discrete field theories and spatial properties of strings*+ abstract - SLAC

..

SLAC-PUB-4727 October 1988 T

-.

DISCRETE FIELD THEORIES AND SPATIAL PROPERTIES OF STRINGS*+

Stanford Linear Accelerator Stanford

University,

Stanford,

Center

California

94309

and

LEONARD i

Physics Department,

Stanford

-

SUSSKIND’

University,

Stanford,

California

94309

ABSTRACT We use the ground-state spatial properties

wave function

of fundamental

strings.

in the light-cone

gauge to study the

We find that, as the cut-off in the param-

eter space is removed, the strings are smooth and have a divergent these properties,

we consider a large-N

size. Guided by

lattice gauge theory which has an unstable

phase where the size of strings diverges. We show that this phase exactly describes free fundamental

strings.

The lattice

spacing does not have to be taken to zero for

this equivalence to hold. Thus, exact rotation

and translation

invariance is restored

in a discrete space. This suggests that the number of fundamental degrees of freedom in string

theory

short-distance

is much smaller than in a conventional

field

theory.

-*

Talk delivered at Superstring Workshop Sirings ‘88, College Park, MD, May 24-28, 1988 + Work supported by the Department of Energy, contract DE-kC03-76SF00515. $ Work supported by NSF PHY 812280

e --

SPATIAL PROPERTIES OF STRINGS

1.

Many systems in statistical citations.

Superconducting

nets, Nielsen-Olesen

mechanics and field theory exhibit

fluxoids,

will refer to as fundamental

mag-

strings.

a field theoretic

interesting

or statistical

string theory can be simulated

There are, however, significant

differences.

question:

can fundamental

mechanical

string behavior occur in

system ? A related question

on a computer

as in lattice gauge theory.

the answer is positive.

Our construction

of the spatial properties are highly

peculiar.

of fundamental

In particular,

of the region occupied

which stores information

is whether in terms of

stability

by a fundamental

but they are unavoidable

one might think

distributed

fundamental

This instability

would

description

of calculating

diverges,

the wave function

hamiltonian 2

strings.

In order to display

string looks like in space-time.

is defined so that the total longitudinal

over g. The light-cone

by this,

All these features may seem surprising,

The points of a closed string are parametrized

The parameter

Motivated

Indeed, as in the conventional

in any theory of fundamental

We will begin by considering

and the average size

system we are looking for must have an in-

and the S-matrix.

them, we consider what the typical

frame.

We will show that these properties

string are divergent.

strings, almost everything

except for the spectrum

_P_

strings.

exclude it as a sensible theory.

of fundamental

A-

of such a theory will be guided by a study

which creates the divergence in the size of strings.

ordinarily

on a

The purpose of this work is to show that

both the average length

we will assume that the field theoretic

In

This

local degrees of freedom and evolves it according to near neighbor interactions discrete lattice,

In

to the idealized objects of string theory, which we

these field theories do not have massless gauge bosons or gravitons.

raises the following

-

in 2+1 dimensional

ex-

vortices and electric flux tubes in QCD are a few examples.

many respects they are similar

particular,

domain boundaries

string-like

of a string in the light-cone by 0 running momentum

for the transverse

from 0 to 1.

P+ is uniformly coordinates

of

-

the string is 1 -.

6

H=;

-.

sxqo)

dXi

6

-

GX+T)

'+

0

The D - 2 transverse

coordinates

(x

2

(1.1)

)>

are free fields with mode expansions

X”(g)= xi, + x(x;

cos(27rnf7) + Xi sin(27rn.g))

(14

n>O

The wave function the superscript

of the ground state of the string has the product

form (dropping

;)

-

B(X(a))= n((z)1’2eXp(-Wn(X:

(l-3)

+X3)/4))

n

with wn = 2mz. Squaring pkition

of the string.

this gives a probability

distribution

We will use this distribution

semble of strings and plot their transverse positions. it is necessary to truncate N.

the mode expansion

This is one of the ways of introducing

string.

Passage to the continuum

limit

cedure can be thought

of as introduction

of transverse

of the string.

light-cone

position

for the transverse

to generate a statistical

To carry this out in practice,

at some maximum

a cut-off in the parameter

A string

wave number space of the

is achieved as N -+ 00. This cut-off proof time averaging into the measurement

Indeed, if the string is ‘photographed’

time exposure 7, then all the modes with wn >> l/r

can be neglected.

en-

with

average to zero and

The cut-off is removed as r + 0.

configuration

is determined

by a sequence of values of Xi

and Xk,

with n = 1, . . . , N and i = 1, . . . , D - 2, sampled with probability

P(Xi)

and similarly 2)-dimensional

for Xz.

=

gp2

eXp(-Wn(Xi)2/2)

Each such configuration

defines a parametrized

space. By necessity we show projection 3

-

(1.4) curve in (D -

of string onto 2 transverse

dimensions. i

,;

Each run consists of choosing 100 x (D - 2) random numbers from their

respective probability

distributions.

to 50, with increments

.

example,

For each run we compute

of 10. We adopt the following

from N = 10 to N = 20, the coefficients

curves with N = 10

method:

as we proceed, for

of the normal

modes with the

first 10 wave numbers are kept the same as for the N = 10 ‘snapshot’. for each run, increasing shorter exposure.

N corresponds

to ‘photographing’

-

the same string

with

The ‘snapshots’ generated by one such run are shown in figure 1.

In fact, we find that different runs are overwhelmingly with the following

Therefore,

likely to produce ‘snapshots’

features.

1)

The size of the region in transverse space occupied by string grows slowly with

N.

A quantitative

measure of this is r, the rms distance of a point on the string

to its center of mass:

T2= ((x’(“) - xQ2)

P-5)

Since there is no preferred point on the closed string, we can arbitrarily

set CY= 0.

=(D-2)&Xz)=y5$ n=l

(1.6) n=l

The rms radius of the string grows with the mode cut-off as &$V. 2)

The plots of total

analytically,

string

length

L vs. N appear to be linear.

To show this

we start with

W) = j WI> da, 0

where v’= dJ?ld 0. By translation

(a)

(1.12)

Therefore,

-

(1.13)

* In other words, the Hausdorff dimension of the fundamental string is infinite. -t1t n wo t ransverse dimensions the expectation value of the extrinsic curvature diverges due to a relatively high likelihood of cusps. 5

,;

We find that (1/v2)

. -.

N l/C2

and (1Zll)

N c, where

g2=en3 =

(~N(N

(1.14)

+ 1,>" N c4

1

It follows that 5)

(K) - c/C2

is independent

The entire string consists of O(N)

roughly

independent

of N. Adding

each loop occupies a fraction

-

-

smooth ‘loops’.

modes simply

l/N

The structure

of a loop is

adds more loops. Furthermore,

of the parameter

space 0. Thus, in a fairly

uniform

manner

P+/N.

To show that no ‘accidents’ occur as we proceed to high values of N, we

have plotted

the individual

of the cut-off!

loops tend to carry a longitudinal

momentum

in figure 2 the section of the string confined between (T = 0 and 2/N

for N = 20 and N = 500. qualitatively

The remarkable

regarded as a statement

similarity

of conformal

between the two can be

invariance

in our approach.

One may wonder if any of the effects discussed above are artifacts mode cut-off. a different

the string

mentum

P+/2N

transverse string P+/2N

which is. quite appealing

into 2N + 1 segments, each one carrying

X”(m).

be subdivided

+ 1. Th e regulated

into pieces with

s

2

-sxym)

diagonalized

g(m) = m/2N

grounds.

s sxym)

longitudinal

longitudinal

+ (Xi(m

string

the a-axis,

the string

‘parton’ m with

momentum

the

less than

(1.15)

+ 1) - X$72))” >

+ 1. The ground state wave function

regularized

mo-

is

by Fourier modes (1.2) evaluated

cies wn = (4N + 2) sin(rn/2N

Imagine

A rough way to describe this is to say that

string hamiltonian

H=2N+1 It is exactly

on physical

+ 1. Each segment is replaced by an indivisible

coordinates

cannot

of the sharp

We have checked that this is not the case. Also, one can introduce

regularization

chopping

-

at discrete positions

is given by (1.3) with frequen-

+ 1). In fig ure 3 we show a typical

(N = 50). Note that,

picture

of the

as more and more discrete points crowd

never becomes continuous 6

in space. We can show that,

as

c --

N +

00, the average distance in space between each pair of neighboring

approaches a constant

value a, which is of the order of the Planck length.

responsible for the linear growth of the total length. rms radius grows as J&3. properties

information

in the regularization

about the spatial

where the string never

in space. This suggests that, if the transverse space is replaced

by a lattice

with spacing of the order of a, the critical

be affected.

In what follows we confirm this expectation

properties

of strings will not

by explicitly

a class of discrete field theories which give rise to the fundamental One may wonder whether servable.

In particular,

the infinite

light-cone of-order

at rest.

r = l/E.

we retain a number particle

‘expands’.

This phenomenon

radius

instantaneous.

The difference the conventional

factor

>> l/r

In the

of the interaction

average to zero.

a mode cut-off

As the resolution

for scattering

exchange.

string

is

Thus,

and gives an

is improved,

the string

Regge behavior of scattering

Thus, the effect is indeed observof strings by strings.

by a local external

On the other

field, then the interaction

coupled to local external

is

Thus, the fundamental fields:

they are either a

or of nothing. between the fundamental

strings and the extended

field theory can be easily exhibited

of the longitudinal

F(q2).

by string

the string must appear infinite.

cannot be consistently

by interaction

frequency

leads to the well-known

is scattered

Therefore,

theory of everything

bution

with

However,

of a high-energy

of energy E the lifetime

satisfied by the dual amplitudes.‘)

hand, if the string

>- _T_

is mediated

N Jii.

able and presents no difficulty

strings

scattering

of modes N E. This introduces

observable

cross-sections

Consider

The interaction

Oscillations

above are ob-

rms radius seems very unphysical.

effect.

frame of the fast string

constructing

string behavior.

any of the strange effects described

it does lead to an observable from a string

This is

It is also easy to show that the

Thus, all the important

of string can be obtained

becomes continuous

partons

with external

momentum

field.

string,

tained by observing that the distribution

in

the spatial distri-

P +. For a hadron, this could be measured

gravitational

F or a fundamental

by studying

objects

The result is a non-singular

an analogous

form factor

form

can be ob-

of P+ is measured by the vertex operator 7

exp( iq . X), w h ere Q is the world sheet index.

&X+d”X+

In the light-cone

gauge

.-.- X+ = r and the form factor reduces in D = 26 to .~

%12) = In the limit that

N +

- exp( - iq2 log N)

da exp(iq . X(a))

(1.16)

>

co the form factor

P+ is smeared uniformly

is non-vanishing

all over transverse

only at q = 0. It follows

space.

This peculiar

property

applies not only to the ground state of the string but also to any finitely state.

For any such state the change in the wave function

relative

excited

to the ground

state concerns only a finite number of normal modes and becomes negligible limit

N +

co. The strange behavior

of the gravitational

connected with the existence of the graviton:

states that, in a Lorentz invariant

mementum satisfy Einstein

tensor, the gravitational

F(q2

# 0) = 0.

gravity

invariant

if the graviton

get around

invariant

must

this theorem?

energy momentum

In

tensor.

string theory the form factors are Lorentz-

and the theorem must apply.

It seems to require that,

is to exist in string theory, it must be smeared all over transverse

space. This suggests that the fundamental fluctuations

energy-

form factor of a massless spin-2 particle

one cannot define a Lorentz

by construction,

A theorem by Weinberg and

theory with a Lorentz invariant

H ow do various theories

On the other hand, in 26-dimensional

form factors is possibly

at least for the massless spin-2 state

it could be foreseen on the basis of general principles. Witten

in the

string theory uses its infinite

to allow the existence of gravitons.

zero-point

2. LATTICE THEORY ON THE LIGHT CONE .. -. In this section we describe a light-cone Pearson and Rabinovici.3) large-N,

limit.

It has string-like

lattice’.gauge excitations

theory due to Bardeen, which become free in the

This theory has a parameter, p, which governs the average length

of string in the ground state. For p greater than the critical length of string is finite in lattice

value pC the average

For p = pC the average length of string

units.

diverges. We will be interested in the range p < pu, where the theory is unstable with respect to infinite growth of strings. This instability of the fundamental

strings discussed in the previous chapter. In fact, we will show

that, for p < pC, the strings in this light-cone -

fundamental

is similar to the behavior

strings.

lattice

theory are identical

to the

This exact equivalence does not require lattice spacing to be

taken to zero. We introduce ~“-~)/fi,

the light-cone

variables Z+ = (x0 + $-I)/&

and xi, where i labels the D - 2 transverse

Bardeen, Pearson and Rabinovici dimensional

lattice L there is a unitary matrix-valued quantization

directions.

= (x0 Following

we replace the transverse space by a (D - 2)-

cubic lattice n’with integer coordinates.

In the light-cone

Z-

On each directed link of the

variable which satisfies U;j(L)

= UT,(-L).

the variable Z+ is treated as time. After passing to

the light-cone gauge A- = 0 the gauge theory light-cone Lagrangian becomes

L=-j/ dx-{c t+pU(L)d’U(-L)) + c

links

g4/dx-‘c

tr(U(&)U(L2)U(L3)U(L4))+

daq

Ix- - x-‘IP(Z,

x-)J-“(?T, x-‘)}

ii where the index p refers to the + and - directions and the lattice spacing has been set to 1 for convenience. J?(S) site Z.

denotes the longitudinal

To make the theory tractable,

ref.

3 relaxes the condition

Thus the matrices U are replaced by gM where M(L) matrices.

In order to restore unitarity,

momentum

current at the of unitarity.

are general N x N complex

ref. 3 introduces the effective potential 9

for

M: i

..

-.

V(M)

= p MijMG

The speculation

of ref. 3 is that the continuum

can be obtained

by tuning

the lattice

(2.2)

+ g2X MijM&MlkMc limit

of the U(N,)

~1 and X along a renormalization

gauge theory

group trajectory,

spacing is being taken to zero. We will not pursue this interesting

as idea

here. Instead we will argue that, with no fine tuning, there exists a broad range of parameters where this theory exactly reproduces fundamental

strings.

For our purposes it is necessary to add to the lagrangian other terms quartic in M which would be trivial tr(M4) -

if gM was unitary.

These are the plaquette-like

terms

shown in figure 4. The trace is taken around all possible loops of length 4

and zero area. In fact the second term in (2.2) is of this type.

All other terms of

this type reside on pairs of links, L and K, beginning at the same vertex: 6L = -,‘A’/

dx- gtr(M(L)M(-L)M(K)M(-K))

(2.3)

, Using standard methods we derive the light-cone hamiltonian k($M(L)M(-L)

+ x M(L)M(-L)M(L)M(-L))+

xl):r(M(L)M(-L)M(r()IM(-K))

Ix- - x-‘IJ_“(Z,

- Ctr(M(L1)M(L2)M(L3)M(L4))daq x-)J-“(?T,

x-‘)} (2.4)

The link fields may be decomposed into creation and annihilation

Mij(x-)

=

-&

?r

J -$=(Aij(k)

exp(-ilcx-)

operators:

+ B&(k) exp(ikx-))

(2.5)

0

which obey the commutation LAijck)7 Ail(q)]

relations = [Bij(k),

Bll(q)] 10

= SikSjlS(k - q)

(2.6)

Each link -

is assigned a direction

longitudinal

momentum

k and points

the assigned direction.

An oriented

bit which

carries

from index i to index j along the link’s

Bijt (k) creates a bit pointing

Similarly,

direction.

creates a string

and At(k)

string

from j to i and opposite

state is defined in the following

Consider an oriented connected loop I consisting

to way.

of links Li, with i running

from

1 to N. The Fock state associated with this loop is tr{#(Ll, where Oij = NF1’“Aij

kl).

or N-‘12B,ti

. . Ot(h,

h)}

(2.7)

IO >

d epending on whether a given string bit points

along or opposite the assigned direction. An important hamiltonian

feature of the N, + 00 limit

(2.4) d oes not split

state of the form

(2.7), we obtain

a linear dynamical Schroedinger

equation

equation.

Makeenkoequation4) One conspicuous

to be formidable.

for one-string

such state.

hamiltonian

Therefore,

states, which is simply

limit,

is that

the Migdal-Makeenko

where strings

about the large-N, equation.

become free.

limit

there exists the light-cone

equation

is non-linear

We believe that

all the

of the theory in (2.4) is contained

It is even difficult

to write it down in an explicit

to a free field hamiltonian.

is quite complicated.

form, since the

Nevertheless,

Fortunately,

string.

this is the fundamental

string the

For example, the term

qwvwL)) plays the role similar

there

can be easily analyzed and shown

For now, let us compare various terms in (2.4) with (l.l),

of a free fundamental

in

On the first glance, solving this equation appears

is a phase of the theory, where the hamiltonian

phase we are after.

it to a one-string

value of the Wilson loop in a gauge theory.

action of various terms in the hamiltonian

equivalent

If we apply

Such an equation should not be confused with the Migdal-

difference

the linear Schroedinger

strings.

another

for the expectation

even in the large-N, useful information

or join

of the theory is that the light-cone

to the second term in (1.1). 11

C

P-8) It accounts for the potential

energy due to the string .=- bits of string.*

kinetic

energy.

Below we will argue that,

theory where strings are infinitely

long, its light-cone

hamiltonian

(2.4) is,

model in one

transverse dimension.

There, each lattice state can be mapped into a function

used in the light-cone

description

longitudinal it with

momentum

X(a

= 0).

and a = kl/kt. in parameter

kt.

Starting

The first

Thereafter,

of the fundamental

X(ai)

which

lattice

by a segment with

momentum.

coordinates

that,

longitudinal

of (2.8) and is N ,Y c;”

for a sufficiently

of the corresponding

pi

site.

ki’.

energy,

It is clear that,

for

many

negative

of links with

p, an instability

links each one carrying

smaller

limit

in parameter

infinitesimal suggesting

as anti-periodic

momentum. in the x-

k.

In fact,

k.

Evidently,

the possibility

it is clear

in this case

of a continuum

limit’ will always refer to

space, and not in real space.

In order to discuss this phase we need a regulator allowed longitudinal

fraction

develops which favors strings of

in a-space. Indeed, from here on ‘continuum

the continuum

a significant

As we decrease p it becomes energetically

the a-axis becomes densely populated

M(x-)

At the point

In this case we do not expect a behavior

strings.

to have a larger number

description

length

can be defined by a linear interpolation.

momentum.

to the fundamental

infinitely

0 = 0

large values of the string tension p, the ground state will be dominated

of the total

favorable

site, we identify

by a segment between

by states with small numbers of links, with each link carrying

similar

X(a)

Let the string carry

of the model is governed by the string potential

comes from the action

sufficiently

an arbitrary

to its longitudinal

is given by the lattice

The phase structure

strings.

each link is represented

space proportional

the function

with

link is represented

Between sites 0; and ai+r, X(a)

.- -1.

displace

in the phase of the

to (1.1). W e will show this in detail for the lattice

in fact, identical

-

The other terms in (2.4) can locally

They are the analogues of the first term in (1.1); i.e., they give

rise to the string lattice

tension.

in the form of a minimum

Following

ref.

5 we will think

direction.

Then the k-space is discretized

* The reader can check this by applying various terms in the hamiltonian string configurations. 12

of the matrices and

(2.4) to simple

e --

the zero-momentum r

multiple kt/k,

links are excluded

of the minimum = N in lattice

km. The total,length

momentum

units.

so that each link carries an odd integer

The limit

of string cannot exceed

N + 0~) defines our lattice

string theory.

3. EFFECTIVE HAMILTONIAN AND EQUIVALENCE TO A FREE FIELD _ We will begin studying ground length

state and low-lying = N.

kt/km

turbation -

excitations

maximum

interaction

allowed length.

string in the transverse Fo’rtunately,

consist

To solve for their

theory in the ‘kinetic

current-current

p t

-oo.

of strings

wave functions,

has vanishing

matrix

to be included string constraint

the

possible

we use degenerate perIt turns out that the

elements between the states of

The only terms in the hamiltonian are the plaquette-like

that act to move the

terms introduced

the model based on these terms is soluble exactly.

If the maximum

In this limit

of maximum

terms’ of the hamiltonian.

dimension

the necessary construction

to think

this model in the limit

above.

Below we describe

in some detail.

allowed length is N then the string configurations

that need

are labeled by series of N pluses and minuses subject to the closed that their sum is zero. This requires N to be even. It is convenient

of these configuration

as series of Ising spins 03(i).

of terms that need to be taken into account.

A’ C

dx-Mij(n)Mjk(n

There are two types

The first one,

-I- l)M&.(n

n J

+ l)Mi(n),

(34

is shown in figure 4b). The second one,

(3.2) xcJdx-Mij(n)Mfj(n)Mkl(n)M~( n

is shown in figure 4~).

It is not hard to show that, 13

to leading order in NC, the

action of (3.1) on states in the spin representation I

r

Yi+

.

n

( 1+ a3(n)a3(n2

+ 1)

where in terms of the standard

+ a+(n)a-(n

to

+ 1, + g.-(n>g+(n

+ 1))

(3.3)

Pauli matrices

01 + ia

c7+ =

is equivalent

2

’

and we have set g2NC = 1. Similarly,

AN

CT- =

x(1 n

icr2

01 -

(3.4)

2

the term (3.2) is represented by

-

as(n)m(n

+

1))

(3.5)

-

-

Let us choose temporarily hamiltonian

2X = X’ = 1. Then, up to an additive

is simply given by the quantum H = N x(c+(n)a-(n

constant,

the

XY model:

+ 1) + a-(n)o+(n

+ 1))

(3.6)

n

Actually,

since we have ignored the center of mass motion

hamiltonian

is only applicable

verse lattice

(have zero lattice momentum).

tonian

to states of finite

show that the quantum

to states that are translation

lattice

momentum

A generalization

of strings, invariant

the above

on the trans-

of the effective hamil-

will be given below.

XY model can be solved by introducing

But first let us anti-commuting

variables

$+(n> = iwn n

a3(++(n>,

(3.7)

a3(m>o-(n>.

(3.8)

ma+(n

+ 1) exP(-2iP/N))

(3.13)

n

To justify

the appearance

a+(n)a-(n to the right.

of phases in the above formula

we note that whenever

+ 1) acts on a string state, the center of mass moves 2/N Similarly,

fact, the interaction vector potential

lattice

units

the conjugate term moves the center of mass to the left. In (3.13) can be obtained

along the string.

from (3.6) by introducing

a constant

A gauge transformation

-

a-(n)

+ o-(n)

exp(--2ipn/N),

a+(n)

(3.14)

+ a+(n) exP(wnlN)

reduces (3.13) to (3.6) at th e expense of a non-trivial e continuum fermionic variables:

boundary

condition

$o( = 1) = -$~(a = O)exp(-2ip).

Upon bosonization,

this condition

changes the constraint

on the

(3.15)

on the values of the center

of mass momentum:

Cm = &h(n + y), where n is an integer.

After inclusion

of string states with non-zero lattice momen-

tum, the zero mode in the boson hamiltonian As a result,

(3.12) is the standard

reason is the fact that,

(3.16)

light-cone

has acquired continuous free string hamiltonian.

as N --+ 00, the possible positions

spectrum.

The physical

of-the string

center of

mass become continuous. -

- Now we would like to argue that relaxing the constraint cients of the plaquette-like

X’ = 2X on the coeffi-

terms does not in general violate the free boson behavior 16

c --

demonstrated ..

above. In fact, this adds a term

_

-cN

c

os(n)os(n

(3.17)

+“l)

n

to the effective

hamiltonian

(3.13).

This term can be interpreted

dependence on the extrinsic

line curvature

on the sign of the coefficient

c, it favors alignment

links.

The continuum

limit

-1 5 c < 1, the Thirring

or anti-alignment

model is equivalent

aligned.

As c +

too strongly

is the Thirring

scale. Therefore,

and the relativistic

Increasing

pairs of adjacent links become more

Our discussion

confirms

that one should be allowed to take the spatial continuum changing the equivalence with fundamental spacing beyond the typical

phase, a transition

size of string

to the lattice-dominated

range of parameters,

the extrinsic

curvature

is favored

-

On the other

is favored so much that the behavior

by the lattice.

For any

c makes string wiggles

free field behavior is no longer valid.

anti-alignment

strings is dominated

the lattice

model.

1 the size of the wiggles diverges. For c 2 1 alignment

hand, for c < -1,

without

of the adjacent

to a free boson field. The only effect

of changing c is resealing of the string tension.

-

8, Depending

into the string hamiltonian.

of this new hamiltonian

look bigger on a fixed lattice

as introducing

of lattice

the simple intuition

limit in our lattice model

strings.

However, if we increase

wiggles in the ‘fundamental’

phase takes place. Thus, for a broad terms do not alter the critical

behavior

of strings. In section 1 we demonstrated lattice

that the spatial behavior and energy levels of our

string theory are completely

free field in a-space.

determined

For example,

this guarantees that the growth

squared radius of string is logarithmic formalism,

in the length of string.

the ground state energy is quadratically

the negative

sign.

Conventionally,

absorbed in renormalization

in the light-cone

of the mean

As in the usual string

divergent. formalism

In our system it has this divergence

is

of the speed of light.

- Next consider the corrections hamiltonian

by the equivalence with a massless

to the limit

which act to decrease the total 17

p +

-co.

Then the terms in the

length of string

become important.

c --

For example, r

menturn.&

the plaquette-like

terms can replace three links of longitudinal

by one link of momentum

string and in the continuum

limit

in a-space. -The only operators

3k,.

Such perturbations

mo-

act locally on the

can be represented by a series of operators

that can affect the critical

behavior

in a-space are

renormalizable

and super-renormalizable.

able operators

that can be induced are just the two terms already present in the

hamiltonian

It turns out that

local

the only renormaliz-

(1.1). Th ere f ore, a range of p must exist in which the effect of the cor-

rections to the limit

/.L+ -KI

is absorbed in resealing of the string tension.

Since

for large and positive p there exists another phase of the theory where strings have finite length, there must be a critical -

A straightforward number of dimensions.

generalization

occurs.

of this lattice field theory can be given in any

If D = 26 then, in the phase where strings are infinite,

the finite energy spectrum string.

value p = pC where a phase transition

is identical

to the spectrum

of the conventional

The same equivalence applies to the expectation

-

all

bosonic

values of products of vertex

operators.

4. CONCLUSIONS Perhaps the most striking pletely equivalent is required. objects.

result of this work is that a discrete theory is com-

to a free string in continuous

This is possible because we are dealing with

a theory

limit

of extended

Since their length diverges, the physics depends only on the critical

erties of the effective

1+1-dimensional

field theory induced on the string.

way, theories with discrete and continuous end up in the same universality

values of transverse

class. The physical

only allows string bits with vanishingly light-cone

space. No spatial continuum

and completely

small longitudinal

space coordinates

momenta.

Typically,

this effect does not depend on the details of any particular

18

the

i.e., the string bits

wash out any memory of the lattice.

expect that there exists a large universality

In this

reason is that the instability

time scale for motion of the i-th string bit is N P+(i),

move very rapidly

prop-

lattice

Evidently,

structure.

We

class of discrete theories which exhibit

Z --

identical r

behavior

ingredient

for a broad range of parameters.

interesting

question

is the connection

above and the NC = 0;) QCD. The original a very special limit the parameters

are carefully

broad range of parameters, The construction

with the fundamental without

suggested by the smoothness structure

amplitudes

group trajectory.

spacing to zero.

reproduces

bosonic

has vastly fewer short distance

degrees

quantum

of the string

field theory.*

pictures

in strings is probably

fall off exponentially

is taken as

string behavior which occurs for a

of a discrete field theory which exactly

of freedom in space than a conventional

scattering e

limit

necessity of taking the lattice

theory

discussed

work of ref. 3 suggested that QCD is

tuned along some renormalization

makes it clear that string

the short-distance

between the theory

of this theory in which the spatial continuum

This should be contrasted

-

share one important

--. the instability.

Another

strings

They

This was already

-

of figure 1. The absence of

the reason why the fixed-angle

at high energy.r’)

; Our model required the number of colors N, to be taken to co. It is interesting to study the new effects induced by a finite N,. The string can then split and join as in the conventional is proportional expansion

picture

to l/N,.

of string interactions.

The string coupling

It should be possible to determine

whether

constant the l/NC

reproduces the bosonic string amplitudes.

Acknowledgements:

Section 1 of this talk is based on joint work with M. Karliner.“)

e --

* This was also suggested on very different grounds in the work of Atick and Witten 19

(ref. 9).

REFERENCES ,z.-

1. Susskmd, L., Phys. Rev. Dl, 2. Weinberg, S. and Witten,

1182 (1970).

E. Phys. Lett. m,

3. Bardeen, W., Pearson, R. and Rabinovici, 4. For a review, see Migdal, 5. Hornbostel,

59 (1980).

E., Phys. Rev. m,

1037 (1980)

A. A., Phys. Rep. 102, 199 (1983).

K., Stanford Ph. D. Thesis

6. Coleman, S. Phys. Rev. D11, 2088 (1975) 7. Mandelstam, 8. Polyakov,

-

9. Atick,

S., Phys. Rev. D11, 3026 (1975)

A. M., Nucl. Phys. B268, 406 (1986)

J. and Witten,

E., Inst. for Advanced Study preprint

10. Gross, D. J. and Mende, P., Princeton

preprint

PUPT-1067

IASSNS(1987)

-

11. Karliner,

M., Klebanov,

I. and Susskind, L., Int. Journal of Mod. Phys. A3,

1981 (1988)

FIGURE CAPTIONS 1)

Projections

of string onto 2 transverse dimensions with mode cut-off varying

from 10 to 50.

2)

The section of string confined between cr = 0 and 2/N for N = 20 (solid line) and N = 500 (dashed line).

3)

A typical

configuration

of the ground state of 101 partons

connected by

springs.

.- _Y_

4)

The plaquette-like

terms which would be trivial

unitary.

20

if the link variables were

5 --

N=IO

r

5

. X2

0

-c J

L

r 5

c

x2

0

s

-5 I

-5

0 Xl

5

5 -

0 .

-5

FIG.

Fig 1

1

..

_

--

N=20 N= 500

\ *\

i i

. --.

Q-88

Xl Fig. 2

i ; I* /’ 4’

6129A3

..

-

. I

.

’

I

-

-

.

5

-5 I

-5

.

-

’

0

1

5 6129A2

8-88

Fig. 3

..

-

0

0

0

0

0

0

.

0

0

0

0

0

.

0

0 c!l

0 (a)

0

0

0

0

0

0

cj

0

0

0

0

0

0

0

.

0

0

a

0

0

l

0

0

0

0

0

0

(c) 0

5998Al

4-88

Fig. 4

-

-