which provides the motivation for this basis choice. In particular, we can
now identify
ξ
and
η
as fields of definite and opposite charge
Q
. Together,
ξ
and
η
constitute a single Dirac spin-1/2 fermion.
Finally, we introduce the concept of
off-shell
degrees of freedom and
contrast this with
on-shell
or physical degrees of freedom. Here, off-shell
[on-shell] is shorthand for off- [on-] mass shell and distinguishes between
virtual and real particle propagation, respectively.
Given a particle
with four-momentum
p
, real particle propagation must satisfy
p
2
=
m
2
(the mass-shell condition), whereas for virtual particles (
e.g.
, particle
exchange inside Feynman diagrams), the four-momentum
p
and the mass
m
are independent.
The field operator corresponding to real particle
propagation satisfies the field equations, whereas the field operator of a
virtual particle is not constrained by the field equations.
For scalar (spin-0) fields, there is no distinction between on-shell and
off-shell degrees of freedom. The on-shell scalar field satisfies the Klein-
Gordon equation, but this does not alter the fact that a real (complex)
scalar field consists of one (two) degree(s) of freedom.
For a spin-
s
field (
s >
0), the number of on-shell degrees of freedom is less than
the number of off-shell degrees of freedom. Although all on-shell spin-
s
fields satisfy the Klein-Gordon equation, the spin-
s
field equations provide
additional constraints that reduce the number of degrees of freedom
originally present.
We illustrate this in the case of free fermion field
theory.
A theory of a neutral self-conjugate fermion is described in terms of
a single complex two-component fermion field
χ
α
(
x
).
This corresponds
initially to four off-shell degrees of freedom, since
χ
α
and
χ
˙
α
≡
(
χ
†
)
α
are independent degrees of freedom.
But, when the field equations are
imposed [see eq. (1.100)],
χ
(
x
) is determined in terms of
χ
(
x
).
Thus,
the number of physical (“on-shell”) degrees of freedom for a neutral self-
conjugate fermion is equal to two.
A theory of a charged fermion is described in terms of a pair of mass-
degenerate two-component fermions. We shall work in a basis, where the
charged fermion is represented by a pair of two-component fermion fields
ξ
and
η
. The number of off-shell degrees of freedom for a charged fermion
is eight. Applying the field equations determines
ξ
in terms of
η
and ¯
η
in terms of
ξ
[see eqs. (1.106) and (1.107)]. Thus, the number of physical
on-shell degrees of freedom for a charged fermion is equal to four.

26
1 Two-component formalism for Spin-1/2 Fermions
Although we have illustrated the counting of degrees of freedom for free
fermion field theory, the same counting applies in an interacting theory.
1.6 The fermion mass-matrix and its diagonalization
We now generalize the discussion of Section 1.5 and consider a collection
of
n
free anti-commuting two-component spin-1/2 fields,
ζ
αi
(
x
), which
transform as (
1
2
,
0) fields under the Lorentz group. Here,
α
is the spinor
index, and
i
is a ﬂavor index (
i
= 1
,
2
, . . . , n
) that labels the distinct fields
of the collection. The free-field Lagrangian is given by