7.2 First-Order bc Ghost System
173
Energy Normal Ordering (Grassmann Odd)
We now turn towards the definition of the energy normal ordering (6.150). Ultimately, it will be found that | ↓↓ is the physical vacuum in string theory. For this
reason, the energy normal ordering
· · ·
is associated with the vacuum | ↓↓ in
order to resolve the ambiguity of the zero-modes. In particular, b 0 is an annihilation
operator in this case, while c 0 is a creation operator. In the rest of this section,
we translate the normal ordering of expressions from the conformal vacuum to the
energy vacuum.
The Virasoro operators L n for n = 0 have no ordering problems since the modes
which compose them commute. The expression of L 0 (7.129) in the energy ordering
becomes
L 0 =
n
n
b −n c n
+ a λ = N
b
+ N
c
+ a λ ,
(7.159)
where a λ is the zero-point energy (7.152) and N b and N c are the ghost mode
numbers (7.127). The contribution of the non-zero modes is denoted by
L 0 = N
b
+ N
c .
(7.160)
The expression can be rewritten to encompass all modes:
L m =
n
n − (1 − λ)m
b m−n c n
+ a λ δ m,0 .
(7.161)
Similarly, the expression of the ghost number is
N gh,L = j 0 =
n
b −n c n
−
q λ
2
+
1
2
(7.162a)
=
n>0
N
c
n − N
b
n
+
1
2
N
c
0 − N
b
0
−
q λ
2
,
(7.162b)
and thus:
j m =
n
b m−n c n
−
q λ
2
+
1
2
δ m,0 .
(7.163)
It is useful to define the ghost number without ghost zero-modes:
N gh,L :=
n>0
N
c
n − N
b
n
.
(7.164)