Absence of negative-norm states

The goal of this section is to show that a spectrum free of negative-norm

states is only possible for certain values of a and the space-time dimension D.

In order to carry out the analysis in a covariant manner, a crucial ingredient

is the Virasoro algebra in Eq. (2.93).

In the quantum theory the values of a and D are not arbitrary. For

some values negative-norm states appear and for other values the physical

Hilbert space is positive de¬nite. At the boundary where positive-norm

states turn into negative-norm states, an increased number of zero-norm

states appear. Therefore, in order to determine the allowed values for a and

D, an e¬ective strategy is to search for zero-norm states that satisfy the

physical-state conditions.

Spurious states

A state |ψ is called spurious if it satis¬es the mass-shell condition and is

orthogonal to all physical states

(L0 ’ a)|ψ = 0 and φ|ψ = 0, (2.106)

where |φ represents any physical state in the theory. An example of a

spurious state is

∞

|ψ = L’n |χn (L0 ’ a + n)|χn = 0.

with (2.107)

n=1

5 J ij generates rotations and J i0 generates boosts.

2.4 Canonical quantization 45

In fact, any such state can be recast in the form

|ψ = L’1 |χ1 + L’2 |χ2 (2.108)

as a consequence of the Virasoro algebra (e.g. L’3 = [L’1 , L’2 ]). Moreover,

any spurious state can be put in this form. Spurious states |ψ de¬ned this

way are orthogonal to every physical state, since

∞ ∞

φ|L’n |χn = χn |Ln |φ

φ|ψ = = 0. (2.109)

n=1 n=1

If a state |ψ is spurious and physical, then it is orthogonal to all physical

states including itself

∞

χn |Ln |ψ = 0.

ψ|ψ = (2.110)

n=1

As a result, such a state has zero norm.

Determination of a

When the constant a is suitably chosen, a class of zero-norm spurious states

has the form

|ψ = L’1 |χ1 (2.111)

with

(L0 ’ a + 1)|χ1 = 0 Lm |χ1 = 0

and m > 0. (2.112)

Demanding that |ψ is physical implies

Lm |ψ = (L0 ’ a)|ψ = 0 for m = 1, 2, . . . (2.113)

The Virasoro algebra implies the identity

L1 L’1 = 2L0 + L’1 L1 , (2.114)

which leads to

L1 |ψ = L1 L’1 |χ1 = (2L0 + L’1 L1 )|χ1 = 2(a ’ 1)|χ1 = 0, (2.115)

and hence a = 1. Thus a = 1 is part of the speci¬cation of the boundary

between positive-norm and negative-norm physical states.

46 The bosonic string

Determination of the space-time dimension

The number of zero-norm spurious states increases dramatically if, in addi-

tion to a = 1, the space-time dimension is chosen appropriately. To see this,

let us construct zero-norm spurious states of the form

|ψ = L’2 + γL2 |χ . (2.116)

’1

This has zero norm for a certain γ, which is determined below. Here |ψ is

spurious if |χ is a state that satis¬es

(L0 + 1)|χ = Lm |χ = 0 for m = 1, 2, . . . (2.117)

Now impose the condition that |ψ is a physical state, that is, L1 |ψ = 0 and

L2 |ψ = 0, since the rest of the constraints Lm |ψ = 0 for m ≥ 3 are then

also satis¬ed as a consequence of the Virasoro algebra. Let us ¬rst evaluate

the condition L1 |ψ = 0 using the relation

L1 , L’2 + γL2 = 3L’1 + 2γL0 L’1 + 2γL’1 L0

’1

= (3 ’ 2γ)L’1 + 4γL0 L’1 . (2.118)

This leads to

L1 |ψ = L1 L’2 + γL2 |χ = [(3 ’ 2γ) L’1 + 4γL0 L’1 ] |χ . (2.119)

’1

The ¬rst term vanishes for γ = 3/2 while the second one vanishes in general,

because

L0 L’1 |χ = L’1 (L0 + 1)|χ = 0. (2.120)

Therefore, the result of evaluating the L1 |ψ = 0 constraint is γ = 3/2. Let

us next consider the L2 |ψ = 0 condition. Using

3 D

L2 , L’2 + L2 = 13L0 + 9L’1 L1 + (2.121)

’1

2 2

gives

3 D

L2 |ψ = L2 L’2 + L2 |χ = ’13 + |χ . (2.122)

2 ’1 2

Thus the space-time dimension D = 26 gives additional zero-norm spurious

states.

2.4 Canonical quantization 47

Critical bosonic theory

The zero-norm spurious states are unphysical. The fact that they are spu-

rious ensures that they decouple from all physical processes. In fact, all

negative-norm states decouple, and all physical states have positive norm.

Thus, the complete physical spectrum is free of negative-norm states when

the two conditions a = 1 and D = 26 are satis¬ed, as is proved in the

next section. The a = 1, D = 26 bosonic string theory is called critical,

and one says that the critical dimension is 26. The spectrum is also free of

negative-norm states for a ¤ 1 and D ¤ 25. In these cases the theory is

called noncritical. Noncritical string theory is discussed brie¬‚y in the next

chapter.

EXERCISES

EXERCISE 2.10

Find the mode expansion for angular-momentum generators J µν of an open

bosonic string.

SOLUTION