31 May 1999

Ž .Physics Letters A 256 1999 89–94

Stochastic motion of an open bosonic string

L.F. Santos a,1, C.O. Escobar b,2

a ´Departamento de Fisica Nuclear, Instituto de Fısica da UniÕersidade de Sao Paulo, C.P. 66318, CEP 05389-970 Sao Paulo, Sao Paulo,´ ˜ ˜ ˜Brazil

b Departamento de Raios Cosmicos e Cronologia, Instituto de Fısica Gleb Wataghin, UniÕersidade Estadual de Campinas, C.P. 6165, CEP´ ´13083-970 Campinas, Sao Paulo, Brazil˜

Received 6 October 1998; received in revised form 29 March 1999; accepted 5 April 1999Communicated by P.R. Holland

Abstract

We show that the classical stochastic motion of an open bosonic string leads to the same results as the standard firstquantization of this system. For this, the diffusion constant governing the process has to be proportional to a

X, the Reggeslope parameter, which is the only constant, along with the velocity of light, needed to describe the motion. q 1999Published by Elsevier Science B.V. All rights reserved.

PACS: 02.50.Ey; 05.40.q jKeywords: Stochastic quantum mechanics; Bosonic string

1. Introduction

w xString theory 1 provides an example of the notalways straight path towards new ideas in physics.From its inception in the framework of dual-reso-

w xnance models 2 to its current position as a candi-date for a unified theory of all interactions, includinggravity, it has gone through several modificationsand increasing sophistication in its mathematical for-mulation. As a consequence of these changes, theoriginal motivation for strings was abandoned andwhat we have now is a description of physics at the

w xPlanck scale 3 , bringing with it all uncertainties on

1 E-mail: lsantos@charme.if.usp.br2 E-mail: escobar@ifi.unicamp.br

the validity of the basic quantum mechanical ideas.For sure we lack a clear physical picture of phenom-ena at such small distances, despite recent efforts by

w xseveral authors 4 who introduce a radical view ofphysics at this scale, expanding the traditional quan-tum mechanical view. Some authors have gone as faras to speculate that at the Planck scale, quantum and

w xthermal fluctuations cannot be distinguished 5 .Motivated by the above considerations we de-

cided to look at a classical string subjected tostochastic motion, very much in the spirit of Nelson’sstochastic approach to the motion of a classical

w xnewtonian particle 6 . This formulation achieves thederivation of a Schrodinger equation for a classical¨non-relativistic particle moving in an external poten-tial by a stochastic version of Newton’s second law

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00231-5

( )L.F. Santos, C.O. EscobarrPhysics Letters A 256 1999 89–9490

of motion. There have been several criticisms tow xNelson’s alternative to quantum mechanics 7,8 and

w xfurther arguments in its favor 9 . While it is not ourpurpose to discuss them, the reader should be awarethat some were made by Nelson himself, such as thelack of separability of two correlated but dynami-

w xcally uncoupled systems 10 . Separability is a fea-ture of quantum mechanics and is essential for areasonable theory. Another severe limitation to thisapproach, which also prevents it from being a viablealternative to quantum theory, is the fact that dynam-ical correlations calculated in stochastic mechanicsfor an EPR situation differ from those obtained in

w xquantum mechanics 10 . However, we think that ifstochastic fluctuations, whose nature are never speci-fied by Nelson and his followers, have any chance ofmanifesting themselves, then the natural place forthem to occur would be at the very small scales like

w xthe Planck scale 5 . Since the objects that are candi-dates to describe physics at this scale are strings, wewill in the following describe the stochastic motionof an open bosonic string, as the simplest such anobject.

This paper is organized as follows. In the nextsection we briefly review Nelson’s stochastic me-chanics and very succinctly describe the stochasticvariational principle, developed by Guerra and his

w xcoworkers 11 . This principle is not free from criti-Ž .cism, as it starts from a Lagrangian, Eq. 19 , which

already depends on " through the diffusion constantn . This clearly indicates that this approach is not anab initio quantization of a classical system in con-trast to Feynman’s Lagrangian approach to quantum

w xtheory 12 , however it will be useful in Section 3,where our stochastic formulation of an open bosonicstring is presented. Among the results obtained, wecan mention the derivation of a wave-functionalequation for the string, the existence of a critical

Ž .dimension Ds26 , obtained through the require-Žment of Lorentz invariance as we work in the

.light-cone gauge and the two-point correlation func-tion for the non-zero normal modes of the string. Weshow that the two-point correlation function calcu-lated using this method agrees with the one obtainedfrom the standard first quantized string, but stressthat the agreement depends on our use of the groundstate wave-functional. Our conclusions are in Section4.

2. Stochastic mechanics

2.1. Nelson’s approach

The starting point of Nelson’s approach is toŽconsider the stochastic motion of a point particle for

.simplicity we treat a one dimensional motion givenby

dq x ,t sÕ q t ,t dtqdw t , 1Ž . Ž . Ž . Ž .Ž .q

where the first term on the right-hand side is deter-ministic and introduces a velocity field for forward

Ž .propagation dt)0 written as

= S x ,tŽ .qÕ x ,t s , 2Ž . Ž .q m

with m the particle mass and S a scalar function.qŽ .The stochastic process is described by dw t , which

satisfies the following averages

² :dw t s0 3Ž . Ž .² :dw t dw t s2n dt . 4Ž . Ž . Ž .

Ž .In 4 n is a diffusion constant to be specifiedlater.

Ž .Given the non-differentiable nature of 1 , Nelsonthen introduces the mean backward and forwardtransport derivatives

² :q tqDt yq tŽ . Ž .D q x ,t s limŽ . Ž .q q DtDt™0

sÕ x ,t 5Ž . Ž .q

² :q t yq tyDtŽ . Ž .D q x ,t s limŽ . Ž .y q DtDt™0

sÕ x ,t , 6Ž . Ž .y

which for a function F of x and t can be written,Ž . Ž .using 1 and 4 as

D F x ,t s E F x ,tŽ . Ž . Ž . Ž ." t

qÕ x ,t = F x ,tŽ . Ž . Ž ."

"n = 2F x ,t . 7Ž . Ž . Ž .Ž . Ž .1 and 4 also imply the Fokker-Planck equation

E p x ,t ; x ,t sy= Õ x ,t p x ,t ; x ,tŽ . Ž . Ž .Ž .t 0 0 q 0 0

qn= 2 p x ,t ; x ,t , 8Ž . Ž .0 0

( )L.F. Santos, C.O. EscobarrPhysics Letters A 256 1999 89–94 91

where p is the transition probability, which, bydefinition, propagates the probability density of anensemble of particles

r x ,t s p x ,t ; x ,t r x ,t dx . 9Ž . Ž . Ž . Ž .H 0 0 0 0 0

With Õ and Õ we can also defineq y

= S1

Õs Õ qÕ s 10Ž . Ž .q y2 m

=r1us Õ yÕ sn 11Ž . Ž .q y2

r

and then obtain the continuity equation

E r x ,t sy= r x ,t Õ x ,t . 12Ž . Ž . Ž . Ž .Ž .t

w xNelson’s formulation of the second law is 8

11 D D qD D q x ,t sy = V x .Ž . Ž . Ž . Ž .q y y q2 m

13Ž .From this equation of motion follows a Madelungtype equation

2= SŽ . 22 2E Sq y2mn = R q= R syV x ,Ž . Ž .t 2m

14Ž .where R is related to the probability density r asfollows:

exp 2 R x ,t sr x ,t . 15Ž . Ž . Ž .Ž .The Madelung equation and the continuity equa-

tion will correspond respectively to the real andimaginary parts of a Schrodinger equation when¨writing the wave function in polar form

S x ,tŽ .c x ,t sexp R x ,t exp ı 16Ž . Ž . Ž .Ž . ž /"

provided the diffusion constant n is identified with" 3w x8 .2m

2.2. Stochastic Õariational principle

In order to avoid the ambiguity of defining theacceleration in Newton’s second law, Guerra and

3 w xAs pointed out by Davidson 8 , different definitions ofacceleration require different diffusion constants.

w xcollaborators 11 formulated stochastic mechanicswith a variational principle. We refer the reader to

w xRef. 11 for more details, since here we only needtheir lagrangian density which will lead to aMadelung equation.

Starting from

1L x ,t s mÕ x ,t Õ x ,t yV x 17Ž . Ž . Ž . Ž . Ž .q y2

and defining an average stochastic action

t² :A t ,t ;r ;Õ s L x ,t r x ,t dx dt , 18Ž . Ž . Ž . Ž .HH0 0 qt0

where r is the initial distribution, another la-0

grangian density depending only on Õ can be intro-qduced

1 2L x ,t s mÕ x ,tŽ . Ž .q q2

qmn = Õ x ,t yV x . 19Ž . Ž . Ž . Ž .q

Ž .It is possible to show that the action in Eq. 18 isthe same as would be obtained replacing L by LqŽ Ž .the extra terms in 19 vanish when taking the

.stochastic average .Ž .Using a smooth field S and B t ,t;r ,S ;Õ1 0 0 1 q

defined as

² :B t ,t ;r ,S ;Õ sA t ,t ;r ;Õ y S q tŽ . Ž . Ž .Ž .0 0 1 q 0 0 q 1 1

sy J x ,t ;t ,S ;ÕŽ .H 0 0 1 1 q

=r x dx , 20Ž . Ž .0 0 0

we obtain

D J x ,t sL x ,t . 21Ž . Ž . Ž . Ž .q q

The variational principle based on dBs0 givesŽ . Ž .Ž . ŽÕ x,t s = S x,t rm where S is J making B

. Ž .stationary and allows the identification of 21 withthe Madelung equation.

The continuity equation is the same as before andwith the above Madelung equation, thus reproduce

Žthe Schrodinger equation imaginary and real parts¨.respectively . Furthermore, the variational approach

leads naturally to a canonical stochastic formulationwith r and S as canonical variables and, as shown

w xby Guerra and Marra 13 , to a redefinition of Pois-son brackets in this context. We will make use ofthis formalism at the end of Section 3.

( )L.F. Santos, C.O. EscobarrPhysics Letters A 256 1999 89–9492

A final remark is important before we proceed tothe stochastic description of the open bosonic string.In this section we summarized the stochastic ap-proach to the non-relativistic point particle. In deal-ing with a string we should use a relativistic general-ization of the Markov process underlying Nelson’sstochastic formalism. This has been achieved in the

w xliterature 14 and in the following we will use thisgeneralization.

3. Stochastic motion of a string

We start from the Nambu–Goto action for astring

1Ssy X2pa

=2 2 2E x E x E x E xt p1

dt ds y q , 22Ž .(H H ž / ž / ž /Et Es Et Est 00

which is proportional to the area of the two dimen-Žsional surface embedded in a D dimensional space-

.time swept by a string. The space-time pointsmŽ .x s ,t on this surface are labeled by s and t , two

dimensionless parameters. aX is the Regge slope

which in string theory is related to the square of aw xfundamental length 3 .

The above action is invariant under reparametriza-Ž .tions of the surface: s™sss s ,t , t™ts˜ ˜ ˜ ˜

Ž .t s ,t , which introduces a gauge freedom into the˜ ˜theory. This leads to difficulties in the canonicalformalism already at the classical level and one isforced to fix the gauge before proceeding. We choose

w xto work in the light-cone gauge 15 in our classicalapproach. This choice of gauge will make Lorentzinvariance not manifest and this will have to beaddressed by the stochastic approach.

In the light-cone system one introduces coordi-nates as follows:

x 0 qx Dy 1qx s 23Ž .'2

x 0 yx Dy 1yx s 24Ž .'2

and fixes the gauge as

xq s ,t spqt . 25Ž . Ž .

The transverse coordinates, the physical degreesof freedom, satisfy the following equation of motion,

x yxXX s0 is1, . . . , Dy2 , 26Ž . Ž .i i

E x X E xwhere xs and x s .˙ Et Es

We can now approach the system described byŽ .26 from a stochastic point of view. We do this by

Ž .expanding x s ,t in normal modesi

`

x s x cosns 27Ž .Ýi n ins0

Žwe are following the standard convention of defin-.ing s between 0 and p . From which follows that

x qn2 x s0 . 28Ž .¨ni ni

We now promote the normal modes to a stochas-w x i w i xtic process 16 q x ,t with ms0, . . . ,` and isn m

1, . . . , Dy2, satisfying

i i i idq x ,t sÕ dtqdw , 29Ž .n m qn n

where dwi obeysn

² i :dw t s0 30Ž . Ž .n

² i j :X Xdw t dw t s2 d d n dt . 31Ž . Ž . Ž .n n i j nn n

Ž .Eq. 29 illustrates the use of t as an evolutionparameter.

Notice that we have not specified the diffusionconstant n and have also allowed, for the sake ofn

generality, a possible dependence on the normalmode index.

Following the steps outlined in Section 2 wederive an equation of continuity

i i i i iE r x ,t sy = r x ,t Õ x ,t , 32Ž .Ž .Ýt m n m n mn

where

Xi i i iÕ x ,t s4a = S x ,t 33Ž .n m n m

and for the zero mode

Xi i i iÕ x ,t s2a = S x ,t . 34Ž .0 m 0 m

( )L.F. Santos, C.O. EscobarrPhysics Letters A 256 1999 89–94 93

We can introduce, as before, the transport deriva-tives, which now read

2i i iD sE q Õ = " = . 35Ž .Ž .Ý Ý" t " n n nn n

In the classical equation for the normal modeamplitudes

i 2 iq qn q s0 36Ž .n n

we use the definition of stochastic acceleration

1 i 2 iD D qD D q syn q . 37Ž . Ž .q y q y n n2

The Madelung equation that follows is

n n2 2 20 nX i i iE Ssya = S q = R q = RŽ . Ž . Ž .Xt 0 0 02a

n n2 2n nX i iy 2a = S q = RŽ . Ž .Ý ÝXm m2am/0 m/0

22 i1 m xŽ .2 miq = R y s0 . 38Ž .Ž . ÝXm 4a 2m

Ž .It satisfies, together with 32 , the wave func-Ž Ž .tional equation for a string using csexp R -

Ž ..exp ıS

2X iıE cs ya =Ž .t 0

22 in xŽ .2 nX iq y2a = q c , 39Ž .Ž .Ý Xnž /8an/0

provided the diffusion constants are

n s2aX n/0 40Ž .n

n saX . 41Ž .0

The difference between n and n comes fromn 0

the convention used for separating the zero mode.Notice the fact that we started this analysis with-

out knowing the diffusion constant governing theŽ .stochastic process 31 . For consistency reasons it

results that in the case of the stochastic motion of astring the diffusion constant is a

X, which accordswith the point of view stressed by Veneziano that astringy world has only two constants c and l, which

X 2 X w xis related to a as l s2a 3 . If we naivelyexpected n to be related to the energy of the system,

like in point particle mechanics, n would be differ-ent for each state of the string. This encouragesspeculations on the indistinguishability of quantumand classical fluctuations at the Planck scale.

Ž .It follows from 38 that we obtain the standardspectrum of an open bosonic string, given by aninfinite set of harmonic oscillators, as expected.

3.1. Lorentz inÕariance

We must now address the question of Lorentzinvariance. In order to do so we make use of thecorrespondence between stochastic Poisson brackets

w xand quantum commutators 13 . To achieve this cor-respondence it is important that r and S, as definedabove, be canonical variables, in which case for apair of dynamical variables A and B, functionals ofr and S,

d A dB d A dB� 4A , B s y dx . 42Ž .Hs ž /dr dS dS dr

w xGuerra e Marra 13 established the correspon-dence

² :� 4A , B s ı A , B , 43Ž .s q q

where A and B are quantum operators and A andq q

B are classical variables. The average value on theŽ .right hand side of 43 is defined as

)² :A , B s c A , B c dx . 44Ž .Hq q q q

This result can be easily extended to our caseŽ i .dxsŁ dq and allows us to examine the issuei,m m

of Lorentz invariance by computing the Poissonbrackets for the generators of the Lorentz group nowformulated in stochastic language. With

M mn s ds x mP n yxnP m , 45Ž . Ž .Hthe critical element of the algebra is

iy jy iy jy² :� 4M , M s ı M , M . 46Ž .q q

In order to close the algebra at a classical, butstochastical level, it is required that Ds26, inagreement with the well known results in string

w xtheory 1 .

( )L.F. Santos, C.O. EscobarrPhysics Letters A 256 1999 89–9494

3.2. Correlation function

In order to evaluate the correctness of our results,we calculate next the two-point correlation functionusing classical stochastic methods. We restrict thecalculation to the non-zero normal modes in order to

w xavoid the well-known infrared divergence 1 . Forthe ground state we have

² i i X : i i X i X X ix t x t s x p x ,t , x ,t xŽ . Ž . Ž .H=r xX i dx i dxX i , 47Ž .Ž .0

which gives

1 exp ntXŽ .

X Xi i² :x t x t s Dy2 2a .Ž . Ž . Ž . Ýn exp ntŽ .n/0

48Ž .

This is the same result obtained from the standardw xfirst quantized string, as can be seen in Ref. 1 , if

we continue t to the Euclidean domain.

4. Conclusion

We have shown in this paper that the classicalstochastic motion of an open bosonic string leads toa wave functional equation that correctly describesthe first quantized string and that for the consistencyof our results requires the string to live in a 26dimensional world. The agreement of our two-pointcorrelation function calculated by classical stochasticmethods, with the one obtained by the standardtreatment, reinforces the soundness of our formula-tion.

We stress once again that it was not our purposehere to promote Nelson’s approach, as it is madeclear in the Introduction, where some criticisms to it

are pointed out. The main goal of this paper was toinvestigate the idea that stochastic fluctuations couldmanifest themselves at the very small scales like thePlanck scale, and this we accomplished as our resultsexhibit the naturalness of the string constant in play-ing the role of the diffusion constant at such smallscales.

Acknowledgements

The authors acknowledge the support of theBrazilian Research Council, CNPq.

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