String Theory Demystified

80 String Theory Demystifi ed
∞
Σ
=
It follows that D/ 2 n= D/ 2( − 1/ 12) =−D/
24.
Another undetermined constant
n 1
piece is missing from the difference between the general expression of L0 and the
normal ordered expression of L0 . This is a normal ordering constant which is
denoted by a. Therefore in any calculation L0 is replaced by L0− a,
where a is a
constant.
The point of all this is to write down the commutation relations for the Virasoro
operators. Using Eq. (4.10), one fi nds that
D 3
[ Lm, Ln] = ( m− n) Lm+
n+ ( m −m)
δm+
n,
12
0
(4.20)
We call this commutation relation the Virasoro algebra with central extension.
The central charge is the space-time dimension D which has shown up in the second
term on the right-hand side. This is also the number of free scalar fi elds on the
worldsheet. It is clear that if m = 0, ± 1 the central extension term will vanish. This
singles out L1, L0, and L−1which
form a closed subalgebra. We call this the SL
(2, R) algebra.
The Virasoro operators can be used to eliminate unphysical states (i.e., negative
norm states) from the theory by requiring that the expectation value of L0a −
vanishes for a physical state ψ . That is, we impose the constraint
ψ L − aδ
ψ = 0
m m,0
for m ≥ 0. The term aδ m,0
takes care of the fact that we only need the normal
ordering constant a in the case of L0 . To eliminate negative norm states, specifi c
conditions must be put on a and D, which is the origin of the “extra dimensions”
in string theory. In particular, it can be shown that negative norm states can be
eliminated if
a= 1 D=
26 (4.21)
The reason that a = 1is chosen is a bit beyond the scope we want to cover in this
book, see the references if interested in the proof.
We can proceed further to obtain a mass operator. First recall that Einstein’s
equation tells us
µ 2 2 µ
p p + m = 0
⇒ m =−p
p
µ
µ