218
GAUGE FIELDS AND STRINGS
because effectively the 7^-constraints remove one component of each
oscillator. However, this is not the end of the story because among the
physical states there are some of zero norm. Suppose first that ^ ^ 26.
The only set of zero norm states is produced by:
|/> =
-h
-h "OIO, p>
p^ll =\-n
(9.297)
This state is physical, when T_i acts on a physical state of zero
dimension. Therefore, the number of physical states with nonzero norm
v^(n) is given by :
v^(n) =
- 1)
v(x) = X v^(n)x"
(9.298)
/ oc I \^ - l
In the critical dimension ^ = 26 many more states have zero norm. We
have already observed this in the case of the second level. For the
general case we have to use the Kac formula. Let us see when some
conformal family with c = ^ = 26 has a physical secondary operator at
some level, with dimension one. Using (9.272) we obtain:
.
25 1
^nm + «'w = — + -
+ mcL_y
25 1
(9.299)
We see that degeneracy (or zero norm states) is possible provided that:
3n = 2m±l
(9.300)
and the corresponding level is given by:
n(3n ± 1)
N = nrn =
(9.301)
The previously found state with N = 2 has n = 1, w = 2. It is clear, that
now, instead of (9.298) we will have:
v(x) = (1 — X —
— •••)
00 I \2
(9.302)