Lectures on String Theory

– 96 –
The two-dimensional Dirac matrices ρ a , a = 0, 1 obey the algebra
{ρ a , ρ b } = 2η ab , η ab =
A particular basis foe the Clifford algebra is given by
ρ 0 =
( ) 0 1
, ρ 1 =
−1 0
( ) −1 0
.
0 +1
( ) 0 1
.
1 0
We also define a matrix ¯ρ
¯ρ = ρ 0 ρ 1 =
( 1 0
)
0 − 1
being a 2dim analogue of the 4dim matrix γ 5 . The charge conjugation matrix can be
taken to be C = ρ 0 . The Majorana spinor is then ψ † ρ 0 = ψ t C = ψ t ρ 0 , i.e. ψ = ψ ∗
which simple means that the spinor is real. Thus, in 2dim Majorana spinor is just a
spinor with real components.
Finally, with the help of the vielbein (“zweibein” in 2dim) one can define the
“curved” ρ-matrices:
ρ α = e α aρ a .
They satisfy the following algebra
{ρ α , ρ β } = 2h αβ .
Spinors in 2dim have various interesting properties. One of them is the so-called
spin-flip identity. If we have two Majorana spinors ψ 1 and ψ 2 , then the following
identity is valid
¯ψ 1 ρ α1 · · · ρ αn ψ 2 = (−1) n ¯ψ2 ρ αn · · · ρ α 1
ψ 1 .
It is proved as follows.
¯ψ 1 ρ α1 · · · ρ α n
ψ 2 = ( ¯ψ 1 ρ α1 · · · ρ α n
ψ 2 ) t = −ψ t 2(ρ α n
) t · · · (ρ α 1
) t (ρ 0 ) t ψ 1
= (−1) n ψ t 2Cρ αn C −1 · · · Cρ α 1
C −1 Cψ 1 = (−1) n ¯ψ2 ρ αn · · · ρ α 1
ψ 1 .
Another identity is
ρ α ρ β ρ α = 0 . (6.1)
Indeed, one has from the Clifford algebra that ρ a ρ α = 2 and, therefore
ρ α ρ β ρ α = −ρ α ρ α ρ β + 2ρ β = −2ρ β + 2ρ β = 0 .