216
GAUGE FIELDS AND STRINGS
Again, among these physical states some are secondary. This implies
that there is some second level gauge invariance in (9.287). To find it, let
us use formulas (9.269) and (9.271) for the case A = — 1 (in this case the
secondary operator will have A = 1). We conclude that the states
l«> = (T-2 + fTli)|0,/>>, / = - 2
(9.288)
and
\b} = T.,X^(p)atJO^pX
= 0
are physical if c = ^ = 26 in (9212). The state \ b} corresponds to the
following gauge transformation:
2 ( P a ^ p + P p K )
(9.289)
with p^À^ = 0. It is easy to see, that the conditions (9.287) are invariant
under (9.289) provided that = —2. This first gauge invariance works
for all
The second one, associated with the state |a> is true only for
^ = 26. After some computations, one finds:
Sap -► + 8(d^p -h 3p^Pp)
(9.290)
In both cases, gauge transformations create states of zero norm, just as
was the case with photons (recall that the state Pf^a^^lO, p} with = 0
is of zero norm). The reason for zero norm is quite simple. If any state
l/> is physical:
r j /> = o, v«>o
and secondary:
then:
l/> = T _ J^ > ; T „ ( T . J g } ) = 0
(9.291)
The meaning of this result is that at each level we have a certain gauge
transformation, which does not change the physics, but creates zero
norm states. It is also clear, that existence of such an invariance implies
a considerable reduction of the physical spectrum. Let us now count the
number of states remaining. Consider the states with p^ = —m^ =
2(1 — n). They are represented in the oscillator formalism as:
l/n) =
S
+ ‘ ‘ *)|0» P>
p^l2 = I - n
(9.292)