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Clifford Algebra Formulation of Electromagnetism

Previously, we discussed both the vector calculus and tensor calculus formulations of electromagnetism. In this section, we will explore the Clifford algebra formulation of electromagnetism, which provides a more geometric approach to the subject.

Table of Contents​

Electromagnetic Field in Cl(3, 0)​

In the vector calculus formulation, we represent the electromagnetic field using two vector fields: the electric field and the magnetic field . In the tensor calculus formulation, we represent the electromagnetic field using the electromagnetic field tensor , which is an antisymmetric rank-2 tensor. In the Clifford algebra formulation, we can represent the electromagnetic field as a multivector.

We shall first use the algebra of physical space (APS) , which is the Clifford algebra of three-dimensional Euclidean space.

First, let denote the electromagnetic field as a multivector in the spacetime algebra. We obviously cannot define as

because and are both vectors, and thus they will mix and be unable to be separated. One way to resolve this is to place the pseudoscalar in front of the magnetic field , so that we have

Then, we just need to multiply by to get the correct units, so we have

Current and Charge Density​

Next, we need to define the current and charge density.

Much like in the tensor formulation, we can combine the current density and the charge density into a single object. This is because they are just spatiotemporal densities, and thus make sense to combine them into a single multivector. We can define the current and charge density as

It is a combination of a scalar and a vector , where the scalar is the charge density and the vector is the current density.

Maxwell's Equations in Cl(3, 0)​

We can now write Maxwell's equations in the Clifford algebra formulation (specifically in the algebra of physical space ). We shall denote the nabla operator as for the ordinary gradient operator in vector calculus.

The equations are as follows:

It is really surprising how simple these equations are in the Clifford algebra formulation. If you do not want to be spoiled, don't click the dropdown below.

Maxwell's Equations in Cl(3, 0)

That's it.

To derive the equation, we shall convert the equations to the language of multivectors. Beginning with the Ampère-Maxwell law, we rearrange it to get

We know that in the APS, the cross product can be expressed as a wedge product along with a pseudoscalar;

Hence the law becomes

Next, we want to remove the coefficients from the electric field term, so we multiply everything by ;

or

This equation is still a vector equation, in that both sides are vectors. The trick we will employ is to convert the other equations to other types of equations, such as scalar or bivector equations. This way, they are all independent of each other and can be combined into a single equation.

Gauss's law is already a scalar equation, so we can leave it as

We shall now convert Faraday's law to a bivector equation. Multiplying everything in Faraday's law by , we get

As , we can rewrite this as

Finally, we can convert Gauss's law for magnetism to a trivector equation by simply multiplying by ;

Now we will just add all of these equations together. We have

Here's where the magic happens. Notice that we have a term and a term. The sum of those two is just the definition of a Clifford product

This leads to the equation

Likewise, we have

so we can rewrite the equation as

Finally, we can define the vector gradient in APS as

and thus we can simplify some terms like this:

Thus we can rewrite the equation as

Now we will invoke the definition of the electromagnetic field as

and the definition of the current and charge density as

to finally get

As such, we can see that Clifford algebra provides a very elegant and concise formulation of Maxwell's equations.