Lectures on String Theory

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4.1.2 Virasoro constraints in quantum theory
At first sight it seems that the natural analog of the classical equations L n = 0 = ¯L n
in quantum theory is to require that a physical state should be annihilated by all
Virasoro generators:
L n |Φ〉 = 0 = ¯L n |Φ〉 , n ∈ Z .
Due to the normal-ordering ambiguity in the definition of L 0 in quantum theory the
classical conditions L 0 = 0 = ¯L 0 are replaced now by
(L 0 − a)|Φ〉 = 0 , (¯L 0 − ā)|Φ〉 = 0 , (4.8)
where in fact the normal-orderings constants a and ā must be equal to each other 8 and
L 0 , ¯L 0 are understood as the normal-ordered generators. It is easy to see, however,
that eqs.(4.8) cannot be consistently imposed for all m. Indeed, if eqs.(4.8) would
be satisfied for all m we would have
[L n , L −n ]|Φ〉 = 2nL 0 |Φ〉 + d 12 n(n2 − 1)|Φ〉 ,
i.e.
0 =
(
2na + d )
12 n(n2 − 1) |Φ〉 for any n .
This is obviously not possible to satisfy unless |Φ〉 = 0. The physical reason for
impossibility to impose in quantum theory the same set of constraints as in the
classical one is an anomaly. Because of the anomaly term the first-class Virasoro
constrains of the classical theory turn upon quantization into the constraints of the
second class!
From the experience with the quantum electrodynamics one can try to impose only
“half” of the constraints, i.e.
(L 0 − a)|Φ〉 = 0 ,
L n |Φ〉 = 0 , n > 0 . (4.9)
The conjugate state then obeys 〈Φ|L −n = 0 for n > 0 and we see that 〈Φ|L n |Φ〉 for
all n ≠ 0, i.e. expectation values of L n vanish for all nonnegative n.
Let us recall that the mass operator is obtained from the constraint L 0 − a = 0, We
have
∞∑
M 2 = −p 2 = 4πT (−a + N) , N = α −nα µ n,µ .
8 This follows from the constraint (L 0 − ¯L 0 )|Φ〉 = 0.
n=1