A Short Introduction to Geometric Algebra – Maxwell’s Equations in Compact Form
A Short Introduction to Geometric Algebra (GA)
Geometric Algebra unifies scalars, vectors, bivectors (oriented areas), trivectors, and higher-grade elements into a single algebraic system, with the geometric product as its fundamental operation. For orthogonal vectors $a,b$, $ab = a\wedge b$ (an oriented area), while for parallel vectors $ab = a\cdot b$ (a scalar). In general: $$ ab \;=\; a\cdot b \;+\; a\wedge b. $$ This allows us to handle directions and magnitudes without extra machinery (no component indices, no $\epsilon_{ijk}$ symbols, no pseudo-vectors).
Spacetime Algebra (STA) and Signature
In relativistic physics we often use spacetime algebra (STA) for Minkowski space with signature $(+,-,-,-)$. Let $\{\gamma_\mu\}_{\mu=0}^3$ be an orthonormal basis with $$ \gamma_\mu\cdot \gamma_\nu = \eta_{\mu\nu}=\mathrm{diag}(+,-,-,-), \qquad \gamma_\mu\gamma_\nu + \gamma_\nu\gamma_\mu = 2\eta_{\mu\nu}. $$ The spacetime pseudoscalar (4-volume element) is $$ I \equiv \gamma_0\gamma_1\gamma_2\gamma_3, \qquad I^2=-1. $$
The Vector Derivative (Dirac Operator)
The fundamental differential operator in GA is the vector derivative $$ \nabla \;\equiv\; \gamma^\mu \partial_\mu, $$ where indices are raised with $\eta^{\mu\nu}$. The product $\nabla X$ splits into an inner (divergence) and outer (curl) part: $$ \nabla X \;=\; \nabla\cdot X \;+\; \nabla\wedge X, $$ just as $ab = a\cdot b + a\wedge b$ for vectors.
Maxwell’s Equations in One Line
The electromagnetic field is represented by the Faraday bivector $F$ (a 2-vector in STA). With the 4-current $J=\rho\,\gamma_0 + \mathbf{J}$ (a vector), Maxwell’s equations in SI units become:
$$\boxed{ \;\nabla F \;=\; \mu_0\, J \; }$$
Since the left-hand side contains both a vector part ($\langle\cdot\rangle_1$) and a trivector part ($\langle\cdot\rangle_3$), and the right-hand side is a pure vector, we automatically get:
- Inhomogeneous equations (sources): $\langle \nabla F\rangle_1 = \nabla\cdot F = \mu_0 J$.
- Homogeneous equations (no magnetic monopoles): $\langle \nabla F\rangle_3 = \nabla\wedge F = 0$.
For an observer with timelike unit vector $u$ ($u^2=+1$), the measurable fields are:
$$\mathbf{E} = F\cdot u, \qquad \mathbf{B} = -\frac{1}{c}\, I\,(F\wedge u),$$ $$F \;=\; u\wedge \mathbf{E} \;+\; I\,c\, (u\wedge \mathbf{B}).$$
For $u=\gamma_0$ and with $c^2=1/(\mu_0\varepsilon_0)$ we recover the four standard equations:
- $\nabla\!\cdot\!\mathbf{E} = \rho/\varepsilon_0$ (Gauss’ law),
- $\nabla\!\times\!\mathbf{B} - \dfrac{1}{c^2}\,\partial_t \mathbf{E} = \mu_0\,\mathbf{J}$ (Ampère–Maxwell law),
- $\nabla\!\cdot\!\mathbf{B} = 0$ (no monopoles),
- $\nabla\!\times\!\mathbf{E} + \partial_t \mathbf{B} = 0$ (Faraday’s induction law).
Unit note. In Heaviside–Lorentz units we simply write $\nabla F = J$. In SI units, the above form with $F = \mathbf{E} + I\,c\,\mathbf{B}$ (relative to $\gamma_0$) reproduces the constants automatically.
Monogenic Functions and the Dirac Operator
In an $n$-dimensional Euclidean GA with orthonormal basis $\{e_i\}_{i=1}^n$, the Dirac operator (vector derivative) is $$ \partial \;\equiv\; \sum_{i=1}^{n} e_i \,\frac{\partial}{\partial x_i}. $$ A multivector-valued function $f:\Omega\subset\mathbb{R}^n \to \mathcal{G}_n$ is called left-monogenic if $$ \partial f = 0 \quad \text{in } \Omega. $$ (Similarly, right-monogenic if $f\,\partial = 0$.)
The key fact is: $$ \partial^2 \;=\; \sum_{i,j} e_i e_j\,\partial_i \partial_j = \sum_{i} \partial_i^2 \;=\; \Delta, $$ i.e. the Dirac operator squared is the Laplacian. Therefore, each component of a monogenic function is harmonic ($\Delta f=0$), just as holomorphic functions in complex analysis have harmonic real and imaginary parts.
2D Example (Cauchy–Riemann in GA Form)
In planar GA ($n=2$) let $i \equiv e_1 e_2$ with $i^2=-1$ and $\partial = e_1 \partial_x + e_2 \partial_y$. Write $f = u + v\,i$ (scalar + bivector). Then $\partial f = 0$ gives the Cauchy–Riemann equations: $$ \partial_x u = \partial_y v, \qquad \partial_y u = -\,\partial_x v. $$ Thus, monogenic functions generalize holomorphy to higher dimensions without introducing tensor notation.
Spacetime Version
In Minkowski STA, replace $\partial$ with $\nabla = \gamma^\mu \partial_\mu$. Then $\nabla^2 = \Box$ (the d’Alembertian). Fields satisfying $$ \nabla \Psi = 0 $$ are monogenic in spacetime; they are thus solutions to the wave equation $\Box \Psi = 0$. Source-free Maxwell fields are exactly monogenic in the sense $\nabla F=0$ (when $J=0$).
Why GA is Conceptually Clean Here
- Maxwell in one line, with grade decomposition yielding all four equations automatically.
- Duality is a simple multiplication by the pseudoscalar $I$ (no external Hodge star needed).
- Monogenic functions give a direct higher-dimensional analogue of complex analysis with the same elegance.
- The observer-dependence is explicit in the choice of $u$ (timelike unit vector) – separating invariant structure ($F$) from observed decomposition ($\mathbf{E},\mathbf{B}$).
Quick Reference
- Geometric product: $ab=a\cdot b + a\wedge b$.
- Pseudoscalar: $I=\gamma_0\gamma_1\gamma_2\gamma_3$, $I^2=-1$.
- Dirac operator: $\nabla=\gamma^\mu \partial_\mu$ (STA), $\partial=e_i\partial_i$ (Euclidean).
- Maxwell: $\nabla F=\mu_0 J$ (SI), with $F = u\wedge \mathbf{E} + I\,c\,(u\wedge \mathbf{B})$.
- Monogenicity: $\partial f=0 \Rightarrow \partial^2 f = \Delta f = 0$ (harmonic); in STA: $\nabla^2=\Box$.
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