π π

µν ™ ™

µν

(X µ X ν ’ X ν X µ )dσ.

J = J0 dσ =T

0 0

Now

1 µ ’im„

X µ („, σ) = xµ + ls pµ „ + ils

2

±e cos(mσ),

mm

m=0

™ ±m e’im„ cos(mσ),

X µ („, σ) = ls pµ + ls

2 µ

m=0

2

and T = 1/(πls ). A short calculation gives

∞

1 µ

µν µν νµ ν ν µ

=x p ’x p ’i ±’m ±m ’ ±’m ±m .

J

m

m=1

2

48 The bosonic string

2.5 Light-cone gauge quantization

As discussed earlier, the bosonic string has residual di¬eomorphism symme-

tries, even after choosing the gauge h±β = ·±β , which consist of all the con-

formal transformations. Therefore, there is still the possibility of making an

additional gauge choice. By making a particular noncovariant gauge choice,

it is possible to describe a Fock space that is manifestly free of negative-norm

states and to solve explicitly all the Virasoro conditions instead of imposing

them as constraints.

Let us introduce light-cone coordinates for space-time6

1

X ± = √ (X 0 ± X D’1 ). (2.123)

2

Then the D space-time coordinates X µ consist of the null coordinates X ±

and the D ’2 transverse coordinates X i . In this notation, the inner product

of two arbitrary vectors takes the form

v · w = vµ wµ = ’v + w’ ’ v ’ w+ + v i wi . (2.124)

i

Indices are raised and lowered by the rules

v ’ = ’v+ , v + = ’v’ , v i = vi .

and (2.125)

Since two coordinates are treated di¬erently from the others, Lorentz invari-

ance is no longer manifest when light-cone coordinates are used.

What simpli¬cation can be achieved by using the residual gauge symme-

try? In terms of σ ± the residual symmetry corresponds to the reparametriza-

tions in Eq. (2.86) of each of the null world-sheet coordinates

σ ± ’ ξ ± (σ ± ). (2.126)

These transformations correspond to

1++

ξ (σ ) + ξ ’ (σ ’ ) ,

„= (2.127)

2

1++

ξ (σ ) ’ ξ ’ (σ ’ ) .

σ= (2.128)

2

This means that „ can be an arbitrary solution to the free massless wave

equation

‚2 ‚2

’ „ = 0. (2.129)

‚σ 2 ‚„ 2

√

6 It is convenient to include the 2 factor in the de¬nition of space-time light-cone coordinates

while omitting it in the de¬nition of world-sheet light-cone coordinates.

2.5 Light-cone gauge quantization 49

Once „ is determined, σ is speci¬ed up to a constant.

In the gauge h±β = ·±β , the space-time coordinates X µ (σ, „ ) also satisfy

the two-dimensional wave equation. The light-cone gauge uses the residual

freedom described above to make the choice

X + (σ, „ ) = x+ + ls p+ „ .

2

(2.130)

This corresponds to setting

+

±n = 0 for n = 0. (2.131)

In the following the tildes are omitted from the parameters „ and σ.

When this noncovariant gauge choice is made, there is a risk that a

quantum-mechanical anomaly could lead to a breakdown of Lorentz in-

variance. So this needs to be checked. In fact, conformal invariance is

essential for making this gauge choice, so it should not be surprising that a

Lorentz anomaly in the light-cone gauge approach corresponds to a confor-

mal anomaly in a covariant gauge that preserves manifest Lorentz invariance.

The light-cone gauge has eliminated the oscillator modes of X + . It is

possible to determine the oscillator modes of X ’ , as well, by solving the

™

Virasoro constraints (X ±X )2 = 0. In the light-cone gauge these constraints

become

1

™ ™

X ’ ± X ’ = + 2 (X i ± X i )2 . (2.132)

2p ls

This pair of equations can be used to solve for X ’ in terms of X i . In terms

of the mode expansion for X ’ , which for an open string is

1 ’ ’in„

X ’ = x’ + ls p’ „ + ils

2

±e cos nσ, (2.133)

nn

n=0

the solution is

D’2 +∞

1 1

’ i i

: ±n’m ±m : ’aδn,0

±n = . (2.134)

p+ ls 2

i=1 m=’∞

Therefore, in the light-cone gauge it is possible to eliminate both X + and

X ’ (except for their zero modes) and express the theory in terms of the

transverse oscillators. Thus a critical string only has transverse excitations,

just as a massless particle only has transverse polarization states. The con-

venient feature of the light-cone gauge in Eq. (2.130) is that it turns the

Virasoro constraints into linear equations for the modes of X ’ .

50 The bosonic string

Mass-shell condition

In the light-cone gauge the open-string mass-shell condition is

D’2

+’

2 µ

p2 = 2(N ’ a)/ls ,

2

M = ’pµ p = 2p p ’ (2.135)

i

i=1

where