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Maxwell's Unification: Causaloid Example

This example demonstrates how to model Maxwell's electromagnetic field unification using DeepCausality's monadic composition with CausalMultiVector.

How to Run

From the root of the deep_causality project, run:

cargo run -p physics_examples --example maxwell_example

Engineering Value

In standard engineering, Electric (E) and Magnetic (B) fields are treated as separate vectors, requiring manual consistency checks. In Geometric Algebra, they are unified into a single Electromagnetic Field Bivector (F) derived from a Vector Potential (A).

Application: 5G/6G Antenna Design (Phased Arrays)

  • Simulate the Interference Pattern of the Vector Potential directly on the antenna mesh
  • Calculate A (4 scalars) is 50% faster than calculating E, B (6 scalars)
  • Numerically more stable (no divergence cleaning)

Physics Background

The Vector Potential Formulation

Instead of working with E and B separately, we use:

F = ∇A (Geometric Product)

Where:

  • A: 4-Vector Potential (φ, A_x, A_y, A_z)
  • ∇: Spacetime Gradient Operator
  • F: Electromagnetic Field Bivector

Extracted Components

The geometric product automatically produces:

  • Scalar (Grade 0): Divergence → Lorenz Gauge Check
  • Bivector e_tx: Electric Field E
  • Bivector e_xz: Magnetic Field B

Plane Wave Example

For a linearly polarized plane wave moving in the Z-direction:

A = (0, cos(ω(t-z)), 0, 0)

The physics verification confirms:

  • |E| = |B| (characteristic of light waves)
  • Divergence ≈ 0 (Lorenz Gauge satisfied)

Causal Chain

PlaneWaveConfig → Potential(A) → EM Field(F = ∇A) → Gauge Check → Results

Each step is a pure function composed monadically via PropagatingEffect::bind.

Reference

For more on Geometric Algebra in electromagnetism, see:

  • Hestenes, D. "Spacetime Algebra" (Gordon and Breach, 1966)