204
GAUGE FIELDS AND STRINGS
This property expresses conformal invariance of a theory in any
number of dimensions. Indeed, generically, the energy momentum
tensor determines the variation of the action S of any system under the
change of variables:
S-^S + Ô S
ÔS=
+
(9.240)
Hence, the tracelessness (9.239) implies that the action is not changed
for such that
or
(9.241)
(9.242)
(n is the dimensionality of the (^-space).
These conditions for n = 2 are just the Cauchy-Riemann equations,
satisfied by any analytic function. Thus we have an infinite dimensional
conformal group in this case. At the same time, for n > 2 the group is
finite, and consists of the ordinary translations and rotations plus
dilations and inversions.
We see that the formula (9.238) is a condensed expression of
conformal invariance of the theory. It is not just conservation of T since
generically one would also have terms in (9.238) containing derivatives. Their absence is crucial for further discussion.
Let us show that correlation functions involving the operator T can
be determined explicitly by the Ward identities.
We shall give first their formal derivation, and then in order to get a
feeling of how they arise, analyze the special wase of free fields.
It is convenient to treat the variables z and z as independent and to
concentrate on the variation of z. Suppose that we have a set of
operators {0„} which transform as
or
(9.243)
S ,0„iz) = 8(z)d,0„iz) -f A„8'(z)0„(z)