**8. Conclusion**

22 Electromagnetic Theory

F quantities S quantities

<sup>F</sup> <sup>=</sup> <sup>−</sup>*e*<sup>0</sup> <sup>∧</sup> *<sup>E</sup>* <sup>+</sup> *<sup>c</sup>*0<sup>B</sup> ← ∗ <sup>→</sup> <sup>G</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> *<sup>H</sup>* <sup>+</sup> *<sup>c</sup>*0<sup>D</sup> ↓ d *Z*<sup>0</sup> ↓ d

(*Z*0*Gαβ* =)*F*˜ *αβ G*˜ *αβ*(= *Y*0*Fαβ*)

−∗

∗

gg

*Fαβ Gαβ*

In Fig. 2, the relativistic quantities are arranged as a diagram, the rows of which correspond to the orders of tensors (*n* = 1, 2, 3, 4). In the left column, the quantities related to the electromagnetic forces (F quantities), and in the right column, the quantities related to the electromagnetic sources (S quantities) are listed. The exterior derivative "d" connects a pair of quantities by increasing the tensor order by one. These differential relations correspond to the definition of (scalar and vector) potentials, the Maxwell's equations, and the charge conservation (See Fig. 1). Hodge's star operator "∗" connects two 2-forms: F and G. This corresponds to the constitutive relations for vacuum and here appears the vacuum impedance

In Fig. 3, various kinds of tensors of order 2 in the relativistic Maxwell equations and their relations are shown. The left column corresponds to the source fields (*D*, *H*), the right column corresponds to the force fields (*E*, *B*). Though not explicitly written, due to the difference in dimension, the conversions associate the vacuum impedance (or admittance). "E" and "E�" represent the conversion by Levi-Civita (or by its conjugate), "∗" represents the conversion by Hodge's operator. Associated with the diagonal arrows, "g�g�", and "gg" represent raising and lowering of the indices with the metric tensors, respectively. The tensors in the upper row are derived from the scalar-vector formalism and those in the lower row are derived from the

∗

g†g†

E E E † E†

−∗

Fig. 3. Various kinds of tensors of order 2 used in the relativistic Maxwell equations

<sup>0</sup> <sup>J</sup> <sup>=</sup> <sup>−</sup>*e*<sup>0</sup> <sup>∧</sup> <sup>J</sup> <sup>+</sup> *<sup>c</sup>*0<sup>R</sup>

↓ d 0

↓ d *Y*<sup>0</sup>

Fig. 2. Relations of electromagnetic field forms in four dimension

**7.6 Summary for relativistic relations**

*Z*<sup>0</sup> = 1/*Y*<sup>0</sup> as the proportional factor.

*<sup>V</sup>* <sup>=</sup> *<sup>φ</sup>e*<sup>0</sup> <sup>−</sup> *<sup>c</sup>*0*<sup>A</sup>*

In this book chapter, we have reformulated the electromagnetic theory. First we have confirmed the role of vacuum impedance *Z*<sup>0</sup> as a fundamental constant. It characterizes the electromagnetism as the gravitational constant *G* characterizes the theory of gravity. The velocity of light *c*<sup>0</sup> in vacuum is the constant associated with space-time, which is a framework in which electromagnetism and other theories are constructed. Then, *Z*<sup>0</sup> is a single parameter characterizing electromagnetism, and *ε*<sup>0</sup> = 1/(*Z*0*c*0) and *μ*<sup>0</sup> = *Z*0/*c*<sup>0</sup> are considered derived parameters.

Next, we have introduced anti-symmetric covariant tensors, or differential forms, in order to represent EM field quantities most naturally. It is a significant departure from the conventional scalar-vector formalism. But we have tried not to be too mathematical by carrying over the conventional notations as many as possible for continuous transition. In this formalism, the various field quantities are defined through the volume form, which is the machinery to calculate the volume of parallelepipedon spanned by three tangential vectors. To be precise, it is a pseudo (twisted) form, whose sign depends on the orientation of basis.

Even though the constitutive relation seems as a simple proportional relation, it associates the conversion by the Hodge dual operation and the change in physical dimensions by the vaccum impedance. We have found that this non-trivial relation is the keystone of the EM theory.

The EM theory has the symmetry with respect to the space inversion, therefore, each field quantity has a definite parity, even or odd. We have shown that the parity is determined by the tensorial order and the pseudoness (twisted or untwisted).

The Maxwell equations can be formulated most naturally in the four dimensional space-time. However, the conventional expression with tensor components (with superscripts or subscripts) is somewhat abstract and hard to read out its geometrical or physical meaning. Moreover, sometimes contravariant tensors are introduced in order to avoid the explicit use of the Hodge dual with sacrificing the beauty of equations. It has been shown that the four-dimensional differential forms (anti-symmetric covariant tensors) are the most suitable tools for expressing the structure of the EM theory.

The structured formulation helps us to advance electromagnetic theories to various areas. For example, the recent development of new type of media called metamaterials, for which we have to deal with electric and magnetic interactions simultaneously, confronts us to reexamine theoretical frameworks. It may also be helpful to resolve problems on the electromagnetic momentum within dielectric media.
