248
GAUGE FIELDS AND STRINGS
The first term here again comes from the unit operator, while the
second is the energy-momentum contribution. Combining (9.418) and
(9.402) we get:
1
(x -
1
=
1 6 ( z - x ) ^ ( z - y ) 2 |x - y r
(9.423)
This is compatible with the conformal Ward identity (which, in
particular, says that the residue of the second order pole at z -► x is
equal to A) provided that we chose A = 1/16. We see that conformal
algebra indeed determines the dimensionality of the spin operator, as
well as its correlation functions.
There is an important representation for (9.412). Take its
oo,
X 0 limit, which can be achieved by a projective transformation. We
obtain:
= -
1 Zi -H Z2 1
2 (ZiZ2)^' Zi — z
(9.424)
This formula has an interesting interpretation. Namely, since we have
introduced spins at 0 and oo the fields {¡/(z) have become double valued
in the z-plane with a cut from 0 to oc. We can expect that they have the
following mode expansion:
+ Z
+ K z"))
(9.425)
(9.426)
and proper anticommutations of if/ requires
{•I'o'Po} = 21^0 = 1
{b„,b:}=d„„
Let us now compute the correlation function;
=
1
1 ® /Z2VI 1 Zl + Zz
(ZiZ2)‘'^ 2 ''' „?,U, ) I 2 (ziZj)''^ ;
(9.427)
which is just what we got from (9.412). In the case when spin operators
and the cut are absent, the mode expansion would be
>l/(z) = z
X (b„z " + h;z")l
(9.428)
L" = i/2
J
and
<'A(Zl)il'(Z2)> =
1
f /z^Y _
1
(ziZj)*'^ „=1,2 VzJ Zi - ;
as it should be.