String Theory Demystified

130 String Theory Demystifi ed
MAJORANA SPINORS
µ µ
The fi elds introduced in the action, ψ = ψ ( σ, τ),
are two-component Majorana
spinors on the worldsheet. Given that they have two components, they are
sometimes written with two indices ψ µ
, where A µ = 01 , , …, D −1is
the space-time
index and A =± is the spinor index. We can write ψ µ
as a column vector in the
A
following way (suppressing the space-time index):
ψ
ψ =
ψ
⎛ ⎞ −
⎝
⎜
⎠
⎟
Under Lorentz transformations, these fi elds transform as vectors in space-time
µ [recall that a contravariant vector fi eld V ( x)
is one that transforms as
µ µ µ ν
µ µ µ ν
µ
V ( x) → V′ ( x′ ) =Λ V ( x)
under ν
x → x′ =Λ x where Λ is a Lorentz
ν
ν
transformation].
Following the convention used with Dirac spinors in quantum fi eld theory, we
have the defi nition:
+
µ † µ 0
ψ = ( ψ ) ρ
Note that the defi nitions used here depend on the basis used to write down the Dirac
matrices Eq. (7.3), and that other conventions are possible. We can also introduce a
matrix you’re familiar with from studies of
third Dirac matrix analogous to the γ 5
the Dirac equation, which in this context we denote by ρ 3 :
3 0 1 ⎛ 1 0⎞
ρ = ρ ρ =
⎝
⎜
0 −1⎠
⎟
It will be of interest to make left movers and right movers manifest. This can be
done by recalling the following defi nitions:
EXAMPLE 7.2
µ α
Show that ψ ρ ∂ ψ = 2( ψ ⋅∂ ψ + ψ ⋅∂ ψ ).
α µ
±
σ = τ ± σ
(7.5)
1
∂ = ( ∂ ±∂ )
±
2
τ σ (7.6)
∂ =∂ + ∂ ∂ =∂ −∂
(7.7)
τ + − σ + −
− + − + − +