String Theory Demystified

142 String Theory Demystifi ed
The Majorana condition is the requirement that the fermionic fi elds are real. This
forces us to take
d d
− n n = µ µ ( )
† (7.29)
Here, the summation index is an integer and so runs n = 0, ± 1, ± 2,….
The NS sector results in different mode expansions, as you might guess, since
this gives rise to different string states. The expansions are
ψ ( σ, τ)
=
µ µ −ir ( τ−σ) −
r
r
ψ ( σ, τ)
=
1
2
1
2
∑
∑
be
be
µ µ −ir
( τ+ σ)
+
r
r
(7.30)
This is more than simple notational gymnastics. The summations in the NS sector
are quite different than those for the R sector, because here we take
1 3 5
r = ± , ± , ± ,…
2 2 2
(7.31)
CLOSED STRING BOUNDARY CONDITIONS
In the case of closed strings, we can apply periodic or antiperiodic boundary
conditions. These are given by
ψ ( σ, τ) = ψ ( σ + π , τ)
(periodic boundary condition)
± ±
ψ ( σ, τ) =− ψ ( σ + π , τ)
(antiperiodic boundary
condition) (7.32)
± ±
CLOSED STRING MODE EXPANSIONS
The boundary conditions in Eq. (7.32) can be applied separately to left and right
movers. The mode expansions are
ψ ( σ, τ)
=
µ µ − 2ir
( τ+ σ)
+
r
µ
−
r
µ −2ir
( τ−σ) ∑d
e r
r
ψ ( σ, τ)
=
∑
d e
(7.33)