202
GAUGE FIELDS AND STRINGS
A convenient notational trick is to introduce two sets of harmonic
oscillators
with the ordinary commutation rules
=
flm.v] =
[«, 2] = 0 and associatc with the
operator (9.227) a vector in their Fock space:
(9.230)
^ m i f i i . . . m k M k , n i v i . . .n iv i
. ^ m k / i k ^ n i v i . . A m v i \ ^ ^
We see, that with these notations at hand we can present
as an
eigenvalue of the operator:
= E
,) - 2
(9.231)
m
The states of these oscillators thus can be interpreted as vibrational
excitations of the string. Sometimes this representation is very useful.
An important point is that not all of the operators (9.227) contribute
to the amplitude. The residues of some of the corresponding poles
vanish. For example, it is clear that operators odd under the change of
orientation
(such as (d^x^ d^x^ — d^x^ d^x^) e'^ *) will not
contribute, since the residue, being proportional to
res j/^(/?i.../?^)
I d^z d^Z3.. .d^
(^,
2,)... U^,(z-^, z^)> (9.232)
i'
would vanish since all {Up] are even under this transformation.
Also, the operators must have conformal spin zero i.e. under the
rotation z -► e‘® z they have to be scalars for the same reason as above.
The first excited state of the string thus corresponds to the operator
(9.233)
and has
«2 = -2-h 1-f 1 = 0
We shall analyse its spin content later.
There is another reason for the cancellation of residues. Consider an
operator:
(9.234)
(where we have used the fact, that d^d^x = 0 since jc is a free field. The
same thing was implicitly used in (9.227)).