Single group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ for even systems

For even fermion numbers we consider the Fock-space operators defined in Sec. 2.1, and restrict them to ${\cal{H}}_+$, Eqs. (17) and (18). Then, the complete multiplication table reads

       

$$\hat{\mathbf{R}}_{k}^2=\hat{\mathbf{S}}_{k}^2={\hat{\mathbf{T}}}^2 \hat{\mathbf{E}},$$ = (20)

$$\left({\hat{\mathbf{R}}_{k}^T}\right)^2=\left({\hat{\mathbf{S}}_{k}^T}\right)^2= {\hat{\mathbf{P}}}^2 \hat{\mathbf{E}},$$ = (21)

$$\hat{\mathbf{R}}_{k}\hat{\mathbf{S}}_{k}=\hat{\mathbf{S}}_{k}\hat{\mathbf{R}}_{k} \hat{\mathbf{P}},$$ = (22)

$$\hat{\mathbf{R}}_{k}^T\hat{\mathbf{S}}_{k}^T=\hat{\mathbf{S}}_{k}^T\hat{\mathbf{R}}_{k}^T \hat{\mathbf{P}},$$ = (23)

$$\hat{\mathbf{R}}_{k}\hat{\mathbf{R}}_{k}^T=\hat{\mathbf{R}}_{k}^T... ...mathbf{S}}_{k}\hat{\mathbf{S}}_{k}^T=\hat{\mathbf{S}}_{k}^T\hat{\mathbf{S}}_{k} \hat{\mathbf{T}},$$ = (24)

$$\hat{\mathbf{R}}_{k}\hat{\mathbf{S}}_{k}^T=\hat{\mathbf{S}}_{k}^T... ...mathbf{R}}_{k}^T\hat{\mathbf{S}}_{k}=\hat{\mathbf{S}}_{k}\hat{\mathbf{R}}_{k}^T \hat{\mathbf{P}}^T,$$ = (25)

for k=x,y,z, and

     

$$\hat{\mathbf{R}}_{k}\hat{\mathbf{R}}_{l} = \hat{\mathbf{S}}_{k}\h... ...{R}}_{k}^T\hat{\mathbf{R}}_{l}^T = \hat{\mathbf{S}}_{k}^T\hat{\mathbf{S}}_{l}^T \hat{\mathbf{R}}_{m},$$ = (26)

$$\hat{\mathbf{R}}_{k}\hat{\mathbf{S}}_{l} = \hat{\mathbf{S}}_{k}\h... ...{R}}_{k}^T\hat{\mathbf{S}}_{l}^T = \hat{\mathbf{S}}_{k}^T\hat{\mathbf{R}}_{l}^T \hat{\mathbf{S}}_{m},$$ = (27)

$$\hat{\mathbf{R}}_{k}\hat{\mathbf{R}}_{l}^T = \hat{\mathbf{R}}_{k}... ...thbf{S}}_{k}\hat{\mathbf{S}}_{l}^T = \hat{\mathbf{S}}_{k}^T\hat{\mathbf{S}}_{l} \hat{\mathbf{R}}_{m}^T,$$ = (28)

$$\hat{\mathbf{R}}_{k}^T\hat{\mathbf{S}}_{l} = \hat{\mathbf{S}}_{k}... ...thbf{R}}_{k}\hat{\mathbf{S}}_{l}^T = \hat{\mathbf{S}}_{k}^T\hat{\mathbf{R}}_{l} \hat{\mathbf{S}}_{m}^T,$$ = (29)

for (k,l,m) being an arbitrary permutation of (x,y,z).

We see that the 16 operators acting in the even-A fermion spaces constitute the Abelian single group which we denote by D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$,

$$\mbox{D$_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize... ...hbf{P}}^T, \hat{\mathbf{R}}_{k}^T, \hat{\mathbf{S}}_{k}^T \},$$ (30)

for k=x,y,z. The half of the elements in Eq. (30) are linear operators and the other half are antilinear.

It follows from the multiplication table of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ 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 { $\hat{\mathbf{T}}$, $\hat{\mathbf{P}}$, $\hat{\mathbf{R}}_{x}$, $\hat{\mathbf{R}}_{y}$} can be used to obtain all the operators that belong to D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$.

The D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ group has two important subgroups, the eight-element Abelian group D $_{\mbox{\rm\scriptsize {2h}}}$ composed of all the linear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ operators,

$$\mbox{D$_{\mbox{\rm\scriptsize {2h}}}$ }: \quad \{ \hat{\ma... ...t{\mathbf{P}}, \hat{\mathbf{R}}_{k}, \hat{\mathbf{S}}_{k} \},$$ (31)

and the four-element Abelian group D $_{\mbox{\rm\scriptsize {2}}}$,

$$\mbox{D$_{\mbox{\rm\scriptsize {2}}}$ }: \quad \{ \hat{\mathbf{E}}, \hat{\mathbf{R}}_{k} \}.$$ (32)

Obviously, the D $_{\mbox{\rm\scriptsize {2h}}}$ subgroup of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, 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, $\hat{\mathbf{P}}$, $\hat{\mathbf{R}}_{x}$ and $\hat{\mathbf{R}}_{y}$.

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 $\hat{\mathbf{R}}_{k}$, 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 $\hat{\mathbf{P}}$.

Since $\hat{\mathbf{T}}$ 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 $\hat{\mathbf{T}}$[15] (see Sec. 2.5 below). In this way we construct eight irreducible one-dimensional corepresentations (ircoreps) of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, 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 $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$, 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 ${\cal{H}}_+$ can also be classified according to the same set of sixteen ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$. All these ircoreps are listed in Table 3 together with explicit transformation properties of several examples of one-particle operators belonging to each ircorep.