Properties of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators

In this section we recall properties of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ 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 $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ 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 $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators are split into two or four subsets, respectively. Below we review the properties of operators in each such subset.

 

Table 1: Properties of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ operators $\hat{\mathbf{U}}$ in even fermion spaces.

	Linear	Antilinear
Hermitian $(\hat{\mathbf{U}}^2$= $\hat{\mathbf{E}})$	$\hat{\mathbf{P}}$, $\hat{\mathbf{R}}_{k}$, $\hat{\mathbf{S}}_{k}$	$\hat{\mathbf{R}}_{k}^T$, $\hat{\mathbf{S}}_{k}^T$, $\hat{\mathbf{T}}$, $\hat{\mathbf{P}}^T$

 

Table 2: Properties of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators $\hat{\cal{U}}$ in odd fermion spaces.

	Linear	Antilinear
Hermitian $(\hat{\cal{U}}^2$= $\hat{\cal{E}})$	$\hat{\cal{P}}$	$\hat{\cal{R}}_{k}^T$, $\hat{\cal{S}}_{k}^T$
Antihermitian $(\hat{\cal{U}}^2$= $-\hat{\cal{E}})$	$\hat{\cal{R}}_{k}$, $\hat{S}_{k}$	$\hat{\cal{T}}$, $\hat{\cal{P}}^T$

For each linear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operator one can attribute quantum numbers to fermion states. These quantum numbers can be equal to $\pm1$ or $\pm{i}$for hermitian (involutive) or antihermitian (antiinvolutive) operators, respectively. Therefore, the parity operators $\hat{\mathbf{P}}$ or $\hat{\cal{P}}$ give the parity quantum numbers, $\pi$=$\pm1$, the signature operators $\hat{\mathbf{R}}_{k}$ give the signature quantum numbers, r=$\pm1$, in even systems and the signature operators $\hat{\cal{R}}_{k}$ give r=$\pm{i}$ in odd systems. Likewise, the simplex operators $\hat{\mathbf{S}}_{k}$and $\hat{\cal{S}}_{k}$give the simplex quantum numbers, s=$\pm1$ and s=$\pm{i}$, 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 $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operator, i.e., for $\hat{\mathbf{R}}_{k}^T$, $\hat{\mathbf{S}}_{k}^T$, $\hat{\mathbf{T}}$, $\hat{\mathbf{P}}^T$, $\hat{\cal{R}}_{k}^T$, or $\hat{\cal{S}}_{k}^T$one can find a basis consisting solely of its eigenstates with the common eigenvalue equal to 1 [15]. Indeed, if state $\vert\Psi\rangle$ is an eigenstate of, e.g., $\hat{R}^T_{k}$, the corresponding eigenvalue must be a phase, i.e., $\hat{R}^T_{k}\vert\Psi\rangle$= $e^{2i\phi}\vert\Psi\rangle$. In such a case, state $\vert\Psi'\rangle$= $e^{i\phi}\vert\Psi\rangle$ is an eigenstate of $\hat{R}^T_{k}$ with eigenvalue 1. This demonstrates explicitly that properties of eigenstates of $\hat{R}^T_{k}$ are, of course, phase-dependent. In the case when state $\vert\Psi\rangle$ is not an eigenstate of $\hat{R}^T_{k}$, one can transform the two linearly-independent states $\vert\Psi\rangle$ and $\hat{R}^T_{k}\vert\Psi\rangle$ into eigenstates of $\hat{R}^T_{k}$ with eigenvalue 1 by symmetrization and antisymmetrization of the two:

 

$$\vert\Psi_s\rangle = \vert\Psi\rangle + \hat{R}^T_{k}\vert\Psi\rangle,$$ (53)

$$\vert\Psi_a\rangle = i\vert\Psi\rangle - i\hat{R}^T_{k}\vert\Psi\rangle,$$ (54)

which also requires a specific phase. Therefore, phase-convention properties of states are essential for a discussion of bases of eigenstates of the hermitian antilinear operators, and in Ref.[14] a special discussion is devoted to this problem.

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 $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups only the $\hat{\cal{T}}$ and $\hat{\cal{P}}^T$ operators in odd systems belong to such a subset (Table 2). For each antihermitian antilinear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operator the space of fermion states can be arranged in pairs of orthogonal states $(\vert\Psi(+)\rangle,\vert\Psi(-)\rangle)$[15], such that, for example,

$$\hat{\cal{T}}\vert\Psi(\pm)\rangle=\pm \vert\Psi(\mp)\rangle ,$$ (55)

or

$$\hat{\cal{P}}^T\vert\Psi(\pm)\rangle=\pm \vert\Psi(\mp)\rangle .$$ (56)

Therefore, the $\hat{\cal{T}}$ and $\hat{\cal{P}}^T$ operators cannot be diagonalized. In particular, there is no odd fermion state which would be invariant with respect to $\hat{\cal{T}}$ or $\hat{\cal{P}}^T$.