String Theory and M-Theory

4.4 Boundary conditions and mode expansions 123
The overall relative sign between ψ µ
+ and ψµ
− is a matter of convention.
Therefore, without loss of generality, one can choose to set
ψ µ
+ |σ=0 = ψ µ
− |σ=0. (4.50)
The relative sign at the other end then becomes meaningful, and there are
two possible cases:
• Ramond boundary condition: In this case one chooses at the second end
of the string
ψ µ
+ |σ=π = ψ µ
− |σ=π. (4.51)
As is shown later, Ramond (or R) boundary conditions give rise to spacetime
fermions. The mode expansion of the fermionic field in the R sector
takes the form
ψ µ 1
− (σ, τ) = √
n∈¢
2
dµ ne −in(τ−σ) , (4.52)
ψ µ
+ (σ, τ) = 1
√
2
n∈¢ dµ ne −in(τ+σ) . (4.53)
The Majorana condition requires these expansions to be real, and hence
d µ
−n
= dµ†
n . The normalization factor is chosen for later convenience.
• Neveu–Schwarz boundary condition: This boundary condition corresponds
to choosing a relative minus sign at the second end of the string, namely
ψ µ
+ |σ=π = −ψ µ
− |σ=π. (4.54)
As is shown later, Neveu–Schwarz (or NS) boundary conditions give rise
to space-time bosons. The mode expansion in the NS sector is
ψ µ
− (σ, τ) = 1 √ 2
ψ µ
+ (σ, τ) = 1 √ 2
r∈¢ +1/2
r∈¢ +1/2
b µ r e −ir(τ−σ) , (4.55)
b µ r e −ir(τ+σ) . (4.56)
In the following, the letters m and n are used for integers while r and s
are used for half-integers, that is,
m, n ∈
while r, s ∈ + 1
. (4.57)
2