214
GAUGE FIELDS AND STRINGS
In order to find this expression we must use the operator product:
T(z)l/p.;(0)
= z
*: -h less singular terms
(9.276)
(9.277)
Recalling the definition of T„ we find:
TnU,,^ = 0, n>l
We see that the condition for Up to be a primary operator is
P^C,(P) = 0
(9.278)
An equivalent way to derive (9.278) is to use the oscillator representation:
^(z) = 9;, + log Z + X
jm ; 2-" + - ^ z"
T(z)= -:i(5,x)^: = i;r„z-"'^n
V ^ ^ n .n “I " ^
^ m ,ti^ m + n,ix
1
+ I
« > 0
z H=1
(9.279)
T _„ = T „ +
=5P^ + Z iimOm
m = 1
Applying Ti to the state C^(p)ai^^\0, p} we obtain (9.278). This is a
useful demonstration of the complete equivalence of the operator
product formalism with the harmonic oscillator one.
Returning to (9.278) let us notice that in order that 1/^ ^ shall have
dimension 1, we must have
= 0. In this case the number of coupled
states will not be ^ — 1 as follows from (9.278) but ^ — 2. In order to
show this, we notice that among the physical operators i^^
exp(ipjc(z)) with p^C|z(p) = 0 there is one which is both physical and
secondary. We have already shown that the family produced by : e*^*:
with
= 0 is degenerate and:
(9.280)