1 two component formalism for spin 12 fermions that

1 Two-component formalism for Spin-1/2 Fermions
That is, the fields χ
1
and χ
2
obey the standard time reversal transforma-
tion laws of a single two-component fermion field [eqs. (1.155)–(1.158)],
but with opposite sign phase factors η
T
in each case. Had one chosen
η
T
= η
∗
T
in eqs. (1.175) and (1.176), then the time reversal transformation
laws of χ
1
and χ
2
would have been more complicated [eqs. (1.179) and
(1.180)]. But as before, one is free to make a further SO(2) rotation
to transform χ
1
and χ
2
into new fields that do exhibit the simpler time
reversal transformation laws [eq. (1.151)].
One can now work out the time reversal properties of bilinear covariants
constructed out of the fields of a charged fermion pair. The results are
listed in Table B.3.
1.10 Charge conjugation of two-component spinors
Charge conjugation was introduced in section 1.7. The charge conjugation
operator is a discrete operator that interchanges particles and their C -
conjugates. Here, conjugation refers to some conserved charge operator
Q , where C Q C
− 1
= − Q . The conjugate of the conjugate field is the
original field, so that C
2
= 1. For a single two-component fermion field
C χ
α
( x ) C
− 1
≡ χ
C
α
( x ) = η
C
χ
α
( x ) , (1.183)
C χ
˙ α
( x ) C
− 1
≡ χ
C
˙ α
( x ) = η
∗
C
χ
˙ α
( x ) , (1.184)
where | η
C
| = 1. In this case, no conserved charge exists, so that charge
conjugation is trivial. In particular, as noted in section 1.7, the invariance
of the mass term χχ + χ χ implies that η
∗
C
= η
C
, and hence η
C
= ± 1.
27
The behavior of the bilinear covariants under C is simple:
χ
C
1
O χ
C
2
= η
C 1
η
C 2
χ
1
O χ
2
, (1.185)
for any O = I
2
, σ
µ
, σ
µ
, σ
µν
and σ
µν
(where bars should appear over the
appropriate χ
i
depending on the choice of O ).
In theories with multiple two-component fermion fields with no global
symmetries, the charge conjugation properties of each fermion is given
by eqs. (1.183) and (1.184), with η
C
= ± 1. When interactions are
included, if there is some choice of the η
C
such that the action is invariant
under charge conjugation, then the theory is charge conjugation invariant.
If there is an internal global symmetry ( χ
i
→ U
ij
χ
j
) under which
the Lagrangian is invariant, then there are relations among the charge
conjugation transformations of the fermion fields. Here, we consider
27
In a theory with just one fermion field, no physical quantity can depend on the sign of
η
C
since the fermion field must appear quadratically in the Lagrangian. So, without
loss of generality, we can take η
C
= 1.

1.11 CP and CPT conjugation of two-component spinors 41
the simplest case of two mass-degenerate fermion fields, χ
1
and χ
2
, and
identify the conserved charge as Q = J
0
d
3
x , where the conserved
current J
µ
is given in eq. (1.108). In terms of the linear combinations
ξ and η [eqs. (1.103) and (1.104)] corresponding to the fields of definite
charge, we define the charge conjugation transformations as follows.