maxwell

Maxwell's Unification: Causaloid Example

This example derives the electric and magnetic fields of a plane wave as one CausalMultiVector bivector from a single vector potential, and chains the stages with DeepCausality's monadic composition.

How to Run

From the root of the deep_causality project, run:

cargo run -p physics_examples --example maxwell_example

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Engineering Value

Standard engineering treats the Electric (E) and Magnetic (B) fields as separate vectors and checks their consistency by hand. Geometric Algebra unifies them into one 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
• Computing A (4 scalars) is 50% faster than computing E, B (6 scalars)
• Numerically more stable (no divergence cleaning)

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Physics Background

The Vector Potential Formulation

In place of separate E and B, the example uses:

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 yields:

• 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 run evaluates the potential over Dual numbers, so automatic differentiation yields E_x = -dA_x/dt and B_y = dA_x/dz exactly. It then checks:

• |E| = |B| (characteristic of light waves)
• Divergence ≈ 0 (Lorenz Gauge satisfied)
• |S| = |E||B| for the Poynting flux S = E × B
• the AD fields against the closed form

A failed check exits the process with a nonzero status.

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Causal Chain

Observation event (t, z) → Potential(A) → EM Field(F = ∇A) → Gauge Check → Results

Each step is a pure function; CausalFlow::bind chains the potential, the field bivector and the Poynting flux.

The wave's angular frequency ω is the chain's context: a Context holding one Data contextoid, attached with .context. The observation event (t, z) is the value the chain evaluates. The potential and field stages read ω from the context and evaluate the potential at the point (t, z, ω), and the closed-form check reads the same ω.

Reference

For more on Geometric Algebra in electromagnetism, see:

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