String Theory Demystified

144 String Theory Demystifi ed
In addition, in superstring theory we have a second generator that arises from the
supercurrent
for the NS sector, while we take
∞
2 π
irσ
G = d e J = ⋅b
r ∫ σ
−
+ ∑ α
(7.37)
m r+ m
π π
m=−∞
F = d
m ∑α ⋅
(7.38)
− n m+ n
for the R sector. Here, m and n are integers while r =± 12 /, ± 32 /,….
Canonical Quantization
n
Now we are ready to quantize the theory, and canonical quantization is not so bad
because fermions are simple to deal with. The condition on the modes for the
bosonic string was the commutator:
⎡
⎣α
, α ⎤ mδ
η
m n⎦ m n,
= + 0
µ ν µν
(7.39)
This relation is supplemented by a similar commutator for the α ′ s in the case of
closed strings. For the supersymmetric theory, we need to supplement Eq. (7.39)
with relations for the fermionic modes. You will recall from your studies of quantum
fi eld theory that fermionic fi elds satisfy anticommutation relations. In our case the
Majorana fi elds will satisfy the equal time anticommutation relation:
µ ν µν
{ ψ ( σ, τ), ψ ( σ′ , τ)} = π η δ δ( σ − σ′
)
(7.40)
A B AB
In terms of the modes, we will have the following sets of anticommutation relations
depending on the sector used:
µ ν µν
{ b , b }= η δ
µ ν µν
{ d , d }= η δ
r s r+ s,
0
m n m+ n,
0
(7.41)
The presence of the Minkowski metric in these equations mean that the theory will
still be plagued by negative norm states that we will have to remove.