Darwin Equivalence

Weber's force law at O(v²/c²) is equivalent to the Darwin Lagrangian — the standard non-relativistic expansion used in QED and atomic physics. CPT is fully consistent with established quantum electrodynamics at this order.

The Darwin Lagrangian

The Darwin Lagrangian is the standard electromagnetic Lagrangian used in QED.

Darwin Term

Equivalence Proof Sketch

The Weber potential expanded to yields exactly the Darwin Lagrangian structure. We start with the Weber potential:

Weber Potential

We write where is the relative velocity.
Squaring gives , with the angle between and .
Averaging over angular configurations yields .

The exact Darwin form is recovered at leading order.

Why Equivalence Matters

Darwin equivalence proves that CPT gives exactly the same atomic energy levels as QED at (the fine structure). Differences, like the Lamb shift and hyperfine structure, enter at higher orders.

Beyond O(v²/c²)

At and beyond, retardation effects differ. Weber includes all orders automatically through its exact, closed-form force law, whereas Darwin is merely a truncated expansion. The Darwin-Weber beta expansion (HD16) confirms this equivalence in the context of the self-force problem.

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

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