QUANTUM STRINGS AND RANDOM SURFACES
215
is, according to (9.265), physical for = 0. The same thing is seen from
the fact that
satisfies:
P%=P^=0
(9.281)
This is a standard feature of the photon: its polarization on shell can be
shifted by a gauge transformation:
C ,(P )-C ,(P ) + ^P,
(9.282)
By the use of this transformation we can make Co(p) = 0 and then
P\(P) = -P%ip) = 0
(9.283)
Hence we find that the photon has ^ — 2 possible polarizations. In our
formalism this was the consequence of two facts. First, we had the
condition for the vertex operator to transform properly under the
conformal group: T„Upj^ = 0,n>0. That has left — 1 possible states.
Second, one of these states was seen to be the secondary operator
generated by 1/^ = :e*^ *: with p^ = 0. As was shown at the beginning
of this section all secondary operators decouple. Thus, we got Q) — 2,
It is remarkable that a similar mechanism works at higher levels. Let
us discuss the second one. The most general vertex will be given by:
Lp(s, 0 = :(5,,i a,x"i
+ C,i a^x^)
+ C ,< ,)|0 ,/7 >
(9.284)
p2= - 2
Let us see what forms of S^Xp)
in arc left by the condition of
conformal covariance:
T„U.(S,O^0, n > 0
(9.285)
It is easier to work out in the oscillator representation. Applying (9.279)
we find:
= 2(p,S,, + CvXv|0,p>=0
TiiSnvKuKv + C^«2m )|0,P>
= (2p,C, + S,,)|0,/,>=0
So, the conditions for physical polarizations are:
P / . +
= 0
Pu^uv + C v = 0. P ^ = - 2
(9.286)
(9.287)