Lectures on String Theory

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Another example is provided by the Lorentz group of the 4dim Minkowski spacetime,
which is O(3, 1). The spinor representation of O(3, 1) is realized on the space
C 4 , which is called a space of 4-component spinors. 18 This representation looks as
( 1
)
g(ω) = exp
2 γab ω ab ,
where γ ab is the anti-symmetric product of the 4dim γ-matrices and ω ab = −ω ba are
parameters of the Lorentz transformation. [ ]
It is important to note that in general
d
dimension d the spinor has 2 2 complex components.
The spinor ¯ψ = ψ † γ 0 is called the Dirac conjugate of ψ. Its importance is explained
by the fact that the quantity ¯ψψ = ψ † γ 0 ψ is an invariant of O(3, 1).
In any dimension one can define the charge conjugation matrix C. Indeed, the
Clifford algebra of γ-matrices transforms into itself under operation of transposition
{γ a , γ b } t = {(γ a ) t , (γ b ) t } = 2η ab ,
therefore by irreducibility of the corresponding representation of the Clifford algebra
there should exists a matrix C which intertwines the original and the transposed
representation of the algebra, namely:
(γ a ) t = −Cγ a C −1 .
Matrix C is called the charge conjugation matrix.
Sometimes (depending on the dimension and signature od space-time) it is possible
to define the notion of Majorana spinor. Majorana conjugate spinor is, by definition,
ψ t C. The Majorana spinor is then the spinor for which the Dirac conjugate is equal
to the Majorana conjugate:
ψ † γ 0 = ψ t C .
Spinor algebra in two dimensions
18 The group O(3, 1) is not connected and it has four connected components, which however are
not simply connected. The component which contains an identity coincides with SO(3, 1), which
are the transformations preserving orientation of the vierbein. The transformations which preserve
the direction of time are called orthochronous and they form the subgroup SO + (3, 1). The quotient
group O(3, 1)/SO + (3, 1) is the Klein four-group Z 2 ×Z 2 , which is the semidirect product of SO + (3, 1)
with an element of the discrete group {1, P, T, P T }, where P and T are the space inversion and
time reversal operators
P = diag(1, −1, −1, −1) , T = diag(−1, 1, 1, 1) .
The covering (or spin) group of SO + (3, 1) coincides with SL(2, C). One can show that SO + (3, 1) =
SL(2, C)/{I, −I} ≡ PSL(2, C). Thus, SL(2, C) is the double-cover of SO + (3, 1).