Eight Types of Symmetrically Distinct Vectorlike Physical Quantities

The Letter draws the attention to the spatiotemporal symmetry of various vectorlike physical quantities. The symmetry is specified by their invariance under the action of symmetry operations of the nonrelativistic space-time rotation group $\mathrm{O}(3)\times \{1,1^{\prime }\}={\mathrm{O}}^{\prime }(3)$, where $1^{\prime }$ is a time-reversal operation, the $\text{symbol}\times \text{stands}$ for the group direct product, and O(3) is a group of proper and improper rotations. It is argued that along with the canonical polar vector, there are another seven symmetrically distinct classes of stationary physical quantities, which can be—and often are—denoted as standard three-component vectors, even though they do not transform as a static polar vector under all operations of ${\mathrm{O}}^{\prime }(3)$. The octet of symmetrically distinct “directional quantities” can be exemplified by two kinds of polar vectors (electric dipole moment $\mathbf{P}$ and magnetic toroidal moment $\mathbf{T}$), two kinds of axial vectors (magnetization $\mathbf{M}$ and electric toroidal moment $\mathbf{G}$), two kinds of chiral “bidirectors” $\mathbf{C}$ and $\mathbf{F}$ (associated with the so-called true and false chirality, respectively) and still another two bidirectors $\mathbf{N}$ and $\mathbf{L}$, achiral ones, transforming as the nematic liquid crystal order parameter and as the antiferromagnetic order parameter of the hematite crystal $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$, respectively.

Figure 1

Pictographs of eight kinds of quantities defined by a sign, a magnitude, and an axis. Letter symbols allow us to identify each pictograph with the symmetry assignment given in Tables 1, 2, 3. Arrows in pictographs drawn with dashed lines should be considered as indicating a stationary current or motion (time inversion operation does change their sense), while arrows in the pictographs drawn with full lines are time irreversible (as, e.g., the electric polarization). Pictographs were inspired by pictures employed for similar purposes in Refs. [5, 14, 15].

Figure 2

Pictographs of same eight kinds of quantities as in Fig. 1, but with an opposite sense.