# Preamble:

*Quantum* Electrodynamics gets a lot of good press, what with

Feyman's

excellent book describing the theory, and in classical

electromagnetism Maxwell's theory soaks up most of the acclaim, leaving this intermediate theory sadly overlooked. All it essentially is is a 'tidying up' of Maxwell's theory which fits in better with the

Special Theory of Relativity - there is no genuinely

*new* science being uncovered. However, there is a very attractive neatness to it, and it provides the simplest example of what it means to unify fields in Physics.

What follows is a mathematical discussion of how to create a new field from the electrical and magnetic fields, and a new formulation of the associated physical laws. To reduce the length of this write up, I'll assume you know about the Maxwell theory already.

# Apology:

The displayed equations below look

*horrible* when set in HTML. I tried my best to make it look readable, but if you're getting confused, it might make more sense if you write the equations out again by hand.

# The Theory:

Let's start with the electromagnetic force law given by the

Lorentz Equation:

m**a**=q**E** + q**v**×**B**

*or*
ma

_{i}=eE

_{i} + qε

_{ijk}v

_{j}B

_{k}, using the

summation convention. We seek a relativistic form of the equation: clearly it should be a

four-vector equation, although there is no four dimensional form of the cross product of 3-vectors, and no obvious time components for possible E

_{μ}, B

_{μ} four-vectors. Instead, note that the

**v**×

**B** term is linear in

**v**, and given that the time component of the velocity four vector is a constant, the addition of the

**E** leaves the equation still linear in four-velocity. From this viewpoint, the simplest generalisation of the force equation is:

ma^{μ}=qF^{μ}_{ν}v^{ν}

*"what's with the raised indicies?"*

For some tensor

(look up the quotient theorem if dubious) F

_{μν}. Looking at the Lorenz equation in component form, we can readily identify F

_{io}=E

_{i} and F

_{ij}=ε

_{ijk}B

_{k}. This gives us most of the tensor:

F_{μν}= (? ? ? ? )
(E_{1} 0 B_{3} -B_{2})
(E_{2} -B_{3} 0 B_{1})
(E_{3} B_{2} -B_{1} 0 )

The Lorentz Equation gives no information about the top row, but if we impose the condition of antisymmetry on the tensor (reasonable, since the lower right hand 3×3 block is), the top row becomes (0 -E

_{1} -E

_{2} -E

_{3}). The resulting tensor is known as the

Maxwell Tensor.

This gives and extra component to the Lorentz Equation - it predicts the conservation of energy (LHS = rate of change of energy, RHS = work done by field = q**E**•**v**).

Given this, let's try and express Maxwell's Equations in terms of this new tensor. Previously:

d_{i}E_{i} = ρ/ε_{0}

ε_{ijk}d_{j}E_{k} = -d_{0}B_{i}

d_{i}B_{i} = 0

ε_{ijk}d_{j}B_{k} = μ_{0}ε_{0} d_{0}E_{i} + μ_{0}j_{i}

with d

_{ρ} meaning partial derivative with respect to x

_{ρ} (so d

_{0} meaning partial derivative with respect to time, since t=x

_{o}).

Looking at the last of these first, we have (noting that units have been chosen such that μ

_{0}ε

_{0}=1):

ε_{ijk}d_{j}B_{k}-d_{0}E_{i}=d_{j}F_{ij}-d_{0}F_{i0}=d_{j}F^{ij}+d_{0}F^{i0}=μ_{0}j^{i}.

In SR, the rule is that lower spatial index <-> upper spatial index keeps same sign, whereas upper time index <-> lower time index changes sign.

Guessing the relativistic form:

d_{μ}F^{νμ}=
-d_{ν}F^{μν}=
-μ_{0}j^{ν},

this gives an extra equation

d_{μ}F^{μ0}=d_{i}F^{i0}=-d_{i}E_{i}=-μ_{0}j_{0}

which gives us back the first equation if j

_{0}=ρ.

The other two equations are best approached from a slightly different angle: first define a new tensor G

^{μν}=½ε

^{μνρσ}F

_{ρσ} (usually called the

dual field strength tensor). It has compnents:

(0 -B_{1} -B_{2} -B_{3})
(B_{1} 0 -E_{3} E_{2})
(B_{2} E_{3} 0 -E_{1})
(B_{3} -E_{2} E_{1} 0 )

Then

d_{μ}G^{μ0}=d_{i}G^{i0} = ½ε^{i0ρσ}d_{i}F_{ρσ} =
½ε^{i0jk}d_{i}F_{jk}=
½ε^{ijk}d_{i}F_{jk}=
½div **B**=0

gives one of Maxwell's equations, and

d_{μ}G^{μi} = d_{j}G^{ji}+d_{j}G^{j0}

= ½ε^{jiρσ}d_{j}F_{ρσ}+½ε^{j0ρσ}d_{0}F_{ρσ}

= ½ε^{jik0}d_{j}F_{ko}+
½ε^{ji0k}d_{j}F_{0k}+
½ε^{j0ik}d_{0}F_{ik}

= ½ε_{ijk}d_{j}E_{k}+
½ε_{ijk}d_{j}E_{k}+
½ε_{ijk}d_{0}F_{ij}

= ε_{ijk}d_{j}E_{k}+d_{0}B_{i},

which is zero by the remaining equation; so we have d

_{μ}G

^{μi} = 0 and d

_{μ}G

^{μ0} = 0. Then Maxwell's equations can now be summarised as:

d_{μ}F^{μν} = -μ_{0}j^{ν}

d_{μ}G^{μν} = 0

which after all the hellish notation above looks quite elegant.

# Meaning what?

Once you have adopted Special Relativity, the notion of four dimensional space, and four-vectors, must become fundamental to all other physical theories you wish to describe. The three-dimensional notions of

**E** and

**B** fields are now no longer valid, but by some seemingly remarkable stroke of luck these two fields can be put together into a single matrix field which behaves properly under

Lorentz Transformations. Even better, when Maxwell's Equations are expressed in terms of this tensor, they come out in a much neater from (which reveals

symmetries which were previously not clear) - eg. now it becomes clear why the time derivative of one field always occurs in connection with the

curl of the other.

As I was saying above, while this may all look far from simple, it is the simplest example of how ideas in Physics can be simplified by adding another dimension.

# I want more pain:

Read about the

vector potential and the

stress-energy tensor.