Fock-space representations

It will be convenient to use the Cartesian representation of the symmetry operators. Let $\hat{I}_k$ for k=x,y,z denote the Cartesian components of the total angular momentum operator (generators of the group of rotations). In the coordinate-space representation these operators read ($\hbar$=1)

$$\hat{I}_k \equiv \sum_{n=1}^A {\hat{j}}_k^{(n)} = \hat{L}_... ...^{(n)} + {\textstyle{\frac{1}{2}}}\hat{\sigma}_k^{(n)}\right),$$ (1)

where ${\hat{j}}_k^{(n)}$, ${\hat{l}}_k^{(n)}$, and ${\textstyle{\frac{1}{2}}}\hat{\sigma}_k^{(n)}$ are operators of the total, orbital, and intrinsic angular momenta, respectively, of particle number n. By definition, operators (1) act in the Hilbert space ${\cal{H}}_A$ of A-particle states, and the number of particles A appears explicitly in their definitions.

One can use another representations of $\hat{I}_k$, the so-called second-quantized, or Fock-space form,

$$\hat{I}_k = \sum_{\mu\nu} \langle\mu\vert\hat{j}_{k}\vert\nu\rangle a^+_\mu a_\nu,$$ (2)

where $\langle\mu\vert\hat{j}_{k}\vert\nu\rangle$ are the matrix elements of the angular-momentum operators in the single-particle basis defined by the fermion creation and annihilation operators $a^+_\mu$ and $a_\nu$. Operators $\hat{I}_k$ in the form of Eq. (2) do not explicitly depend on A, and act simultaneously in all the A-particle spaces, i.e., they act in the Fock space ${\cal{H}}$,

$${\cal{H}} \equiv {\cal{H}}_0\oplus{\cal{H}}_1\oplus\ldots \oplus{\cal{H}}_A\oplus\ldots.$$ (3)

In each subspace ${\cal{H}}_A$, operators (2) are equal to (1). Since both act in different domains, one should, in principle, denote them with different symbols. However, one usually understands definition (1) as a prescription to construct $\hat{I}_k$ for all values of A simultaneously (adjoined by $\hat{I}_k$$\equiv$0 for A=0). With this extension, operators (1) and (2) are equal. In this section we understand that all operators act in the Fock space (3), while the corresponding definitions are given in the coordinate-space representation.

We introduce three standard transformations of rotation around three mutually perpendicular axes, ${\cal{O}}x$, ${\cal{O}}y$ and ${\cal{O}}z$, through the angles of $\pi$as

$$\hat{R}_{k} \equiv e^{-i\pi\hat{I}_k} = \bigotimes_{n=1}^A e^{-i\pi{\hat{j}}_k^{(n)}}.$$ (4)

Similarly, we introduce three operators of reflection in planes yz, zx, and xy, for k=x,y,z, respectively, which can be written as

$$\hat{S}_{k} \equiv \hat{P}\hat{R}_{k} ,$$ (5)

where the inversion operator is denoted by $\hat{P}$. The order of operators in Eq. (5) is unimportant because

$$\lbrack\hat{P}, \hat{R}_{k}\rbrack=0.$$ (6)

Finally, the (antilinear) time-reversal operator in the coordinate-space representation is defined as [15]:

$$\hat{T}\equiv \bigotimes_{n=1}^A\left(-i\hat{\sigma}_y^{(n)}\right) \hat{K},$$ (7)

where $\hat{K}$ is the complex conjugation operator associated with the coordinate representation.

In what follows it will be convenient to denote with separate symbols the products of $\hat{T}$ with $\hat{P}$, $\hat{R}_{k}$, and $\hat{S}_{k}$ [16], i.e., the seven additional (apart from $\hat{T}$ itself) antilinear operators read

    

$$\hat{P}^T \equiv \hat{P}\hat{T},$$ (8)

$$\hat{R}^T_{k} \equiv \hat{R}_{k}\hat{T},$$ (9)

$$\hat{S}^T_{k} \equiv \hat{S}_{k}\hat{T}.$$ (10)

The order of multiplications in the above definitions is irrelevant since

$$\lbrack\hat{P}, \hat{T}\rbrack = \lbrack\hat{R}_{k},\hat{T}\rbrack = \lbrack\hat{S}_{k},\hat{T}\rbrack=0.$$ (11)

In nuclear physics applications the linear operators $\hat{P}$, $\hat{R}_{k}$, and $\hat{S}_{k}$ are usually referred to as inversion, signature, and simplex. The antilinear operators $\hat{P}^T$, $\hat{R}^T_{k}$, and $\hat{S}^T_{k}$ will be called T-inversion, T-signature, and T-simplex, respectively.

For completeness, yet two another operators must be added to the above symmetry operators. One of them is, of course, the identity operator, $\hat{E}$, which can be treated as the rotation through angle equal to 0 or $4\pi$ about an arbitrary axis. The second one is the rotation through angle $2\pi$ about an arbitrary axis, i.e.,

$$\bar{E}\equiv e^{-i2\pi\hat{I}_k} =e^{-i2\pi\hat{L}_k}\bigotimes_{n=1}^A\left(-\hat{\sigma}_0^{(n)}\right) = (-1)^A\hat{E}$$ (12)

where $\hat{\sigma}_0$ is the unity $2\times 2$ matrix. We see that only for even systems $\bar{E}$ is equal to identity, while for odd systems it is equal to the minus identity. We should keep in mind, that in the group theory there is no such a notion as a change of sign. Operators like (-1)^A may appear in representations, like here they do appear in the Fock-space representation, however, one cannot use them when defining the group structures in Secs. 2.2 and 2.3 below.

To investigate multiplication rules of the symmetry operators introduced above one explicitly calculates products of them. For example, the products of two signatures are

 

$$\hat{R}_{k}\hat{R}_{m} e^{-i\pi\hat{L}_k}e^{-i\pi\hat{L}_m}\bigotimes_{n=1}^A \left(-\hat{\sigma}_k^{(n)}\hat{\sigma}_m^{(n)}\right) ,$$ = (13)

and the square of the time reversal reads

$${\hat{T}}^2= \bigotimes_{n=1}^A\left(-({\hat{\sigma}}_y^{(n)})^2\right)=\bar{E}.$$ (14)

It is obvious that these results depend on whether A is even or odd.

Therefore, in what follows we introduce notation which explicates whether the operators act in even or odd fermion spaces, ${\cal{H}}_+$ or ${\cal{H}}_-$,

   

$${\cal{H}}_+ \equiv {\cal{H}}_0\oplus{\cal{H}}_2\oplus\ldots \oplus{\cal{H}}_{A=2p}\oplus\ldots,$$ (15)

$${\cal{H}}_- \equiv {\cal{H}}_1\oplus{\cal{H}}_3\oplus\ldots \oplus{\cal{H}}_{A=2p+1}\oplus\ldots.$$ (16)

Any Fock-space operator $\hat{U}$:${\cal{H}}$ $\longrightarrow$${\cal{H}}$, which conserves the particle number, is split into two parts with different domains, i.e.,

$$\hat{U}= {\hat{\mathbf{U}}} + {\hat{\cal{U}}},$$ (17)

where the bold symbols denote operators which act in the even-A spaces, while the script symbols denote those acting in the odd-A spaces, i.e.,

   

$${\hat{\mathbf{U}}} {\cal{H}}_+\longrightarrow{\cal{H}}_+,$$ : (18)

$${\hat{\cal{U}}} {\cal{H}}_-\longrightarrow{\cal{H}}_-.$$ : (19)

With these definitions we are now in a position to investigate the group structures appearing for the introduced operators.