Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

intensity " emerges in the small noise limit. To

explain this, let us focus on the basic indicator for

periodic transitions – the time the Brownian particle

needs to exit from the starting well, say the left one.

In the ‘‘frozen’’ case, that is, if the time variation of

the potential term is eliminated just by freezing it at

some time s, the asymptotics of the exit time is

derived from the classical large deviation theory of

randomly perturbed dynamical systems (see Freidlin

and Wentzell 1998). Let us assum e that U is locally

Lipschitz. We denote by D

(resp. D

) the domai n

corresponding to the left (resp. right) well and 

their common boundary. The law of the first exit

time 

= inf {t 0, X

=2D

} is described by some

particular functional related to large deviation. For

t > 0, we introduce the ‘‘action functional’’ on the

space of continuous functions C([0, t]) on [0, t]by

ð’Þ¼

’

ðs;’



du; if ’ is abs:

continuous

þ1 otherwise

(

which is non-negative and vanishes on the set

of solutions of the ordinary differential equation

x = (@U=@x)(s, x). Let x and y 2R. In relation with

the action functional, we define the quasipotential

ðx; yÞ¼ inf f S

ð’Þ: ’ 2Cð½0; tÞ;’

¼x;’

¼y; t 0g

It represents the minimal work the diffusion starting

in x has to do in order to reach y. To switch wells,

the Brownian particle starting in the left well ’s

bottom x

has to overcome the barrier. So we let

¼ inf

y 2

ðx

; yÞ

This minimal work needed to exit from the left well

can be computed explicitly, and is seen to equal to

twice its depth. The asymptotic behavior of the exit

time is expressed by

lim

" !0

" ln E½

¼V

and

lim

" !0

ðV

Þ="

<

< e

ðV

þ="Þ



¼ 1

for any >0

The prefactor for the exponential rate, derived by

Freidlin and Wentzell (1998), was first given by

Eyring and Kramers and then by Bovier et al. (2004).

Let us now assume that the left well is the deeper

one at time s. If the Brownian particle has enough

time to cross the barrier, that is, if T > e

=

, then

whatever the starting point is, Freidlin (2000) proved

that it should stay near x

in the following sense:

ðt 2½0; 1 : jX

 x

j >Þ!0

in probability as " !0. Here  denotes Lebesgue

measure on R.IfT < e

=

, the time left is not long

enough for crossings: the particle stays in the

starting well, near the stable equilibrium point:

ðt 2½0; 1: j X

ðx

fx 2D

þx

fx 2D

Þj >Þ!0

This observation is at the basis of Freidlin’s law of

quasideterministic periodic motion discussed in the

subsequent section. The lesson it teaches is this: to

observe switching of the position to the energetically

most favorable well, T should be larger than some

critical level e

=

. Measuring time in exponential

scales by  through the equation T

= e

="

, the

condition becomes >.

Stochastic Resonance for Landscapes,

Frozen on Half-Periods

This particular case has analytical advantages, since

it allows one to employ classical techniques of

semigroup and operator theory. The situation is the

following: let U be a double-well potential with

minima x

= 1andx

= 1 and a saddle point at

the origin. We assume that U(x) !1 as jxj!1

and U(1) = V=2 = 

=2, U(1) = v=2 = V

=2,

U(0) = 0, and 0 < v < V. We define the 1-periodic

potential by U(t, x) = U(t þ1=2, x). Hence on each

half-period the corresponding diffusion is time homo-

geneous. The critical level  is then easily defined by

 = v, that is, twice the depth of the shallow well. By

letting

ðtÞ¼

1 for t 2½k; k þ

1 for t 2½k þ

; k þ 1Þ; k ¼ 0; 1; 2; ...

(

the periodic function which describes the location of

the global minimum of the potential, we get in the

small noise limit

ðt 2½0; 1 : jX

 ðtÞj >Þ!0

in probability as " !0. This result expresses

Freidlin’s law of quasideterministic motion: for

large periods, the trajectories of the particle

approach a periodic deterministic function. But the

sense in which this notion measures periodicity does

not take into account that for large periods short

excursions to the wrong well may occur in an erratic

way without counting much for Lebesgue measure

of time. In fact, if the period is too large, that is,

>V, the time availab le in one period permits the

exit of not only the shallow well but also that of the

deep well. So, whatever the starting position of

Stochastic Resonance 89

the particle is, the number of observed transitions in

one half period becomes very large. Indeed the first

time  the particle starting in x

hits again x

after

visiting the position x

satisfies

EðÞ¼e

v="

þ e

V="

< T

¼ e

="

The motion of the particle appears more chaotical

than periodic: noise intensity is too large compared

to period length. We avoid this range of chaotic

spontaneous transitions by defining the resonance

interval I

= [v, V], as the range of admissible energy

parameters  for randomly periodic behavior. In this

regime, the trajecto ries possess periodicity proper-

ties. In these terms the resonance point describes the

tuning rate 

for which the stochastic response

to weak external periodic forcing is optimal. To

make sense, this point has to refer to some measure

of quality for periodicity of random trajectories. In

the huge physics literature concerning resonance,

two families of criteria can be distinguished. The

first one is based on invariant measures and spectral

properties of the infini tesimal generator associated

with the diffusion X

. Now, X

is not Markovian

and consequently does not admit invar iant mea-

sures. But by taking into account deterministic

motion of time in the interval of periodicity and

considering the process Z

= (t mod(T

), X

), we

obtain a Markov process with an invariant measure



(x)dx. In other words, the law of X

 

(x)dx and

the law of X

tþT

 

tþT

(x)dx, under this measure,

are the same for all t  0. Let us present the most

important ones:



the spectral power amplification (SPA) which

plays an eminent role in the physics literature

describes the energy carried by the spectral

component of the averaged trajectories of X

corresponding to the period:

SPA

ð"; TÞ¼



½X

e

2is





the SPA-to-noise ratio, giving the ratio of the

amplitude of the response and the noise intensity,

which is also related to the signal-to-noise ratio:

SPN

ð"; TÞ¼M

SPA

ð"; TÞ="



the total energy of the averaged trajectories

ð"; TÞ¼

ðE



½X

Þ

The second family of criteria is more probabilistic.

It refers to quality measures based on transition

times between the domains of attraction of the local

minima, residence times distributions measuring the

time spent in one well between two transitions, or

interspike times. This family is certainly less popular

in the physics community.

However, measures related to invariant measures

may suffer from robustness deficiency (Imkeller and

Pavlyukevich 2002). To explain what we mean by

robustness, let us introduce a model reduction first

discussed by McNamara and Wie senfeld (1989).

Instead of studying the diffu sion X

in the double-

well landscap e, they introduce a two-state Markov

chain Y

(Figure 5) the dynamics of which just

takes account of the domain of attraction the diffusion

is in, and therefore with state space {1, 1}. A

reasonable choice of the infinitesimal generator should

retain the dynamics of the diffusion’s transitions

characterized by Kramers’ rate. We may take

QðtÞ¼

’’





; 0  t 

QðtÞ¼



’ ’



;

 t < T

periodically continued on R

. Here, ’ = pe

V="

and

= qe

v="

. The prefactors of subexponential order

are beyond the scope of large deviation theory. They

are related to the curvature of the potential in the

0 T 2T 3T 4T

–1

0 T 2T 3T 4T

–1

0 T 2T 3T 4T

–1

Figure 5 Resonance pictures for Markov chain.

90 Stochastic Resonance

minima and the saddle point of the landscape and

given by

p ¼

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð1ÞjU

ð0Þj

q ¼

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð1ÞjU

ð0Þj

On the intervals [kT=2, (k þ 1)T=2[, k  0, the

Markov chain Y

is time-homogeneous and its

transition probabilities can be expres sed in terms

of ’ and . For instance, the probability with which

the chain jumps from state 1 to state þ1 in the time

window [t, t þ h] equals ’h þ o(h), if this time

interval is contained in [kT=2, (k þ 1)T=2[ for

some even k. The stationar y measure of the Markov

chain denoted by  can be explicitly calculated, and

so can the classical quality measures based on the

spectral notions. For instance, the spectral power

amplification coefficient equals

SPA

ð"; TÞ¼



½Y

e

2is





ð’  Þ

ð’ þ Þ

þ 

This simple expression admits asymptotically a

unique maximum which exhibits the resonance

point:

opt



ﬃﬃﬃﬃﬃﬃﬃﬃ

2pq

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

V  v

ðVþvÞ=2"

1 þOðe

ðVvÞ="

The optimal period is then exponentially large – as

was suggested by large deviation theory – and the

growth rate is the sum of the two wells’ depths. The

simple Markov chain model is popular since the

usual physical quantities are easy computable and

since it is believed to mimic the dynamics of a

Brownian particle in the corresponding double-well

landscape. However, the models are not as similar

as expected (Freidlin 2003). Indeed, in a reasonably

large time window around the resonance point for

, the tuning picture of the spectral power

amplification for the diffusion is different. Under

weak regularity conditions on the potential, it

exhibits strict monotonicity in the window. Hence,

optimal tuning points for diffusion and Markov

chain differ essentially. In other words, the SPA

tuning behavior of the diffusion is not robust for

passage to the reduced model. This strange defi-

ciency is difficult to explain. The main reason of this

subtle effect appears to be that the diffusive nature

of the Brownian particle is neglected in the reduced

model. In order to point out this feature, we may

compute the SPA coefficient of g(X

), where g is a

particular function designed to cut out the small

fluctuations of the diffusion in the neighborhood of

the bottoms of the wells, by identif ying all states

there. So g(x) = 1 (resp. 1) in some neighborhood

of 1 (resp. 1) and otherwise g is the identity. This

results in

SPA

ð"; TÞ¼



gðX

Þe

2is



In the small noise limit this quality function admits a

local maximum close to the resonance point of the

reduced model: the growth rate of T

opt

is also given

by the sum of the wells’ depths. So the lack of

robustness seems to be due to the small fluctuations

of the particle in the wells’ bottoms. In any case, this

clearly calls for other quality measures to be used to

transfer properties of the reduced model to the

original one. Our discussion indicates that due to

their emphasis on the pure transition dynamics, the

second family of quality measures should be used.

For these notions there is no need to restrict to

landscapes frozen in time-independent potential

states on half-period intervals.

Stochastic Resonance for Continuously

Varying Landscapes

From now on the potential U(t, x) is supposed to be

continuously varying in (t, x). For simplicity, its

local minima are assumed to be located at 1, and

its only saddle point at 0, indepen dently of time. So

the only meta-stable states on the whole time axis

are 1. Let us denote by 



(t) (resp. 

(t)) the

depth of the left (resp. right) well at time t. Together

with U, these functions are continuous and

1-periodic. Assume that they are strictly monoto-

nous between their global extrema. Let us now come

back to the motion of a Brownian particle in this

landscape. The exit time law by Eyring–Kramers–

Freidlin entails that trajectories get close to the

global minimum, if the period is large enough.

Stated as before in exponential rates T = e

="

, with

  max

i = 

sup

t0

2



(t), that is,  exceeds the

maximal work needed to cross the barrier, the

particle often switches between the two wells and

should stay close to the deepest position in the

landscape. This position being described by the

function (t) = 21

{

(t)>



(t)}

1, we get in the small

noise limit

ðt 2½0; 1 : jX

 ðtÞj >Þ!0

in probability. But on these long timescales, many

short excursions to the wrong well are observed, and

trajectories look chaotical instead of periodic. So we

Stochastic Resonance 91

have to look at smaller periods even at the cost that

the particle may not stay close to the global

minimum. Let us study the transition dynamics.

Assume that the starting point is 1 corresponding

to the bottom of the deep well. If the depth of the well

is always larger than  = " log T

, the particle has too

little time during one period to climb the barrier, and

should stay in the starting well. If, on the contrary,

the minimal work to leave the starting well, given by

2



(s), becomes smaller than  at some time s, then

the transition can and will happen. More formally,

for  2[ inf

t0

2



(t), sup

t0

2



(t)], we define

(Figure 6).





ðsÞ¼infft  s : 2



ðtÞg

The first transition time from 1 to 1 denoted 

has the following asymptotic behavior as

" !0: 





(0). At the second transition the

particle returns to the starting well. If a



is defi ned

analogously with respect to the depth function 

this transition will occur near the deterministic time







(s))T

. In order to observe periodicity, and to

exclude chaoticity from all parts of its trajectories,

the particle has to stay for some time in the other

well before returning. This will happen under the

assumption 2



(0)) >, that is, the right well is

the deep one at transition time. In fact, we can

define the resonance interval I

(Figure 7), as the set

of all scales  for which trajectories exhibit

periodicity in the small noise limit, by

max

i¼

inf

t0

2

ðtÞ; inf

t0

max

i¼

2

ðtÞ

On this interval they get clos e to deterministic

periodic ones. Again, periodicity is quantified by a

quality measure, to be maximized in order to obtain

resonance as the best possible response to periodic

forcing. One interesting measure is based on the

probability that random transitions happen in some

small time window around a deterministic time, in

the small noise limit (Herrmann and Inkeller 2005).

Formally, for h > 0, the meas ure gives

ð"; TÞ¼min

i¼

ð



2½a



 h; a



þ hÞ

where P

is the law of the diffusion starting in i.In

the small noise limit, this quality measure tends to 1,

and optimal tuning can be related to the exponential

rate at which this happens. This is due to the

following large deviations principle:

lim

" !0

" logð1  M

ð"; TÞÞ ¼ max

i¼

f  2

ða



 hÞg

for  2I

, with uniform convergence on each

compact subset of I

. The result is established

using classical large-deviation techniques app lied to

locally time homogeneous approximations of the

diffusion. Maximizing the transition probability in

the time window position means minimizing the

default rate obtained by the large deviations

principle. This can be easily achieved. In fact, if the

window length 2h is small, then   2



 h) 

2h



), since 2



) =  by definition. The value





) is negative, so we have to find the position

where its absolute value is maximal. In this position

the depth of the starting well has the most rapid

drop under the level , characterizing the link

between the noise intensity and the period. So the

transition time is best concent rated around it.

It is clear that a good candidate for the resonance

point is given by the eventually existing limit of the

global minimizer 

(h) as the window length h tends

to 0. This limit is therefore called the resonance

point of the diffusion with time-periodic landscape

U. Let us note that for sinusoidal depth functions



ðtÞ¼

V þ v

V  v

cosð2tÞ

and



ðtÞ¼



ðt þ Þ

the optimal tuning is given by T

= exp 

=" with



= (v þ V)=2. This optimal rate is equivalent to

the optimal rate given by the SPA coeff icient of the

reduced dynamics’ Markov chain in the preceding

section.

The big advantage of the quality measure M

is its

robustness. Indeed, consider the reduced model

2Δ

–

(t )

(0)

–

Figure 6 Definition of a





2Δ

–

(t )

2Δ

(t )

Figure 7 Resonance interval.

92 Stochastic Resonance

consisting of a two-state Markov chain with

infinitesimal generator

QðtÞ¼

’ðtÞ ’ðtÞ

ðt Þ ðtÞ



where ’(t) = exp 2



(t=T)=" and (t) =

exp 2

(t=T)=". The law of transition times of

this Markov chain is readily computed from Laplace

transforms. Normalized by T

it converges to a



This calculation even reveals a rigorous underlying

pattern for the second- and higher-order transition

times interpreting the interspike dist ributions of

the physics literature. The dynamics of diffusion

and Markov chain are similar. Resonance points

provided by M

for the diffusion and its analog for

the Markov chain agree.

Related Notions: Synchronization

In the preceding sections, we interpreted stochastic

resonance as optimal response of a randomly

perturbed dynamical system to weak periodic forcing,

in the spirit of the physics literature (see Gammaitoni

et al. (1998)). Our crucial assumption concerned the

barrier heights a Brownian particle has to overcome

in the potential landscape of the dynamical system: it

is uniformly lower bounded in time. Measures for the

quality of tuning were based on essentially two

concepts: one concerning spectral criteria, with the

spectral power amplification as most prominent

member, the other one concerning the pure transi-

tions dynamics between the domains of attraction of

the local minima. A number of different criteria can

be used to create an optimal tuning between the

intensity of the noise perturbation and the large

period of the dynamical system. The relations have to

be of an exponential type T = exp =", since the

Brownian particle needs exponentially long times to

cross the barrier separating the wells according to the

Eyring–Kramers–Freidlin transition law. Our barrier

height assumption seems natural in many situations,

but can fail in others. If it becomes small periodically,

and eventually scales with the noise-intensity para-

meter, the Brownian particle does not need to wait an

exponentially long time to climb it. So periodicity

obtains for essentially smaller timescales. In this

setting, the slowness of periodic forcing may also be

assumed to be essentially subexponential in the noise

intensity.

If it is fast enough to allow for substantial changes

before large deviation effects can take over, we are

in the situation of Berglund and Gentz (2002). They

in fact consider the case in which the barrier

between the wells becomes low twice per period,

to the effect of modulating periodically a bifurcation

parameter: at time zero the right-hand well becomes

almost flat, and at the same time the bottom of the

well and the saddle approach each other; half a

period later, a spatially symmetric scenario is

encountered. In this situation, there is a threshold

value for the noise intensity under which transitions

become unlikely. Above this threshold, the trajec-

tories typically contain two transitions per period.

Results are formulated in terms of concentration

properties for random trajectories. The intuitive

picture is this: with overwhelming probability,

sample paths will be concentrated in spacetime sets

scaling with the small parameters of the problem. In

higher dimensions, these sets may be given by

adiabatic or center manifolds of the deterministic

system, which allow model reduction of higher-

dimensional systems to lower-dimensional ones.

Asymptotic results hold for any choice of the small

parameters in a whole parameter region. A passage

to the small noise limit as for optimal tuning in the

preceding sections is not needed.

Related problems studied by Berglund and Gentz

in the multidimensional case concern the noise-

induced passage through periodic orbits, where

unexpected phenomena arise. Here, as opposed to

the classical Freidlin–Wentzell theory, the distribu-

tion of first-exit points depends nontrivially on the

noise intensity. Again aiming at results valid for

small but nonvanishing parameters in subexponen-

tial scale ranges, they investigate the density of first-

passage times in a large regime of parameter values,

and obtain insight into the transition from the

stochastic resonance regime into the synchronization

regime.

See also: Dynamical Systems in Mathematical Physics:

An Illustration from Water Waves; Magnetic Resonance

Imaging; Spectral Theory for Linear Operators;

Stochastic Differential Equations.

Further Reading

Benzi R, Sutera A, and Vulpiani A (1981) The mechanism of

stochastic resonance. Journal of Physics A 14: L453–L457.

Berglund N and Gentz B (2002) A sample-paths approach to

noise-induced synchronization: stochastic resonance in a

double-well potential. Annals of Applied Probability 12(4):

1419–1470.

Bovier A, Eckhoff M, Gayrard V, and Klein M (2004) Meta-

stability in reversible diffusion processes. I. Sharp asymptotics

for capacities and exit times. Journal of the European

Mathematical Society 6(4): 399–424.

Freidlin MI (2000) Quasi-deterministic approximation,

metastability and stochastic resonance. Physica D 137(3–4):

333–352.

Stochastic Resonance 93

Freidlin M (2003) Noise sensitivity of stochastic resonance and

other problems related to large deviations. In: Sri

Namachchivaya and Lin YK (eds.) IUTAM Symposium on

Nonlinear Stochastic Dynamics, Solid Mech. Appl, vol. 110,

pp. 43–55. Dordrecht: Kluwer Academic.

Freidlin MI and Wentzell AD (1998) Random Perturbations of

Dynamical Systems, 2nd edn. New York: Springer.

Gammaitoni L, Ha¨nggi P, Jung P, and Marchesoni F (1998)

Stochastic resonance. Reviews of Modern Physics 70(1):

223–287.

Gardiner CW (2004) Handbook of Stochastic Methods for

Physics, Chemistry and the Natural Sciences. Springer Series

in Synergetics, 3rd edn., vol. 13. Berlin: Springer.

Herrmann S and Imkeller P (2005) The exit problem for

diffusions with time-periodic drift and stochastic resonance.

Annals of Applied Probability 15(1A): 39–68.

Imkeller P and Pavlyukevich I (2002) Model reduction and

stochastic resonance. Stochastics and Dynamics 2(4):

463–506.

Mazo RM (2002) Brownian Motion: Fluctuations, Dynamics,

and Applications. International Series of Monographs on

Physics, vol. 112. New York: Oxford University Press.

McNamara B and Wiesenfeld K (1989) Theory of stochastic

resonance. Physical Review A 39(9): 4854–4869.

Nelson E (1967) Dynamical Theories of Brownian Motion.

Princeton, NJ: Princeton University Press.

Schweitzer F (2003) Brownian Agents and Active Particles,

Springer Series in Synergetics, (Collective dynamics in the

natural and social sciences, With a foreword by J. Doyne

Farmer). Berlin: Springer.

Strange Attractors see Lyapunov Exponents and Strange Attractors

String Field Theory

L Rastelli, Princeton University, Princeton, NJ, USA

Introduction

String field theory (SFT) is the second-quantized

approach to string theory. In the usual, first-

quantized, formulation of string perturbation the-

ory, one postulates a recipe for the string S-matrix in

terms of a sum over two-di mensional (2D) world

sheets embedded in spacetime. Very schematically,

hhV

ðk

Þ...V

ðk

Þii

topologies



½d



hV

ðk

Þ...V

ðk

Þi

f



½1

Here the left-hand side stands for the S-matrix of the

physical string states {V

)}. The symbol h...i

{



}

denotes a correlation function on the 2D world sheet,

which is a punctured Riemann surface of Euler number

 and given moduli {



}. In SFT, one aims to recover

this standard prescription from the Feynman rules of a

second-quantized spacetime action S[]. The string

field , the fundamental dynamical variable, can be

thought of as an infinite-dimensional array of space-

time fields {



)}, one field for each basis state in the

Fock space of the first-quantized string.

The most straightforward way to construct S[]

uses the unitary light-cone gauge. Light-cone SFT is

an almost immediate transcription of Mandelstam’s

light-cone diagrams in a second-quantized language.

While often useful as a bookkeeping device, light-

cone SFT seems unlikely to represent a real

improvement over the first-quantized approach. By

contrast, from our experience in ordinary quantum

field theory, we should expect Poincare´ -covariant

SFTs to give important insights into the issues of

vacuum selection, background independence and the

nonperturbative definition of string theory.

Covariant SFT actions are well established for the

open (Witten 1986), closed (Zwiebach 1993)and

open/closed (Zwiebach 1998) bosonic string. These

theories are based on the BRST formalism, where the

world sheet variables include the bc ghosts intro-

duced in gauge-fixing the world sheet metric to the

conformal gauge g



. (An alternative approach

(Hata et al.), based on covariantizing light-cone SFT,

will not be described in this article.) Much less is

presently known for the superstring: classical actions

have been established for the Neveu-Schwarz sector

of the open superstring (Berkovits 2001)andforthe

heterotic string (Berkovits et al. 2004).

During the first period of intense activity in SFT

(1985–1992), the covariant bosonic actions were

constructed and shown to pass the basic test of

reproducing the S-matrix [1] to each order in the

perturbative expansion. The more recent revival of

the subject (since 1999) was triggered by the

realization that SFT contains nonperturbative infor-

mation as well: D-branes emerge as solitonic

solutions of the classical equations of motion in

94 String Field Theory

open SFT (OSFT). We can hope that the nonpertur-

bative string dualities will also be understood in the

framework of SFT, once covarian t SFTs for the

superstring are better developed.

In this article, we review the basic formalism of

covariant SFT, using for illustration purposes the

simplest model – cubic bosonic OSFT. We then

briefly sketch the generalization to bosonic SFTs

that include closed strings. Finally, we turn to the

subjects of classical solutions in OSFT and the

physics of the open-string tachyon.

Open Bosonic SFT

The standard formulation of string theory starts with

the choice of an on-shell spacetime background where

strings propagate. In the bosonic string, the closed

string background is described by a conformal field

theory of central charge 26 (the ‘‘matter’’ CFT). The

total world sheet CFT is the direct sum of this matter

CFT and of the universal ghost CFT, of central charge

26. To describe open strings, we must further specify

boundary conditions for the string endpoints. The

open-string background is encoded in a boundary CFT

(BCFT), a CFT defined in the upper-half plane, with

conformal boundary conditions on the real axis

(see Boundary Conformal Field Theory in this encyclo-

pedia). In modern language, the choice of BCFT

corresponds to specifying a D-brane state.

In classical OSFT, we fix the closed-string back-

ground (the bulk CFT) and consider varying the

D-brane configuration (the boundary conditions).

To lowest order in g

, we can neglect the back-

reaction of the D-brane on the closed-string fields,

since this is a quantum effect from the open-string

viewpoint. Let us prepare the ground by recalling

the standard -model philosophy. To describe off-

shell open-string configurations, we should allow for

general (not necessarily conformal) boundary condi-

tions. We can imagine to proceed as follows:

1. We choose an initial open-string background, a

reference BCFT that we shall call BCFT

. For

example, a Dp brane in flat 26 dimensions

(Neumann boundary conditions on p þ 1 coordi-

nates, Dirichlet on 25  p coordinates).

2. We then write a basis of boundary perturbations

around this background. Taking, for example,

BCFT

to be a D25 brane in flat space, the world

sheet action S

takes the schematic form

2

UHP



Tðx





ðx



Þ@X



ðx



Þ@



þ ½2

Here to the standard free bulk action (integrated

over the upper-ha lf complex plane UHP) we have

added perturbation localized on the real axis R.

Notice that the basis of perturbations depends on

the chosen BCFT

3. We interpret the coefficients {





)} of the

perturbations as spacetime fields. (The tilde on



(x) serves as a reminder that these fields are not

quite the same as the fields 

(x) that will appear

in the OSFT action). We are after a spacetime

action S[{



}] such that solut ions of its classical

equations of motion correspond to conformal

boundary conditions:

S





¼ 0ðspacetimeÞ

$ 

½f



g ¼ 0 ðworld sheetÞ½3

We recognize in [2] the familiar open-string

tachyon

T(x) and gauge field



(x), which are the

lowest modes in an infinite tower of fields. Relevant

perturbations on the world sheet (with conformal

dimension h < 1) correspond to tachyonic fields in

spacetime (m

< 0), whereas marginal world sheet

perturbations (h = 1) give massless spacetime fields.

To achieve a complete description, we must include

all the higher massive open-string modes as well,

which correspond to nonrenormalizable boundary

perturbations (h > 1). In the traditional -model

approach, this appears like a daunting task. The

formalism of OSFT will automatically circumvent

this difficulty.

The Open-String Field

In covariant SFT the reparametrization ghosts play

a crucial role. The ghost CFT consists of the

Grassmann odd fields b(z), c(z),



z),



z), of dimen-

sions (2, 0), (1, 0), (0, 2), (0, 1), respectively. The

boundary conditions on the real axis are

b =



b, c =



c. The state space H

BCFT

of the full

matter þghost BCFT can be broken up into

subspaces of definite ghost number,

BCFT

G¼1

ðGÞ

BCFT

½4

We use conventions where the SL(2, R) vacuum j0i

carries zero ghost number, G(j0i) = 0, while

G(c) = þ1 and G(b) = 1. As is familiar from the

first-quantized treatment, physical open-string states

are identified with G = þ1 cohomology classes of

the BRST operator,

QjV

phys

i¼0; jV

phys

ijV

phys

iþQji

GðjV

phys

iÞ ¼ þ1

½5

String Field Theory 95

where the nilpotent BRST operator Q has the

standard expression

Q ¼

2i

matter

þ : bc@c :ðÞ½6

Though not a priori obvious, it turns out that the

simplest form of the OSFT action is achieved by

taking as the fundamental off-shell variable an

arbitrary G = þ1 element of the first-quantized

Fock space,

ji2H

ð1Þ

BCFT

½7

By the usual state–operator correspondence of CFT,

we can also represent ji as a local (boundary)

vertex operator acting on the vacuum,

ji¼ð0Þj0i½8

The open-string field ji is really an infinite-

dimensional array of spacetime fields. We can

make this transparent by expanding it as

ji¼

pþ1

kj

ðkÞi

ðk



Þ½9

where {j

(k)i} is some convenient basis of H

(1)

BCFT

that diagonalizes the momentum k



. The fields



are a priori complex. This is remedied by

imposing a suitable reality condition on the string

field, which will be stated momentarily. Notice that

there are many more elements in {j

(k)i} than in the

physical subspace (the cohomology clas ses of Q).

Some of the extra fields will turn out to be

nondynamical and could be integrated out, but at

the price of making the OSFT action look much

more complicated.

It is often useful to think of the string field in

terms of its Schro¨ dinger representation, that is, as a

functional on the configuration space of open

strings. Consider the unit half-disk in the upper-

half plane, D

 {jzj1, =z  0}, with the vertex

operator (0) inserted at the origin. Impose BCFT

open string boundary conditions for the fields X(z,



on the real axis (here X(z,



z) is a short-hand notation

for all matter and ghost fields), and boundary

conditions X() = X

() on the curved boundary of

, z = exp (i), 0    . The path integral over

X(z,



z) in the interior of the half-disk assigns a

complex number to any given X

(), so we obtain a

functional [X

()]. This is the Schro¨ dinger wave

function of the state (0)j0i. Thus, we can think of

open-string functionals [X

()] as the fundamental

variables of OSFT. This is as it should be: the

first-quantized wave functions are promoted to

dynamical fields in the second-quantized theory.

Finally, let us quote the reality condition for the

string field, which takes a compact form in the

Schro¨ dinger representation:

½X



ðÞ; bðÞ; cðÞ

¼ 



½X



ð  Þ; bð  Þ; cð  Þ ½10

where the superscript  denotes complex conjugation.

The Classical Action

With all the ingredients in place, it is immediate to

write the quadratic part of the OSFT action. The

linearized equations of motion must reproduce the

physical-state condition [5]. This suggests

S hjQji½11

Here hji is the usual BPZ inner product of BCFT

which is defined in terms of a two-point correlator on

the disk, as we review below. The ghost anomaly

implies that on the disk we must have G

tot

= þ3,

which happily is the case in [11]. Moreover, since the

inner product is nondegenerate, variation of [11] gives

Qji¼0 ½12

as desired. The equivalence relation jV

phys

i

phys

iþQji is interpreted in the second-quantized

language as the spacetime gauge invariance





ji¼Qji; ji2H

ð0Þ

BCFT

½13

valid for the general off-shell field. This equation is

a very compact generalization of the linearized

gauge invariance for the mass less gauge field.

Indeed, focusing on the level-zero components,

jiA



(x)(c@X



)(0)j0i and ji(x)j0i, we find

A



(x) = @



(x). It is then plausible to guess that the

nonlinear gauge invariance should take the form





ji¼Qjiþjij ijiji½14

where  is some suitable product operation that

conserves ghost number

 : H

ðnÞ

BCFT

H

ðmÞ

BCFT

ðnþmÞ

BCFT

½15

Based on a formal analogy with 3D nonabelian

Chern–Simons theory, Witten proposed the cubic

action

S ¼

hjQjiþ

hj  i



½16

The string field ji is analogous to the Chern–

Simons gauge potential A = A

, the  product to

the ^ product of differential forms, Q to the exterior

derivative d, and the ghost number G to the degree

96 String Field Theory

of the form. The analogy also suggests a number of

algebraic identities:

¼ 0

hQAjBi ¼ ð1Þ

GðAÞ

hAjQBi

QðA  BÞ¼ðQAÞB þð1Þ

GðAÞ

A ðQBÞ

hAjBi¼ð1Þ

GðAÞGðBÞ

hBjAi

hAjB  Ci¼hBjC  Ai

A ðB  CÞ¼ðA  BÞC

½17

Note in particular the associativity of the -product.

It is straightforward to check that this algebraic

structure implies the gauge invariance of the cubic

action under [14].A-product satisfying all required

formal properties can indeed be defined. The most

intuitive prese ntation is in the functional language.

Given an open-st ring curve X(), 0    ,we

single out the string mid-point  = =2 and define

the left and right ‘‘half-string’’ curves

ðÞXðÞ for 0   



ðÞXð  Þ for



   

½18

A functional [X()] can, of course, be regarded as

a functional of the two half-strings, [X] !

[X

, X

]. We define

ð

 

Þ½X

; X



½dY

½X

; Y

½Y; X

½19

where

[dY] is meant as the functional integral over

the space of half-strings Y(), with Y(=2) =

(=2) = X

(=2). Figure 1a shows two open

strings interacting (to form a single ope n string) if

and only if the right half of the first string precisely

overlaps with the left half of the second string.

Associativity is transparent (Figure 1b).

We can now trans late this formal construction in

the precise CFT language. Very generally, an n -point

vertex of open strings can be defined by specifying

an n-punctured disk, that is, a disk with marked

points on the boundary (punctures) and a choice of

local coordinates around each puncture. Local

coordinates are essential since we are dealing with

off-shell open-string states. The BPZ inner product

(two-point vertex) is given by

h

j

ihI  

ð0Þ 

ð0Þi

UPH

IðzÞ¼

½20

The symbol f  (0), where f is a complex map,

means the conformal transform of (0) by f. For

example, if  is a dimension-d primary field, then

f  (0) = f

(0)

(f (0)). If  is nonprimary, the

transformation rule will be more complicated and

involve extra terms with higher derivatives of f.By

performing the SL(2, C) transformation

w ¼ hðzÞ

1 þ iz

1  iz

½21

we can represent the two-point vertex as a corre-

lator on the unit disk D = {jwj1},

h

j

i¼hf

 

ð0Þ; f

 

ð0Þi

ðz

Þ¼hðz

Þ; f

ðz

Þ¼hðz

½22

The vertex operators are inserted as w = 1 and

w = þ1onD (see Figure 2a) and correspond to the

two open strings at (Euclidean) world sheet time

 = 1 (we take z = exp (i þ )). The left half of

D is the world sheet of the first open string; the right

half of D is the worl d sheet of the second string. The

two strings meet at  = 0 on the imaginary w axis.

The three-point Witten vertex is given by

h

; 

hg

 

ð0Þg

 

ð0Þg

 

ð0Þi

½23

where

ðz

Þ¼e

2i=3

1 þiz

1 iz



2=3

ðz

Þ¼

1 þiz

1 iz



2=3

ðz

Þ¼e

2i=3

1 þiz

1 iz



2=3

½24

(a) (b)

π/2

Figure 1 Midpoint overlaps of open strings.

(a) (b)

Figure 2 Representation of the quadratic and cubic vertices as

2- and 3-punctured unit disks.

String Field Theory 97

The 3-punctured disk is depicted in Figure 2b, and

describes the symmetric mid-point overlap of the

three strings at  = 0. Finally, the relation between

the three-point vertex and the -product is

h

j

 

ih

; 

i½25

Knowledge of the right-hand side (RHS) in [25] for

all  allows to reconstruct the -product. All formal

properties [17] are easily shown to hold in the CFT

language. This complet es the definition of the OSFT

action.

Evaluation of the classical action is completely

algorithmic and can be carried out for arbitrary

massive states, with no fear of divergences, since in

all required correlators the operators are inserted

well apart from each other.

Quantization

Quantization is defined by the path integral over the

second-quantized string field. The first step is to deal

with the gauge invariance [14] of the classical action.

The gauge symmetry is reducible: not all gauge

parameters 

(0)

(the superscript labels ghost number)

lead to a gauge transformation. This is clear at the

linearized level; indeed, if 

(0)

= Q

(1)

,then





(0)



(1)

= Q



(0)

= 0. Thus, the set {

(0)

}givesa

redundant parametrization of the gauge group.

Characterizing this redundancy is somewhat subtle,

since fields of the form 

(1)

= Q

(2)

do not really

lead to a redundancy in 

(0)

, and so on, ad infinitum.

It is clear that we need to introduce an infinite tower

of (second-quantized) ghosts for ghosts.

The Batalin–Vilkovisky formalism is a powerful way

to handle the problem. The basic object is the master

action S(

, 



), which is a function of the ‘‘fields’’ 

and of the ‘‘antifields’’ 



. Each field is paired with a

corresponding antifield of opposite Grassmanality.

(‘‘Grassmanality’ ’ is defined to be even or odd: a

Grassmann even (odd) field is a commuting (antic-

ommuting) field). The master action must obey the

boundary condition of reducing to the classical action

when the antifields are set to zero. (Note that in general

the set of fields 

will be larger than the set of fields 

that appear in the classical action). Independence of the

S-matrix on the gauge-fixing procedure is equivalent to

the BV master equation

fS; Sg¼hS ½26

The antibracket { , } and the  operator are defined as

fA; Bg

@





@



@

 

@



½27

where @

and @

are derivatives from the left and

from the right. It is often convenient to expand S in

powers of h, S = S

þ hS

þ h

þ, with

; S

g¼0

; S

gþfS

; S

g¼2hS

; ...

½28

With these definitions in place, we shall simply

describe the answer, which is extremely elegant. In

OSFT the full set of fields and antifields is packaged

in a single string field ji of unrestricted ghost

number. If we write

ji¼j



iþj

with G ð



Þ1 and Gð

Þ2

½29

all the fields are contained in j



i and all the

antifields in j

i. To make the pairing explicit, we

pick a basis {j

i}ofH

BCFT

, and define a conjugate

basis {j

i}by

h

j

i¼

½30

Clearly, G(

) þ G(

) = 3. Then

j



i¼

Gð

Þ1

j

i

; j

i¼

Gð

Þ1

j

i



½31

Basis states j

i with even (odd) ghost number

G(

) are defined to be Grassmann even (odd). The

full string field ji is declared to be Grassmann

odd. It follows that 

is Grassmann even (od d) for

G(

) odd (even), and that the corresponding

antifield 



has the opposite Grassmanality of 

as it must be. With this understanding of ji , the

classical master action S

is identical in form to the

Witten action [16]! The boundary condition is

satisfied; indeed, setting j

i= 0, the ghost number

anomaly implies that only the terms with G = þ1

survive. The equation {S

, S

} = 0 follows from

straightforward manipulations using the algebraic

identities [17]. On the other hand, the issue of

whether S

= 0, or whether instead quantum

corrections are needed to satisfy full BV master

equation, is more subtle and has never been fully

resolved. The  operator receives singular contri-

butions from the same region of moduli space

responsible for the appearance of closed-string

poles, which are discussed below. (See Thorn

(1989) for a classic statement of this issue). It

seems possible to choose a basis in H

BCFT

such that

there are no quantum corrections to S

(Erler and

Gross 2004). In the following we shall derive the

Feynman rules implied by S

alone.

98 String Field Theory