**1. Introduction**

18 Will-be-set-by-IN-TECH

20 Trends in Electromagnetism – From Fundamentals to Applications

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van Kampen, P. (2008). Lorentz contraction and current-carrying wires. Eur. J. Phys. 29,

van Kampen, P. (2010) Reply to 'Comment on "Lorentz contraction and current-carrying

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Transactions of the Royal Society of London 155, 459–512.

Long Grove.

31, L25–L27.

York.

York.

879–883.

0-070-04908-4, New York.

wires" '. Eur. J. Phys. 31, L25–L27.

Classical electromagnetism is a well-established discipline. However, there remains some confusions and misunderstandings with respect to its basic structures and interpretations. For example, there is a long-lasting controversy on the choice of unit systems. There are also the intricate disputes over the so-called EH or EB formulations. In some textbooks, the authors respect the fields *E* and *B* as fundamental quantities and understate *D* and *H* as auxiliary quantities. Sometimes the roles of *D* and *H* in a vacuum are totally neglected.

These confusions mainly come from the conventional formalism of electromagnetism and also from the use of the old unit systems, in which distinction between *E* and *D*, or *B* and *H* is blurred, especially in vacuum. The standard scalar-vector formalism, mainly due to Heaviside, greatly simplifies the electromagnetic (EM) theory compared with the original formalism developed by Maxwell. There, the field quantities are classified according to the number of components: vectors with three components and scalars with single component. But this classification is rather superficial. From a modern mathematical point of view, the field quantities must be classified according to the tensorial order. The field quantities *D* and *B* are the 2nd-order tensors (or 2 forms), while *E* and *H* are the 1st-order tensors (1 forms). (The anti-symmetric tensors of order *n* are called *n*-forms.)

The constitutive relations are usually considered as simple proportional relations between *E* and *D*, and between *B* and *H*. But in terms of differential forms, they associate the conversion of tensorial order, which is known as the Hodge dual operation. In spite of the simple appearance, the constitutive relations, even for the case of vacuum, are the non-trivial part of the EM theory. By introducing relativistic field variables and the vacuum impedance, the constitutive relation can be unified into a single equation.

The EM theory has the symmetry with respect to the space inversion, therefore, each field quantity has a definite parity, even or odd. In the conventional scalar-vector notation, the parity is assigned rather by hand not from the first principle: the odd vectors *E* and *D* are named the polar vectors and the even vectors *B* and *H* are named the axial vectors. With respect to differential forms, the parity is determined by the tensorial order and the pseudoness (twisted or untwisted). The pseudoness is flipped under the Hodge dual operation. The way of parity assignment in the framework of differential forms is quite natural in geometrical point of view.

permeability *μ* can be varied from values for vacuum and thereby the phase velocity *v*ph = 1/√*εμ* and the wave impedance *Z* = *μ*/*�* can be adjusted independently. With the control of wave impedance the reflection at the interfaces of media can be reduced or suppressed.

Reformulation of Electromagnetism with Differential Forms 23

In this section, we show some examples for which *Z*<sup>0</sup> plays important roles (Kitano (2009)). The impedance (resistance) is a physical quantity by which voltage and current are related. In the SI system, the unit for voltage is V(= J/C) (volt) and the unit for current is A(= C/s) (ampere). We should note that the latter is proportional to and the former is inversely proportional to the unit of charge, C (coulomb). Basic quantities in electromagnetism can be

*φ*, *A*, *E*, *B* Force quantities ∝ V,

The quantities in the former categories contain V in their units and are related to electromagnetic forces. On the other hand, the quantities in the latter contain A and are related to electromagnetic sources. The vacuum impedance *Z*<sup>0</sup> (or the vacuum admittance

*Y*<sup>0</sup> = 1/*Z*0) plays the role to connect the quantities of the two categories.

 *E c*0*B* = *Z*<sup>0</sup> *c*0*D H* 

We know that the scalar potential Δ*φ* induced by a charge Δ*q* = *�*Δ*v* is

Δ *φ c*0*A* <sup>=</sup> *<sup>Z</sup>*<sup>0</sup> 4*πr*

(current times length) *J*Δ*v* generates the vector potential

The relations (5) and (6) are unified as

resultant fields Δ(*φ*, *c*0*A*) in a unified manner.

The electric relation and magnetic relation are united under the sole parameter *Z*0.

<sup>Δ</sup>*<sup>φ</sup>* <sup>=</sup> <sup>1</sup> 4*πε*<sup>0</sup>

<sup>Δ</sup>*<sup>A</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> 4*π*

where *r* is the distance between the source and the point of observation. The charge is presented as a product of charge density *�* and a small volume Δ*v*. Similarly a current moment

*�*Δ*v*

*J*Δ*v*

 *c*0*� J* 

We see that the vacuum impedance *Z*<sup>0</sup> plays the role to relate the source (*J*, *c*0*�*)Δ*v* and the

The constitutive relations for vacuum, *D* = *ε*0*E* and *H* = *μ*−<sup>1</sup>

*D*, *H*, *P*, *M*, *�*, *J* Source quantities ∝ A. (3)

<sup>0</sup> *B*, can be simplified by using

. (4)

*<sup>r</sup>* , (5)

*<sup>r</sup>* . (6)

Δ*v*. (7)

**2.1 Roles of the vacuum impedance**

classified into two categories as

**2.1.1 Constitutive relation**

**2.1.2 Source-field relation**

the relativistic pairs of variables as

It is well understood that the Maxwell equations can be formulated more naturally in the four dimensional spatio-temporal (Minkowski) space. However, the conventional expression with tensor components (with superscripts or subscripts) is somewhat abstract and hard to read out its geometrical or physical meaning. Here it will be shown that the four-dimensional differential forms are the most suitable method for expressing the structure of the EM theory. We introduce two fundamental, relativistic 2-forms, which are related by the four-dimensional Hodge's dual operation and the vacuum impedance.

In this book chapter, we reformulate the EM theory with the differential forms by taking care of physical perspective, the unit systems (physical dimensions), and geometric aspects, and thereby provide a unified and clear view of the solid and beautiful theory.

Here we introduce notation for dimensional consideration. When the ratio of two quantities *X* and *Y* is dimensionless (just a pure number), we write *X* SI ∼ *Y* and read "*X* and *Y* are dimensionally equivalent (in SI)." For example, we have *c*0*t* SI ∼ *x*. If a quantity *X* can be measured in a unit u, we can write *X* SI <sup>∼</sup> u. For example, for *<sup>d</sup>* <sup>=</sup> 2.5 <sup>m</sup> we can write *<sup>d</sup>* SI ∼ m.

#### **2. The vacuum impedance as a fundamental constant**

The vacuum impedance was first introduced explicitly in late 1930's (Schelkunoff (1938)) in the study of EM wave propagation. It is defined as the amplitude ratio of the electric and magnetic fields of plane waves in vacuum, *Z*<sup>0</sup> = *E*/*H*, which has the dimension of electrical resistance.

It is also called the characteristic impedance of vacuum or the wave resistance of vacuum. Due to the historical reasons, it has been recognized as a special parameter for engineers rather than a universal physical constant. Compared with the famous formula (Maxwell (1865)) representing the velocity of light *c*<sup>0</sup> in terms of the vacuum permittivity *ε*<sup>0</sup> and the vacuum permeability *μ*0,

$$\omega\_0 = \frac{1}{\sqrt{\mu\_0 \varepsilon\_0}},\tag{1}$$

the expression for the vacuum impedance

$$Z\_0 = \sqrt{\frac{\mu\_0}{\varepsilon\_0}},\tag{2}$$

is used far less often. In fact the term is rarely found in index pages of textbooks on electromagnetism.

As we will see, the pair of constants (*c*0, *Z*0) can be conveniently used in stead of the pair (*ε*0, *μ*0) for many cases. However, conventionally the asymmetric pairs (*c*0, *μ*0) or (*c*0,*ε*0) are often used and SI equations become less memorable.

In this section, we reexamine the structure of electromagnetism in view of the SI system (The International System of Units) and find that *Z*<sup>0</sup> plays very important roles as a universal constant.

Recent development of new type of media called metamaterials demands the reconsideration of wave impedance. In metamaterials (Pendry & Smith (2004)), both permittivity *ε* and permeability *μ* can be varied from values for vacuum and thereby the phase velocity *v*ph = 1/√*εμ* and the wave impedance *Z* = *μ*/*�* can be adjusted independently. With the control of wave impedance the reflection at the interfaces of media can be reduced or suppressed.

#### **2.1 Roles of the vacuum impedance**

2 Electromagnetic Theory

It is well understood that the Maxwell equations can be formulated more naturally in the four dimensional spatio-temporal (Minkowski) space. However, the conventional expression with tensor components (with superscripts or subscripts) is somewhat abstract and hard to read out its geometrical or physical meaning. Here it will be shown that the four-dimensional differential forms are the most suitable method for expressing the structure of the EM theory. We introduce two fundamental, relativistic 2-forms, which are related by the four-dimensional

In this book chapter, we reformulate the EM theory with the differential forms by taking care of physical perspective, the unit systems (physical dimensions), and geometric aspects, and

Here we introduce notation for dimensional consideration. When the ratio of two quantities

The vacuum impedance was first introduced explicitly in late 1930's (Schelkunoff (1938)) in the study of EM wave propagation. It is defined as the amplitude ratio of the electric and magnetic fields of plane waves in vacuum, *Z*<sup>0</sup> = *E*/*H*, which has the dimension of electrical

It is also called the characteristic impedance of vacuum or the wave resistance of vacuum. Due to the historical reasons, it has been recognized as a special parameter for engineers rather than a universal physical constant. Compared with the famous formula (Maxwell (1865)) representing the velocity of light *c*<sup>0</sup> in terms of the vacuum permittivity *ε*<sup>0</sup> and the vacuum

> *<sup>c</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> √*μ*0*ε*<sup>0</sup>

*Z*<sup>0</sup> =

*μ*<sup>0</sup> *ε*0

is used far less often. In fact the term is rarely found in index pages of textbooks on

As we will see, the pair of constants (*c*0, *Z*0) can be conveniently used in stead of the pair (*ε*0, *μ*0) for many cases. However, conventionally the asymmetric pairs (*c*0, *μ*0) or (*c*0,*ε*0) are

In this section, we reexamine the structure of electromagnetism in view of the SI system (The International System of Units) and find that *Z*<sup>0</sup> plays very important roles as a universal

Recent development of new type of media called metamaterials demands the reconsideration of wave impedance. In metamaterials (Pendry & Smith (2004)), both permittivity *ε* and

∼ *Y* and read "*X* and *Y* are

∼ *x*. If a quantity *X* can be

, (1)

, (2)

∼ m.

<sup>∼</sup> u. For example, for *<sup>d</sup>* <sup>=</sup> 2.5 <sup>m</sup> we can write *<sup>d</sup>* SI

thereby provide a unified and clear view of the solid and beautiful theory.

*X* and *Y* is dimensionless (just a pure number), we write *X* SI

dimensionally equivalent (in SI)." For example, we have *c*0*t* SI

**2. The vacuum impedance as a fundamental constant**

Hodge's dual operation and the vacuum impedance.

measured in a unit u, we can write *X* SI

the expression for the vacuum impedance

often used and SI equations become less memorable.

resistance.

permeability *μ*0,

electromagnetism.

constant.

In this section, we show some examples for which *Z*<sup>0</sup> plays important roles (Kitano (2009)). The impedance (resistance) is a physical quantity by which voltage and current are related. In the SI system, the unit for voltage is V(= J/C) (volt) and the unit for current is A(= C/s) (ampere). We should note that the latter is proportional to and the former is inversely proportional to the unit of charge, C (coulomb). Basic quantities in electromagnetism can be classified into two categories as

$$\begin{array}{ccccccccc}\phi, & \text{A}, & \text{E}, & \text{B} & & & \text{Force quantities} & \propto \text{V}\_{\text{t}}\\\mathbf{D}, & \text{H}, & \text{P}, & \text{M}, & \text{g}, & \text{J} & & \text{Source quantities} & \propto \text{A}.\end{array} \tag{3}$$

The quantities in the former categories contain V in their units and are related to electromagnetic forces. On the other hand, the quantities in the latter contain A and are related to electromagnetic sources. The vacuum impedance *Z*<sup>0</sup> (or the vacuum admittance *Y*<sup>0</sup> = 1/*Z*0) plays the role to connect the quantities of the two categories.

#### **2.1.1 Constitutive relation**

The constitutive relations for vacuum, *D* = *ε*0*E* and *H* = *μ*−<sup>1</sup> <sup>0</sup> *B*, can be simplified by using the relativistic pairs of variables as

$$
\begin{bmatrix} E \\ c\_0 \mathbf{B} \end{bmatrix} = Z\_0 \begin{bmatrix} c\_0 \mathbf{D} \\ H \end{bmatrix}. \tag{4}
$$

The electric relation and magnetic relation are united under the sole parameter *Z*0.

#### **2.1.2 Source-field relation**

We know that the scalar potential Δ*φ* induced by a charge Δ*q* = *�*Δ*v* is

$$
\Delta\phi = \frac{1}{4\pi\varepsilon\_0} \frac{\varrho\Delta v}{r},
\tag{5}
$$

where *r* is the distance between the source and the point of observation. The charge is presented as a product of charge density *�* and a small volume Δ*v*. Similarly a current moment (current times length) *J*Δ*v* generates the vector potential

$$
\Delta A = \frac{\mu\_0}{4\pi} \frac{J \Delta v}{r} . \tag{6}
$$

The relations (5) and (6) are unified as

$$
\Delta \begin{bmatrix} \phi \\ c\_0 \mathbf{A} \end{bmatrix} = \frac{Z\_0}{4\pi r} \begin{bmatrix} c\_0 \wp \\ J \end{bmatrix} \Delta v. \tag{7}
$$

We see that the vacuum impedance *Z*<sup>0</sup> plays the role to relate the source (*J*, *c*0*�*)Δ*v* and the resultant fields Δ(*φ*, *c*0*A*) in a unified manner.

which represents a small spatial displacement from *r* to *r* + *x*. We have chosen an arbitrary

Reformulation of Electromagnetism with Differential Forms 25

1 (*i* = *j*)

SI ∼ 1.

<sup>∼</sup> <sup>m</sup>2.

(*d*1*φ*<sup>1</sup> + *d*2*φ*2)(*x*) = *d*1*φ*1(*x*) + *d*2*φ*2(*x*) (11)

*φ*(*x*)=(*a*1*ν*<sup>1</sup> + *a*2*ν*<sup>2</sup> + *a*3*ν*3)(*x*1*e*<sup>1</sup> + *x*2*e*<sup>2</sup> + *x*3*e*3) (12)

*aixjνi*(*ej*) = *x*1*a*<sup>1</sup> + *x*2*a*<sup>2</sup> + *x*3*a*3. (13)

*a* · *x* = *φ*(*x*). (14)

*<sup>i</sup>* = *ni*. Due to these incidental

Such vectors form a linear space which is called a tangential space at *r*. The inner product of

We consider a linear function *φ*(*x*) on the tangential space. For any *c*1, *c*<sup>2</sup> ∈ **R**, and any vectors *x*<sup>1</sup> and *x*2, *φ*(*c*1*x*<sup>1</sup> + *c*2*x*2) = *c*1*φ*(*x*1) + *c*1*φ*(*x*2) is satisfied. Such linear functions form a linear space, because the (weighted) sum of two functions *d*1*φ*<sup>1</sup> + *d*2*φ*<sup>2</sup> with *d*1, *d*<sup>2</sup> ∈ **R** defined with

is also a linear function. This linear space is called a dual space. The dimension of the dual space is three. In general, the dimension of dual space is the same that for the original linear space. We can introduce a basis {*ν*1, *ν*2, *ν*3}, satisfying *νi*(*ej*) = *δij*. Such a basis, which is dependent on the choice of the original basis, is called a dual basis. Using the dual basis, the action of a dual vector *φ*()= *a*1*ν*1()+ *a*2*ν*2()+ *a*3*ν*3( ), *a*1, *a*2, *a*<sup>3</sup> ∈ **R** can be written simply as

Here we designate an element of dual space with vector notation as *a* rather as a function *φ*( )

We call *a* as a dual vector or a *covector*. The dual basis {*ν*1, *ν*2, *ν*3} are rewritten as {*n*1, *n*2, *n*3} with *n<sup>i</sup>* · *e<sup>j</sup>* = *δij*. The dot product *a* · *x* and the inner product (*x*, *y*) should be distinguished. Here bold-face letters *x*, *y*, *z*, and *e* represent tangential vectors and other bold-face letters

A covector *a* can be related to a vector *z* uniquely using the relation, *a* · *x* = (*z*, *x*) for any *x*. The vector *z* and the covector *a* are called conjugate each other and we write *z* = *a*� and *a* = *z*�. In terms of components, namely for *a* = ∑*<sup>i</sup> ain<sup>i</sup>* and *z* = ∑*<sup>i</sup> ziei*, *ai* = *zi* (*i* = 1, 2, 3)

relations, we tend to identify *n<sup>i</sup>* with *ei*. Thus a covector *a* is identified with its conjugate *a*� mostly. However, we should distinguish a covector as a different object from vectors since it

The inner product for covectors are defined with conjugates as (*a*, *b*)=(*a*�, *b*�). We note the

*<sup>i</sup>* , *n*�

bears different functions and geometrical presentation (Weinreich (1998)).

*<sup>i</sup>* = *ei*, *e*�

*<sup>j</sup>* )=(*ei*, *ej*) = *δij*.

<sup>0</sup> (*<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*) (10)

orthonormal basis {*e*1, *e*2, *e*3} with inner products (*ei*, *ej*) = *δij*, where

is Kronecker's delta. We note that *xi*

vectors *x* and *y* is (*x*, *y*) = *x*1*y*<sup>1</sup> + *x*2*y*<sup>2</sup> + *x*3*y*<sup>3</sup> SI

= 3 ∑ *i*=1

in order to emphasize its vectorial nature, i.e.,

For the case of orthonormal basis, we note that *n*�

dual basis is also orthonormal, since (*ni*, *nj*)=(*n*�

represent covectors.

are satisfied

3 ∑ *j*=1 *δij* =

∼ m and *e<sup>i</sup>*

SI

#### **2.1.3 Plane waves**

We know that for linearly polarized plane waves propagating in one direction in vacuum, a simple relation *E* = *c*0*B* holds. If we introduce *H* (= *μ*−<sup>1</sup> <sup>0</sup> *B*) instead of *B*, we have *E* = *Z*0*H*. This relation was introduced by Schelkunoff (Schelkunoff (1938)). The reason why *H* is used instead of *B* is as follows. A dispersive medium is characterized by its permittivity *ε* and and permeability *μ*. The monochromatic plane wave solution satisfies *E* = *vB*, *H* = *vD*, and *E*/*H* = *B*/*D* = *Z*, where *v* = 1/√*εμ* and *Z* = *μ*/*ε*. The boundary conditions for magnetic fields at the interface of media 1 and 2 are *H*1t = *H*2t (tangential) and *B*1n = *B*2n (normal). For the case of normal incidence, which is most important practically, the latter condition becomes trivial and cannot be used. Therefore the pair of *E* and *H* is used more conveniently. The energy flow is easily derived from *E* and *H* with the Poynting vector *S* = *E* × *H*. In the problems of EM waves, the mixed use of the quantities (*E* and *H*) of the force and source quantities invites *Z*0.

#### **2.1.4 Magnetic monopole**

Let us compare the force between electric charges *q* ( SI ∼ A s = C) and that between magnetic monopoles *g* ( SI ∼ V s = Wb). If these forces are the same for equal distance, *r*, i.e., *q*2/(4*πε*0*r*2) = *g*2/(4*πμ*0*r*2), we have the relation *g* = *Z*0*q*. This means that a charge of 1 C corresponds to a magnetic charge of *Z*<sup>0</sup> × 1 C ∼ 377Wb.

With this relation in mind, the Dirac monopole *g*<sup>0</sup> (Sakurai (1993)), whose quantization condition is *g*0*e* = *h*, can be beautifully expressed in terms of the elementary charge *e* as

*<sup>g</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup> <sup>e</sup>* <sup>=</sup> *<sup>h</sup> <sup>Z</sup>*0*e*<sup>2</sup> (*Z*0*e*) = *<sup>Z</sup>*0*<sup>e</sup>* <sup>2</sup>*<sup>α</sup>* , (8)

where *h* = 2*πh*¯ is Planck's constant. The dimensionless parameter *α* = *Z*0*e*2/2*h* = *<sup>e</sup>*2/4*πε*0*hc*¯ <sup>0</sup> <sup>∼</sup> 1/137 is called the fine-structure constant, whose value is independent of unit systems and characterizes the strength of electromagnetic interaction.

#### **2.1.5 The fine-structure constant**

We have seen that the fine-structure constant itself can be represented more simply with the use of *Z*0. Further, by introducing the von Klitzing constant (the quantized Hall resistance) (Klitzing et al. (1980)) *<sup>R</sup>*<sup>K</sup> = *<sup>h</sup>*/*e*2, the fine-structure constant can be expressed as *<sup>α</sup>* = *<sup>Z</sup>*0/2*R*<sup>K</sup> (Hehl & Obukhov (2005)). We have learned that the use of *Z*<sup>0</sup> helps to keep SI-formulae in simple forms.

#### **3. Dual space and differential forms**

#### **3.1 Covector and dual space**

We represent a (tangential) vector at position *r* as

$$\mathbf{x} = \mathbf{x}\_1 \mathbf{e}\_1 + \mathbf{x}\_2 \mathbf{e}\_2 + \mathbf{x}\_3 \mathbf{e}\_3 \stackrel{\text{sl}}{\sim} \mathbf{m},\tag{9}$$

4 Electromagnetic Theory

We know that for linearly polarized plane waves propagating in one direction in vacuum, a

This relation was introduced by Schelkunoff (Schelkunoff (1938)). The reason why *H* is used instead of *B* is as follows. A dispersive medium is characterized by its permittivity *ε* and and permeability *μ*. The monochromatic plane wave solution satisfies *E* = *vB*, *H* = *vD*, and *E*/*H* = *B*/*D* = *Z*, where *v* = 1/√*εμ* and *Z* = *μ*/*ε*. The boundary conditions for magnetic fields at the interface of media 1 and 2 are *H*1t = *H*2t (tangential) and *B*1n = *B*2n (normal). For the case of normal incidence, which is most important practically, the latter condition becomes trivial and cannot be used. Therefore the pair of *E* and *H* is used more conveniently. The energy flow is easily derived from *E* and *H* with the Poynting vector *S* = *E* × *H*. In the problems of EM waves, the mixed use of the quantities (*E* and *H*) of the force and source

SI

∼ V s = Wb). If these forces are the same for equal distance, *r*, i.e.,

*<sup>Z</sup>*0*e*<sup>2</sup> (*Z*0*e*) = *<sup>Z</sup>*0*<sup>e</sup>*

*q*2/(4*πε*0*r*2) = *g*2/(4*πμ*0*r*2), we have the relation *g* = *Z*0*q*. This means that a charge of

With this relation in mind, the Dirac monopole *g*<sup>0</sup> (Sakurai (1993)), whose quantization condition is *g*0*e* = *h*, can be beautifully expressed in terms of the elementary charge *e* as

where *h* = 2*πh*¯ is Planck's constant. The dimensionless parameter *α* = *Z*0*e*2/2*h* = *<sup>e</sup>*2/4*πε*0*hc*¯ <sup>0</sup> <sup>∼</sup> 1/137 is called the fine-structure constant, whose value is independent of unit

We have seen that the fine-structure constant itself can be represented more simply with the use of *Z*0. Further, by introducing the von Klitzing constant (the quantized Hall resistance) (Klitzing et al. (1980)) *<sup>R</sup>*<sup>K</sup> = *<sup>h</sup>*/*e*2, the fine-structure constant can be expressed as *<sup>α</sup>* = *<sup>Z</sup>*0/2*R*<sup>K</sup> (Hehl & Obukhov (2005)). We have learned that the use of *Z*<sup>0</sup> helps to keep SI-formulae in

*x* = *x*1*e*<sup>1</sup> + *x*2*e*<sup>2</sup> + *x*3*e*<sup>3</sup> SI

<sup>0</sup> *B*) instead of *B*, we have *E* = *Z*0*H*.

∼ A s = C) and that between magnetic

<sup>2</sup>*<sup>α</sup>* , (8)

∼ m, (9)

simple relation *E* = *c*0*B* holds. If we introduce *H* (= *μ*−<sup>1</sup>

Let us compare the force between electric charges *q* (

1 C corresponds to a magnetic charge of *Z*<sup>0</sup> × 1 C ∼ 377Wb.

*<sup>g</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup>*

systems and characterizes the strength of electromagnetic interaction.

*<sup>e</sup>* <sup>=</sup> *<sup>h</sup>*

**2.1.3 Plane waves**

quantities invites *Z*0.

monopoles *g* (

simple forms.

**2.1.4 Magnetic monopole**

SI

**2.1.5 The fine-structure constant**

**3. Dual space and differential forms**

We represent a (tangential) vector at position *r* as

**3.1 Covector and dual space**

which represents a small spatial displacement from *r* to *r* + *x*. We have chosen an arbitrary orthonormal basis {*e*1, *e*2, *e*3} with inner products (*ei*, *ej*) = *δij*, where

$$
\delta\_{ij} = \begin{cases} 1 & (i=j) \\ 0 & (i \neq j) \end{cases} \tag{10}
$$

is Kronecker's delta. We note that *xi* SI ∼ m and *e<sup>i</sup>* SI ∼ 1.

Such vectors form a linear space which is called a tangential space at *r*. The inner product of vectors *x* and *y* is (*x*, *y*) = *x*1*y*<sup>1</sup> + *x*2*y*<sup>2</sup> + *x*3*y*<sup>3</sup> SI <sup>∼</sup> <sup>m</sup>2.

We consider a linear function *φ*(*x*) on the tangential space. For any *c*1, *c*<sup>2</sup> ∈ **R**, and any vectors *x*<sup>1</sup> and *x*2, *φ*(*c*1*x*<sup>1</sup> + *c*2*x*2) = *c*1*φ*(*x*1) + *c*1*φ*(*x*2) is satisfied. Such linear functions form a linear space, because the (weighted) sum of two functions *d*1*φ*<sup>1</sup> + *d*2*φ*<sup>2</sup> with *d*1, *d*<sup>2</sup> ∈ **R** defined with

$$(d\_1\phi\_1 + d\_2\phi\_2)(\mathbf{x}) = d\_1\phi\_1(\mathbf{x}) + d\_2\phi\_2(\mathbf{x})\tag{11}$$

is also a linear function. This linear space is called a dual space. The dimension of the dual space is three. In general, the dimension of dual space is the same that for the original linear space. We can introduce a basis {*ν*1, *ν*2, *ν*3}, satisfying *νi*(*ej*) = *δij*. Such a basis, which is dependent on the choice of the original basis, is called a dual basis. Using the dual basis, the action of a dual vector *φ*()= *a*1*ν*1()+ *a*2*ν*2()+ *a*3*ν*3( ), *a*1, *a*2, *a*<sup>3</sup> ∈ **R** can be written simply as

$$\phi(\mathbf{x}) = (a\_1\nu\_1 + a\_2\nu\_2 + a\_3\nu\_3)(\mathbf{x}\_1\mathbf{e}\_1 + \mathbf{x}\_2\mathbf{e}\_2 + \mathbf{x}\_3\mathbf{e}\_3) \tag{12}$$

$$\mathbf{x} = \sum\_{i=1}^{3} \sum\_{j=1}^{3} a\_i \mathbf{x}\_j \boldsymbol{\nu}\_i(\mathbf{e}\_j) = \mathbf{x}\_1 a\_1 + \mathbf{x}\_2 a\_2 + \mathbf{x}\_3 a\_3. \tag{13}$$

Here we designate an element of dual space with vector notation as *a* rather as a function *φ*( ) in order to emphasize its vectorial nature, i.e.,

$$
\mathfrak{a} \cdot \mathfrak{x} = \mathfrak{\phi}(\mathfrak{x}).\tag{14}
$$

We call *a* as a dual vector or a *covector*. The dual basis {*ν*1, *ν*2, *ν*3} are rewritten as {*n*1, *n*2, *n*3} with *n<sup>i</sup>* · *e<sup>j</sup>* = *δij*. The dot product *a* · *x* and the inner product (*x*, *y*) should be distinguished. Here bold-face letters *x*, *y*, *z*, and *e* represent tangential vectors and other bold-face letters represent covectors.

A covector *a* can be related to a vector *z* uniquely using the relation, *a* · *x* = (*z*, *x*) for any *x*. The vector *z* and the covector *a* are called conjugate each other and we write *z* = *a*� and *a* = *z*�. In terms of components, namely for *a* = ∑*<sup>i</sup> ain<sup>i</sup>* and *z* = ∑*<sup>i</sup> ziei*, *ai* = *zi* (*i* = 1, 2, 3) are satisfied

For the case of orthonormal basis, we note that *n*� *<sup>i</sup>* = *ei*, *e*� *<sup>i</sup>* = *ni*. Due to these incidental relations, we tend to identify *n<sup>i</sup>* with *ei*. Thus a covector *a* is identified with its conjugate *a*� mostly. However, we should distinguish a covector as a different object from vectors since it bears different functions and geometrical presentation (Weinreich (1998)).

The inner product for covectors are defined with conjugates as (*a*, *b*)=(*a*�, *b*�). We note the dual basis is also orthonormal, since (*ni*, *nj*)=(*n*� *<sup>i</sup>* , *n*� *<sup>j</sup>* )=(*ei*, *ej*) = *δij*.

where the last sum is taken for (*i*, *j*)=(1, 2),(2, 3),(3, 1). A general anti-symmetric bicovector

Reformulation of Electromagnetism with Differential Forms 27

We see that the 2-form has three independent components; *T*<sup>12</sup> = −*T*21, *T*<sup>23</sup> = −*T*32, *T*<sup>31</sup> =

· *xx* <sup>+</sup> <sup>T</sup> ·

An anti-symmetric multi-covector of order *n* are often called an *n*-form. A scalar and a covector are called a 0-form and a 1-form, respectively. The order *n* is bounded by the dimension of the vector space, *d* = 3, in our case. An *n*-form with *n* > *d* vanishes due to

Geometrical interpretations of *n*-forms are given in the articles (Misner et al. (1973); Weinreich

Field quantities in electromagnetism can be naturally represented as differential forms (Burke (1985); Deschamps (1981); Flanders (1989); Frankel (2004); Hehl & Obukhov (2003)). A good example of 2-form is the current density. Let us consider a distribution of current that flows through a parallelogram spanned by two tangential vectors *x* and *y* at *r*. The current *I*(*x*, *y*) is bilinearly dependent on *x* and *y*. The antisymmetric relation *I*(*y*, *x*) = −*I*(*x*, *y*) can understood naturally considering the orientation of parallelograms with respect to the current

∼ <sup>A</sup>, <sup>J</sup> = ∑

The charge density can be represented by a 3-form R. The charge *Q* contained in a

Thus electromagnetic field quantities are represented as *n*-forms (*n* = 0, 1, 2, 3) as shown in Table 1, while in the conventional formalism they are classified into two categories, scalars and vectors, according to the number of components. We notice that a quantity that is represented *n*-form contains physical dimension with m−*<sup>n</sup>* in SI. An *n*-form takes *n* tangential vectors,

In this article, 1-forms are represented by bold-face letters, 2-forms sans-serif letters, and

The nabla operator ∇ can be considered as a kind of covector because a directional derivative ∇ · *u*, which is a scalar, is derived with a tangential vector *u*. It acts as a differential operator

(*i*,*j*)

<sup>∼</sup> <sup>C</sup>, <sup>R</sup> <sup>=</sup> *<sup>R</sup>*123*n*<sup>1</sup> <sup>∧</sup> *<sup>n</sup>*<sup>2</sup> <sup>∧</sup> *<sup>n</sup>*<sup>3</sup> SI

*Jijn<sup>i</sup>* ∧ *n<sup>j</sup>*

SI

<sup>∼</sup> <sup>A</sup>/m2. (21)

<sup>∼</sup> <sup>C</sup>/m3. (22)

*Tijn<sup>i</sup>* ∧ *nj*. (20)

· *yy*.

· *xx* <sup>=</sup> 0 for any *<sup>x</sup>*, then it is anti-symmetric. It is easily seen from

· *yx* <sup>+</sup> <sup>T</sup> ·

· *xy* <sup>+</sup> <sup>T</sup> ·

T = ∑ (*i*,*j*)

<sup>−</sup>*T*13, and others are zero. The norm of <sup>T</sup> is ||<sup>T</sup> || = (T, <sup>T</sup> )1/2 <sup>=</sup> <sup>∑</sup>(*i*,*j*) *TijTij*.

· (*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*)(*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*) = <sup>T</sup> ·

flow. Thus the current density can be represented by a 2-form J as

· *xy* <sup>=</sup> *<sup>I</sup>*(*x*, *<sup>y</sup>*) SI

parallelepipedon spanned by three tangential vectors *x*, *y*, and *z*:

each of which has dimension of length and is measured in m (meters).

· *xyz* <sup>=</sup> *<sup>Q</sup>*(*x*, *<sup>y</sup>*, *<sup>z</sup>*) SI

J ·

3-forms calligraphic letters as shown in Table 1.

R ·

**3.5 Exterior derivative**

can be written as

If a bicovector T satisfies T ·

**3.4 Field quantities as** *n***-forms**

the relation: 0 = T ·

the anti-symmetries.

(1998)).

A good example of covector is the electric field at a point *r*. The electric field is determined through the gained work *W* when an electric test charge *q* at *r* is displaced by *x*. A function *φ* · · *<sup>x</sup>* �→ *<sup>W</sup>*/*<sup>q</sup>* is linear with respect to *<sup>x</sup>* if <sup>|</sup>*x*<sup>|</sup> is small enough. Therefore *<sup>φ</sup>*( ) is considered as a covector and normally written as *<sup>E</sup>*, i.e., *<sup>φ</sup>*(*x*) = *<sup>E</sup>* · *<sup>x</sup>* SI ∼ V. Thus the electric field vector can be understood as a covector rather than a vector. It should be expanded with the dual basis as

$$E = E\_1 \mathfrak{n}\_1 + E\_2 \mathfrak{n}\_2 + E\_3 \mathfrak{n}\_3 \stackrel{\text{Sl}}{\sim} \text{V/m.} \tag{15}$$

The norm is given as ||*E*|| = (*E*, *<sup>E</sup>*)1/2 <sup>=</sup> *E*2 <sup>1</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup> <sup>3</sup>. We note that *n<sup>i</sup>* SI ∼ 1 and *Ei* SI ∼ V/m.

#### **3.2 Higher order tensors**

Now we introduce a tensor product of two covectors *a* and *b* as T = *ab*, which acts on two vectors and yield a scalar as

$$\mathcal{T} . \mathbf{x}y = (ab) : \mathbf{x}y = (\mathbf{a} \cdot \mathbf{x})(\mathbf{b} \cdot \mathbf{y}). \tag{16}$$

It can be considered as a bi-linear functions of vectors, i.e., T · · *xy* <sup>=</sup> *<sup>Φ</sup>*(*x*, *<sup>y</sup>*) with

$$\begin{aligned} \Phi(c\_1\mathbf{x}\_1 + c\_2\mathbf{x}\_2, \mathbf{y}) &= c\_1\Phi(\mathbf{x}\_1, \mathbf{y}) + c\_1\Phi(\mathbf{x}\_2, \mathbf{y}),\\ \Phi(\mathbf{x}, c\_1\mathbf{y} + c\_2\mathbf{y}\_2) &= c\_1\Phi(\mathbf{x}, \mathbf{y}\_1) + c\_1\Phi(\mathbf{x}, \mathbf{y}\_2), \end{aligned} \tag{17}$$

where *c*1, *c*<sup>2</sup> ∈ **R**. We call it a bi-covector.

We can define a weighted sum of bi-covectors T = *d*1T<sup>1</sup> + *d*2T2, *d*1, *d*<sup>2</sup> ∈ **R**, which is not necessarily written as a tensor product of two covectors but can be written as a sum of tensor products. Especially, it can be represented with the dual basis as

$$\mathcal{T} = \sum\_{i=1}^{3} \sum\_{j=1}^{3} T\_{ij} n\_i n\_j \tag{18}$$

where *Tij* = T · · *<sup>e</sup>ie<sup>j</sup>* is the (*i*, *<sup>j</sup>*)-component of <sup>T</sup>.

Similarly we can construct a tensor product of three covectors as T = *abc*, which acts on three vectors linearly as T · · *xyz*. Weighted sums of such products form a linear space, an element of which is called a tri-covector. Using a tensor product of *n* covectors, a multi-covector or an *n*-covector is defined.

#### **3.3 Anti-symmetric multi-covectors —** *n***-forms**

If a bicovector T satisfies T · · *yx* <sup>=</sup> <sup>−</sup><sup>T</sup> · · *xy* for any vectors *<sup>x</sup>* and *<sup>y</sup>*, then it is called antisymmetric. Anti-symmetric bicovectors form a subspace of the bicovector space. Namely, a weighted sum of anti-symmetric bicovector is anti-symmetric. It contains an anti-symmetrized tensor product, *a* ∧ *b* := *ab* − *ba*, which is called a *wedge* product. In terms of basis, we have

$$\mathbf{a} \wedge \mathbf{b} = \sum\_{i=1}^{3} a\_i \mathfrak{n}\_i \wedge \sum\_{j=1}^{3} b\_j \mathfrak{n}\_j = \sum\_{(i,j)} (a\_i b\_j - a\_j b\_i) \mathfrak{n}\_i \wedge \mathfrak{n}\_j \tag{19}$$

6 Electromagnetic Theory

A good example of covector is the electric field at a point *r*. The electric field is determined through the gained work *W* when an electric test charge *q* at *r* is displaced by *x*. A function

· *<sup>x</sup>* �→ *<sup>W</sup>*/*<sup>q</sup>* is linear with respect to *<sup>x</sup>* if <sup>|</sup>*x*<sup>|</sup> is small enough. Therefore *<sup>φ</sup>*( ) is considered as

be understood as a covector rather than a vector. It should be expanded with the dual basis as

Now we introduce a tensor product of two covectors *a* and *b* as T = *ab*, which acts on two

*Φ*(*c*1*x*<sup>1</sup> + *c*2*x*2, *y*) = *c*1*Φ*(*x*1, *y*) + *c*1*Φ*(*x*2, *y*),

We can define a weighted sum of bi-covectors T = *d*1T<sup>1</sup> + *d*2T2, *d*1, *d*<sup>2</sup> ∈ **R**, which is not necessarily written as a tensor product of two covectors but can be written as a sum of tensor

> 3 ∑ *j*=1

Similarly we can construct a tensor product of three covectors as T = *abc*, which acts on three

of which is called a tri-covector. Using a tensor product of *n* covectors, a multi-covector or an

antisymmetric. Anti-symmetric bicovectors form a subspace of the bicovector space. Namely, a weighted sum of anti-symmetric bicovector is anti-symmetric. It contains an anti-symmetrized tensor product, *a* ∧ *b* := *ab* − *ba*, which is called a *wedge* product. In terms

> *bjn<sup>j</sup>* = ∑ (*i*,*j*)

<sup>2</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup>

<sup>3</sup>. We note that *n<sup>i</sup>*

*Φ*(*x*, *c*1*y* + *c*2*y*2) = *c*1*Φ*(*x*, *y*1) + *c*1*Φ*(*x*, *y*2), (17)

· *xyz*. Weighted sums of such products form a linear space, an element

· *xy* = (*<sup>a</sup>* · *<sup>x</sup>*)(*<sup>b</sup>* · *<sup>y</sup>*). (16)

· *xy* <sup>=</sup> *<sup>Φ</sup>*(*x*, *<sup>y</sup>*) with

*Tijninj*, (18)

· *xy* for any vectors *<sup>x</sup>* and *<sup>y</sup>*, then it is called

(*aibj* − *ajbi*)*n<sup>i</sup>* ∧ *nj*, (19)

*E* = *E*1*n*<sup>1</sup> + *E*2*n*<sup>2</sup> + *E*3*n*<sup>3</sup> SI

 *E*2 <sup>1</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup> ∼ V. Thus the electric field vector can

∼ V/m. (15)

SI ∼ 1 and *Ei*

SI ∼ V/m.

a covector and normally written as *<sup>E</sup>*, i.e., *<sup>φ</sup>*(*x*) = *<sup>E</sup>* · *<sup>x</sup>* SI

T ·

It can be considered as a bi-linear functions of vectors, i.e., T ·

products. Especially, it can be represented with the dual basis as

· *<sup>e</sup>ie<sup>j</sup>* is the (*i*, *<sup>j</sup>*)-component of <sup>T</sup>.

**3.3 Anti-symmetric multi-covectors —** *n***-forms**

*a* ∧ *b* =

3 ∑ *i*=1

*ain<sup>i</sup>* ∧

3 ∑ *j*=1

· *xy* = (*ab*) ·

T =

· *yx* <sup>=</sup> <sup>−</sup><sup>T</sup> ·

3 ∑ *i*=1

The norm is given as ||*E*|| = (*E*, *<sup>E</sup>*)1/2 <sup>=</sup>

where *c*1, *c*<sup>2</sup> ∈ **R**. We call it a bi-covector.

**3.2 Higher order tensors**

where *Tij* = T ·

vectors linearly as T ·

*n*-covector is defined.

of basis, we have

If a bicovector T satisfies T ·

vectors and yield a scalar as

*φ* ·

where the last sum is taken for (*i*, *j*)=(1, 2),(2, 3),(3, 1). A general anti-symmetric bicovector can be written as

$$\mathcal{T} = \sum\_{(i,j)} T\_{ij} \mathfrak{n}\_i \wedge \mathfrak{n}\_j. \tag{20}$$

We see that the 2-form has three independent components; *T*<sup>12</sup> = −*T*21, *T*<sup>23</sup> = −*T*32, *T*<sup>31</sup> = <sup>−</sup>*T*13, and others are zero. The norm of <sup>T</sup> is ||<sup>T</sup> || = (T, <sup>T</sup> )1/2 <sup>=</sup> <sup>∑</sup>(*i*,*j*) *TijTij*.

If a bicovector T satisfies T · · *xx* <sup>=</sup> 0 for any *<sup>x</sup>*, then it is anti-symmetric. It is easily seen from the relation: 0 = T · · (*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*)(*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*) = <sup>T</sup> · · *xx* <sup>+</sup> <sup>T</sup> · · *xy* <sup>+</sup> <sup>T</sup> · · *yx* <sup>+</sup> <sup>T</sup> · · *yy*.

An anti-symmetric multi-covector of order *n* are often called an *n*-form. A scalar and a covector are called a 0-form and a 1-form, respectively. The order *n* is bounded by the dimension of the vector space, *d* = 3, in our case. An *n*-form with *n* > *d* vanishes due to the anti-symmetries.

Geometrical interpretations of *n*-forms are given in the articles (Misner et al. (1973); Weinreich (1998)).

#### **3.4 Field quantities as** *n***-forms**

Field quantities in electromagnetism can be naturally represented as differential forms (Burke (1985); Deschamps (1981); Flanders (1989); Frankel (2004); Hehl & Obukhov (2003)). A good example of 2-form is the current density. Let us consider a distribution of current that flows through a parallelogram spanned by two tangential vectors *x* and *y* at *r*. The current *I*(*x*, *y*) is bilinearly dependent on *x* and *y*. The antisymmetric relation *I*(*y*, *x*) = −*I*(*x*, *y*) can understood naturally considering the orientation of parallelograms with respect to the current flow. Thus the current density can be represented by a 2-form J as

$$J.\,\text{x}\,\text{y} = I(\mathbf{x},\mathbf{y}) \stackrel{\text{si}}{\sim} \text{A}, \quad J = \sum\_{(i,j)} I\_{\text{ij}} \mathfrak{n}\_{\text{i}} \wedge \mathfrak{n}\_{\text{j}} \stackrel{\text{si}}{\sim} \text{A}/\text{m}^2. \tag{21}$$

The charge density can be represented by a 3-form R. The charge *Q* contained in a parallelepipedon spanned by three tangential vectors *x*, *y*, and *z*:

$$\mathcal{R}: \mathbf{x}yz = \mathbf{Q}(\mathbf{x}, \mathbf{y}, \mathbf{z}) \stackrel{\text{Si}}{\sim} \mathbf{C}, \quad \mathcal{R} = \mathbf{R}\_{123} \mathfrak{n}\_1 \wedge \mathfrak{n}\_2 \wedge \mathfrak{n}\_3 \stackrel{\text{Si}}{\sim} \mathbf{C}/\mathfrak{m}^3. \tag{22}$$

Thus electromagnetic field quantities are represented as *n*-forms (*n* = 0, 1, 2, 3) as shown in Table 1, while in the conventional formalism they are classified into two categories, scalars and vectors, according to the number of components. We notice that a quantity that is represented *n*-form contains physical dimension with m−*<sup>n</sup>* in SI. An *n*-form takes *n* tangential vectors, each of which has dimension of length and is measured in m (meters).

In this article, 1-forms are represented by bold-face letters, 2-forms sans-serif letters, and 3-forms calligraphic letters as shown in Table 1.

#### **3.5 Exterior derivative**

The nabla operator ∇ can be considered as a kind of covector because a directional derivative ∇ · *u*, which is a scalar, is derived with a tangential vector *u*. It acts as a differential operator

0-form

1-form

2-form

· *xyz* is the energy contained in the

<sup>0</sup> (∗B) − *M*. (29)

0 0 3-form

*∂t*

R

<sup>∼</sup> <sup>J</sup>/m<sup>3</sup> corresponds to the energy density and can be represented by

·

J

d d

−*∂<sup>t</sup>*

operator "∗" called the Hodge star operator. In terms of components, *D*<sup>1</sup> = *D*23, *D*<sup>2</sup> = *D*31,

The charge density form can be written as R = *�n*<sup>1</sup> ∧ *n*<sup>2</sup> ∧ *n*<sup>3</sup> = *�*E with the conventional scalar charge density *�*. The relation can be expressed as R = ∗*�* or *�* = ∗R. Similarly, we

Equations (24) and (25) are related to the conventional notations; ∗(∇ ∧ *E*) = ∇ × *E* and

In electromagnetism, the Hodge duality and the constitutive relations are closely related. We know that the electric field *E* and the electric flux density D are proportional. However we cannot compare them directly because they have different tensorial orders. Therefore we utilize the Hodge dual and write D = *ε*0(∗*E*). Similarly, the magnetic relation can be written

be trivial relations just describing proportionality. Especially in the Gaussian unit system, they tend to be considered redundant relations. But in the light of differential forms we understand

With the polarization P and the magnetization *M*, the constitutive relations in a medium are

<sup>D</sup> <sup>=</sup> *<sup>ε</sup>*0(∗*E*) + <sup>P</sup>, *<sup>H</sup>* <sup>=</sup> *<sup>μ</sup>*−<sup>1</sup>

<sup>0</sup> (∗B). Generally speaking, the constitutive relations in vacuum are considered to

<sup>2</sup> (*E*, *D*)E, because U ·

0

*μ*−<sup>1</sup> 0 ∗

*ε*0∗

Reformulation of Electromagnetism with Differential Forms 29

B D

*∂t*

Fig. 1. Relations of electromagnetic field forms in three dimension

*D*<sup>3</sup> = *D*<sup>12</sup> for *D* = ∗D with *D* = ∑*<sup>i</sup> Din<sup>i</sup>* and D = ∑(*i*,*j*) *Dijn<sup>i</sup>* ∧ *nj*.

d d d

*A E H*

d

<sup>2</sup> *<sup>E</sup>* <sup>∧</sup> <sup>D</sup> <sup>=</sup> <sup>1</sup>

**3.7 The Hodge duality and the constitutive equation**

that they are the keystones in electromagnetism.

parallelepipedon spanned by *x*, *y*, and *z*.

*φ*

Physically, (*E*, *D*) SI

the 3-form <sup>U</sup> <sup>=</sup> <sup>1</sup>

have E = ∗1 and 1 = ∗E.

as *H* = *μ*−<sup>1</sup>

expressed as follows:

∗(∇ ∧ J) = ∇ · *J*, respectively.

−d

−*∂<sup>t</sup>*


Table 1. Electromagnetic field quantities as *n*-forms

and also as a covector. Therefore it can be written as

$$
\nabla = \mathfrak{n}\_1 \frac{\partial}{\partial \mathfrak{x}\_1} + \mathfrak{n}\_2 \frac{\partial}{\partial \mathfrak{x}\_2} + \mathfrak{n}\_3 \frac{\partial}{\partial \mathfrak{x}\_3} \stackrel{\text{S}}{\sim} 1/\text{m}. \tag{23}
$$

The wedge product of the nabla ∇ and a 1-form *E* yields a 2-form;

$$\nabla \wedge \mathbf{E} = \sum\_{(i,j)} \left( \frac{\partial E\_j}{\partial \mathbf{x}\_i} - \frac{\partial E\_i}{\partial \mathbf{x}\_j} \right) \mathbf{n}\_i \wedge \mathbf{n}\_{j\prime} \tag{24}$$

which corresponds to ∇ × *E* in the scalar-vector formalism. Similarly a 2-form J are transformed into a 3-form as

$$\nabla \wedge \mathbf{J} = \left(\frac{\partial \mathbf{J}\_{23}}{\partial \mathbf{x}\_1} + \frac{\partial \mathbf{J}\_{31}}{\partial \mathbf{x}\_2} + \frac{\partial \mathbf{J}\_{12}}{\partial \mathbf{x}\_3}\right) \mathfrak{n}\_1 \wedge \mathfrak{n}\_2 \wedge \mathfrak{n}\_3 \tag{25}$$

which corresponds to ∇ · *J*.

#### **3.6 Volume form and Hodge duality**

We introduce a 3-form, called the volume form, as

$$\mathcal{E} = \mathfrak{n}\_1 \wedge \mathfrak{n}\_2 \wedge \mathfrak{n}\_3 = \sum\_{i=1}^3 \sum\_{j=1}^3 \sum\_{k=1}^3 \mathfrak{c}\_{ijk} \mathfrak{n}\_i \mathfrak{n}\_j \mathfrak{n}\_k \stackrel{\text{SI}}{\sim} \mathbf{1},\tag{26}$$

where

$$
\varepsilon\_{ijk} = \begin{cases}
1 & \text{(\$i\$, j\$, \$k\$: cyclic)} \\
0 & \text{(others)}
\end{cases}.
\tag{27}
$$

It gives the volume of parallelepipedon spanned by *x*, *y*, and *z*;

$$V(x, y, z) = \mathcal{E} : \mathbf{x}yz \quad \stackrel{\text{sl}}{\sim} \text{m}^3. \tag{28}$$

Using the volume form we can define a relation between *n*-forms and (*d* − *n*)-forms, which is call the Hodge dual relation. First we note that *n*-forms and (*d* − *n*)-forms have the same degrees of freedom (the number of independent components), *dCn* = *dCd*−*n*, and there could be a one-to-one correspondence between them. In our case of *d* = 3, there are two cases: (*n*, *d* − *n*)=(0, 3) and (1, 2). We consider the latter case. With a 1-form *E* and a 2-form D, we can make a 3-form *E* ∧ D = *f*(*E*, D)E. The scalar factor *f*(*E*, D) is bilinearly dependent on *E* and D. Therefore, we can find a covector (a 1-form) *D* that satisfies (*E*, *D*) = *f*(*E*, D) for any *E*. Then, *D* is called the Hodge dual of D and we write *D* = ∗D or D = ∗*D* using a unary 8 Electromagnetic Theory

Table 1. Electromagnetic field quantities as *n*-forms

and also as a covector. Therefore it can be written as

transformed into a 3-form as

which corresponds to ∇ · *J*.

where

**3.6 Volume form and Hodge duality**

∇ = *n*<sup>1</sup>

∇ ∧ J =

We introduce a 3-form, called the volume form, as

The wedge product of the nabla ∇ and a 1-form *E* yields a 2-form;

∇ ∧ *<sup>E</sup>* = ∑

� *∂J*<sup>23</sup> *∂x*<sup>1</sup> + *∂J*<sup>31</sup> *∂x*<sup>2</sup> + *∂J*<sup>12</sup> *∂x*<sup>3</sup>

E = *n*<sup>1</sup> ∧ *n*<sup>2</sup> ∧ *n*<sup>3</sup> =

*�ijk* =

It gives the volume of parallelepipedon spanned by *x*, *y*, and *z*;

⎧ ⎪⎨

⎪⎩

*V*(*x*, *y*, *z*) = E ·

*∂ ∂x*<sup>1</sup> + *n*<sup>2</sup> *∂ ∂x*<sup>2</sup>

> �*∂Ej ∂xi*

which corresponds to ∇ × *E* in the scalar-vector formalism. Similarly a 2-form J are

3 ∑ *i*=1

3 ∑ *j*=1

1 (*i*, *j*, *k* : cyclic) −1 (anti-cyclic) 0 (others)

> · · *xyz* SI

Using the volume form we can define a relation between *n*-forms and (*d* − *n*)-forms, which is call the Hodge dual relation. First we note that *n*-forms and (*d* − *n*)-forms have the same degrees of freedom (the number of independent components), *dCn* = *dCd*−*n*, and there could be a one-to-one correspondence between them. In our case of *d* = 3, there are two cases: (*n*, *d* − *n*)=(0, 3) and (1, 2). We consider the latter case. With a 1-form *E* and a 2-form D, we can make a 3-form *E* ∧ D = *f*(*E*, D)E. The scalar factor *f*(*E*, D) is bilinearly dependent on *E* and D. Therefore, we can find a covector (a 1-form) *D* that satisfies (*E*, *D*) = *f*(*E*, D) for any *E*. Then, *D* is called the Hodge dual of D and we write *D* = ∗D or D = ∗*D* using a unary

3 ∑ *k*=1

*�ijkninjn<sup>k</sup>*

SI

(*i*,*j*)

rank quantities (unit) scalar/vector 0-form *φ* (V) scalar 1-form *A* (Wb/m), *E* (V/m), *H* (A/m), *M* (A/m) vector 2-form B (Wb/m2), D (C/m2), P (C/m2), J (A/m2) vector 3-form <sup>R</sup> (C/m3) scalar

> + *n*<sup>3</sup> *∂ ∂x*<sup>3</sup> SI

<sup>−</sup> *<sup>∂</sup>Ei ∂xj* �

�

∼ 1/m. (23)

*n<sup>i</sup>* ∧ *nj*, (24)

*n*<sup>1</sup> ∧ *n*<sup>2</sup> ∧ *n*3, (25)

∼ 1, (26)

. (27)

<sup>∼</sup> <sup>m</sup>3. (28)

Fig. 1. Relations of electromagnetic field forms in three dimension

operator "∗" called the Hodge star operator. In terms of components, *D*<sup>1</sup> = *D*23, *D*<sup>2</sup> = *D*31, *D*<sup>3</sup> = *D*<sup>12</sup> for *D* = ∗D with *D* = ∑*<sup>i</sup> Din<sup>i</sup>* and D = ∑(*i*,*j*) *Dijn<sup>i</sup>* ∧ *nj*.

Physically, (*E*, *D*) SI <sup>∼</sup> <sup>J</sup>/m<sup>3</sup> corresponds to the energy density and can be represented by the 3-form <sup>U</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> *<sup>E</sup>* <sup>∧</sup> <sup>D</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> (*E*, *D*)E, because U · · · *xyz* is the energy contained in the parallelepipedon spanned by *x*, *y*, and *z*.

The charge density form can be written as R = *�n*<sup>1</sup> ∧ *n*<sup>2</sup> ∧ *n*<sup>3</sup> = *�*E with the conventional scalar charge density *�*. The relation can be expressed as R = ∗*�* or *�* = ∗R. Similarly, we have E = ∗1 and 1 = ∗E.

Equations (24) and (25) are related to the conventional notations; ∗(∇ ∧ *E*) = ∇ × *E* and ∗(∇ ∧ J) = ∇ · *J*, respectively.

#### **3.7 The Hodge duality and the constitutive equation**

In electromagnetism, the Hodge duality and the constitutive relations are closely related. We know that the electric field *E* and the electric flux density D are proportional. However we cannot compare them directly because they have different tensorial orders. Therefore we utilize the Hodge dual and write D = *ε*0(∗*E*). Similarly, the magnetic relation can be written as *H* = *μ*−<sup>1</sup> <sup>0</sup> (∗B). Generally speaking, the constitutive relations in vacuum are considered to be trivial relations just describing proportionality. Especially in the Gaussian unit system, they tend to be considered redundant relations. But in the light of differential forms we understand that they are the keystones in electromagnetism.

With the polarization P and the magnetization *M*, the constitutive relations in a medium are expressed as follows:

$$D = \varepsilon\_0(\*E) + \mathcal{P}, \quad \mathbf{H} = \mu\_0^{-1}(\*\mathcal{B}) - \mathbf{M}.\tag{29}$$

Assume that Alice adopts the basis *Σ* ∈ *C* and Bob adopts *Σ*� ∈ *C*�

B = ∑ *i* ∑ *j B*� *ijn*� *in*� *<sup>j</sup>* = ∑ *k* ∑ *l*

with the change of components *Bkl* = ∑*<sup>i</sup>* ∑*<sup>j</sup> RikRjlB*�

T = *T*�

untwisted (twisted) form to a twisted (untwisted) form.

are untwisted while the source fields are twisted in general.

123*n*� <sup>1</sup> ∧ *n*�

Similarly, in the case of 3-forms, we have

**5.2 Twisted forms**

*<sup>i</sup>* = −*ni*.

with *T*<sup>123</sup> = *T*�

hand, *H* = *μ*−<sup>1</sup>

**5.3 Source densities**

**5.3.1 Polarization**

can define

can be treated in the same manner.

as *n*�

can be expressed in *Σ* and *Σ*� as

call a *twisted* (untwisted) form.

parallelepipedon by specifying an ordered triple of vectors (*x*, *y*, *z*) and ask each of them to measure its volume, their answers will always be different in the sign. It seems inconvenient but there is no principle to choose one over the other. It is only a customary practice to use the right-handed basis to avoid the confusion. Fleming's left-hand rule (or right-hand rule) seems

Reformulation of Electromagnetism with Differential Forms 31

Tensors (or forms) are independent of the choice of basis. For example, a second order tensor

<sup>2</sup> ∧ *n*�

*�*<sup>123</sup> = (det *R*)*�*�

to have E� = −E in the case of reverse of orientation. Therefore, the volume form is called a *pseudo* form in order to distinguish from an ordinary form. The pseudo (normal) form are also

In electromagnetism, some quantities are defined in reference to the volume form or to the Hodge star operator. Therefore, they could be twisted or untwisted. First of all, *φ*, *A*, *E*, and B are not involved with the volume form, they are all untwisted forms. On the other

The quantities *M*, P, J, and R (charge density), which represent volume densities of electromagnetic sources, are also twisted as shown below. We have found that the force fields

Here we look into detail why the quantities representing source densities are represented by twisted forms. As examples, we deal with polarization and charge density. Other quantities

We consider two tangential vectors d*x*, d*y* at a point *P*. Together with the volume form E, we

<sup>d</sup>*<sup>S</sup>* <sup>=</sup> <sup>E</sup> : <sup>d</sup>*x*d*<sup>y</sup>* SI

<sup>0</sup> (∗B) and D = *ε*0(∗*E*) are twisted forms. The Hodge operator transforms an

<sup>123</sup>. However, for the volume form the components must be changed as

to break the symmetry but it implicitly relies upon the use of the right-handed basis.

. When we pose a

*Bklnknl*, (33)

*ij*. We note the dual basis has been flipped

<sup>3</sup> = *T*123*n*<sup>1</sup> ∧ *n*<sup>2</sup> ∧ *n*<sup>3</sup> (34)

<sup>123</sup>, (35)

<sup>∼</sup> <sup>m</sup>2, (36)

### **4. The Maxwell equations in the differential forms**

With differential forms, we can rewrite the Maxwell equations and the constitutive relations as,

$$
\nabla \wedge \mathcal{B} = 0, \quad \nabla \wedge \mathcal{E} + \frac{\partial \mathcal{B}}{\partial t} = 0,
$$

$$
\nabla \wedge \mathcal{D} = \mathcal{R}, \quad \nabla \wedge \mathcal{H} - \frac{\partial \mathcal{D}}{\partial t} = \mathcal{J},
$$

$$
\mathcal{D} = \varepsilon\_0 \mathcal{E} \cdot \mathcal{E} + \mathcal{P}, \quad \mathcal{H} = \frac{1}{2} \mu\_0^{-1} \mathcal{E} : \mathcal{B} - \mathcal{M}.\tag{30}
$$

In the formalism of differential forms, the spatial derivative ∇ ∧ is simply denoted as d . Together with the Hodge operator "∗", Eq. (30) is written in simpler forms;

$$\begin{aligned} \mathbf{d}B &= 0, \quad \mathbf{d}E + \partial\_l B = 0, \\ \mathbf{d}D &= \mathcal{R}, \quad \mathbf{d}H - \partial\_l D = J, \\ D &= \varepsilon\_0 (\*\mathbf{E}) + \mathcal{P}, \quad H = \mu\_0^{-1} (\*\mathbf{B}) - M, \end{aligned} \tag{31}$$

where *∂<sup>t</sup>* = *∂*/*∂t*.

In Fig. 1, we show a diagram corresponding Eq. (31) and related equations (Deschamps (1981)). The field quantities are arranged according to their tensor order. The exterior derivative "d" connects a pair of quantities by increasing the tensor order by one, while time derivative *∂<sup>t</sup>* conserves the tensor order. *E* (B) is related to D (*H*) with the Hodge star operator and the constant *ε*<sup>0</sup> (*μ*0). The definitions of potentials and the charge conservation law

$$E = -\clubsuit \phi - \partial\_t A, \quad \mathcal{B} = \mathsf{d}A, \quad \mathsf{d}J + \partial\_t \mathcal{R} = 0 \tag{32}$$

are also shown in Fig. 1. We can see a well-organized, perfect structure. We will see the relativistic version later (Fig. 2).

#### **5. Twisted forms and parity**

#### **5.1 Twist of volume form**

We consider two bases *Σ* = {*e*1, *e*2, *e*3} and *Σ*� = {*e*� <sup>1</sup>, *e*� <sup>2</sup>, *e*� <sup>3</sup>}. They can be related as *e*� *<sup>i</sup>* = ∑*<sup>j</sup> Rije<sup>j</sup>* by a matrix *R* = [*Rij*] with *Rij* = (*e*� *i* , *ej*). It is orthonormal and therefore det *R* = ±1. In the case of det *R* = 1, the two bases have the same orientation and they can be overlapped by a continuous transformation. On the other hand, for the case of det *R* = −1, they have opposite orientation and an operation of reversal, for example, a diagonal matrix diag(−1, 1, 1) is needed to make them overlapped with rotations. Thus we can classify all the bases according to the orientation. We denote the two classes by *C* and *C*� , each of which contains all the bases with the same orientation. The two classes are symmetric and there are no *a priori* order of precedence, like for i and −i.

We consider a basis *Σ* = {*e*1, *e*2, *e*3} ∈ *C* and the reversed basis *Σ*� = {*e*� <sup>1</sup>, *e*� <sup>2</sup>, *e*� <sup>3</sup>} = {−*e*1, −*e*2, −*e*3}, which belongs to *C*� . The volume form E in *Σ* is defined so as to satisfy E · · · *<sup>e</sup>*1*e*2*e*<sup>3</sup> = +1, i.e., the volume of the cube defined by *<sup>e</sup>*1, *<sup>e</sup>*2, and *<sup>e</sup>*<sup>3</sup> should be <sup>+</sup>1. Similarly, the volume form E� in *Σ*� is defined so as to satisfy E� · · · *e*� 1*e*� 2*e*� <sup>3</sup> = +1. We note that E� · · · *<sup>e</sup>*1*e*2*e*<sup>3</sup> <sup>=</sup> −E� · · · *e*� 1*e*� 2*e*� <sup>3</sup> = −1, namely, E� = −E. Thus we have two kinds of volume forms E and E� (= −E) and use either of them depending on the orientation of basis.

Assume that Alice adopts the basis *Σ* ∈ *C* and Bob adopts *Σ*� ∈ *C*� . When we pose a parallelepipedon by specifying an ordered triple of vectors (*x*, *y*, *z*) and ask each of them to measure its volume, their answers will always be different in the sign. It seems inconvenient but there is no principle to choose one over the other. It is only a customary practice to use the right-handed basis to avoid the confusion. Fleming's left-hand rule (or right-hand rule) seems to break the symmetry but it implicitly relies upon the use of the right-handed basis.

#### **5.2 Twisted forms**

10 Electromagnetic Theory

With differential forms, we can rewrite the Maxwell equations and the constitutive relations

In the formalism of differential forms, the spatial derivative ∇ ∧ is simply denoted as d .

In Fig. 1, we show a diagram corresponding Eq. (31) and related equations (Deschamps (1981)). The field quantities are arranged according to their tensor order. The exterior derivative "d" connects a pair of quantities by increasing the tensor order by one, while time derivative *∂<sup>t</sup>* conserves the tensor order. *E* (B) is related to D (*H*) with the Hodge star operator and the constant *ε*<sup>0</sup> (*μ*0). The definitions of potentials and the charge conservation law

are also shown in Fig. 1. We can see a well-organized, perfect structure. We will see the

det *R* = ±1. In the case of det *R* = 1, the two bases have the same orientation and they can be overlapped by a continuous transformation. On the other hand, for the case of det *R* = −1, they have opposite orientation and an operation of reversal, for example, a diagonal matrix diag(−1, 1, 1) is needed to make them overlapped with rotations. Thus we can classify all

contains all the bases with the same orientation. The two classes are symmetric and there are

· *<sup>e</sup>*1*e*2*e*<sup>3</sup> = +1, i.e., the volume of the cube defined by *<sup>e</sup>*1, *<sup>e</sup>*2, and *<sup>e</sup>*<sup>3</sup> should be <sup>+</sup>1.

(= −E) and use either of them depending on the orientation of basis.

the bases according to the orientation. We denote the two classes by *C* and *C*�

We consider a basis *Σ* = {*e*1, *e*2, *e*3} ∈ *C* and the reversed basis *Σ*� = {*e*�

Similarly, the volume form E� in *Σ*� is defined so as to satisfy E� ·

*∂*B *<sup>∂</sup><sup>t</sup>* <sup>=</sup> 0,

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>J</sup>,

*E* = −d*φ* − *∂tA*, B = <sup>d</sup>*A*, <sup>d</sup>J + *∂t*R = 0 (32)

<sup>1</sup>, *e*� <sup>2</sup>, *e*�

. The volume form E in *Σ* is defined so as to satisfy

<sup>3</sup> = −1, namely, E� = −E. Thus we have two kinds of volume

· · *e*� 1*e*� 2*e*�

*i*

<sup>0</sup> E : B − *M*. (30)

<sup>0</sup> (∗B) − *M*, (31)

<sup>3</sup>}. They can be related as

, each of which

<sup>1</sup>, *e*� <sup>2</sup>, *e*� <sup>3</sup>} =

<sup>3</sup> = +1. We note

, *ej*). It is orthonormal and therefore

<sup>2</sup> *<sup>μ</sup>*−<sup>1</sup>

∇ ∧ B = 0, ∇ ∧ *E* +

<sup>∇</sup> <sup>∧</sup> <sup>D</sup> <sup>=</sup> <sup>R</sup>, <sup>∇</sup> <sup>∧</sup> *<sup>H</sup>* <sup>−</sup> *<sup>∂</sup>*<sup>D</sup>

<sup>D</sup> <sup>=</sup> *<sup>ε</sup>*0E · *<sup>E</sup>* <sup>+</sup> <sup>P</sup>, *<sup>H</sup>* <sup>=</sup> <sup>1</sup>

dB = 0, d*E* + *∂t*B = 0, <sup>d</sup>D = R, <sup>d</sup>*H* − *∂t*D = J, <sup>D</sup> <sup>=</sup> *<sup>ε</sup>*0(∗*E*) + <sup>P</sup>, *<sup>H</sup>* <sup>=</sup> *<sup>μ</sup>*−<sup>1</sup>

Together with the Hodge operator "∗", Eq. (30) is written in simpler forms;

**4. The Maxwell equations in the differential forms**

as,

where *∂<sup>t</sup>* = *∂*/*∂t*.

relativistic version later (Fig. 2).

**5. Twisted forms and parity**

We consider two bases *Σ* = {*e*1, *e*2, *e*3} and *Σ*� = {*e*�

*<sup>i</sup>* = ∑*<sup>j</sup> Rije<sup>j</sup>* by a matrix *R* = [*Rij*] with *Rij* = (*e*�

no *a priori* order of precedence, like for i and −i.

· · *e*� 1*e*� 2*e*�

{−*e*1, −*e*2, −*e*3}, which belongs to *C*�

· *<sup>e</sup>*1*e*2*e*<sup>3</sup> <sup>=</sup> −E� ·

**5.1 Twist of volume form**

*e*�

E · ·

that E� · ·

forms E and E�

Tensors (or forms) are independent of the choice of basis. For example, a second order tensor can be expressed in *Σ* and *Σ*� as

$$B = \sum\_{i} \sum\_{j} B\_{ij}' \mathfrak{n}\_i' \mathfrak{n}\_j' = \sum\_{k} \sum\_{l} B\_{kl} \mathfrak{m}\_k \mathfrak{m}\_l \tag{33}$$

with the change of components *Bkl* = ∑*<sup>i</sup>* ∑*<sup>j</sup> RikRjlB*� *ij*. We note the dual basis has been flipped as *n*� *<sup>i</sup>* = −*ni*.

Similarly, in the case of 3-forms, we have

$$\mathcal{T} = T\_{123}' \mathfrak{n}\_1' \wedge \mathfrak{n}\_2' \wedge \mathfrak{n}\_3' = T\_{123} \mathfrak{n}\_1 \wedge \mathfrak{n}\_2 \wedge \mathfrak{n}\_3 \tag{34}$$

with *T*<sup>123</sup> = *T*� <sup>123</sup>. However, for the volume form the components must be changed as

$$
\epsilon\_{123} = (\det R)\epsilon'\_{123\prime} \tag{35}
$$

to have E� = −E in the case of reverse of orientation. Therefore, the volume form is called a *pseudo* form in order to distinguish from an ordinary form. The pseudo (normal) form are also call a *twisted* (untwisted) form.

In electromagnetism, some quantities are defined in reference to the volume form or to the Hodge star operator. Therefore, they could be twisted or untwisted. First of all, *φ*, *A*, *E*, and B are not involved with the volume form, they are all untwisted forms. On the other hand, *H* = *μ*−<sup>1</sup> <sup>0</sup> (∗B) and D = *ε*0(∗*E*) are twisted forms. The Hodge operator transforms an untwisted (twisted) form to a twisted (untwisted) form.

The quantities *M*, P, J, and R (charge density), which represent volume densities of electromagnetic sources, are also twisted as shown below. We have found that the force fields are untwisted while the source fields are twisted in general.

#### **5.3 Source densities**

Here we look into detail why the quantities representing source densities are represented by twisted forms. As examples, we deal with polarization and charge density. Other quantities can be treated in the same manner.

#### **5.3.1 Polarization**

We consider two tangential vectors d*x*, d*y* at a point *P*. Together with the volume form E, we can define

$$\mathbf{d}\mathbf{S} = \mathcal{E} : \mathbf{d}\mathbf{x} \mathbf{d}y \quad \stackrel{\text{Sh}}{\sim} \mathbf{m}^2 \tag{36}$$

untwist/twist rank quantities parity polar/axial scalar/vector untwisted 0-form *φ* even – scalar untwisted 1-form *A*, *E* odd polar vector untwisted 2-form B even axial vector twisted 1-form *H*, *M* even axial vector twisted 2-from D, P, J odd polar vector twisted 3-form R even – scalar

Reformulation of Electromagnetism with Differential Forms 33

Parity is the eigenvalues for a spatial inversion transformation. It takes *p* = ±1 depending on the types of quantities. The quantity with eigenvalue of +1 (−1) is called having even (odd) parity. In the three dimensional case, the spatial inversion can be provided by simply flipping the basis vectors; P*e<sup>i</sup>* = −*e<sup>i</sup>* (*i* = 1, 2, 3). The dual basis covectors are also flipped; P*n<sup>j</sup>* = −*n<sup>j</sup>*

A scalar (0-form) *φ* is even because P*φ* = *φ*. The electric field *E* is a 1-form and transforms as

and, therefore, it is odd. The magnetic flux density B is a 2-form and even since it transforms

The additional minus sign is due to the change in the orientation of basis. If *Σ* ∈ *C*, then

In the conventional vector-scalar formalism, the parity is introduced rather empirically. We have found that 1-forms and twisted 2-forms are unified as polar vectors, 2-forms and twisted 1-forms as axial vectors, and 0-forms and twisted 3-forms as scalars. Thus we have unveiled

Combining a three dimensional orthonormal basis {*e*1, *e*2, *e*3} and a unit vector *e*<sup>0</sup> representing the time axis, we have a four-dimensional basis {*e*0, *e*1, *e*2, *e*3}. With the basis,

the real shapes of electromagnetic quantities as twisted and untwisted *n*-forms.

, and *vice versa*. The twisted 3-form has even parity. In general, the parity of a twisted

*Bijn<sup>i</sup>* ∧ *<sup>n</sup>j*) = ∑

*Ei*P*n<sup>i</sup>* = − ∑

(*i*,*j*)

*i*

PE = P(*V*123*n*<sup>1</sup> ∧ *n*<sup>2</sup> ∧ *n*3) = −*V*123P*n*<sup>1</sup> ∧ P*n*<sup>2</sup> ∧ P*n*<sup>3</sup> = E. (45)

*x* = (*c*0*t*)*e*<sup>0</sup> + *xe*<sup>2</sup> + *ye*<sup>2</sup> + *ze*<sup>3</sup> = *xαeα*, (46)

*Ein<sup>i</sup>* = −*E*, (43)

*Bij*P*n<sup>i</sup>* ∧ P*n<sup>j</sup>* = B. (44)

Table 2. Electromagnetic field quantities as twisted and untwisted *n*-forms

P*E* = P(∑

PB = P(∑

It is easy to see that the parity of an *<sup>n</sup>*-forms is *<sup>p</sup>* = (−1)*n*.

The volume form is transformed as

*<sup>n</sup>*-form is *<sup>p</sup>* = (−1)(*n*+1).

**6. Relativistic formulae**

**6.1 Metric tensor and dual basis**

a four (tangential) vector can be written

*i*

(*i*,*j*)

*Eini*) = ∑ *i*

**5.4 Parity**

(*j* = 1, 2, 3).

as

P*Σ* ∈ *C*�

which is a pseudo 1-form. (Conventionally, it is written as d*S* = *x* × *y*.) In fact, for a tangential vector *ζ* at *P*, the volume d*S* · *ζ* = E · · · <sup>d</sup>*x*d*<sup>y</sup> <sup>ζ</sup>*, spanned by the three vectors is a linear function of *ζ*. We choose d*z* that is perpendicular to the plane spanned by d*x* and d*y*, i.e., (d*z*, d*x*) = (d*z*, d*y*) = 0. We assume |d*z*|�|d*x*| and |d*z*|�|d*y*|. d*V* = d*S* · d*z* is the volume of thin parallelogram plate.

When a charge +*q* is displaced by *l* from the other charge −*q*, they form an electric dipole *p* = *ql*. We consider an electric dipole moment at a point *P* in d*V*. The displacement *l* can be considered as a tangential vector at *<sup>P</sup>*, to which <sup>d</sup>*<sup>S</sup>* acts as <sup>d</sup>*<sup>S</sup>* · *<sup>l</sup>* <sup>=</sup> *<sup>q</sup>*−1d*<sup>S</sup>* · *<sup>p</sup>*. Then

$$q' = \frac{\text{dS} \cdot \text{p}}{\text{dS} \cdot \text{d} \mathbf{z}} = q \frac{\text{dS} \cdot \text{l}}{\text{d} \text{V}} \tag{37}$$

is the surface charge that is contributed by *p*. In the case of d*S* · *p* = 0, there are no surface charge associated with *p*. If d*z* and *p* are parallel, *q*� d*z* = *ql* = *p* holds.

Next we consider the case where many electric dipoles *p<sup>i</sup>* = *qil<sup>i</sup>* are spatially distributed. The total surface charge is given as

$$\mathbf{Q}' = \sum\_{i \in \mathrm{d}V} q'\_i = \sum\_{i \in \mathrm{d}V} \frac{\mathrm{d}\mathbf{S} \cdot \boldsymbol{\mathcal{p}}\_i}{\mathrm{d}V}$$

$$= \left(\mathrm{d}V\right)^{-1} \sum\_{i \in \mathrm{d}V} \mathcal{E} \cdot \boldsymbol{\mathcal{p}}\_i : \mathrm{d}\mathbf{x} \mathrm{d}y = \mathcal{P} : \mathrm{d}\mathbf{x} \mathrm{d}y,\tag{38}$$

where the sum is taken over the dipoles contained in d*V*. The pseudo 2-form

$$\mathcal{P} := (\mathrm{d}V)^{-1} \sum\_{i \in \mathrm{d}V} \mathcal{E} \cdot \mathfrak{p}\_i \quad \stackrel{\mathrm{S1}}{\sim} \mathrm{C}/\mathrm{m}^2 \tag{39}$$

corresponds to the polarization (the volume density of electric dipole moments).

#### **5.3.2 Charge density**

The volume d*V* spanned by tangential vectors d*x*, d*y*, d*z* at *P* is

$$\mathrm{d}V = \mathcal{E} : \mathrm{d}\mathbf{x} \mathrm{d}y \mathrm{d}\mathbf{z} \quad \stackrel{\mathrm{si}}{\sim} \mathrm{m}^3. \tag{40}$$

For distributed charges *qi*, the total charge in d*V* is given as

$$\begin{split} Q &= \sum\_{i \in \mathrm{d}V} q\_i = \sum\_{i \in \mathrm{d}V} \frac{q\_i \mathrm{d}V}{\mathrm{d}V} \\ &= (\mathrm{d}V)^{-1} \sum\_{i \in \mathrm{d}V} q\_i \mathcal{E} : \mathrm{d}\mathbf{x} \mathrm{d}y \mathrm{d}z = \mathcal{R} : \mathrm{d}\mathbf{x} \mathrm{d}y \mathrm{d}z. \end{split} \tag{41}$$

The pseudo 3-form

$$\mathcal{R} := (\mathrm{d}V)^{-1} \sum\_{i \in \mathrm{d}V} q\_i \mathcal{E} \quad \overset{\mathrm{si}}{\sim} \mathrm{C}/\mathrm{m}^3 \tag{42}$$

gives the charge density.


Table 2. Electromagnetic field quantities as twisted and untwisted *n*-forms

#### **5.4 Parity**

12 Electromagnetic Theory

which is a pseudo 1-form. (Conventionally, it is written as d*S* = *x* × *y*.) In fact, for a tangential

of *ζ*. We choose d*z* that is perpendicular to the plane spanned by d*x* and d*y*, i.e., (d*z*, d*x*) = (d*z*, d*y*) = 0. We assume |d*z*|�|d*x*| and |d*z*|�|d*y*|. d*V* = d*S* · d*z* is the volume of thin

When a charge +*q* is displaced by *l* from the other charge −*q*, they form an electric dipole *p* = *ql*. We consider an electric dipole moment at a point *P* in d*V*. The displacement *l* can be

<sup>d</sup>*<sup>S</sup>* · <sup>d</sup>*<sup>z</sup>* <sup>=</sup> *<sup>q</sup>*

is the surface charge that is contributed by *p*. In the case of d*S* · *p* = 0, there are no surface

Next we consider the case where many electric dipoles *p<sup>i</sup>* = *qil<sup>i</sup>* are spatially distributed. The

d*S* · *p<sup>i</sup>* d*V*

E · *p<sup>i</sup>*

· <sup>d</sup>*x*d*y*d*<sup>z</sup>* SI

· <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>=</sup> <sup>R</sup> ·

*qi*<sup>E</sup> SI

·

*qi*d*V* d*V*

*i*∈d*V*

*qi*E · · SI

d*S* · *l*

d*z* = *ql* = *p* holds.

considered as a tangential vector at *<sup>P</sup>*, to which <sup>d</sup>*<sup>S</sup>* acts as <sup>d</sup>*<sup>S</sup>* · *<sup>l</sup>* <sup>=</sup> *<sup>q</sup>*−1d*<sup>S</sup>* · *<sup>p</sup>*. Then

*<sup>q</sup>*� <sup>=</sup> <sup>d</sup>*<sup>S</sup>* · *<sup>p</sup>*

· <sup>d</sup>*x*d*<sup>y</sup> <sup>ζ</sup>*, spanned by the three vectors is a linear function

<sup>d</sup>*<sup>V</sup>* (37)

E · *p<sup>i</sup>* : d*x*d*y* = P : d*x*d*y*, (38)

<sup>∼</sup> <sup>C</sup>/m<sup>2</sup> (39)

<sup>∼</sup> <sup>m</sup>3. (40)

· <sup>d</sup>*x*d*y*d*z*. (41)

<sup>∼</sup> <sup>C</sup>/m<sup>3</sup> (42)

·

vector *ζ* at *P*, the volume d*S* · *ζ* = E ·

total surface charge is given as

**5.3.2 Charge density**

The pseudo 3-form

gives the charge density.

charge associated with *p*. If d*z* and *p* are parallel, *q*�

*Q*� = ∑ *i*∈d*V q*� *<sup>i</sup>* = ∑ *i*∈d*V*

= (d*V*)−<sup>1</sup> ∑

*i*∈d*V*

corresponds to the polarization (the volume density of electric dipole moments).

*i*∈d*V*

where the sum is taken over the dipoles contained in d*V*. The pseudo 2-form

<sup>P</sup> := (d*V*)−<sup>1</sup> ∑

d*V* = E · ·

*qi* = ∑ *i*∈d*V*

*i*∈d*V*

<sup>R</sup> := (d*V*)−<sup>1</sup> ∑

= (d*V*)−<sup>1</sup> ∑

The volume d*V* spanned by tangential vectors d*x*, d*y*, d*z* at *P* is

For distributed charges *qi*, the total charge in d*V* is given as

*Q* = ∑ *i*∈d*V*

parallelogram plate.

Parity is the eigenvalues for a spatial inversion transformation. It takes *p* = ±1 depending on the types of quantities. The quantity with eigenvalue of +1 (−1) is called having even (odd) parity. In the three dimensional case, the spatial inversion can be provided by simply flipping the basis vectors; P*e<sup>i</sup>* = −*e<sup>i</sup>* (*i* = 1, 2, 3). The dual basis covectors are also flipped; P*n<sup>j</sup>* = −*n<sup>j</sup>* (*j* = 1, 2, 3).

A scalar (0-form) *φ* is even because P*φ* = *φ*. The electric field *E* is a 1-form and transforms as

$$\mathcal{P}\mathbf{E} = \mathcal{P}(\sum\_{i} E\_{i}\mathfrak{n}\_{i}) = \sum\_{i} E\_{i}\mathcal{P}\mathfrak{n}\_{i} = -\sum\_{i} E\_{i}\mathfrak{n}\_{i} = -\mathbf{E}\_{\prime} \tag{43}$$

and, therefore, it is odd. The magnetic flux density B is a 2-form and even since it transforms as

$$\mathcal{P}\mathcal{B} = \mathcal{P}(\sum\_{(i,j)} B\_{ij}\mathfrak{n}\_{i} \wedge \mathfrak{n}\_{j}) = \sum\_{(i,j)} B\_{ij}\mathcal{P}\mathfrak{n}\_{i} \wedge \mathcal{P}\mathfrak{n}\_{j} = \mathcal{B}.\tag{44}$$

It is easy to see that the parity of an *<sup>n</sup>*-forms is *<sup>p</sup>* = (−1)*n*.

The volume form is transformed as

$$\mathcal{P}\mathcal{E} = \mathcal{P}(V\_{123}\mathfrak{n}\_1 \wedge \mathfrak{n}\_2 \wedge \mathfrak{n}\_3) = -V\_{123}\mathfrak{P}\mathfrak{n}\_1 \wedge \mathfrak{P}\mathfrak{n}\_2 \wedge \mathfrak{P}\mathfrak{n}\_3 = \mathcal{E}.\tag{45}$$

The additional minus sign is due to the change in the orientation of basis. If *Σ* ∈ *C*, then P*Σ* ∈ *C*� , and *vice versa*. The twisted 3-form has even parity. In general, the parity of a twisted *<sup>n</sup>*-form is *<sup>p</sup>* = (−1)(*n*+1).

In the conventional vector-scalar formalism, the parity is introduced rather empirically. We have found that 1-forms and twisted 2-forms are unified as polar vectors, 2-forms and twisted 1-forms as axial vectors, and 0-forms and twisted 3-forms as scalars. Thus we have unveiled the real shapes of electromagnetic quantities as twisted and untwisted *n*-forms.

#### **6. Relativistic formulae**

#### **6.1 Metric tensor and dual basis**

Combining a three dimensional orthonormal basis {*e*1, *e*2, *e*3} and a unit vector *e*<sup>0</sup> representing the time axis, we have a four-dimensional basis {*e*0, *e*1, *e*2, *e*3}. With the basis, a four (tangential) vector can be written

$$
\underline{\mathbf{x}} = (c\_0 t)\mathbf{e}\_0 + \mathbf{x}\mathbf{e}\_2 + y\mathbf{e}\_2 + z\mathbf{e}\_3 = \boldsymbol{\pi}^\alpha \mathbf{e}\_{\mathfrak{U}} \tag{46}
$$

We have (*z*�)� = *z*, (*a*�)� = *a*, namely, �� = 1.

*�αβγδ* =

where we introduced, *�μνστ* = *�αβγδgαμgβν gγσgδτ*.

From the relation between covariant and contravariant components

*�αβγδ�αβγτ* <sup>=</sup> <sup>−</sup>6*δ<sup>δ</sup>*

*�αβγδ�αβστ* <sup>=</sup> <sup>−</sup>2(*<sup>δ</sup>*

*�αβγδ�ανστ* <sup>=</sup> <sup>−</sup>6*<sup>δ</sup>*

note <sup>A</sup> <sup>∧</sup> *<sup>B</sup>* <sup>=</sup> *<sup>A</sup>αβBγe<sup>α</sup>* <sup>∧</sup> *<sup>e</sup><sup>β</sup>* <sup>∧</sup> *<sup>e</sup><sup>γ</sup>* <sup>=</sup> <sup>6</sup>*A*[*αβBγ*]*eαeβeγ*.

The conjugate of the metric tensor is given by

We note that

**6.2 Levi-Civita symbol**

Here we note *�*<sup>0123</sup> <sup>=</sup> <sup>−</sup>1.

With respect to contraction, we have

*ij*0*k*, *ijk*0 is opposite to that of *ijk*.

⎧ ⎪⎨

⎪⎩

We introduce the four dimensional completely anti-symmetric tensor of order 4 as

<sup>E</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> *<sup>e</sup>*<sup>1</sup> <sup>∧</sup> *<sup>e</sup>*<sup>2</sup> <sup>∧</sup> *<sup>e</sup>*<sup>3</sup>

Reformulation of Electromagnetism with Differential Forms 35

The components *�αβγδ*, which are called the Levi-Civita symbol2 can be written explicitly as

<sup>E</sup>� = (*e*0)� <sup>∧</sup> (*e*1)� <sup>∧</sup> (*e*2)� <sup>∧</sup> (*e*3)�

= *�αβγδgαμgβν gγσgδτeμeνeσe<sup>τ</sup>*

Here we will confirm some properties of the completely anti-symmetric tensor of order 4.

and *<sup>g</sup>*<sup>00</sup> <sup>=</sup> <sup>−</sup>1, we see that *�*<sup>0123</sup> <sup>=</sup> <sup>−</sup>*�*<sup>0123</sup> and similar relations hold for other components.

*γ <sup>σ</sup> δ<sup>δ</sup> <sup>τ</sup>* − *δ γ <sup>τ</sup> δ<sup>δ</sup>*

*β* [*νδ γ σδδ τ*]

where [ ] in the subscript represents the anti-symmetrization with respect to the indices. For example, we have *A*[*αβBγ*] = (*AαβB<sup>γ</sup>* + *AβγB<sup>α</sup>* + *AγαB<sup>β</sup>* − *AβαB<sup>γ</sup>* − *AγβB<sup>α</sup>* − *AαγBβ*)/6. We

<sup>2</sup> In the case of three dimension, the parity of a permutation can simply be discriminated by the cyclic or anti-cyclic order. In the case of four dimension, the parity of 0*ijk* follows that of *ijk* and those of *i*0*jk*,

−1 (an odd permutation)

= (−*e*0) ∧ *e*<sup>1</sup> ∧ *e*<sup>2</sup> ∧ *e*<sup>3</sup>

0 (other cases)

1 (*αβγδ* is an even permutation of 0123)

= *�αβγδeαeβeγeδ*. (54)

= *�μνστeμeνeσeτ*, (56)

*�αβγδ* = *gαμgβν gγσgδτ�μνστ*, (58)

*<sup>τ</sup>* (60)

. (62)

*<sup>τ</sup>*] (61)

*γ* [*σδδ*

*�αβγδ�αβγδ* <sup>=</sup> <sup>−</sup><sup>24</sup> (= <sup>−</sup>4!) (59)

*<sup>σ</sup>*) = −4*δ*

g� = *gαβ*(*eα*)�(*eβ*)� = *gαβgαμgβνeμe<sup>ν</sup>* = *gμνeμeν*. (57)

. (55)

where the summation operator ∑<sup>3</sup> *<sup>α</sup>*=<sup>0</sup> is omitted in the last expression according to the Einstein summation convention. The vector components are represented with variables with superscripts. The sum is taken with respect to the Greek index repeated once as superscript and once as subscript. With the four-dimensional basis, the Lorentz-type inner product can be represented as

$$\mathbf{x}(\underline{\mathbf{x}},\underline{\mathbf{x}}) = -(\mathbf{c}\_0 \mathbf{t})^2 + \mathbf{x}^2 + y^2 + z^2 = \mathbf{x}^\mathbf{a} \mathbf{x}^\mathbf{\beta} (\mathbf{e}\_\mathbf{a}, \mathbf{e}\_\mathbf{\beta}) = \mathbf{x}^\mathbf{a} g\_{\mathbf{a}\boldsymbol{\beta}} \mathbf{x}^\mathbf{\beta} = \mathbf{x}\_\mathbf{\beta} \mathbf{x}^\mathbf{\beta} \tag{47}$$

where we set *<sup>x</sup><sup>β</sup>* <sup>=</sup> *<sup>x</sup>αgαβ* and (*eα*, *<sup>e</sup>β*) = *<sup>g</sup>αβ* with *<sup>g</sup>αβ* <sup>=</sup> 0 (*<sup>α</sup>* �<sup>=</sup> *<sup>β</sup>*), <sup>−</sup>*g*<sup>00</sup> <sup>=</sup> *gii* <sup>=</sup> 1 (*<sup>i</sup>* <sup>=</sup> 1, 2, 3).

We introduce the corresponding dual basis as {*e*0, *<sup>e</sup>*1, *<sup>e</sup>*2, *<sup>e</sup>*3} with *<sup>e</sup><sup>μ</sup>* · *<sup>e</sup><sup>ν</sup>* <sup>=</sup> *<sup>δ</sup> μ <sup>ν</sup>* , where *δ μ <sup>ν</sup>* = 0 (*<sup>μ</sup>* �<sup>=</sup> *<sup>ν</sup>*), *<sup>δ</sup>*<sup>0</sup> <sup>0</sup> = *<sup>δ</sup><sup>i</sup> <sup>i</sup>* = 1 (*i* = 1, 2, 3). The dual basis covector has a superscript, while the components have subscripts. A four covector can be expressed with the dual basis as

$$
\underline{a} = a\_{\alpha} \underline{e}^{\alpha}.\tag{48}
$$

Then the contraction (by dot product) can be expressed systematically as

$$\mathbf{a} \cdot \mathbf{x} = a\_{\mathfrak{a}} \mathbf{e}^{\mathfrak{a}} \cdot \mathbf{x}^{\mathfrak{b}} \mathbf{e}\_{\mathfrak{B}} = a\_{\mathfrak{a}} \mathbf{x}^{\mathfrak{b}} \mathbf{e}^{\mathfrak{a}} \cdot \mathbf{e}\_{\mathfrak{B}} = a\_{\mathfrak{a}} \mathbf{x}^{\mathfrak{b}} \delta^{\mathfrak{a}}\_{\mathfrak{B}} = a\_{\mathfrak{a}} \mathbf{x}^{\mathfrak{a}}.\tag{49}$$

We note that the dual and the inner product (metric) are independent concepts. Especially the duality can be introduced without the help of metric.

Customary, tensors which are represented by components with superscripts (subscripts) are designated as contravariant (covariant) tensors. With this terminology, a vector (covector) is a contravariant (covariant) tensor.

The symmetric second order tensor g = *gαβeαe<sup>β</sup>* is called a metric tensor. Its components are

$$\mathcal{g}\_{\alpha\beta} = \begin{cases} -1 & (\alpha = \beta = 0) \\ 1 & (\alpha = \beta \neq 0) \\ 0 & (\text{other cases}) \end{cases} . \tag{50}$$

For a fixed four vector *z*, we can find a four covector *a* = *aβe<sup>β</sup>* that satisfy

$$
\underline{\mathbf{a}} \cdot \underline{\mathbf{x}} = (\underline{\mathbf{z}}, \underline{\mathbf{x}}) \tag{51}
$$

for any *x*. The left and right hand sides can be written as

$$
\underline{\mathbf{a}} \cdot \underline{\mathbf{x}} = a\_{\beta} \mathbf{x}^{\alpha} \mathbf{e}^{\beta} \cdot \mathbf{e}\_{\alpha} = a\_{\beta} \mathbf{x}^{\alpha} \delta\_{\alpha}^{\beta} = a\_{\beta} \mathbf{x}^{\beta},
$$

$$
\mathbf{e}(\underline{\mathbf{z}}, \underline{\mathbf{z}}) = z^{\alpha} \mathbf{x}^{\beta} (\mathbf{e}\_{\alpha}, \mathbf{e}\_{\beta}) = z^{\alpha} g\_{\alpha \beta} \mathbf{x}^{\beta}.
\tag{52}
$$

respectively. By comparing these, we obtain *a<sup>β</sup>* = *zαgαβ*. We write this covector *a* determined by *z* as

$$\underline{\mathbf{a}} = \underline{\mathbf{z}}^{\top} = \underline{z}^{\kappa} \mathbf{g}\_{\alpha\beta} \mathbf{e}^{\beta} = z\_{\beta} \mathbf{e}^{\beta},\tag{53}$$

which is called the conjugate of *<sup>z</sup>*. We see that (*e*0)� <sup>=</sup> <sup>−</sup>*e*0, (*ei*)� <sup>=</sup> *<sup>e</sup><sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, 2, 3), i.e., (*eα*)� = *gαβe<sup>β</sup>* 1. With *gαβ* = (*eα*, *eβ*), the conjugate of a covector *a* can be defined similarly with *z<sup>α</sup>* = *gαβa<sup>β</sup>* as *z* = *a*�.

<sup>1</sup> An equation *e<sup>α</sup>* = *gαβeβ*, which we may write carelessly, is not correct.

We have (*z*�)� = *z*, (*a*�)� = *a*, namely, �� = 1.

We introduce the four dimensional completely anti-symmetric tensor of order 4 as

$$\begin{aligned} \mathfrak{E} &= \mathfrak{e}^0 \wedge \mathfrak{e}^1 \wedge \mathfrak{e}^2 \wedge \mathfrak{e}^3 \\ &= \mathfrak{e}\_{\mathfrak{a}\mathfrak{F}\gamma\delta} \mathfrak{e}^{\mathfrak{a}} \mathfrak{e}^{\mathfrak{f}} \mathfrak{e}^{\gamma} \mathfrak{e}^{\delta}. \end{aligned} \tag{54}$$

The components *�αβγδ*, which are called the Levi-Civita symbol2 can be written explicitly as

$$\epsilon\_{a}\rho\_{\gamma\delta} = \begin{cases} 1 & \text{( $a\beta\gamma\delta$  is an even permutation of  $0123$ )} \\ -1 & \text{(an odd permutation)} \\ 0 & \text{(other cases)} \end{cases} \tag{55}$$

We note that

14 Electromagnetic Theory

Einstein summation convention. The vector components are represented with variables with superscripts. The sum is taken with respect to the Greek index repeated once as superscript and once as subscript. With the four-dimensional basis, the Lorentz-type inner product can be

where we set *<sup>x</sup><sup>β</sup>* <sup>=</sup> *<sup>x</sup>αgαβ* and (*eα*, *<sup>e</sup>β*) = *<sup>g</sup>αβ* with *<sup>g</sup>αβ* <sup>=</sup> 0 (*<sup>α</sup>* �<sup>=</sup> *<sup>β</sup>*), <sup>−</sup>*g*<sup>00</sup> <sup>=</sup> *gii* <sup>=</sup> 1 (*<sup>i</sup>* <sup>=</sup> 1, 2, 3).

We introduce the corresponding dual basis as {*e*0, *<sup>e</sup>*1, *<sup>e</sup>*2, *<sup>e</sup>*3} with *<sup>e</sup><sup>μ</sup>* · *<sup>e</sup><sup>ν</sup>* <sup>=</sup> *<sup>δ</sup>*

Then the contraction (by dot product) can be expressed systematically as

*gαβ* =

For a fixed four vector *z*, we can find a four covector *a* = *aβe<sup>β</sup>* that satisfy

⎧ ⎪⎨

⎪⎩

*<sup>a</sup>* · *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>βxαe<sup>β</sup>* · *<sup>e</sup><sup>α</sup>* <sup>=</sup> *<sup>a</sup>βxαδ*

respectively. By comparing these, we obtain *a<sup>β</sup>* = *zαgαβ*. We write this covector *a* determined

which is called the conjugate of *<sup>z</sup>*. We see that (*e*0)� <sup>=</sup> <sup>−</sup>*e*0, (*ei*)� <sup>=</sup> *<sup>e</sup><sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, 2, 3), i.e., (*eα*)� = *gαβe<sup>β</sup>* 1. With *gαβ* = (*eα*, *eβ*), the conjugate of a covector *a* can be defined similarly

duality can be introduced without the help of metric.

for any *x*. The left and right hand sides can be written as

<sup>1</sup> An equation *e<sup>α</sup>* = *gαβeβ*, which we may write carelessly, is not correct.

components have subscripts. A four covector can be expressed with the dual basis as

*<sup>a</sup>* · *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>αe<sup>α</sup>* · *<sup>x</sup>βe<sup>β</sup>* <sup>=</sup> *<sup>a</sup>αx<sup>β</sup> <sup>e</sup><sup>α</sup>* · *<sup>e</sup><sup>β</sup>* <sup>=</sup> *<sup>a</sup>αxβδα*

We note that the dual and the inner product (metric) are independent concepts. Especially the

Customary, tensors which are represented by components with superscripts (subscripts) are designated as contravariant (covariant) tensors. With this terminology, a vector (covector) is a

The symmetric second order tensor g = *gαβeαe<sup>β</sup>* is called a metric tensor. Its components are

−1 (*α* = *β* = 0) 1 (*α* = *β* �= 0) 0 (other cases)

*β <sup>α</sup>* = *aβxβ*, (*z*, *x*) = *zαxβ*(*eα*, *eβ*) = *zαgαβxβ*, (52)

(*x*, *<sup>x</sup>*) = <sup>−</sup>(*c*0*t*)<sup>2</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>αxβ*(*eα*, *<sup>e</sup>β*) = *<sup>x</sup>αgαβx<sup>β</sup>* <sup>=</sup> *<sup>x</sup>βxβ*, (47)

*<sup>i</sup>* = 1 (*i* = 1, 2, 3). The dual basis covector has a superscript, while the

*<sup>α</sup>*=<sup>0</sup> is omitted in the last expression according to the

*a* = *aαeα*. (48)

*a* · *x* = (*z*, *x*) (51)

*a* = *z*� = *zαgαβe<sup>β</sup>* = *zβeβ*, (53)

*μ*

*<sup>β</sup>* <sup>=</sup> *<sup>a</sup>αxα*. (49)

. (50)

*<sup>ν</sup>* , where *δ*

*μ <sup>ν</sup>* = 0

where the summation operator ∑<sup>3</sup>

represented as

(*<sup>μ</sup>* �<sup>=</sup> *<sup>ν</sup>*), *<sup>δ</sup>*<sup>0</sup>

by *z* as

with *z<sup>α</sup>* = *gαβa<sup>β</sup>* as *z* = *a*�.

<sup>0</sup> = *<sup>δ</sup><sup>i</sup>*

contravariant (covariant) tensor.

$$\begin{split} \mathfrak{E}^{\top} &= (\mathfrak{e}^{0})^{\top} \wedge (\mathfrak{e}^{1})^{\top} \wedge (\mathfrak{e}^{2})^{\top} \wedge (\mathfrak{e}^{3})^{\top} \\ &= (-\mathfrak{e}\_{0}) \wedge \mathfrak{e}\_{1} \wedge \mathfrak{e}\_{2} \wedge \mathfrak{e}\_{3} \\ &= \mathfrak{e}\_{\mathfrak{A}\mathfrak{F}\mathfrak{F}} \mathfrak{g}^{\mathfrak{A}\mathfrak{F}} \mathfrak{g}^{\mathfrak{A}\mathbb{V}} \mathfrak{g}^{\gamma\sigma} \mathfrak{g}^{\mathfrak{T}\mathfrak{F}} \mathfrak{g}\_{\mathfrak{A}} \mathfrak{e}\_{\mathfrak{V}} \mathfrak{e}\_{\mathfrak{C}} \mathfrak{e}\_{\mathfrak{C}} \\ &= \mathfrak{e}^{\mathsf{H}\mathbb{V}\mathfrak{C}\mathsf{T}} \mathfrak{e}\_{\mathfrak{A}} \mathfrak{e}\_{\mathfrak{V}} \mathfrak{e}\_{\mathfrak{C}} \mathfrak{e}\_{\mathfrak{T}} \end{split} \tag{56}$$

where we introduced, *�μνστ* = *�αβγδgαμgβν gγσgδτ*.

The conjugate of the metric tensor is given by

$$\underline{\mathbf{g}}^{\top} = \mathbf{g}\_{a\not\!\! } (\mathbf{e}^a)^\top (\mathbf{e}^\beta)^\top = \mathbf{g}\_{a\not\!\! } \mathbf{g}^{a\mu} \mathbf{g}^{\beta\nu} \mathbf{e}\_{\mu} \mathbf{e}\_{\nu} = \mathbf{g}^{\mu\nu} \mathbf{e}\_{\mu} \mathbf{e}\_{\nu}.\tag{57}$$

#### **6.2 Levi-Civita symbol**

Here we will confirm some properties of the completely anti-symmetric tensor of order 4. From the relation between covariant and contravariant components

$$
\epsilon^{a\beta\gamma\delta} = \mathcal{g}^{a\mu} \mathcal{g}^{\beta\nu} \mathcal{g}^{\gamma\sigma} \mathcal{g}^{\delta\tau} \epsilon\_{\mu\nu\sigma\tau\iota} \tag{58}
$$

and *<sup>g</sup>*<sup>00</sup> <sup>=</sup> <sup>−</sup>1, we see that *�*<sup>0123</sup> <sup>=</sup> <sup>−</sup>*�*<sup>0123</sup> and similar relations hold for other components. Here we note *�*<sup>0123</sup> <sup>=</sup> <sup>−</sup>1.

With respect to contraction, we have

$$
\epsilon^{a\beta\gamma\delta}\epsilon\_{a\beta\gamma\delta} = -24 \quad (= -4!)\tag{59}
$$

$$
\epsilon^{\mathfrak{a}\beta\gamma\delta}\epsilon\_{\mathfrak{a}\beta\gamma\tau} = -6\delta^{\delta}\_{\tau} \tag{60}
$$

$$
\epsilon^{a\beta\gamma\delta}\epsilon\_{a\beta\sigma\tau} = -2(\delta^{\gamma}\_{\sigma}\delta^{\delta}\_{\tau} - \delta^{\gamma}\_{\tau}\delta^{\delta}\_{\sigma}) = -4\delta^{\gamma}\_{[\sigma}\delta^{\delta}\_{\tau]} \tag{61}
$$

$$
\epsilon^{\mu\beta\gamma\delta}\epsilon\_{\alpha\nu\upsilon\tau} = -6\delta^{\beta}\_{[\nu}\delta^{\gamma}\_{\sigma}\delta^{\delta}\_{\tau]}.\tag{62}
$$

where [ ] in the subscript represents the anti-symmetrization with respect to the indices. For example, we have *A*[*αβBγ*] = (*AαβB<sup>γ</sup>* + *AβγB<sup>α</sup>* + *AγαB<sup>β</sup>* − *AβαB<sup>γ</sup>* − *AγβB<sup>α</sup>* − *AαγBβ*)/6. We note <sup>A</sup> <sup>∧</sup> *<sup>B</sup>* <sup>=</sup> *<sup>A</sup>αβBγe<sup>α</sup>* <sup>∧</sup> *<sup>e</sup><sup>β</sup>* <sup>∧</sup> *<sup>e</sup><sup>γ</sup>* <sup>=</sup> <sup>6</sup>*A*[*αβBγ*]*eαeβeγ*.

<sup>2</sup> In the case of three dimension, the parity of a permutation can simply be discriminated by the cyclic or anti-cyclic order. In the case of four dimension, the parity of 0*ijk* follows that of *ijk* and those of *i*0*jk*, *ij*0*k*, *ijk*0 is opposite to that of *ijk*.

(*∂β*) =

we have a relativistic equation

On the other hand, the force equations

order, antisymmetric tensors *F*˜*αβ*, *G*˜ *αβ*.

where *Z*<sup>0</sup> = 1/*Y*<sup>0</sup> = �*μ*0/*ε*<sup>0</sup> SI

2nd-order tensor is written as

handedness of the basis4 and the metric.

pseudo form, and a pseudo form into a normal form.

are rearranged with

as

related as

⎡ ⎢ ⎢ ⎣

∇ × *E* +

(*F*˜*αβ*) =

In vacuum, the constitutive relations *D* = *ε*0*E*, *H* = *μ*−<sup>1</sup>

*c*−<sup>1</sup> <sup>0</sup> *∂<sup>t</sup> ∂x ∂y ∂z*

⎤ ⎥ ⎥ ⎦ SI

<sup>∼</sup> <sup>1</sup>/m, (˜*<sup>J</sup>*

Reformulation of Electromagnetism with Differential Forms 37

*∂βG*˜ *αβ* <sup>=</sup> *<sup>G</sup>*˜ *αβ*

derivative with respect to the following spatial component, e.g., *H*2,1 = (*∂*/*∂x*1)*H*2.

*∂ ∂*(*c*0*t*)

⎡ ⎢ ⎢ ⎣

We append "˜", by the reason described later. The suffix 0 represents the time component, and the suffixes 1, 2, 3 correspond to *x*, *y*, *z*-components. The commas in suffixes "," means the

> 0 *c*0*Bx c*0*By c*0*Bz* −*c*0*Bx* 0 −*Ez Ey* −*c*0*By Ez* 0 −*Ex* −*c*0*Bz* −*Ey Ex* 0

*∂βF*˜*αβ* <sup>=</sup> *<sup>F</sup>*˜*αβ*

Thus the four electromagnetic field quantities *E*, *B*, *D*, and *H* are aggregated into two second

∼ Ω is the vacuum impedance. The operator ∗ is the four-dimensional Hodge's star operator. From Eq. (64), the action for a

Equation (74) is a relativistic version of constitutive relations of vacuum and carries two roles. First it connect dimensionally different tensors *G*˜ and *F*˜ with the vacuum impedance *Z*0. Secondly it represents the Hodge's dual relation. The Hodge operator depends both on the

<sup>4</sup> We note *�αβγδ* is a pseudo form rather than a form. Therefore, the Hodge operator makes a form into a

where *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>* (*i*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3) are cyclic. We note that ∗∗ <sup>=</sup> <sup>−</sup>1, i.e., <sup>∗</sup>−<sup>1</sup> <sup>=</sup> −∗ holds.

*<sup>α</sup>*) =

,*<sup>β</sup>* <sup>=</sup> ˜*<sup>J</sup>*

⎡ ⎢ ⎢ ⎣

*c*0*� Jx Jy Jz*

> ⎤ ⎥ ⎥ ⎦ SI

*<sup>G</sup>*˜ *αβ* <sup>=</sup> *<sup>Y</sup>*0(∗*F*˜)*αβ*, or *<sup>F</sup>*˜*αβ* <sup>=</sup> <sup>−</sup>*Z*0(∗*G*˜)*αβ*, (74)

(∗*A*)*ij* <sup>=</sup> *<sup>A</sup>*0*k*, (∗*A*)0*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*Ajk*, (75)

⎤ ⎥ ⎥ ⎦ SI

<sup>∼</sup> <sup>A</sup>/m2, (69)

*<sup>α</sup>*. (70)

∼ V/m, (72)

,*<sup>β</sup>* = 0. (73)

<sup>0</sup> *B* hold, therefore, these tensors are

(*c*0*B*) = 0, ∇ · (*c*0*B*) = 0, (71)

#### **6.3 Hodge dual of anti-symmetric 2nd-order tensors**

The four-dimensional Hodge dual (∗*A*)*αβ* of a second order tensor *Aαβ* is defined to satisfy

$$(\left(\underline{\ast}A\right)\_{[a\beta}B\_{\gamma\delta]} = \frac{1}{2}(A^{\mu\nu}B\_{\mu\nu})\varepsilon\_{a\beta\gamma\delta\prime} \tag{63}$$

for an arbitrary tensor *Bγδ* of order (*d* − 2) (Flanders (1989)). This relation is independent of the basis 3.

Here, we will show that

$$(\left(\underline{\ast}A\right)\_{\alpha\beta} = \frac{1}{2}\epsilon\_{\alpha\beta}{}^{\mu\upsilon}A\_{\mu\upsilon}.\tag{64}$$

Substituting into the left hand side of Eq. (63) and contracting with *�αβγδ*, we have

$$\epsilon^{a\mathcal{B}\gamma\delta}\frac{1}{2}A\_{\mu\nu}\epsilon\_{[a\not\!\! ]}{}^{\mu\nu}B\_{\gamma\delta]} = 3\epsilon^{a\not\!\! \gamma\gamma\delta}\epsilon\_{a\not\!\! ]}{}^{\mu\nu}A\_{\mu\nu}B\_{\gamma\delta} = 3\epsilon^{a\not\!\! \gamma\gamma\delta}\epsilon\_{a\not\!\! ]}A^{\mu\nu}B\_{\gamma\delta} = -12A^{\gamma\delta}B\_{\gamma\delta}.\tag{65}$$

With Eq. (59), the right hand side of Eq. (63) yields <sup>−</sup>12*AμνBμν* with the same contraction. We also note

$$\begin{split} (\underline{\ast}\,A)\_{a\beta} &= \frac{1}{4} \mathfrak{e}\_{a\beta}{}^{\gamma\delta} \mathfrak{e}\_{\gamma\delta}{}^{\mu\nu} A\_{\mu\nu} = \frac{1}{4} \mathfrak{e}\_{a\beta\gamma\delta}{}^{\gamma\delta\mu} A\_{\mu\nu} \\ &= -\frac{1}{2} (\delta^{a}\_{\mu}\delta^{\beta}\_{\nu} - \delta^{a}\_{\nu}\delta^{\beta}\_{\mu}) A\_{\mu\nu} = -\frac{1}{2} (A\_{a\beta} - A\_{\beta a}) = -A\_{a\beta} \end{split} \tag{66}$$

i.e., ∗∗ = −1, which is different from the three dimensional case; ∗∗ = 1.

#### **7. Differential forms in Minkowski spacetime**

#### **7.1 Standard formulation**

According to Jackson's textbook (Jackson (1998)), we rearrange the ordinary scalar-vector form of Maxwell's equation in three dimension into a relativistic expression. We use the SI system and pay attention to the dimensions. We start with the source equations

$$
\nabla \times \mathbf{H} - \frac{\partial}{\partial (\mathbf{c}\_0 t)} (\mathbf{c}\_0 \mathbf{D}) = \mathbf{J}, \quad \nabla \cdot (\mathbf{c}\_0 \mathbf{D}) = \mathbf{c}\_0 \varrho. \tag{67}
$$

Combining field quantities and differential operators as four-dimensional tensors and vectors as

$$(\tilde{G}^{\text{a}\oplus}) = \begin{bmatrix} 0 & c\_0 D\_X \ c\_0 D\_y & c\_0 D\_z \\ -c\_0 D\_X & 0 & H\_z & -H\_y \\ -c\_0 D\_y & -H\_z & 0 & H\_x \\ -c\_0 D\_z & H\_y & -H\_x & 0 \end{bmatrix} \stackrel{\text{SI}}{\sim} \text{A/m} \tag{68}$$

<sup>3</sup> With the four-dimensional volume form <sup>E</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> *<sup>e</sup>*<sup>1</sup> <sup>∧</sup> *<sup>e</sup>*<sup>2</sup> <sup>∧</sup> *<sup>e</sup>*3, the Hodge dual for *<sup>p</sup>* form (*<sup>p</sup>* <sup>=</sup> 1, 2, 3) can be defined as (∗*A*) ∧ *B* = (*A*, *B*)E , (∗A) ∧ B = (A, B)E, and (∗A) ∧ B = (A, B)E. The inner product for *p* forms is defined as (*a*<sup>1</sup> ∧···∧ *ap*, *b*<sup>1</sup> ∧···∧ *bp*) = det(*ai*, *bj*). (Flanders (1989))

$$(\boldsymbol{\partial}\_{\boldsymbol{\beta}}) = \begin{bmatrix} c\_0^{-1} \boldsymbol{\partial}\_t \\ \boldsymbol{\partial}\_X \\ \boldsymbol{\partial}\_y \\ \boldsymbol{\partial}\_z \end{bmatrix} \stackrel{\text{si}}{\sim} 1/\text{m}, \quad (\tilde{\boldsymbol{f}}^{\text{a}}) = \begin{bmatrix} c\_0 \boldsymbol{\varrho} \\ \boldsymbol{f}\_x \\ \boldsymbol{f}\_y \\ \boldsymbol{f}\_z \end{bmatrix} \stackrel{\text{si}}{\sim} \text{A}/\text{m}^2,\tag{69}$$

we have a relativistic equation

$$
\partial\_{\beta} \tilde{G}^{a\beta} = \tilde{G}^{a\beta}{}\_{,\beta} = \tilde{f}^{a}. \tag{70}
$$

We append "˜", by the reason described later. The suffix 0 represents the time component, and the suffixes 1, 2, 3 correspond to *x*, *y*, *z*-components. The commas in suffixes "," means the derivative with respect to the following spatial component, e.g., *H*2,1 = (*∂*/*∂x*1)*H*2.

On the other hand, the force equations

$$
\nabla \times \mathbf{E} + \frac{\partial}{\partial (c\_0 t)} (c\_0 \mathbf{B}) = 0, \quad \nabla \cdot (c\_0 \mathbf{B}) = 0,\tag{71}
$$

are rearranged with

$$(\tilde{F}^{u\emptyset}) = \begin{bmatrix} 0 & c\_0 B\_x \ c\_0 B\_y \ c\_0 B\_z \\ -c\_0 B\_x & 0 & -E\_z & E\_y \\ -c\_0 B\_y & E\_z & 0 & -E\_x \\ -c\_0 B\_z & -E\_y & E\_x & 0 \end{bmatrix} \stackrel{\text{S1}}{\sim} \mathcal{V}/\text{m},\tag{72}$$

as

16 Electromagnetic Theory

The four-dimensional Hodge dual (∗*A*)*αβ* of a second order tensor *Aαβ* is defined to satisfy

2

for an arbitrary tensor *Bγδ* of order (*d* − 2) (Flanders (1989)). This relation is independent of

2 *� μν*

With Eq. (59), the right hand side of Eq. (63) yields <sup>−</sup>12*AμνBμν* with the same contraction. We

4

According to Jackson's textbook (Jackson (1998)), we rearrange the ordinary scalar-vector form of Maxwell's equation in three dimension into a relativistic expression. We use the SI

Combining field quantities and differential operators as four-dimensional tensors and vectors

0 *c*0*Dx c*0*Dy c*0*Dz* −*c*0*Dx* 0 *Hz* −*Hy* −*c*0*Dy* −*Hz* 0 *Hx* −*c*0*Dz Hy* −*Hx* 0

<sup>3</sup> With the four-dimensional volume form <sup>E</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> *<sup>e</sup>*<sup>1</sup> <sup>∧</sup> *<sup>e</sup>*<sup>2</sup> <sup>∧</sup> *<sup>e</sup>*3, the Hodge dual for *<sup>p</sup>* form (*<sup>p</sup>* <sup>=</sup> 1, 2, 3) can be defined as (∗*A*) ∧ *B* = (*A*, *B*)E , (∗A) ∧ B = (A, B)E, and (∗A) ∧ B = (A, B)E. The inner

product for *p* forms is defined as (*a*<sup>1</sup> ∧···∧ *ap*, *b*<sup>1</sup> ∧···∧ *bp*) = det(*ai*, *bj*). (Flanders (1989))

*<sup>μ</sup>*)*Aμν* <sup>=</sup> <sup>−</sup><sup>1</sup>

*�αβγδ�γδμνAμν*

2

(*AμνBμν*)*�αβγδ*, (63)

*αβ Aμν*. (64)

(*Aαβ* − *Aβα*) = −*Aαβ*, (66)

∼ A/m, (68)

*αβ <sup>A</sup>μνBγδ* <sup>=</sup> <sup>3</sup>*�αβγδ�αβμνAμνBγδ* <sup>=</sup> <sup>−</sup>12*AγδBγδ*. (65)

(*c*0*D*) = *J*, ∇ · (*c*0*D*) = *c*0*�*. (67)

⎤ ⎥ ⎥ ⎦ SI

(∗*A*)[*αβBγδ*] <sup>=</sup> <sup>1</sup>

(∗*A*)*αβ* <sup>=</sup> <sup>1</sup>

Substituting into the left hand side of Eq. (63) and contracting with *�αβγδ*, we have

*γδ <sup>A</sup>μν* <sup>=</sup> <sup>1</sup>

i.e., ∗∗ = −1, which is different from the three dimensional case; ∗∗ = 1.

system and pay attention to the dimensions. We start with the source equations

*∂*(*c*0*t*)

⎡ ⎢ ⎢ ⎣

**6.3 Hodge dual of anti-symmetric 2nd-order tensors**

[*αβ <sup>B</sup>γδ*] <sup>=</sup> <sup>3</sup>*�αβγδ� μν*

4 *� γδ αβ � μν*

<sup>=</sup> <sup>−</sup><sup>1</sup> 2 (*δ<sup>α</sup> μδ β <sup>ν</sup>* <sup>−</sup> *<sup>δ</sup><sup>α</sup> ν δ β*

<sup>∇</sup> <sup>×</sup> *<sup>H</sup>* <sup>−</sup> *<sup>∂</sup>*

(*G*˜ *αβ*) =

**7. Differential forms in Minkowski spacetime**

(∗∗*A*)*αβ* <sup>=</sup> <sup>1</sup>

the basis 3.

*�αβγδ* <sup>1</sup>

also note

as

Here, we will show that

<sup>2</sup> *<sup>A</sup>μν� μν*

**7.1 Standard formulation**

$$
\partial\_{\beta} \mathcal{F}^{a\beta} = \mathcal{F}^{a\beta}{}\_{,\beta} = 0. \tag{73}
$$

Thus the four electromagnetic field quantities *E*, *B*, *D*, and *H* are aggregated into two second order, antisymmetric tensors *F*˜*αβ*, *G*˜ *αβ*.

In vacuum, the constitutive relations *D* = *ε*0*E*, *H* = *μ*−<sup>1</sup> <sup>0</sup> *B* hold, therefore, these tensors are related as

$$\tilde{G}^{a\beta} = Y\_0 (\underline{\ast} \tilde{F})^{a\beta} \prime \quad \text{or} \quad \tilde{F}^{a\beta} = -Z\_0 (\underline{\ast} \tilde{G})^{a\beta} \prime \tag{74}$$

where *Z*<sup>0</sup> = 1/*Y*<sup>0</sup> = �*μ*0/*ε*<sup>0</sup> SI ∼ Ω is the vacuum impedance.

The operator ∗ is the four-dimensional Hodge's star operator. From Eq. (64), the action for a 2nd-order tensor is written as

$$(\underline{\ast}A)^{ij} = A^{0k}, \quad (\underline{\ast}A)^{0i} = -A^{jk}, \tag{75}$$

where *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>* (*i*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3) are cyclic. We note that ∗∗ <sup>=</sup> <sup>−</sup>1, i.e., <sup>∗</sup>−<sup>1</sup> <sup>=</sup> −∗ holds.

Equation (74) is a relativistic version of constitutive relations of vacuum and carries two roles. First it connect dimensionally different tensors *G*˜ and *F*˜ with the vacuum impedance *Z*0. Secondly it represents the Hodge's dual relation. The Hodge operator depends both on the handedness of the basis4 and the metric.

<sup>4</sup> We note *�αβγδ* is a pseudo form rather than a form. Therefore, the Hodge operator makes a form into a pseudo form, and a pseudo form into a normal form.

**7.3 Differential forms**

*δ μ*

equations

are unified as

Eq. (83) is

as

*<sup>G</sup>αβ* <sup>=</sup> <sup>1</sup> 2

can be written as

Similarly,

where

The tensor *Gαβ* can be written as

Here we start with the Maxwell equations (31) in three-dimensional differential forms. We introduce a basis {*e*0, *<sup>e</sup>*1, *<sup>e</sup>*2, *<sup>e</sup>*3}, and the corresponding dual basis {*e*0, *<sup>e</sup>*1, *<sup>e</sup>*2, *<sup>e</sup>*3}, i.e., *<sup>e</sup><sup>μ</sup>* · *<sup>e</sup><sup>ν</sup>* <sup>=</sup>

Reformulation of Electromagnetism with Differential Forms 39

where ∧ represent the anti-symmetric tensor product or the wedge product. In components,

The covariant tensors (forms) *<sup>G</sup>αβ* and *<sup>J</sup>αβγ* are related to *<sup>G</sup>*˜ *αβ* and ˜*J<sup>α</sup>* in the previous subsection

⎡ ⎢ ⎢ ⎣

The covariant tensor (form) *Fαβ* is related to *F*˜*αβ* in the previous subsection as

*<sup>δ</sup>*, or *<sup>G</sup>*˜ *αβ* <sup>=</sup> <sup>−</sup><sup>1</sup>

⎡ ⎢ ⎢ ⎣ <sup>0</sup> *<sup>∂</sup>te*<sup>0</sup> <sup>+</sup> <sup>∇</sup>, the source

<sup>⎦</sup> . (85)

*�αβγδ Jβγδ*. (86)

(*c*0D) = J, ∇ ∧ (*c*0D) = R, (82)

∇ ∧ G = J , (83)

⎤ ⎥ ⎥

> *<sup>α</sup>* <sup>=</sup> <sup>1</sup> 6

*∂*[*γGαβ*] = *G*[*αβ*,*γ*] = *Jαβγ*/3. (84)

0 *Hx Hy Hz* −*Hx* 0 *c*0*Dz* −*c*0*Dy* −*Hy* −*c*0*Dz* 0 *c*0*Dx* −*Hz c*0*Dy* −*c*0*Dx* 0

2

0 −*Ex* −*Ey* −*Ez Ex* 0 *c*0*Bz* −*c*0*By Ey* −*c*0*Bz* 0 *c*0*Bx Ez c*0*By* −*c*0*Bx* 0

*�αβγδGγδ*, ˜*J*

(*c*0B) = 0, ∇ ∧ (*c*0B) = 0, (87)

∇ ∧ F = 0, (88)

⎤ ⎥ ⎥

<sup>⎦</sup> . (90)

*∂*[*γFαβ*] = *F*[*αβ*,*γ*] = 0, (89)

*<sup>ν</sup>* . With <sup>G</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> *<sup>H</sup>* <sup>+</sup> *<sup>c</sup>*0D, <sup>J</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−J) + *<sup>c</sup>*0R, and <sup>∇</sup> <sup>=</sup> *<sup>c</sup>*−<sup>1</sup>

*∂*(*c*0*t*)

<sup>∇</sup> <sup>∧</sup> *<sup>H</sup>* <sup>−</sup> *<sup>∂</sup>*

(*Gαβ*)=(*G* : *eαeβ*) =

*�αβγδG*˜ *γδ*, *<sup>J</sup>αβγ* <sup>=</sup> <sup>−</sup>*�αβγδ* ˜*<sup>J</sup>*

with <sup>F</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−*E*) + *<sup>c</sup>*0B. In components,

∇ ∧ *E* +

(*Fαβ*)=(*F* : *eαeβ*) =

*∂ ∂*(*c*0*t*)

Finally, the Maxwell equations can be simply represented as

$$
\partial\_{\beta} \tilde{G}^{a\beta} = \tilde{f}^{a}, \quad \partial\_{\beta} \tilde{F}^{a\beta} = 0, \quad \tilde{G}^{a\beta} = Y\_0 (\pm \tilde{F})^{a\beta}. \tag{76}
$$

This representation, however, is quite unnatural in the view of two points. First of all, the components of field quantity should be covariant and should have lower indices. Despite of that, here, all quantities are contravariant and have upper indices in order to contract with the spatial differential operator *∂α* with a lower index. Furthermore, it is unnatural that in Eqs. (68) and (72), *D* and *B*, which are 2-forms with respect to space, have indices of time and space, and *E* and *H* have two spatial indices.

The main reason of this unnaturalness is that we have started with the conventional, scalar-vector form of Maxwell equations rather than from those in differential forms.

#### **7.2 Bianchi identity**

In general textbooks, the one of equations in Eq. (76) is further modified by introducing a covariant tensor *<sup>F</sup>αβ* = <sup>1</sup> <sup>2</sup> *�αβγδF*˜*γδ*. Solving it as *<sup>F</sup>*˜*αβ* <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>2</sup> *�αβγδFγδ* and substituting into Eq. (73), we have

$$0 = \partial\_{\beta} \epsilon^{\alpha \beta \gamma \delta} F\_{\gamma \delta} = \epsilon^{\alpha \beta \gamma \delta} \partial\_{\beta} F\_{\gamma \delta}. \tag{77}$$

Considering *α* as a fixed parameter, we have six non-zero terms that are related as

$$\begin{split} 0 &= \partial\_{\beta} (F\_{\gamma\delta} - F\_{\delta\gamma}) + \partial\_{\gamma} (F\_{\delta\beta} - F\_{\beta\delta}) + \partial\_{\delta} (F\_{\beta\gamma} - F\_{\gamma\beta}) \\ &= 2 \left( \partial\_{\beta} F\_{\gamma\delta} + \partial\_{\gamma} F\_{\delta\beta} + \partial\_{\delta} F\_{\beta\gamma} \right) \quad (\beta, \gamma, \delta = 0, \dots, 3). \end{split} \tag{78}$$

Although there are many combinations of indices, this represents substantially four equations. To be specific, we introduce the matrix representation of *Fαβ* as

$$(F\_{a\notin}) = \begin{bmatrix} 0 & -E\_{\mathcal{X}} & -E\_{\mathcal{Y}} & -E\_{z} \\ E\_{\mathcal{X}} & 0 & c\_{0}B\_{z} & -c\_{0}B\_{\mathcal{Y}} \\ E\_{\mathcal{Y}} & -c\_{0}B\_{z} & 0 & c\_{0}B\_{\mathcal{X}} \\ E\_{z} & c\_{0}B\_{\mathcal{Y}} & -c\_{0}B\_{\mathcal{X}} & 0 \end{bmatrix}. \tag{79}$$

Comparing this with *G*˜ *αβ* in Eq. (68) and considering the constitutive relations, we find that the signs of components with indices for time "0" are reversed. Therefore, with the metric tensor, we have

$$
\tilde{G}^{a\beta} = \mathcal{Y}\_0 \mathcal{g}^{a\gamma} \mathcal{g}^{\beta \delta} F\_{\gamma \delta} = \mathcal{Y}\_0 F^{a\beta}.\tag{80}
$$

Substitution into Eq. (70) yields *Y*0*∂βFαβ* = ˜*Jα*. After all, relativistically, the Maxwell equations are written as

$$
\partial\_{\beta} F^{\alpha \beta} = Z\_0 \tilde{f}^{\alpha}, \quad \partial\_{\alpha} F\_{\beta \gamma} + \partial\_{\beta} F\_{\gamma \alpha} + \partial\_{\gamma} F\_{a \beta} = 0. \tag{81}
$$

Even though this common expression is simpler than that for the non relativistic version, symmetry is somewhat impaired. The covariant and contravariant field tensors are mixed. The reason is that the constitutive relations, which contains the Hodge operator, is eliminated.

#### **7.3 Differential forms**

18 Electromagnetic Theory

This representation, however, is quite unnatural in the view of two points. First of all, the components of field quantity should be covariant and should have lower indices. Despite of that, here, all quantities are contravariant and have upper indices in order to contract with the spatial differential operator *∂α* with a lower index. Furthermore, it is unnatural that in Eqs. (68) and (72), *D* and *B*, which are 2-forms with respect to space, have indices of time and

The main reason of this unnaturalness is that we have started with the conventional,

In general textbooks, the one of equations in Eq. (76) is further modified by introducing a

0 = *∂β*(*Fγδ* − *Fδγ*) + *∂γ*(*Fδβ* − *Fβδ*) + *∂δ*(*Fβγ* − *Fγβ*)

Although there are many combinations of indices, this represents substantially four equations.

Comparing this with *G*˜ *αβ* in Eq. (68) and considering the constitutive relations, we find that the signs of components with indices for time "0" are reversed. Therefore, with the metric

Substitution into Eq. (70) yields *Y*0*∂βFαβ* = ˜*Jα*. After all, relativistically, the Maxwell equations

Even though this common expression is simpler than that for the non relativistic version, symmetry is somewhat impaired. The covariant and contravariant field tensors are mixed. The reason is that the constitutive relations, which contains the Hodge operator, is eliminated.

�

0 −*Ex* −*Ey* −*Ez Ex* 0 *c*0*Bz* −*c*0*By Ey* −*c*0*Bz* 0 *c*0*Bx Ez c*0*By* −*c*0*Bx* 0

<sup>2</sup> *�αβγδF*˜*γδ*. Solving it as *<sup>F</sup>*˜*αβ* <sup>=</sup> <sup>−</sup><sup>1</sup>

Considering *α* as a fixed parameter, we have six non-zero terms that are related as

*∂βFγδ* + *∂γFδβ* + *∂δFβγ*

⎡ ⎢ ⎢ ⎣

scalar-vector form of Maxwell equations rather than from those in differential forms.

*<sup>α</sup>*, *∂βF*˜*αβ* <sup>=</sup> 0, *<sup>G</sup>*˜ *αβ* <sup>=</sup> *<sup>Y</sup>*0(∗*F*˜)*αβ*. (76)

0 = *∂β�αβγδFγδ* = *�αβγδ∂βFγδ*. (77)

⎤ ⎥ ⎥

*<sup>G</sup>*˜ *αβ* = *<sup>Y</sup>*0*gαγgβδFγδ* = *<sup>Y</sup>*0*Fαβ*. (80)

*<sup>α</sup>*, *∂αFβγ* + *∂βFγα* + *∂γFαβ* = 0. (81)

<sup>2</sup> *�αβγδFγδ* and substituting into

(*β*, *γ*, *δ* = 0, . . . , 3). (78)

<sup>⎦</sup> . (79)

Finally, the Maxwell equations can be simply represented as

*∂βG*˜ *αβ* = ˜*<sup>J</sup>*

space, and *E* and *H* have two spatial indices.

= 2 �

To be specific, we introduce the matrix representation of *Fαβ* as

(*Fαβ*) =

*∂βFαβ* = *Z*<sup>0</sup> ˜*J*

**7.2 Bianchi identity**

Eq. (73), we have

tensor, we have

are written as

covariant tensor *<sup>F</sup>αβ* = <sup>1</sup>

Here we start with the Maxwell equations (31) in three-dimensional differential forms. We introduce a basis {*e*0, *<sup>e</sup>*1, *<sup>e</sup>*2, *<sup>e</sup>*3}, and the corresponding dual basis {*e*0, *<sup>e</sup>*1, *<sup>e</sup>*2, *<sup>e</sup>*3}, i.e., *<sup>e</sup><sup>μ</sup>* · *<sup>e</sup><sup>ν</sup>* <sup>=</sup> *δ μ <sup>ν</sup>* . With <sup>G</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> *<sup>H</sup>* <sup>+</sup> *<sup>c</sup>*0D, <sup>J</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−J) + *<sup>c</sup>*0R, and <sup>∇</sup> <sup>=</sup> *<sup>c</sup>*−<sup>1</sup> <sup>0</sup> *<sup>∂</sup>te*<sup>0</sup> <sup>+</sup> <sup>∇</sup>, the source equations

$$
\nabla \wedge \mathbf{H} - \frac{\partial}{\partial(c\_0 t)}(c\_0 D) = \mathcal{J}, \quad \nabla \wedge (c\_0 \mathcal{D}) = \mathcal{R}, \tag{82}
$$

are unified as

$$
\underline{\nabla} \wedge \underline{\mathsf{G}} = \underline{\mathcal{L}} \tag{83}
$$

where ∧ represent the anti-symmetric tensor product or the wedge product. In components, Eq. (83) is

$$\partial\_{[\gamma} G\_{a\beta]} = G\_{[a\beta,\gamma]} = J\_{a\beta\gamma} / 3. \tag{84}$$

The tensor *Gαβ* can be written as

$$(\mathbf{G}\_{a\boldsymbol{\beta}}) = (\underline{\mathbf{G}} : \mathbf{e}\_a \mathbf{e}\_{\boldsymbol{\beta}}) = \begin{bmatrix} 0 & H\_{\mathbf{x}} & H\_{\mathbf{y}} & H\_{\mathbf{z}} \\ -H\_{\mathbf{x}} & 0 & c\_0 D\_{\mathbf{z}} & -c\_0 D\_{\mathbf{y}} \\ -H\_{\mathbf{y}} & -c\_0 D\_{\mathbf{z}} & 0 & c\_0 D\_{\mathbf{x}} \\ -H\_{\mathbf{z}} & c\_0 D\_{\mathbf{y}} & -c\_0 D\_{\mathbf{x}} & 0 \end{bmatrix}. \tag{85}$$

The covariant tensors (forms) *<sup>G</sup>αβ* and *<sup>J</sup>αβγ* are related to *<sup>G</sup>*˜ *αβ* and ˜*J<sup>α</sup>* in the previous subsection as

$$\mathbf{G}\_{a\mathfrak{F}} = \frac{1}{2} \varepsilon\_{a\mathfrak{F}\gamma\delta} \tilde{\mathbf{G}}^{\gamma\delta}, \quad \mathbf{J}\_{a\mathfrak{F}\gamma} = -\varepsilon\_{a\mathfrak{F}\gamma\delta} \tilde{\mathbf{f}}^{\delta}, \quad \text{or} \quad \tilde{\mathbf{G}}^{a\mathfrak{F}} = -\frac{1}{2} \varepsilon^{a\mathfrak{F}\gamma\delta} \mathbf{G}\_{\gamma\delta\epsilon} \quad \tilde{\mathbf{f}}^{\mathfrak{u}} = \frac{1}{6} \varepsilon^{a\mathfrak{F}\gamma\delta} \mathbf{J}\_{\not\mathfrak{F}\gamma\delta}. \tag{86}$$

Similarly,

$$
\nabla \wedge \mathbf{E} + \frac{\partial}{\partial(c\_0 t)}(c\_0 \mathbf{B}) = 0, \quad \nabla \wedge (c\_0 \mathbf{B}) = 0,\tag{87}
$$

can be written as

$$
\underline{\nabla} \wedge \underline{\mathsf{F}} = 0,\tag{88}
$$

with <sup>F</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−*E*) + *<sup>c</sup>*0B. In components,

$$\left[\partial\_{\left[\gamma\right.} F\_{a\beta\right]}\right] = F\_{\left[a\beta,\gamma\right]} = 0,\tag{89}$$

where

$$(F\_{a\mathcal{G}}) = (\underline{F} : \mathfrak{e}\_{\mathcal{A}} \mathfrak{e}\_{\mathcal{B}}) = \begin{bmatrix} 0 & -E\_{\mathcal{X}} & -E\_{\mathcal{Y}} & -E\_{\mathcal{Z}} \\ E\_{\mathcal{X}} & 0 & c\_0 B\_{\mathcal{Z}} & -c\_0 B\_{\mathcal{Y}} \\ E\_{\mathcal{Y}} & -c\_0 B\_{\mathcal{Z}} & 0 & c\_0 B\_{\mathcal{X}} \\ E\_{\mathcal{Z}} & c\_0 B\_{\mathcal{Y}} & -c\_0 B\_{\mathcal{X}} & 0 \end{bmatrix}. \tag{90}$$

The covariant tensor (form) *Fαβ* is related to *F*˜*αβ* in the previous subsection as

**7.4 Potentials and the conservation of charge**

Then we have ∇ ∧ *V* = −F, or

force equation becomes very trivial,

since ∇ ∧ ∇ = 0 or dd = 0 holds.

The conservation of charge is also straightforward;

**7.5 Relativistic representation of the Lorentz force**

By introducing the four dimensional velocity *u<sup>α</sup>* =

factor. The change in action Δ*S* can be written

−*E*/*c*0, *px*, *py*, *pz*

which yields J � = J .

electromagnetic field are

momentum *p<sup>α</sup>* =

We introduce a four-dimensional vector potential *<sup>V</sup>* <sup>=</sup> *<sup>φ</sup>e*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*0(−*A*), i.e.,

(*Vα*)=(*V* · *eα*)

which is a relation between the potential and the field strength. Utilizing the potential, the

Reformulation of Electromagnetism with Differential Forms 41

The freedom of gauge transformation with a 0-form *Λ* can easily be understood; *V*� = *V* + d*Λ* gives no difference in the force quantities, i.e., F� = F. A similar degree of freedom exist for the source fields (Hirst (1997)). With a 1-form *L*, we define the transformation G� = G + d*L*,

0 = ∇ ∧ ∇ ∧ G = ∇ ∧ J

Changes in the energy *E* and momentum *p* of a charged particle moving at velocity *u* in an

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>q</sup><sup>E</sup>* · *<sup>u</sup>*,

d*E*

d*p*

d*p<sup>α</sup>*

where <sup>d</sup>*<sup>τ</sup>* <sup>=</sup> <sup>d</sup>*t*/*<sup>γ</sup>* the proper time of moving charge, and *<sup>γ</sup>* = (<sup>1</sup> <sup>−</sup> *<sup>u</sup>*2/*c*<sup>2</sup>

= (*φ*, −*c*0*Ax*, −*c*0*Ay*, −*c*0*Az*). (99)

*∂*[*αVβ*] = −*Fαβ*/2, (100)

0 = ∇ ∧ (∇ ∧ *V*) = ∇ ∧ F, (101)

<sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (*∂t*<sup>R</sup> <sup>+</sup> <sup>∇</sup> <sup>∧</sup> <sup>J</sup>). (102)

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>q</sup><sup>E</sup>* <sup>+</sup> *<sup>u</sup>* <sup>×</sup> *<sup>B</sup>*. (103)

<sup>d</sup>*<sup>τ</sup>* <sup>=</sup> *qFαβuβ*, (104)

<sup>Δ</sup>*<sup>S</sup>* <sup>=</sup> *<sup>p</sup>α*Δ*x<sup>α</sup>* <sup>=</sup> <sup>−</sup>*E*Δ*<sup>t</sup>* <sup>+</sup> *<sup>p</sup>* · <sup>Δ</sup>*x*. (105)

, and the four dimensional

<sup>0</sup>)−1/2 is the Lorentz

*c*0*γ*, *ux*, *uy*, *uz*

, we have the equation of motion


Table 3. Four dimensional electromagnetic field quantities as twisted and untwisted *n*-forms

$$F\_{a\beta} = \frac{1}{2} \epsilon\_{a\beta\gamma\delta} \mathbf{f}^{\gamma\delta}{}\_{\prime} \quad \text{or,} \quad \mathbf{f}^{a\beta} = -\frac{1}{2} \epsilon^{a\beta\gamma\delta} F\_{\gamma\delta}.\tag{91}$$

The Hodge operator acts on a four-dimensional two form as

$$\underline{\mathbf{e}}(\mathbf{e}^{0}\wedge\mathbf{X}+\mathbf{Y}) = \mathbf{e}^{0}\wedge(-(\ast\mathbf{Y})) + (\ast\mathbf{X}) = \mathbf{e}^{0}\wedge(-\mathbf{Y}) + \mathbf{X}\tag{92}$$

where *X* and Y are a three-dimensional 1-form and a three-dimensional 2-form, and ∗ is the three-dimensional Hodge operator5. Now the constitutive relations <sup>D</sup> <sup>=</sup> *<sup>ε</sup>*0(∗*E*) and *<sup>H</sup>* <sup>=</sup> *μ*−<sup>1</sup> <sup>0</sup> (∗B) are four-dimensionally represented as

$$
\underline{\mathbf{G}} = -Y\_0(\underline{\mathbf{\*}} \underline{\mathbf{F}}), \quad \text{or} \quad \underline{\mathbf{F}} = Z\_0(\underline{\mathbf{\*}} \underline{\mathbf{G}}).\tag{93}
$$

With components, these are represented as

$$G\_{a\mathcal{B}} = -Y\_0(\underline{\ast}F)\_{a\mathcal{B}\prime} \quad \text{or} \quad F\_{a\mathcal{B}} = Z\_0(\underline{\ast}G)\_{a\mathcal{B}\prime} \tag{94}$$

with the action of Hodge's operator on anti-symmetric tensors of rank 2:

$$(\underline{\ast}A)\_{a\beta} = \frac{1}{2} \epsilon\_{a\beta}{}^{\gamma\delta} A\_{\gamma\delta} = \frac{1}{2} g\_{a\mu} g\_{\beta\nu} \varepsilon^{\mu\nu\gamma\delta} A\_{\gamma\delta}.\tag{95}$$

Now we have the Maxwell equations in the four-dimensional forms with components:

$$
\partial\_{[\gamma} G\_{a\beta]} = f\_{a\beta\gamma} / \mathfrak{Z}, \quad \partial\_{[\gamma} F\_{a\beta]} = 0, \quad F\_{a\beta} = Z\_0 (\underline{\ast} \mathcal{G})\_{a\beta\prime} \tag{96}
$$

and in basis-free representations:

$$
\underline{\nabla} \wedge \underline{\mathsf{G}} = \underline{\mathcal{J}} \quad \underline{\nabla} \wedge \underline{\mathsf{F}} = 0, \quad \underline{\mathsf{F}} = \mathsf{Z}\_0(\underline{\ast \mathsf{G}}) , \tag{97}
$$

or

$$
\underline{\sf d} \underline{\sf G} = \underline{\sf C}, \quad \underline{\sf d} \underline{\sf F} = 0, \quad \underline{\sf F} = Z\_0(\underline{\sf \*} \underline{\sf G}).\tag{98}
$$

with the four-dimensional exterior derivative <sup>d</sup> = ∇ ∧ . These are much more elegant and easier to remember compared with Eqs. (76) and (81). A similar type of reformulation has been given by Sommerfeld (Sommerfeld (1952)).

<sup>5</sup> We note the similarity with the calculation rule for complex numbers: <sup>i</sup>(*<sup>X</sup>* <sup>+</sup> <sup>i</sup>*Y*) = <sup>−</sup>*<sup>Y</sup>* <sup>+</sup> <sup>i</sup>*X*. If we can formally set as *G* = *H* + i*c*0*D* and *F* = −*E* + i*c*0*B*, we have *G* = −i*Y*0*F* and *H* = i*Z*0*G*.

#### **7.4 Potentials and the conservation of charge**

We introduce a four-dimensional vector potential *<sup>V</sup>* <sup>=</sup> *<sup>φ</sup>e*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*0(−*A*), i.e.,

$$(V\_{\mathfrak{A}}) = (\underline{\mathbf{V}} \cdot \mathfrak{e}\_{\mathfrak{A}})$$

$$= (\mathfrak{e}\_{\prime} - \mathfrak{c}\_{0}A\_{\mathrm{X}\prime} - \mathfrak{c}\_{0}A\_{\mathrm{y}\prime} - \mathfrak{c}\_{0}A\_{\mathrm{z}}).\tag{99}$$

Then we have ∇ ∧ *V* = −F, or

20 Electromagnetic Theory

untwisted 1-form *<sup>V</sup>* <sup>=</sup> *<sup>φ</sup>e*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*0(−*A*) untwisted 2-form <sup>F</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−*E*) + *<sup>c</sup>*0<sup>B</sup> twisted 2-form <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> twisted 2-form <sup>I</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−*M*) + *<sup>c</sup>*0<sup>P</sup> twisted 3-form <sup>J</sup> <sup>=</sup> *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−J) + *<sup>c</sup>*0<sup>R</sup>

Table 3. Four dimensional electromagnetic field quantities as twisted and untwisted *n*-forms

*�αβγδF*˜*γδ*, or, *<sup>F</sup>*˜*αβ* <sup>=</sup> <sup>−</sup><sup>1</sup>

where *X* and Y are a three-dimensional 1-form and a three-dimensional 2-form, and ∗ is the three-dimensional Hodge operator5. Now the constitutive relations <sup>D</sup> <sup>=</sup> *<sup>ε</sup>*0(∗*E*) and *<sup>H</sup>* <sup>=</sup>

*αβ <sup>A</sup>γδ* <sup>=</sup> <sup>1</sup>

with the four-dimensional exterior derivative <sup>d</sup> = ∇ ∧ . These are much more elegant and easier to remember compared with Eqs. (76) and (81). A similar type of reformulation has

<sup>5</sup> We note the similarity with the calculation rule for complex numbers: <sup>i</sup>(*<sup>X</sup>* <sup>+</sup> <sup>i</sup>*Y*) = <sup>−</sup>*<sup>Y</sup>* <sup>+</sup> <sup>i</sup>*X*. If we can

formally set as *G* = *H* + i*c*0*D* and *F* = −*E* + i*c*0*B*, we have *G* = −i*Y*0*F* and *H* = i*Z*0*G*.

Now we have the Maxwell equations in the four-dimensional forms with components:

2

2

G = −*Y*0(∗F), or F = *Z*0(∗G). (93)

*Gαβ* = −*Y*0(∗*F*)*αβ*, or *Fαβ* = *Z*0(∗*G*)*αβ*, (94)

*∂*[*γGαβ*] = *Jαβγ*/3, *∂*[*γFαβ*] = 0, *Fαβ* = *Z*0(∗*G*)*αβ*, (96)

∇ ∧ G = J , ∇ ∧ F = 0, F = *Z*0(∗G), (97)

<sup>d</sup>G = J , <sup>d</sup>F = 0, F = *Z*0(∗G). (98)

<sup>∗</sup>(*e*<sup>0</sup> <sup>∧</sup> *<sup>X</sup>* <sup>+</sup> <sup>Y</sup> ) = *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−(∗<sup>Y</sup> )) + (∗*X*) = *<sup>e</sup>*<sup>0</sup> <sup>∧</sup> (−*Y*) + <sup>X</sup>, (92)

*�αβγδFγδ*. (91)

*gαμgβν�μνγδAγδ*. (95)

twisted/untwisted order quantities

*<sup>F</sup>αβ* <sup>=</sup> <sup>1</sup> 2

<sup>0</sup> (∗B) are four-dimensionally represented as

With components, these are represented as

and in basis-free representations:

been given by Sommerfeld (Sommerfeld (1952)).

*μ*−<sup>1</sup>

or

The Hodge operator acts on a four-dimensional two form as

with the action of Hodge's operator on anti-symmetric tensors of rank 2:

2 *� γδ*

(∗*A*)*αβ* <sup>=</sup> <sup>1</sup>

$$
\partial\_{[\!a}V\_{\beta]} = -F\_{\mathbf{a}\beta}/\mathfrak{2},\tag{100}
$$

which is a relation between the potential and the field strength. Utilizing the potential, the force equation becomes very trivial,

$$0 = \underline{\nabla} \wedge (\underline{\nabla} \wedge \underline{\mathbf{V}}) = \underline{\nabla} \wedge \underline{\mathbf{E}} \tag{101}$$

since ∇ ∧ ∇ = 0 or dd = 0 holds.

The freedom of gauge transformation with a 0-form *Λ* can easily be understood; *V*� = *V* + d*Λ* gives no difference in the force quantities, i.e., F� = F. A similar degree of freedom exist for the source fields (Hirst (1997)). With a 1-form *L*, we define the transformation G� = G + d*L*, which yields J � = J .

The conservation of charge is also straightforward;

$$0 = \underline{\nabla} \wedge \underline{\nabla} \wedge \underline{\mathcal{G}} = \underline{\nabla} \wedge \underline{\mathcal{J}}$$

$$= \mathfrak{e}^0 \wedge (\partial\_l \mathcal{R} + \nabla \wedge \mathcal{J}).\tag{102}$$

#### **7.5 Relativistic representation of the Lorentz force**

Changes in the energy *E* and momentum *p* of a charged particle moving at velocity *u* in an electromagnetic field are

$$\frac{\mathrm{d}E}{\mathrm{d}t} = q\mathbf{E} \cdot \boldsymbol{\mu},$$

$$\frac{\mathrm{d}p}{\mathrm{d}t} = q\mathbf{E} + \boldsymbol{\mu} \times \mathbf{B}.\tag{103}$$

By introducing the four dimensional velocity *u<sup>α</sup>* = *c*0*γ*, *ux*, *uy*, *uz* , and the four dimensional momentum *p<sup>α</sup>* = −*E*/*c*0, *px*, *py*, *pz* , we have the equation of motion

$$\frac{\mathrm{d}p\_{\alpha}}{\mathrm{d}\tau} = qF\_{\alpha\beta}\mu^{\beta},\tag{104}$$

where <sup>d</sup>*<sup>τ</sup>* <sup>=</sup> <sup>d</sup>*t*/*<sup>γ</sup>* the proper time of moving charge, and *<sup>γ</sup>* = (<sup>1</sup> <sup>−</sup> *<sup>u</sup>*2/*c*<sup>2</sup> <sup>0</sup>)−1/2 is the Lorentz factor. The change in action Δ*S* can be written

$$
\Delta S = p\_d \Delta \mathbf{x}^d = -E \Delta t + \mathbf{p} \cdot \Delta \mathbf{x}. \tag{105}
$$

differential forms. In order to avoid the use of the Hodge operator, the diagonal pair *Fαβ*, *Fαβ*

Reformulation of Electromagnetism with Differential Forms 43

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

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,

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

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 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

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

I thank Yosuke Nakata for helpful discussions. This research was supported by a Grant-in-Aid for Scientific Research on Innovative Areas (No. 22109004) and the Global COE program

"Photonics and Electronics Science and Engineering" at Kyoto University.

it is a pseudo (twisted) form, whose sign depends on the orientation of basis.

the tensorial order and the pseudoness (twisted or untwisted).

tools for expressing the structure of the EM theory.

momentum within dielectric media.

**9. Acknowledgment**

(= *Z*0*G*˜ *αβ*) are used conventionally and the symmetry is sacrificed.

**8. Conclusion**

parameters.

theory.

$$\begin{array}{ccc} \underline{\bf F} \text{quantities} & & \underline{\bf S} \text{quantities} \\ \underline{\bf V} = \phi \mathbf{e}^0 - c\_0 \mathbf{A} & & \\ & \downarrow \underline{\bf } & \mathbf{Y}\_0 \\ \underline{\bf E} = -\mathbf{e}^0 \wedge \mathbf{E} + c\_0 \mathbf{B} & \leftarrow \underline{\bf } \xrightarrow{} \underline{\bf } & \underline{\bf G} = \mathbf{e}^0 \wedge \mathbf{H} + c\_0 \mathbf{D} \\ & \downarrow \underline{\bf } & \mathbf{Z}\_0 \\ & \mathbf{0} & & \downarrow \underline{\bf } \\ & & & \downarrow \underline{\bf } \\ & & & 0 \\ \end{array}$$

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

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

#### **7.6 Summary for relativistic relations**

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

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 differential forms. In order to avoid the use of the Hodge operator, the diagonal pair *Fαβ*, *Fαβ* (= *Z*0*G*˜ *αβ*) are used conventionally and the symmetry is sacrificed.
