For even fermion numbers we consider the Fock-space operators
defined in Sec. 2.1, and restrict them to
,
Eqs. (17) and (18).
Then, the complete multiplication table reads
We see that the 16 operators acting in the even-A fermion
spaces constitute the Abelian single group which we denote by D
,
It follows from the multiplication table of the D
operators,
Eqs. (20) and (26),
that the whole group can be generated by its four elements only.
These elements are called the group generators.
Various possibilities of choosing the generators
are discussed in Ref.[14]; here we only mention that,
e.g., the subset {
,
,
,
}
can be used to obtain all the operators that belong to D
.
The D
group has two important subgroups, the eight-element Abelian
group D
composed of all the linear D
operators,
Obviously, the D
subgroup of D
,
being an Abelian group of
order eight, has eight one-dimensional irreps. These can be labelled by
eigenvalues equal to either +1 or -1 of three of its generators,
say,
,
and
.
We introduce names for the (one-dimensional) bases of the associated
irreps according to the following convention. First, a basis is
called invariant if it remains unchanged (belongs to eigenvalue
+1) under all three signature operators
,
and it is called
either x-, or y-, or z-covariant if it transforms under
the signature operators like the x, or y, or z coordinates,
respectively. Secondly, prefix pseudo is added for bases which
are odd, i.e., belong to eigenvalue -1 with respect to the
inversion
.
Since
is an antilinear operator and also an involutive operator
[i.e., its square is equal to
identity, Eq. (20)], we can always choose the phases of all the basis
states so that they belong to the eigenvalue T=+1 of
[15] (see Sec. 2.5 below).
In this way we construct eight irreducible one-dimensional
corepresentations (ircoreps) of D
,
all being even with respect
to the time reversal. After Wigner [8], we call the
representations of a group containing antilinear operators corepresentations to emphasize the fact that they are not the
representations in the usual sense (see Appendix for details).
By a suitable change of phases of the basis states we can obtain
another set of eight ircoreps of D
,
all of them odd (i.e., belonging
to the eigenvalue T=-1) with respect to the time reversal. We use
prefix anti to name these time-odd ircoreps. Obviously,
time-even and time-odd ircoreps are pairwise equivalent
(cf. Ref.[3]).
Note that all operators acting in the even fermion spaces
can also be classified according to the same set of
sixteen ircoreps of D
.
All these ircoreps are listed in Table
3 together with explicit transformation properties of
several examples of one-particle operators belonging to each ircorep.