Lectures on String Theory

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One can correctly anticipate that the problem with negative norm states arose
because we did not take into account the Virasoro constraints. The covariant approach
to quantization consists in defining the subspace of physical states in the
original Hilbert space which obey the Virasoro constraints. One can further show
that in a special dimension of space-time (d = 26) the negative norm states decouple
from the physical Hilbert space.
In the classical theory we have the constraints L m = 0 = ¯L m . However, in
quantum theory expressions for L m and ¯L m are quadratic in oscillators and might
involve operators (quantum oscillators) which do not commute with each other! From
all L m a constraint which suffers from ordering ambiguity is L 0 as
L 0 = 1 2
∞∑
n=−∞
α µ nα −n,µ (4.5)
and oscillators α n µ and α µ −n do not commute with each other. The standard way
to deal with this ambiguity in quantum field theory is to use the normal ordering
prescription
L m = 1 2
∞∑
n=−∞
The normal ordering prescription means that
: α m1 . . . α mk := α n1 . . . α np
} {{ }
all creation
: α µ m−nα n,µ : . (4.6)
α s1 . . . α sr
} {{ }
all annihilation
in the operators are ordered in such a fashion that all annihilation operators are
put on the right from all the creation operators. The order of the creation (or
annihilation) operators between themselves does not matter because these operators
commute between themselves and therefore their expression does not have ordering
ambiguity. In particular, for L 0 we have
L 0 = 1 2 α2 0 +
∞∑
α −nα µ n,µ − a , (4.7)
n=1
where we include a so far unknown normal ordering constant a. As to the zero modes,
the normal-ordering prescription here is
: p µ x ν := x ν p µ .
Since the ground state obeys ˆp µ |p〉 = p µ |p〉 it can be regarded as the usual quantummechanical
eigenstate of the momentum operator ˆp µ = −i ∂
∂x µ
which is in the momentum
representation has the form of the plane-wave
|p〉 ≡ e ipµ x µ
|0〉 ,