Lamb Shift

In quantum electrodynamics, the Lamb shift refers to a small difference in energy between the $2S_{1/2}$ and $2P_{1/2}$ states of hydrogen—levels that are degenerate in the Dirac equation.

The standard explanation invokes virtual photon exchange, vacuum polarisation, and loop corrections.

In modal dynamics, the Lamb shift arises from a distortion in modal anchoring structure near a dense coherence source—namely, the atomic nucleus.

No virtual particles are involved.
The effect is real, structural, and deterministic.

Anchored Electrons in a Coherence Field

In the modal picture:

• The nucleus is a compact, high-coherence emitter that defines a steep local coherence field $B(x)$.
• The electron is a coherence-anchored mode orbiting within this gradient.
• Anchoring occurs by minimising cost in the presence of this field.

Different electronic modes (e.g. $s$ vs. $p$ orbitals) experience different anchoring geometries:

• $s$-states penetrate closer to the nucleus, where $B(x)$ is steep and nonlinear
• $p$-states anchor further out, where the gradient is smoother

This leads to non-identical anchoring cost landscapes even if the quantum numbers would suggest degeneracy.

Modal Distortion

The key insight is that the anchoring cost is not symmetric for all coherent configurations:

• The $2S_{1/2}$ state incurs a higher cost due to tight phase compression near the core
• The $2P_{1/2}$ state avoids the steepest region of $B(x)$, reducing anchoring stress

This small difference in anchoring cost creates a measurable energy shift:

This is the Lamb shift—not a radiative correction, but a structural asymmetry in phase accommodation within a gradient field.

No Loops, No Renormalisation

In modal dynamics:

• There are no vacuum fluctuations
• There is no need to cancel infinities
• The shift arises from the real structure of the mode under phase constraint

Every mode obeys the same cost functional. The differences arise not from quantum randomness, but from coherence geometry under boundary distortion.

Why It's a Signature

The Lamb shift confirms that:

• Electron modes are not point particles
• Anchoring depth and field steepness matter
• Degeneracy is broken by real structure, not field interaction

This is a strong test of modal mechanics. The shift is small—but entirely predictable from the anchoring rules and phase profiles involved.

(See Appendices/Appendix AJ — Lamb Shift from Coherence Overlap.)

The Lamb shift is not a quantum correction.
It is a modal footprint left by anchoring near coherence saturation.