String Theory and M-Theory

4.1 Ramond–Neveu–Schwarz strings 111
This changes after quantization, of course.
The spinor ψ µ has two components ψ µ
A , A = ±,
ψ µ µ
ψ
= . (4.6)
−
ψ µ
+
Here, and in the following, we define the Dirac conjugate of a spinor as
¯ψ = ψ † β, β = iρ 0 , (4.7)
which for a Majorana spinor is simply ψ T β. Since the Dirac matrices are
purely real, Eq. (4.4) is a Majorana representation, and the Majorana spinors
ψ µ are real (in the sense appropriate to Grassmann numbers)
ψ ⋆ + = ψ+ and ψ ⋆ − = ψ−. (4.8)
In this notation the fermionic part of the action is (suppressing the Lorentz
index)
Sf = i
π
d 2 σ (ψ−∂+ψ− + ψ+∂−ψ+) , (4.9)
where ∂± refer to the world-sheet light-cone coordinates σ ± introduced in
Chapter 2. The equation of motion for the two spinor components is the
Dirac equation, which now takes the form
∂+ψ− = 0 and ∂−ψ+ = 0. (4.10)
These equations describe left-moving and right-moving waves. For spinors
in two dimensions, these are the Weyl conditions. Thus the fields ψ± are
Majorana–Weyl spinors. 3
EXERCISES
EXERCISE 4.1
Show that one can rewrite the fermionic part of the action in Eq. (4.2) in
the form in Eq. (4.9).
SOLUTION
Taking ∂± = 1
2 (∂0 ± ∂1) and the explicit form of the two-dimensional Dirac
3 Group theoretically, they are two inequivalent real one-dimensional spinor representations of
the two-dimensional Lorentz group Spin(1, 1).