According to the normal-ordering prescription the lowering operators always

appear to the right of the raising operators. In particular, L0 becomes

∞

12

±’n · ±n .

L0 = ±0 + (2.92)

2

n=1

Actually, this is the only Virasoro operator for which normal-ordering mat-

ters. Since an arbitrary constant could have appeared in this expression,

one must expect a constant to be added to L0 in all formulas, in particular

the Virasoro algebra.

42 The bosonic string

µ

Using the commutators for the modes ±m , one can show that in the quan-

tum theory the Virasoro generators satisfy the relation

c

[Lm , Ln ] = (m ’ n)Lm+n + m(m2 ’ 1)δm+n,0 , (2.93)

12

where c = D is the space-time dimension. The term proportional to c is a

quantum e¬ect. This means that it appears after quantization and is absent

in the classical theory. This term is called a central extension, and c is called

a central charge, since it can be regarded as multiplying the unit operator,

which when adjoined to the algebra is in the center of the extended algebra.

SL(2, ) subalgebra

¡

The Virasoro algebra contains an SL(2, ) subalgebra that is generated by

¡

L0 , L1 and L’1 . This is a noncompact form of the familiar SU (2) algebra.

Just as SU (2) and SO(3) have the same Lie algebra, so do SL(2, ) and ¡

SO(2, 1). Thus, in the case of closed strings, the complete Virasoro algebra

of both left-movers and right-movers contains the subalgebra SL(2, ) — ¡

SL(2, ) = SO(2, 2). This is a noncompact version of the Lie algebra

¡

identity SU (2) — SU (2) = SO(4). The signi¬cance of this subalgebra will

become clear in the next chapter.

Physical states

As was mentioned above, in the quantum theory a constant may need to be

added to L0 to parametrize the arbitrariness in the ordering prescription.

Therefore, when imposing the constraint that the zero mode of the energy“

momentum tensor should vanish, the only requirement in the case of the

open string is that there exists some constant a such that

(L0 ’ a)|φ = 0. (2.94)

Here |φ is any physical on-shell state in the theory, and the constant a will

be determined later. Similarly, for the closed string

(L0 ’ a)|φ = (L0 ’ a)|φ = 0. (2.95)

Mass operator

The constant a contributes to the mass operator. Indeed, in the quantum

theory Eq. (2.94) corresponds to the mass-shell condition for the open string

∞

2

±’n · ±n ’ a = N ’ a,

±M = (2.96)

n=1

2.4 Canonical quantization 43

where

∞ ∞

na† · an ,

±’n · ±n =

N= (2.97)

n

n=1 n=1

is called the number operator, since it has integer eigenvalues. For the ground

state, which has N = 0, this gives ± M 2 = ’a, while for the excited states

± M 2 = 1 ’ a, 2 ’ a, . . .

For the closed string

∞ ∞

1

± M2 = ±’n · ±n ’ a = ±’n · ±n ’ a = N ’ a = N ’ a. (2.98)

4

n=1 n=1

Level matching

The normal-ordering constant a cancels out of the di¬erence

(L0 ’ L0 )|φ = 0, (2.99)

which implies N = N . This is the so-called level-matching condition of the

bosonic string. It is the only constraint that relates the left- and right-

moving modes.

Virasoro generators and physical states

In the quantum theory one cannot demand that the operator Lm annihilates

all the physical states, for all m = 0, since this is incompatible with the

Virasoro algebra. Rather, a physical state can only be annihilated by half

of the Virasoro generators, speci¬cally

Lm |φ = 0 m > 0. (2.100)

Together with the mass-shell condition

(L0 ’ a)|φ = 0, (2.101)

this characterizes a physical state |φ . This is su¬cient to give vanishing

matrix elements of Ln ’ aδn,0 , between physical states, for all n. Since

L’m = L† , (2.102)

m

the hermitian conjugate of Eq. (2.100) ensures that the negative-mode Vi-

rasoro operators annihilate physical states on their left

φ|Lm = 0 m < 0. (2.103)

44 The bosonic string

There are no normal-ordering ambiguities in the Lorentz generators5

∞

1µν

J µν = xµ pν ’ xν pµ ’ i ν µ

±’n ±n ’ ±’n ±n , (2.104)

n

n=1

and therefore they can be interpreted as quantum operators without any

quantum corrections. Using this expression, it is possible to check that

[Lm , J µν ] = 0, (2.105)

which implies that the physical-state condition is invariant under Lorentz

transformations. Therefore, physical states must appear in complete Lorentz

multiplets. This follows from the fact that, the formalism being discussed