In the same way, one can get the result for the modes of the energy“

momentum tensor of the open string. Comparing with the Hamiltonian,

results in the expression

+∞

1 1

(±’n · ±n + ±’n · ±n ) ,

H = L0 + L0 = (2.75)

2 2 n=’∞

for a closed string, while for an open string

+∞

1

±’n · ±n .

H = L0 = (2.76)

2 n=’∞

The above results hold for the classical theory. Again, in the quantum theory

one needs to resolve ordering ambiguities.

Mass formula for the string

Classically the vanishing of the energy“momentum tensor translates into the

vanishing of all the Fourier modes

m = 0, ±1, ±2, . . .

Lm = 0 for (2.77)

The classical constraint

L0 = L0 = 0, (2.78)

can be used to derive an expression for the mass of a string. The relativistic

mass-shell condition is

M 2 = ’pµ pµ , (2.79)

40 The bosonic string

where pµ is the total momentum of the string. This total momentum is

given by

π

™

µ

dσ X µ (σ),

p =T (2.80)

0

™

so that only the zero mode in the mode expansion of X µ (σ, „ ) contributes.

For the open string, the vanishing of L0 then becomes

∞ ∞

12

±’n · ±n + ± p2 = 0,

±’n · ±n + ±0 =

L0 = (2.81)

2

n=1 n=1

which gives a relation between the mass of the string and the oscillator

modes. For the open string one gets the relation

∞

1

2

±’n · ±n .

M= (2.82)

±

n=1

For the closed string one has to take the left-moving and right-moving modes

into account, and then one obtains

∞

22

(±’n · ±n + ±’n · ±n ) .

M= (2.83)

±

n=1

These are the mass-shell conditions for the string, which determine the mass

of a given string state. In the quantum theory these relations get slightly

modi¬ed.

The Virasoro algebra

Classical theory

In the classical theory the Virasoro generators satisfy the algebra

[Lm , Ln ]P.B. = i(m ’ n)Lm+n . (2.84)

The appearance of the Virasoro algebra is due to the fact that the gauge

choice Eq. (2.23) has not fully gauge ¬xed the reparametrization symmetry.

Let ξ ± be an in¬nitesimal parameter for a reparametrization and let Λ be an

in¬nitesimal parameter for a Weyl rescaling. Then residual reparametriza-

tion symmetries satisfying

‚ ± ξ β + ‚ β ξ ± = Λ· ±β , (2.85)

still remain. These are the reparametrizations that are also Weyl rescalings.

If one de¬nes the combinations ξ ± = ξ 0 ± ξ 1 and σ ± = σ 0 ± σ 1 , then one

2.4 Canonical quantization 41

¬nds that Eq. (2.85) is solved by

ξ ’ = ξ ’ (σ ’ ).

ξ + = ξ + (σ + ) and (2.86)

The in¬nitesimal generators for the transformations δσ ± = ξ ± are given by

1± ± ‚

V± = ξ (σ ) ± , (2.87)

2 ‚σ

and a complete basis for these transformations is given by

±

ξn (σ ± ) = e2inσ

±

n∈ . (2.88)

±

The corresponding generators Vn give two copies of the Virasoro algebra.

In the case of open strings there is just one Virasoro algebra, and the in-

¬nitesimal generators are

‚ ’‚

+

Vn = einσ + einσ n∈ . (2.89)

‚σ ’

‚σ +

In the classical theory the equation of motion for the metric implies the

vanishing of the energy“momentum tensor, that is, T++ = T’’ = 0, which

in terms of the Fourier components of Eq. (2.73) is

+∞

1

±m’n · ±n = 0 m∈ .

Lm = for (2.90)

2 n=’∞

In the case of closed strings, there are also corresponding Lm conditions.

Quantum theory

In the quantum theory these operators are de¬ned to be normal-ordered,

that is,

∞

1

: ±m’n · ±n : .

Lm = (2.91)

2 n=’∞