*5.1. Preliminary*

#### - Polar and Axial Vectors

Here, we introduce two different kinds of vectors. Consider a line segmen<sup>t</sup> in a three-dimensional space. As discussed in Section 2.3, we can set an inner or outer orientation of the line. An inner-oriented line is called a *polar vector* and is represented by an arrow depicted in Figure 37a. The totality of polar vectors forms a vector space, in which we define the sum of vectors and scalar multiplication of a vector. An electric field is represented by a polar-vector field. On the other hand, an outer-oriented line can be considered as shown in Figure 37b. Such an outer-oriented line segmen<sup>t</sup> is called an *axial*

*vector*. Axial vectors also form a vector space. As we saw in Section 2.3, an outer-oriented object can be represented by two-different inner-oriented ones. Here, we can represent the orientation of the space by a helix, of which the winding direction is right- or left-handed. The helix winding direction is arranged to be the same as the outer orientation of the axial vector, then the advance direction of the corresponding screw induces the inner orientation of the line as shown in Figure 38. In other words, when fingers of the left or right hand representing the helix handedness are curled in the direction of the outer orientation, the thumb points out the inner orientation. Then, an axial vector is denoted by *a* = {*<sup>a</sup>o*|*<sup>o</sup>* ∈ O} with polar vectors satisfying *a*−*<sup>o</sup>* = <sup>−</sup>*ao* for the set O of two spatial orientations. Importantly, an axial vector is a geometric object independent of the orientation of the space, although the polar vector obtained from the axial vector depends on the spatial orientation.

**Figure 37.** (**a**) polar vector; (**b**) axial vector.

**Figure 38.** Representation of an axial vector by polar vectors depending on the space orientation. A spatial orientation is entered in the subscript position and its output is a polar vector which depends on the spatial orientation.

#### - Magnetic Field as Axial Vectors

To explain the necessity of the axial vectors, we consider Ampère's law. The conventional image of a magnetic field induced by a current flowing along a line (*x* = *y* = 0) is depicted in Figure 39a, where the direction of the magnetic field is determined by the so-called right-hand rule. However, there are three unnatural points in this illustration: (i) Why is the right-hand rule required? (ii) Mirror reflection with respect to *x* = 0 or *y* = 0 does not change the current flow, but it alters the direction of the vector field. Here, mirror reflection M*x* with respect to *x* = 0 operates as [M*x<sup>v</sup>*](*<sup>x</sup>*, *y*, *z*) = <sup>−</sup>*v<sup>x</sup>*(−*x*, *y*, *<sup>z</sup>*)*<sup>e</sup>x* + *<sup>v</sup><sup>y</sup>*(−*x*, *y*, *<sup>z</sup>*)*<sup>e</sup>y* + *<sup>v</sup><sup>z</sup>*(−*x*, *y*, *<sup>z</sup>*)*<sup>e</sup>z* for a polar vector field *<sup>v</sup>*(*<sup>x</sup>*, *y*, *z*) = *<sup>v</sup><sup>x</sup>*(*<sup>x</sup>*, *y*, *<sup>z</sup>*)*<sup>e</sup>x* + *<sup>v</sup><sup>y</sup>*(*<sup>x</sup>*, *y*, *<sup>z</sup>*)*<sup>e</sup>y* + *<sup>v</sup><sup>z</sup>*(*<sup>x</sup>*, *y*, *<sup>z</sup>*)*<sup>e</sup>z*. (iii) Mirror reflection with respect to *z* = 0 changes the direction of the current, but the vector field is unchanged under the operation. These three problems are resolved when we consider magnetic fields as axial vector fields. A proper illustration of the magnetic field is shown in Figure 39b, where the magnetic line is outer-oriented. In this representation, we do not need the right-hand rule. The field in Figure 39b is symmetric with respect to *x* = 0 or *y* = 0, while it is antisymmetric with respect to *z* = 0. Figure 39a is now interpreted as the right-hand component for Figure 39b.

**Figure 39.** Ampère's law represented by (**a**) polar and (**b**) axial magnetic lines.

## - Vector Product

Another important operation that can generate an axial vector is the vector product of polar vectors. Consider the vector product between polar vectors *A* and *B*. Clearly, *A* and *B* are invariant under a mirror reflection with respect to the plane spanned by *A* and *B*. Therefore, it is natural to consider *A* × *B* as an axial vector as shown in Figure 40a. Then, the mirror symmetry is kept. The representation by polar vectors is also depicted in Figure 40b. The vector product between a polar vector and an axial vector is also defined to give a polar vector.

**Figure 40.** (**a**) axial vector obtained from a vector product of polar vectors; (**b**) its polar-vector representation.

#### - Scalar and Pseudoscalar

For a scalar, we can consider an outer-oriented object. The codimension for a scalar is three in this three-dimensional space. Therefore, an outer-oriented or twisted scalar can be represented by a helix in this space. An outer-oriented scalar is often called a pseudoscalar. A pseudoscalar *s*ˇ is represented by two scalars *s*ˇ = {*s*<sup>ˇ</sup>*o*|*<sup>o</sup>* ∈ O} with *<sup>s</sup>*<sup>ˇ</sup>−*<sup>o</sup>* = <sup>−</sup>*s*<sup>ˇ</sup>*o*.

For example, magnetic charge density *ρ*m = ∇ · *B* is a pseudoscalar because magnetic flux density *B* is an axial vector. Therefore, a magnetic charge should be a pseudoscalar if it exists. The difference between electric and magnetic charges is illustrated in Figure 41.

**Figure 41.** (**a**) electric charges as a scalar; (**b**) magnetic charges as a pseudoscalar.

#### *5.2. Formulation of Electromagnetic Duality*
