## Concise Formulation of Maxwell Equations

The Maxwell equations without sources can be written concisely as

$\triangledown\cdot (\mathbf{E}+i\mathbf{B})=0$

$\triangledown\wedge (\mathbf{E}+i\mathbf{B})=i\frac{\partial}{\partial t}(\mathbf{E}+i\mathbf{B})$

Compare this set to the following set of equations in the traditional notation

$\triangledown\cdot\mathbf{E}=0$

$\triangledown\cdot\mathbf{B}=0$

$\triangledown\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

$\triangledown\times\mathbf{B}=\frac{\partial\mathbf{E}}{\partial t}$

The dot $\cdot$ is the vector inner product , the $\times$ is the vector cross product. Maxwell equations can be written in integral form as well. I will not discuss the integral form in this article. In these equations the physical units of $\mathbf{E}$  and $\mathbf{B}$ are defined such that the speed of light $c=1$.

In the traditional formulation, the $\triangledown\cdot\mathbf{E}$  is called the divergence of $\mathbf{E}$ and $\triangledown\cdot\mathbf{B}$  is called the divergence of $\mathbf{B}$ . Divergence is a scalar quantity. It is related to the source that creates the field. The $\triangledown\cdot\mathbf{E}=0$ means that in vacuum there is no source (electric charge) to create the electric field. Similarly $\triangledown\cdot\mathbf{B}=0$ means that in vacuum there is no source (magnetic charge) to create the magnetic field.

The $\triangledown\times\mathbf{E}$  is called the curl of the vector field $\mathbf{E}$  and $\triangledown\times\mathbf{B}$  is called the curl of the vector field $\mathbf{B}$ . The curl is a vector indicating the axis and magnitude of the rotation of the vector field. The $\triangledown\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$ is telling us that a time-varying magnetic field creates a circular (rotating) electric field around the axis of the time-varying magnetic field vector. Similarly  $\triangledown\times\mathbf{B}=\frac{\partial\mathbf{E}}{\partial t}$    is telling us that a time-varying electric field creates a circular (rotating) magnetic field around the axis of the time-varying electric field.

In the first set of equations (concise formulation) the modern notation of outer (wedge) product is used in place of the vector cross product. One of the motivations for the modern notation is to make the physics equations independent of coordinate frames.The modern notation takes advantage of the geometric product which is

$\mathbf{a}\mathbf{b}=\mathbf{a}\cdot\mathbf{b}+\mathbf{a}\wedge\mathbf{b}$

For a discussion of the modern notation in physics education please see Hestenes [1]. Let’s just mention that the outer product $\mathbf{a}\wedge\mathbf{b}$  is called a bivector, and it can be interpreted geometrically as oriented plane segment. The $\mathbf{a}\wedge\mathbf{b}$ differs from $\mathbf{a}\times\mathbf{b}$ in being intrinsic to the plane containing $\mathbf{a}$ and $\mathbf{b}$, independent of the dimension of any vector space in which the plane lies [1].

I should also mention that the outer product is the geometric dual of the vector cross product. The geometric duality is defined as the multiplication by the pseudoscalar $i$.

$\mathbf{a}\wedge\mathbf{b}=i (\mathbf{a}\times\mathbf{b})$

In the concise formulation the number of equations is reduced from 4 to 2 at the expense of the introduction of a complex vector field. This should not be seen as an expense because $\mathbf{E}+i\mathbf{B}$ gives us new insights.

$\mathbf{E}$  and $\mathbf{B}$  are the electric and magnetic field vectors, respectively.

$\mathbf{E}=(E_x,E_y,E_z)$

$\mathbf{E}=(E_x,E_y,E_z)$

Electric field accelerates an electric charge parallel to its direction. The magnetic field accelerates an electric charge perpendicular to its direction, in other words, the magnetic field curves the path of an electric charge.

The electric charge is both the source and the responder of the electric field. The magnetic charge (monopole) must exist but has not been observed yet.

What is the new insight that we gain from the concise formulation?

The Maxwell equations stay the same (preserve the form) under the duality transformation

$\mathbf{E}\rightarrow\mathbf{B}$ and $\mathbf{B}\rightarrow\mathbf{-E}$

This property was noticed by Heaviside [2] in 1893.

The Heaviside duality transformation mentioned above can be generalized to a continuous transformation

$(\mathbf{E}+i\mathbf{B})\rightarrow e^\phi (\mathbf{E}+i\mathbf{B})$

where  $\phi$  is any angle. Transformations of this kind belong to the U(1) group. David Olive calls this “duality rotation”. Note that U(1) group contains continuous transformations. This is an important point because through the Noether theorem for continuous symmetries   $(\mathbf{E}+i\mathbf{B})\rightarrow e^\phi (\mathbf{E}+i\mathbf{B})$    implies the conservation of electric charge.

Historical Note: The generalization of the Heaviside duality to continuous duality transformation was actually done for the first time by Larmor [3] who expressed it as follows:

$\mathbf{E}_\theta=\mathbf{E}cos\theta-\mathbf{B}sin\theta$

$\mathbf{B}_\theta=\mathbf{E}sin\theta+\mathbf{B}cos\theta$

$0 \le \theta \leq \frac{\pi}{2}$

Larmor’s formulation is equivalent to

$(\mathbf{E}+i\mathbf{B})\rightarrow e^\phi (\mathbf{E}+i\mathbf{B})$

[1] David Hestenes, “Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics”, http://en.wikipedia.org/wiki/David_Hestenes

[2] G. Heaviside, Phil. Trans. Roy. Soc. (London), A 183, 423 (1893)

[3] I. Larmor, “Collected Papers” (London) (1928)