T’’ = ‚’ X µ ‚’ Xµ = 0, (2.37)

while T+’ = T’+ = 0 expresses the vanishing of the trace, which is auto-

matic. The general solution of the wave equation (2.35) is given by

µ µ

X µ (σ, „ ) = XR („ ’ σ) + XL („ + σ), (2.38)

which is a sum of right-movers and left-movers. To ¬nd the explicit form of

XR and XL one should require X µ (σ, „ ) to be real and impose the constraints

(‚’ XR )2 = (‚+ XL )2 = 0. (2.39)

The quantum version of these constraints will be discussed in the next sec-

tion.

Closed-string mode expansion

The most general solution of the wave equation satisfying the closed-string

boundary condition is given by

1 12 i 1 µ ’2in(„ ’σ)

µ

XR = xµ + ls pµ („ ’ σ) + ls ±e , (2.40)

nn

2 2 2

n=0

1 µ 12µ i 1 µ ’2in(„ +σ)

µ

XL = x + ls p („ + σ) + ls ±e , (2.41)

nn

2 2 2

n=0

where xµ is a center-of-mass position and pµ is the total string momentum,

describing the free motion of the string center of mass. The exponential

terms represent the string excitation modes. Here we have introduced a new

parameter, the string length scale ls , which is related to the string tension

T and the open-string Regge slope parameter ± by

1 12

T= and l =±. (2.42)

2s

2π±

µ µ

The requirement that XR and XL are real functions implies that xµ and pµ

are real, while positive and negative modes are conjugate to each other

µ µ

µ µ

±’n = (±n ) and ±’n = (±n ) . (2.43)

2.3 String sigma-model action: the classical theory 35

µ µ

The terms linear in σ cancel from the sum XR + XL , so that closed-string

boundary conditions are indeed satis¬ed. Note that the derivatives of the

expansions take the form

+∞

µ

±m e’2im(„ ’σ)

µ

‚’ XR = ls (2.44)

m=’∞

+∞

µ

±m e’2im(„ +σ) ,

µ

‚+ XL = ls (2.45)

m=’∞

where

1µ

µ µ

±0 = ±0 = ls p . (2.46)

2

These expressions are useful later. In order to quantize the theory, let us

¬rst introduce the canonical momentum conjugate to X µ . It is given by

δS ™

P µ (σ, „ ) = = T X µ. (2.47)

™µ

δX

With this de¬nition of the canonical momentum, the classical Poisson brack-

ets are

P µ (σ, „ ), P ν (σ , „ ) = X µ (σ, „ ), X ν (σ , „ ) = 0, (2.48)

P.B. P.B.

P µ (σ, „ ), X ν (σ , „ ) = · µν δ(σ ’ σ ). (2.49)

P.B.

™

In terms of X µ

™ = T ’1 · µν δ(σ ’ σ ).

X µ (σ, „ ), X ν (σ , „ ) (2.50)

P.B.

™

Inserting the mode expansion for X µ and X µ into these equations gives the

Poisson brackets satis¬ed by the modes3

µ ν µ ν

= im· µν δm+n,0

±m , ± n = ±m , ± n (2.51)

P.B. P.B.

and

µ ν

±m , ± n = 0. (2.52)

P.B.

3 The derivation of the commutation relations for the modes uses the Fourier expansion of the

Dirac delta function

+∞

X

1

e2in(σ’σ ) .

δ(σ ’ σ ) =

π n=’∞

36 The bosonic string

2.4 Canonical quantization

The world-sheet theory can now be quantized by replacing Poisson brackets

by commutators

[. . . ]P.B. ’ i [. . . ] . (2.53)

This gives

[±m , ±n ] = [±m , ±n ] = m· µν δm+n,0 ,

µ ν µ ν µ ν

[±m , ±n ] = 0. (2.54)

De¬ning

1µ 1µ

aµ = √ ± m aµ† = √ ±’m

and for m > 0, (2.55)

m m

m m

the algebra satis¬ed by the modes is essentially the algebra of raising and

lowering operators for quantum-mechanical harmonic oscillators

[aµ , aν† ] = [aµ , aν† ] = · µν δm,n for m, n > 0. (2.56)

mn mn

There is just one unusual feature: the commutators of time components

have a negative sign, that is,

a0 , a0† = ’1. (2.57)

mm

This results in negative norm states, which will be discussed in a moment.

The spectrum is constructed by applying raising operators on the ground

state, which is denoted |0 . By de¬nition, the ground state is annihilated

by the lowering operators:

aµ |0 = 0 for m > 0. (2.58)