Maxwells equations are usually written as a system of four equations. I've read somewhere that when Maxwell first wrote out his equations they were as a system of twenty equations though I think managed to reduce them by using quaternions.

With differential forms they can be written as just two equations. I've read that with Geometric Algebra it's possible to write them as just one equation.

Q. Is this claim true, and does it present any real advantages or is the unity achieved here a mere chimera?

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    $\begingroup$ Useful? en.wikipedia.org/wiki/… $\endgroup$ – Avantgarde Nov 9 '18 at 0:02
  • $\begingroup$ @Avantegarde: That's not really telling me anything new. Most of the real work that has been done with Maxwells equations and it's generalisations - which in fact include the weak and string force, and even gravity in its LQG formulation - use the language of differential forms. What I'm asking about is how useful is the geometric algebra formulation. I've read at least one response from a very well respected physicist that this formulation is superficial. This is why I'm asking this question. $\endgroup$ – Mozibur Ullah Nov 9 '18 at 0:52
  • $\begingroup$ I don't know about the formulation so I can't say whether it's useful or not. In any case, you're looking for its usefulness in what sense, exactly? $\endgroup$ – Avantgarde Nov 9 '18 at 2:03

One way to write Maxwell's equations is \begin{align*} \partial_a F^{ab} &\propto J^b \tag{1a} \\ F_{ab} &=\partial_a A_b-\partial_b A_a. \tag{1b} \end{align*} I'm using the usual summation convention for repeated indices. I'm assuming flat spacetime with the usual Minkowski metric $\eta_{ab}$, and I'm using this metric to raise and lower indices. Equation (1b) says that the EM field bivector $F_{ab}=-F_{ba}$ can be written in terms of a gauge field $A_a$ as shown, and then equation (1a) may be regarded as the equation of motion for $A_a$ with the given current $J^b$.

If we don't want to use the gauge field $A_a$, we can also write Maxwell's equations like this: \begin{align*} \partial_a F^{ab} &\propto J^b \tag{2a} \\ \partial_a F_{bc}+ \partial_b F_{ca}+ \partial_c F_{ab} &= 0. \tag{2b} \end{align*} Equation (1b) is a solution of equation (2b). If the spacetime is topologically trivial, then this is the most general solution. (If we want to allow non-trivial topology, then equation (2b) is more general than (1b).)

Equation (2b) is completely antisymmetric in all three subscripts. This fact is important in the next way of writing the equations.

Geometric Algebra is another name for what mathematicians and physicists usually call Clifford Algebra. Dirac matrices are a matrix representation of the Clifford algebra associated with Minkowski spacetime. This Clifford algebra is an associative algebra in which the basis vectors $\gamma^a$ satisfy $$ \gamma^a\gamma^b + \gamma^b\gamma^a = 2\eta^{ab}. \tag{3} $$ (I'm using the conventional notation for Dirac matrices because it's familiar, but we don't really need any matrix representation here. We only need the associative algebra.) Using this algebra, Maxwells equations (2a)-(2b) may be written as a single equation like this: $$ \partial F\propto J \tag{4} $$ with $$ F = \gamma^a\gamma^b F_{ab} \hskip2cm J=\gamma^a J_a \hskip2cm \partial=\gamma^a\partial_a. \tag{5} $$ When equation (5) is expanded in the given basis, it has two parts:

  • A vector part, which is a linear combination of individual $\gamma^a$s. This part gives equation (2a).

  • A trivector part, which is a linear combination of products $\gamma^a\gamma^b\gamma^c$ with all three indices distinct from each other. This part gives equation (2b).

In my experience, for most purposes, equations (1) or (2) are easier to use than equation (4). However, equation (4) does have a few nice uses. Here's one example: if $J=0$, then we can apply $\partial$ to equation (4) on the left to get $\partial\partial F = 0$, and we can use associativity combined with the identity $xx = x_a x^a$ to get the wave equation $\partial_a\partial^a F_{bc} = 0$. This isn't much of an advantage (if any), because the wave equation can also be derived just as easily by contracting equation (2b) with $\partial^a$ and then using the $J=0$ version of equation (2a). The best use for the Clifford-algebra formulation might be for studying how $F$ transforms under Lorentz transformations. The basic idea is described in another post. (That post is written for Euclidean signature instead of Lorentzian signature, but the idea is the same.)

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  • $\begingroup$ For the case that the electric and the magnetic field component are shifted by 90° in phase one symmetrical equation seems to be possible. But how in the case of the usually claimed in-phase oscillation of both field components? $\endgroup$ – HolgerFiedler Nov 9 '18 at 5:53
  • $\begingroup$ @HolgerFiedler Equation (4) is equivalent to equations (2a)-(2b). If the in-phase versus out-of-phase question can be answered using equations (2a)-(2b), then this equivalence gives an answer that uses equation (4). Is there some doubt that equation (4) really is equivalent to equations (2a)-(2b)? $\endgroup$ – Chiral Anomaly Nov 21 '18 at 4:17
  • $\begingroup$ Dan, going back from your notation to sine and cosine, is there a solution with shifted by 90° in phase magnetic and electric field component? $\endgroup$ – HolgerFiedler Nov 22 '18 at 5:08
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    $\begingroup$ @HolgerFiedler When $J=0$, every term in Maxwell's equation(s) has a single derivative, with respect to either the time-coordinate or a space-coordinate. In conventional notation, $\dot E\propto\nabla\times B$ and $\dot B\propto\nabla\times E$. For a uniform plane wave traveling in the $+x$-direction, the solution must depend on $t$ and $x$ only via the combination $t-x$ (in units with $c=1$). This implies that every partial derivative may be rewritten as $\pm\partial/\partial t$. Altogether, this implies that $E$ and $B$ are in-phase. (Fig 8.13 in Griffiths' EM book, 2nd ed.) $\endgroup$ – Chiral Anomaly Dec 2 '18 at 4:12
  • $\begingroup$ Understood. One last question: It is possible to change Maxwells equation in a way that the EM induction (transformer device, Lenz law, antenna radiation in the near field, ...) will be described? $\endgroup$ – HolgerFiedler Dec 2 '18 at 6:31

It's easily possible to express them as a single equation in the absence of sources: first, consider the electromagnetic field $(\mathbf{E},\mathbf{H})$ as a single variable. This reduces four equations to two, one dynamical and one constraint equation. The constraint equation $(\nabla \cdot \mathbf{E} , \nabla \cdot \mathbf{H}) = (0,0)$ then enters into the definition of the vector space the dynamical equations are defined on.

When sources are present, you could use the dynamical equation and local charge conservation as your two equations plus the requirement that the initial condition satisfies the constraint equation. The solutions then automatically satisfy the constraint equation.

Note that also when you write Maxwell's equations using differential forms, you need charge conservation as a condition placed on the charge and current densities. So also here, you'd have three equations, not just two.

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