Maxwell's Equations in Geometric Algebra | Multivector Form

The Lagrangian formulation is another way of describing the dynamics of Ψ, V, and A fields. It maintains local gauge invariance and is compatible with special relativity. This formulation describes both how charges affect electromagnetism and how electromagnetism affects charges - it contains everything seen before. From this Lagrangian, one can derive the Dirac equation (and thus Schrödinger equation) as well as the four Maxwell equations.

La formulación lagrangiana para el electromagnetismo es L = -1/4 F_mu_nu F^mu_nu, que es la densidad lagrangiana del campo electromagnético. La ecuación de movimiento de este lagrangiano es d_mu F^mu_nu = 0, que es la ecuación de Maxwell sin cargas. Para incluir cargas, se necesita un lagrangiano de materia que acople el campo electromagnético a las cargas. La ecuación de movimiento se obtiene generalizando la ecuación de Euler-Lagrange para campos: la derivada mu de la variación de L con respecto a d_mu A_nu es igual a la variación de L con respecto a A_nu. En el caso electromagnético, el lagrangiano no depende explícitamente de A_nu, solo de F_mu_nu, y al variar se obtiene que d_mu F^mu_nu = 0.

The Lagrangian for a charged particle is L = -mc²√(1-β²) - (e/c)Aμ(dxμ/dt). The electromagnetic field Lagrangian is L_field = (1/16πc)∫FμνFμνd⁴x, where Fμν = ∂μAν - ∂νAμ. The total action is S = S_particle + S_field + S_interaction. The Lagrangian density is quadratic in fields, leading to linear Maxwell equations. In electrostatics, L_field = (1/8π)∫(E² - φρ)d³x, showing field energy and interaction energy.

The Lagrangian density for electromagnetism is L = -1/4 F_μνF^μν, the simplest gauge-invariant and Lorentz-invariant expression from the field tensor. This single expression contains all information needed to derive Maxwell's equations. The Lagrangian is quadratic in field derivatives, similar to the scalar field Lagrangian. It has the form L = T - V, where T contains time derivative terms (kinetic energy) and V contains space derivative terms (potential energy).

For the electromagnetic field with Lagrangian density L = -1/(16πc) F_μν F^μν, the Euler-Lagrange equations are ∂L/∂A_ν - ∂_μ (∂L/∂(∂_μ A_ν)) = 0. Computing these derivatives: ∂L/∂A_ν = 0 (since L depends only on derivatives of A), and ∂L/∂(∂_μ A_ν) = -1/(8πc) F_μν. Thus the equation becomes -∂_μ F_μν = 0, which are the homogeneous Maxwell equations. The derivatives of the Lagrangian density with respect to the field derivatives are computed as follows: ∂L/∂(∂_μ A_ν) = -1/(8πc) F_μν. This is because L = -1/(16πc) F_αβ F^αβ, and ∂F_αβ/∂(∂_μ A_ν) = δ_α^μ δ_β^ν - δ_β^μ δ_α^ν. The antisymmetry of F_αβ ensures that only the terms with α = μ and β = ν contribute, giving the result.

For a ball of radius R, Green's function is constructed using the inverse point x* = (R²/|x|²)x. By similar triangles, |x-y|/|x-x*| = |x|/R. This geometric relationship ensures G(x,y) = 0 on ∂Ω. The function is G(x,y) = -1/(4π|x-y|) + 1/(4π|x-x*|), where the second term is the harmonic correction.

Two perspectives extend Green's work: geometrically, Green's rational function represents a specific instance of a polyhedral cone valuation; algebraically, starting from a poset, one can associate an affine semigroup ring and compute its Hilbert series, from which Green's rational function naturally emerges through algebraic manipulation. These perspectives allow researchers to characterize when the function vanishes and determine minimal denominators.

Green's function is fundamentally determined by the geometry of the domain Ω. The function v(M, P) in G(M, P) = 1/(4πr) + v(M, P) must be harmonic in P for each fixed M and satisfy the boundary condition G(M, P) = 0 for all P on σ. This means that the entire structure of Green's function is dictated by the shape and size of the domain, making it a geometric object associated with Ω rather than depending on the specific differential equation being solved.

This section presents the method of descent, a powerful technique for computing Green's functions in higher dimensions by reducing them to lower-dimensional problems. For the D'Alembert operator in n dimensions, we can obtain the Green's function by integrating the (n-1)-dimensional Green's function over the extra variable. This works because the operator has a product structure that allows separation of variables. In the method of descent, we consider a source independent of one spatial variable, say f(t,x) = f(t,x₁,x₂)δ(x₃). The solution u(t,x) will then be independent of x₃, so we write u(t,x) = v(t,x₁,x₂). The PDE reduces to the (n-1)-dimensional D'Alembert equation for v. The Green's function for the n-dimensional operator is obtained by integrating the (n-1)-dimensional Green's function over the extra variable: G_n(t,x) = ∫ G_{n-1}(t,x',x₃) dx₃. The Green's function for the D'Alembert operator can be expressed as a surface integral over the light cone: G(t,x) = (1/(4πa²)) ∫_{|z| = a|t|} δ(z - x) dS(z) / |z|. This representation shows that the Green's function is supported on the light cone and is given by the delta function restricted to this surface. The surface integral form is equivalent to the delta-function form but provides geometric insight into wave propagation. The Green's function represents the propagation of waves from a point source, with the support being the set of points reachable by a signal traveling at speed a from the origin. The factor 1/(4πa²|x|) represents the spreading of wave amplitude as it propagates outward. This geometric picture is fundamental to understanding wave propagation. The Green's function can be expressed as a delta-sequence approximation of the identity, meaning that as ε → 0+, Gε converges to G in the sense of distributions. This delta-sequence property is essential for the Green's function to correctly reproduce the source term when convolved with arbitrary sources.

This section demonstrates how gauge theories reduce to quantum mechanical systems through canonical quantization. By fixing axial gauge and applying Gauss's law, the full field theory reduces to a quantum mechanical system with compact variables φ and k (the holonomy). This clock-shift system has n states with operators e^(iφ) (clock) and e^(ik) (shift). The system exhibits multiple interpretations: Z_N gauge theory, global Z_N symmetry with order parameter e^(iφ), and spontaneous symmetry breaking in infinite volume where the Hilbert space splits into n superselection sectors. Domain walls implement symmetry transformations across boundaries, showing how extended operators encode vacuum structure.

Compactification transforms the total space into a complex symplectic manifold - the complexification of a classical phase space. This establishes correspondence between four-dimensional gauge theories and quantum mechanical systems. The compactified gauge theory corresponds to the phase space of n particles on a line with periodic Toda chain Hamiltonian - particles with repulsive exponential interactions and periodic boundary conditions. Methods developed for supersymmetric gauge theories (localization, supersymmetric indices) enable exact computation of spectra and wavefunctions for these quantum mechanical systems. This bidirectional correspondence connects decades-old quantum mechanical problems with modern field theory techniques.

This section traces the historical development of gauge theory from Hermann Weyl's 1918 attempt to extend relativity to explain electromagnetism to its modern quantum mechanical formulation. Weyl proposed that if measurement standards could vary across space, a new field would be needed to regulate these differences. His initial attempt failed but was revived 30 years later with quantum mechanics. In quantum mechanics, particle states are described by wave functions with phases that are not directly observable. While global phase changes don't affect physics, local phase changes create mathematical inconsistencies. The solution is introducing a gauge field (A_μ) that compensates for local phase changes, creating a covariant derivative that maintains symmetry. This field is the vector potential describing electromagnetic force, and when quantized, becomes the photon. This framework later extends to explain weak and strong forces, with all forces being results of nature's efforts to maintain symmetry.

Physical descriptions contain excess degrees of freedom—equivalent configurations describe the same reality. Fiber bundles provide the mathematical framework: objects require coordinate bases at each spacetime point, with equivalent configurations differing by gauge basis transformations. The gauge group consists of smoothly varying functions mapping spacetime points to Lie group elements. Constant-parameter transformations (global) behave like rigid rotations, affecting path derivatives uniformly. Variable-parameter transformations (local) produce qualitatively different derivative effects. The covariant derivative incorporates gauge fields to maintain invariance. Gauge fixing eliminates redundant configurations by choosing specific gauges, enabling practical calculations. Anomalies arise from mishandling gauge dependence in boundary conditions. Quantum field theories (QED, Standard Model) share gauge covariant actions but handle gauge redundancies differently—simple QFTs use gauge fixing while sophisticated non-abelian theories employ BRST quantization. Beyond physics, gauge theory techniques transformed mathematics: Donaldson's work revealed exotic four-manifold smooth structures, while Witten and Seiberg's 1994 supersymmetric methods enabled calculating topological invariants.

Gauge theory describes physical forces through fields like electromagnetism and gravity. Gauge transformations shift configurations without changing measurable outcomes, demonstrating gauge invariance. In quantum mechanics, particles exhibit wave-like behavior with probabilities determined by interference patterns. The Aharonov-Bohm effect shows electromagnetic potentials affect quantum systems even when fields are zero, causing phase differences that alter interference patterns. Gauge bosons mediate fundamental forces, with their nature determined by gauge transformations. The Standard Model encompasses all fundamental interactions except gravity.