QUANTUM STRINGS AND RANDOM SURFACES
203
Owing to the fact that (9.234) is a total derivative its contribution to
the residue (9.232) is equal to zerof, or in other words the amplitude for
emission of the massless tensor state (9.233) is purely transverse. This is
a very nice fact because otherwise in the Minkowski continuation of our
theory we would inevitably have negative norm states, the time-like
gravitons.
There are actually many more cancellations. To find them we have to
exploit the conformal properties of the underlying two-dimensional
theory.
9.9 The Energy-Momentum Tensor in Conformal
Quantum Field Theory
There exists a remarkable mechanism which transforms the conformal
Ward identities of the two-dimensional field theory of a string into
certain transversality conditions, generalizing the decoupling of the
states similar to (9.234). This mechanism will be important not only for
^ = 26, but for all ^ and for that reason we shall investigate it in detail.
Let us begin with the derivation of the Ward identities in the case
when only the free jc-field is present ( ^ = 26). The energy-momentum
tensor of such a field is given by:
Ta/i) =
- ^ô^p(d;,xŸ
and due to the equations of motion it is conserved:
dT^ = 0
It is very helpful to use a complex notation and to introduce:
T(z, z) = Til - 722 -I- 2iTi2 = d^X'd^x
The conservation law now reads
djT(z, z) = Id^x • d^d^x = 0
(9.235)
(9.236)
(9.237)
(9.238)
implying that T is an analytic function of z. In the derivation of (9.238)
we have used not only (9.236) but also the following important property
of (9.235):
7,« = 0
(9.239)
t Because integrals are analytic in Pi Pj and there is always a region in the space of
momenta pj where all boundary terms vanish.