In this section we recall properties of the D
and D
operators
when they are represented in the fermion Fock space.
Within representations,
apart from the corresponding multiplication tables,
Eqs. (20)-(26) and (35)-(41),
these operators are characterized
by their hermitian-conjugation properties. Since
all the Fock-space representations of the D
and D
operators are unitary,
they are hermitian
or antihermitian depending on whether they are involutive
or antiinvolutive, respectively. Properties
of these operators are very different depending on whether they
are linear or antilinear. These characteristics are summarized
in Tables 1 and 2, where the D
and D
operators are split into two or four subsets, respectively. Below we
review the properties of operators in each such subset.
For each linear D
or D
operator one can attribute quantum numbers to
fermion states. These quantum numbers can be equal to
or
for hermitian (involutive) or antihermitian (antiinvolutive) operators, respectively.
Therefore, the parity operators
or
give the
parity quantum numbers,
=
,
the signature operators
give the signature quantum numbers, r=
,
in even systems and
the signature operators
give
r=
in odd systems. Likewise, the simplex operators
and
give the
simplex quantum numbers, s=
and s=
,
respectively.
Antilinear operators do not give good quantum numbers, and their role is very different, depending on whether they are hermitian or antihermitian, Tables 1 and 2.
For each hermitian antilinear D
or D
operator, i.e.,
for
,
,
,
,
,
or
one can find a basis consisting solely of its eigenstates with
the common eigenvalue equal to 1 [15]. Indeed, if
state
is an eigenstate of, e.g.,
,
the corresponding eigenvalue must be a phase, i.e.,
=
.
In such a
case, state
=
is an
eigenstate of
with eigenvalue 1. This
demonstrates explicitly that properties of eigenstates of
are, of course, phase-dependent. In the case when
state
is not an eigenstate of
,
one can transform the two linearly-independent states
and
into eigenstates of
with eigenvalue 1 by symmetrization and antisymmetrization
of the two:
One should also remember, that only linear combinations of basis states with real coefficients remain eigenstates of any hermitian antilinear operator. This is in contrast to properties of linear operators, for which a linear combination of eigenstates, corresponding to the same eigenvalue, with arbitrary coefficients, is also an eigenstate with the same eigenvalue.
Very special properties characterize the antihermitian antilinear
operators. Within the D
or D
groups only the
and
operators in odd systems belong to such a subset
(Table 2). For each antihermitian antilinear D
operator the space of fermion states can be
arranged in pairs of orthogonal states
[15], such that, for example,