SO�8� p; 1�. In order to preserve this symme-try, it was proposed to ’’smear out’’ the localization of thebrane by a homogeneous distribution along x?: Thus thedensity b%? is given by

%? !?d8�px?; (5)

where !? !0��gH8�p

p !0a

8�p? , !0 is a constant and

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TACHYON DRIVEN QUANTUM COSMOLOGY IN STRING THEORY PHYSICAL REVIEW D 71, 063517 (2005)

gH8�p is the determinant of the metric of the hyperbolicspace perpendicular to the brane. The �p� 2�-form fieldstrength Fp�2 is given in terms of the �p� 1�-form RRpotential Cp�1, which is chosen in a gauge in which theonly nonvanishing component is C12��p�1 C�t�;

F2p�2 �N�2 _C2p�1 �N�2 _C2: (6)

In order to preserve homogeneity, the tachyon field is afunction only of time T T�t�. Hence the tachyon couplesto RR fields in the following form

dT ^ Cp�1 _TCdp�2xk: (7)

We can simplify the Lagrangian introducing the coordi-nates �1; �2 defined as,

�1 1

9��p� 1��k � �8� p��?�; (8)

�2 �k � �?; (9)

where �k lnak and �? lna?: Also the space volumeis given by VS

116G10

Rdp�1xkd8�px?, so S

Rd10xL;

with L VSRdtL:

In order to describe the influence of the tachyon matterat the quantum level, in driving a simple cosmologicalmodel we consider the simplest case given by takingboth curvature constants kk and k? to be zero. This casewill allow us to give some insight about the tachyon matterat the quantum level. Other cases with kk and k? takingdifferent values among f0; 1;�1g can be analyzed follow-ing the same procedure. However, with this choice it isexpected that the R-symmetry along x? breaks down toSO�8� p�.

In�1 and�2 coordinates and with the ansatz (4) we havethe Lagrangian

L �e9�1

N

�72 _�21 �

�p� 1��8� p�9

_�22 �1

2_�2

�ea�

2�p� 2�!_C2�� $e9�1�a�=2V�T�

��N2e�=2 � _T2

p� $e�8�p��1��1=9��p�1��2�F �T� _TC: (10)

We can formulate an equivalent theory if we introduce aLagrange multiplier � [25] into the Lagrangian (10) asfollows,

L �e9�1

N

�72 _�21 �

�p� 1��8� p�9

_�22 �1

2_�2

�ea�

2�p� 2�!_C2��1

2��1�N2e�=2 � _T2�

�1

2$2e18�1�a�V2�T��

� $e�8�p��1��1=9��p�1��2�F �T� _TC; (11)

063517

where $ �!0: As usual, the variation of this action withrespect to �, @L

@� 0, and substituting � from it intoLagrangian (11) the Lagrangian (10) follows.

III. THE CANONICAL HAMILTONIAN

In this section we discuss the canonical Hamiltonianformalism of the Lagrangian (11). As a consequence ofreparametrization invariance of the theory we expect theassociated Hamiltonian constraint, which will be used inthe next section to give the corresponding Wheeler-deWittequation. The canonical momenta obtained from theLagrangian (11) are given by,

P1 @L

@ _�1 �

144

Ne9�1 _�1;

P2 @L

@ _�22

9

�p� 1��8� p�N

e9�1 _�2;

P� @L

@ _�e9�1

N_�; PC

@L

@ _C

e9�1�a�

N�p� 2�!_C;

PT @L

@ _T ��1 _T � $e�8�p��1��1=9��p�1��2�F �T�C:

(12)

The first class constraints are P� PN 0. When weimplement these constraints, the Hamiltonian is given by

H _�1P1 � _�2P2 � _�P� � _CPC � _TPT � L

N2

��1

144e�9�1P21 �

9e�9�1

2�p� 1��8� p�P22

� e�9�1P2� � �p� 2�!e��9�1�a��P2C

�$2

2V2�T�e18�1�a��

N2��1

2e�=2

��

2

�PT � $e�8�p��1��1=9��p�1��2�F �T�C

�2: (13)

After elimination of � by its equation of motion@H=@� 0; the Hamiltonian gets the form H NH0,where,

H0 �1

144e�9�1P21 �

9e�9�1

2�p� 1��8� p�P22

� e�9�1P2� � �p� 2�!e��9�1�a��P2C

� 2e�=4f$2V2�T�e18�1�a�

� �PT � $e�8�p��1��1=9��p�1��2�F �T�C�2g1=2

0; (14)

is the first class Hamiltonian constraint associated with theinvariance under time reparametrizations.

It is worth noticing that when this constraint is applied atthe quantum level, the resulting Wheeler-deWitt equationdoes not provide a time evolution of the system, and thecorresponding wave function is not normalizable. The

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spectrum of the whole system (bulk-brane) has zero en-ergy. This is known as the ‘‘time problem’’ [26]. However,as we will see in the next section, in the late-time limitwhen the tachyon potential and the RR fields are negli-gible, we can find a ’’time’’ dependent Wheeler-deWittequation.

IV. CANONICAL QUANTIZATION

Exact expressions for the potential V�T� and the cou-pling factor F �T� are not known. However, their asymp-totic universal form V�T� e��T=2 and F �T� sign�T�e��T=2 as T ! 1; are known from string theory[11–14]. Here � 1 for bosonic string theory, and � ��2

pfor superstring theory. We only assume that V�T� has a

maximum at T 0 and a minimum at T ! 1, whereV�T� 0. Also, we see that in this limit the tachyondecouples also from the RR fields as F �T� ! 0. Becausewe want to interpret tachyon as time, we restrict to the late-time noninteracting sector where tachyon decouples fromthe other fields. The ’’strong’’ (near the top of the potential)coupling sector, does not allow to interpret tachyon as timedirectly, and therefore the Wheeler-deWitt equation doesnot lead in a simple way to a solvable wave function.

A. Late-Time Limit

In the sector where V�T� � 0; the canonicalHamiltonian (14) takes the form

H0 �1

144e�9�1P21 �

9

2

e�9�1

�p� 1��8� p�P22 � e�9�1P2�

� �p� 2�!e��9�1�a��P2C � 2e�=4PT

0: (15)

The resulting equation is the Wheeler-deWitt equationwhich in the minisuperspace approach is given by

bH 0� 0; (16)

where bH0 is given by (15), with P1 �i @@�1

; P2 �i @

@�2; PC �i @@C and PT �i @@T : As it is known, the

quantum analysis of the Wheeler-deWitt equation has theproblem of the factor ordering. In order to solve thisproblem there are some proposals in the literature [27],however none of them seems to be satisfactory. This prob-lem is beyond the scope of this paper and here we onlyassume a particular factor ordering. Assuming that thedilaton field is given by its vacuum expectation value inthe late-time, i.e., gs eh�i, where gs is the string couplingconstant, then P� 0, and we have

e�9�1�C1@2�

@�21� C2

@2�

@�22� C3

@2�

@C2

� i

@�@T

; (17)

where C1 1288g

�1=4s ; C2

94�p�1��8�p� g

�1=4s ; C3

063517

��p� 2�!g��a�1=4�s �=2: Now, we see that the Wheeler-

deWitt Eq. (16) leads to a Schrodingerlike equation.Thus in this limit, the tachyon is an usual scalar field whichprovides a useful parametrization of time, because thetachyon momentum enters linearly in (15). This can beinterpreted as a ‘‘matter clock’’[28]. This matter accom-panies gravitation (Sbulk) in a natural and consistent man-ner. So it seems that at least some of the criticisms andproblems related to a ‘‘matter clock’’ can be in this caseavoided. Moreover, Sen [13] showed that for large valuesof time x0, the classical tachyon solution goes as T ’ x0thus, this result provides us another way to recognize T as atime.

The solution of (17) is straightforward. Assuming sepa-ration of variables for � is of the form: � �1��1� �2��2� T�T� C�C� we can rewrite Eq. (17) as

e�9�1�C1 00�1

�1� C2

00�2

�2� C3

00C

C

� i

0T

T �'; (18)

where ' is a separation constant, which we take to be real.Thus, the tachyon wave function is given by

T�T� ei'T: (19)

Similarly, we find for the other field components

�2��2� e�i��(C2

p�2 ; (20)

C�C� e�i

��)C3

qC; (21)

with $ )� ( � 0. Thus the remaining equation isgiven by

C1 00�1

�1�'e9�1 � $ 0: (22)

This equation has as solution the modified Bessel function

�1��1� Ki+

�8

3

��'

pe�9=2��1

�; (23)

where + 29

��$C1

q: The general solutions are then,

�� N ei'Te�i

��)C3

qCe�i

��(C2

p�2Ki+

�8

3

��'

pe�9=2��1

�; (24)

where N is a normalization constant. This is a plane wave,that represents a free particle, with respect to the variablesC and �2 and with T playing the role of time. Also noticethat the constant ' � E enters as an energy when tachyonis interpreted as time. This energy would correspond to thelowest level of the closed string spectrum. In terms of theradii ak and a? we have,

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0

0.5

1

1.5

2

a ||

0

0.5

1

1.5

2

|a _

0 0.5 1

Ψ2

0

0.5

1

1.5

0 0

||

FIG. 1. The figure shows the probability density j�j2 for the‘‘physical’’ case of a SD3-brane (p 3) (with ' 0:1 and + 0:7) and its variation with respect to the radii a? and ak. Themaxima of the quantum solution� determines a trajectory in thea? � ak plane. These maxima satisfy an inverse relation be-tween both radii as shown in Eq. (26).

0

0.5

1

1.5

2

0

0.5

1

1.5

2

|a _

0 0.5 1

Ψ2

0

0.5

1

1.5a ||

0 0

||

FIG. 2. The probability density j�j2 (also with ' 0:1 and+ 0:7) for one extreme case with p 1. The solution showsthat for this case, the maxima of j�j2 determines an evolutionwhich is almost projected on the axis a?, that is, for large valuesof a? it is almost independent on ak.


�� N ei'Te�iga=2�1=8s

��2)=�p�2�!

pC

�

�aka?

��2i=3�g1=8s

��p�1��8�p�(

p

� Ki+

�8

3

��'ap�1

ka8�p?

q �: (25)

This wave function represents the state of the system in theasymptotic limit T ! 1: In this sector the brane hascompletely decayed, therefore, this wave function can beassociated with the stable final state of the system whereonly closed string spectrum exists.

We can compute the expectation value of ak, for acertain constant value of a?

haki N 0Z 1

0��ak�dak

N 0Z 1

0

�Ki+

�8

3

��'ap�1

ka8�p?

q ��2akdak; (26)

where � is a linear combination of �� and N 0 is anoverall normalization constant. Here �� denotes complexconjugate of �: From (26) we get

haki N 0��

p �� 2p�1��2

p�1� i+�� 2p�1� i+�

�3� p�� 3�p2�p�1��

�

�9

64'ha?i8�p

�2=p�1

: (27)

This relation can also be written as a sort of uncertaintyrelation between the two radii haki � ha?i�2�8�p�=�p�1�,where the proportionality factor is, for + 1, of the orderof 10�2 N 0 and decreases exponentially as + increases.Note that the denominator in (27) does not diverge at p

3 due to the properties of the Gamma function, in fact �3�

p�� 3�p2�p�1�� 2�p� 1�� 5�p2�p�1��.In Fig. 1, we plotted the probability density j�j2 for the

physical (‘‘realistic’’) case of p 3, where it is shown acontinuum of maxima in the a? � ak plane, following apath in minisuperspace and showing an inverse relationbetween the two radii given by Eq. (27). Figure 2 and 3show two extreme cases with p 1 and p 8, respec-tively. These figures also show that, for p 1, the proba-bility density is almost projected on the a?-axis, that is, forlarge values of ak, it is almost independent on it. Figure 3,is the case for p 8 and it shows that the probabilitydensity j�j2 is independent on a? and therefore is pro-jected on the ak-axis. All these figures are plotted for thespecific values of the parameters given by ' 0:1 and+ 0:7.

B. First order corrections

Once the tachyon arrives at the minimum of the poten-tial, the radiation is emitted into the bulk and the systembecomes stable. We can therefore calculate corrections

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around this stable minimum. Consider the next leadingorder of the approximation of the Hamiltonian (14), inwhich V2�T� is neglected with respect to the V�T� orF �T�. As we mentioned in the introduction, this approxi-mation corresponds to the first order correction from thelate-time decoupling limit with T still large but nonvanish-ing V�T� and F �T�. This configuration was consideredpreviously by Sen in Refs. [12–14]. In this case we have,

-5

0

0.5

1

1.5

a ||

0

0.5

1

1.5

|a _

0 0.5 1

Ψ2

0

0.5

1

0 0

||

FIG. 3. The figure corresponds with the other extreme casewith p 8 which shows that the probability density j�j2 isindependent on a?. This corresponds with a wave function �describing an evolution whose maxima are localized aroundfixed values of ak.


H0 2e�=4PT �

1

144e�9�1P21 �

9e�9�1

2�p� 1��8� p�P22

� e�9�1P2� � �p� 2�!e��9�1�a��P2C

� 2$e�=4e�8�p��1��1=9��p�1��2��T=2C

0: (28)

After quantization, we obtain from this Hamiltonian againa Schrodinger equation, now with a time-dependent poten-tial for the RR field. It is interesting to note that in this case,the term coming from the RR coupling still allows tointerpret the tachyon field as time, because its moment stillappears linearly, but in this case Eq. (28) leads to aSchrodinger equation with a time-dependent potential. Ofcourse in the absence of RR and dilaton fields, the tachyonfield is coupled only to gravity and we recover the situationdiscussed by Sen in Refs. [12–14].

One way to solve Eq. (28), could be by trying the time-dependent term as a perturbation. In this case we couldlook for a solution of the form,

�T;�1; �2; C� ��T;�1; �2; C�

� e��T=2��i=2�'T 1��1; �2; C�: (29)

However, when substituted into (28), it gives an equationfor 1 which does not seem to have exact solutions. Thenormalized probability density obtained from (29) j j2 ’j�j2 � 2Re�e��1=2��i'�T�� 1�, contains time-dependentinterference terms corresponding to interactions of thetachyon matter (open strings) with background fields(closed strings). This interference represents a manifesta-tion of the quantum backreaction of the tachyon field bythe background. The energy now has a perturbative‘‘stringy’’ correction E0 E�1�O�� . . .�: coming

063517

from the open string spectrum. So in spite of the complex-ity, it seems reasonable to make pertubative calculations (atleast at first order) around the minimum of the potential.

V. INCLUSION OF ELECTROMAGNETIC FIELDS

We want to see in this section how the tachyon dynamicsis modified in the presence of electromagnetic fields. Let usconsider the case in which electric and magnetic fields f��are included. The brane action Sbrane from Eq. (3) ismodified as follows [29–31],

Sbrane �

16G10

Zdp�2xk%?��V�T�e

��A

p�

��

16G10

Z%?F �T�dT ^ Cp�1 ^ e

f; (30)

where now the tachyon metric is A�� g��e�=2 �@�T@�T � f�� and f f��dx� ^ dx�. For simplicity,we will consider only one nonvanishing component forthe electric and magnetic fields, E f01 @0A1 and B f12 �@2A1. With this choice, the exponential ef in thelast term of action (30) contributes only with a factor one.This can be obtained by direct calculation or following [5],taking into account the ansatz ISO�p� 1� � SO�8�p; 1�. After integration of the space coordinates, we getthe Lagrangian

Lbrane $e�8�p��1��1=9��p�1��2�F �T� _TC

� $e9�1�a�=2V�T��N2e�=2 � _T2�

� �1� e�2��1��8�p�=9��2��=4�B2� � E2�1=2: (31)

Making the same procedure of introducing a Lagrangemultiplier � we found in the late-time limit (V�T� ! 0as jTj ! 1) that the relevant part of the Hamiltonian turnsout to be,

HbranejTj!1 2e�=4��1� e�2��1��8�p�=9��2��=4�B2�P2T

�"2�1=2; (32)

where " is the momentum conjugated to A1. From thisexpression, we see that the tachyon would decouple only if" vanishes. Thus, under the presence of electromagneticfields, the tachyon cannot be identified with time in thesense of a Schrodinger-type equation even in the late-timelimit.

VI. CONCLUSIONS

In this work, we have provided an exact solution to thecanonical quantization of the effective tachyon model [3–6,22]. For this effective action, a Wheeler-deWitt equation(for a particular factor ordering) in minisuperspace ap-proach has been obtained from the Hamiltonian analysis.Following Ref. [25], the square root in the tachyonic matteraction (10) was eliminated by the introduction of a

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Lagrange multiplier �. From the resulting action theHamiltonian (14) has been computed and the decouplinglate-time limit (T ! 1) has been done. Even though wehave considered the canonical quantization of the effectiveaction with a maximally symmetric metric (4), the quanti-zation of this field theory and, in particular, of the modelunder consideration is interesting in its own right [14].Although the minisuperspace approach reduces the num-ber of degrees of freedom of the full string theory, it stillcontains information of s-wave approximation of the genu-ine tachyonic quantum dynamics [17]. Moreover it maygive some insight to describe beyond the classical limit, byusing the methods of quantum cosmology, the dynamicaldegrees of freedom for the decay region where V�T� andF �T� are different from zero.

Further we show that the proposal by Sen, concerningthe interpretation of the tachyon as time, in the late-timedecoupling limit, is still valid for this model. In this limitwe find an exact wave function for the correspondingSchrodinger equation. The decoupled limit certainly be-haves different than the quantum cosmology associated tothe closed string sector without an initial decaying non-BPS configuration. Thus, the quantum cosmological de-scription with tachyons and without them is different, evenin this decoupling limit. The associated probability densityis a finite and continuous function of the radii ak and a?, itshows (a nonsingular) continuum of maxima along a defi-nite trajectory, in such a way that if the mean value of oneof the radii increases, the mean value of the other onedecreases, as shown in Fig. 1 for p 3 and in Fig. 2 and

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3 for the extreme cases of p 1 and p 8, respectively.We have also considered the situation beyond the late-timedecoupling limit in which still T is large but V�T� andF �T� are nonvanishing. The coupling of the tachyon withthe RR fields allows us still to interpret the tachyon as time.In this case the Wheeler-deWitt Eq. (28) leads to aSchrodinger equation with a time-dependent potentialfrom which we can calculate corrections in the energyspectrum, at least, at first order. This situation has beenalready discussed in Refs. [12–14] at the classical level. Ifquantum corrections of the string theory have to be takeninto account and if the open-closed duality holds (seeremarks of review, [17]), it would be very interesting toexplore if solutions of the Schrodinger Eq. (28), or itsgeneralizations (representing open-closed states), corre-spond to a description (at higher level) of the physics ofthe quantum string theory associated to SDp-branes. Someof this work is under current investigation.

Finally, we have also shown that in the presence ofelectromagnetic fields, the interpretation of the tachyonas time seems to be spoiled. Indeed, as can be seen fromEq. (32) even in the late-time limit the tachyon does notdecouple from the electric field.

ACKNOWLEDGMENTS

This work was supported in part by CONACyT MexicoGrants Nos. 37851E and 41993F, PROMEP and Gto.University Projects.

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Tachyon driven quantum cosmology in string theory