One can also specify the momentum k µ carried by a state |φ ,

|φ = aµ11† aµ22† · · · aµn † |0; k , (2.59)

mm mn

which is the eigenvalue of the momentum operator pµ ,

pµ |φ = k µ |φ . (2.60)

It should be emphasized that this is ¬rst quantization, and all of these states

(including the ground state) are one-particle states. Second quantization

requires string ¬eld theory, which is discussed brie¬‚y at the end of Chapter 3.

The states with an even number of time-component operators have pos-

itive norm, while those that are constructed with an odd number of time-

2.4 Canonical quantization 37

component operators have negative norm.4 A simple example of a negative-

norm state is given by

a0† |0 0|a0 a0† |0 = ’1,

with norm (2.61)

m mm

where the ground state is normalized as 0|0 = 1. In order for the theory

to be physically sensible, it is essential that all physical states have positive

norm. Negative-norm states in the physical spectrum of an interacting the-

ory would lead to violations of causality and unitarity. The way in which the

negative-norm states are eliminated from the physical spectrum is explained

later in this chapter.

Open-string mode expansion

The general solution of the string equations of motion for an open string

with Neumann boundary conditions is given by

1 µ ’im„

X µ („, σ) = xµ + ls pµ „ + ils

2

±e cos(mσ). (2.62)

mm

m=0

Mode expansions for other type of boundary conditions are given as home-

µ

work problems. Note that, for the open string, only one set of modes ±m

appears, whereas for the closed string there are two independent sets of

µ µ

modes ±m and ±m . The open-string boundary conditions force the left- and

right-moving modes to combine into standing waves. For the open string

∞

™ ±m e’im(„ ±σ) ,

2‚± X = X µ ± X µ = ls

µ µ

(2.63)

m=’∞

µ

where, ±0 = ls pµ .

Hamiltonian and energy“momentum tensor

As discussed above, the string sigma-model action is invariant under various

symmetries.

Noether currents

Recall that there is a standard method, due to Noether, for constructing a

conserved current J± associated with a global symmetry transformation

φ ’ φ + δµ φ, (2.64)

4 States that have negative norm are sometimes called ghosts, but we reserve that word for the

ghost ¬elds that are arise from covariant BRST quantization in the next chapter.

38 The bosonic string

where φ is any ¬eld of the theory and µ is an in¬nitesimal parameter. Such

a transformation is a symmetry of the theory if it leaves the equations of

motion invariant. This is the case if the action changes at most by a surface

term, which means that the Lagrangian density changes at most by a total

derivative. The Noether current is then determined from the change in the

action under the above transformation

L ’ L + µ‚± J ± . (2.65)

When µ is a constant, this change is a total derivative, which re¬‚ects the

fact that there is a global symmetry. Then the equations of motion imply

that the current is conserved, ‚± J ± = 0. The Poincar´ transformations

e

δX µ = aµ ν X ν + bµ , (2.66)

are global symmetries of the string world-sheet theory. Therefore, they give

rise to conserved Noether currents. Applying the Noether method to derive

the conserved currents associated with the Poincar´ transformation of X µ ,

e

one obtains

P± = T ‚ ± X µ ,

µ

(2.67)

J± = T (X µ ‚± X ν ’ X ν ‚± X µ ) ,

µν

(2.68)

where the ¬rst current is associated with the translation symmetry, and the

second one originates from the invariance under Lorentz transformations.

Hamiltonian

World-sheet time evolution is generated by the Hamiltonian

π π

T 2

™µ ™

X2 + X

Xµ P0 ’ L dσ =

H= dσ, (2.69)

2

0 0

where

δS

µ ™

= T X µ,

P0 = (2.70)

™µ

δX

was previously called P µ (σ, „ ). Inserting the mode expansions, the result

for the closed-string Hamiltonian is

+∞

(±’n · ±n + ±’n · ±n ) ,

H= (2.71)

n=’∞

while for the open string the corresponding expression is

+∞

1

±’n · ±n .

H= (2.72)

2 n=’∞

2.4 Canonical quantization 39

These results hold for the classical theory. In the quantum theory there are

ordering ambiguities that need to be resolved.

Energy momentum tensor

Let us now consider the mode expansions of the energy“momentum tensor.

Inserting the closed-string mode expansions for XL and XR into the energy“

momentum tensor Eqs (2.36), (2.37), one obtains

+∞ +∞

’2im(„ ’σ)

Lm e’2im(„ +σ) ,

2 2

T’’ = 2 ls Lm e and T++ = 2 ls

m=’∞ m=’∞

(2.73)

where the Fourier coe¬cients are the Virasoro generators

+∞ +∞

1 1

±m’n · ±n ±m’n · ±n .

Lm = and Lm = (2.74)