Appendix O — Derivation 15: Photon Internal Structure and Polarisation

Overview

In classical electromagnetism, a photon is a quantised oscillation of the EM field. In quantum mechanics, it is a point-like boson with spin-1 and transverse polarisation.

In modal dynamics, the photon is a latency-preserving coherence mode: it has no mass, no anchoring, and no associated field—but it possesses a rich internal structure.

This appendix derives:

The modal geometry of the photon
The origin of polarisation from phase topology
The distinction between linear and circular polarisation as structural states

1. # Photon as a Latent Phase Mode

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In PBG a photon is the simplest coherence mode:

envelope cost term $β | ψ |^{2}$ → set to zero (no anchoring)
phase field follows the massless wave equation $γ \partial_{t}^{2} Φ = α \nabla^{2} Φ$ (see Foundational Definitions).

Its generic form:

ψ_{γ} (x, t) = f (x - c t \hat{z}) \exp [i ϕ (x, t)],

where $f$ is a slowly varying envelope (keeps the beam localised) and the
internal phase $ϕ$ encodes polarisation.

1 Phase surface & transverse geometry

Let the carrier move along $+ z$ with $ω = c k$ :

ϕ (x, y, z, t) = k z - ω t + θ (x, y) .

$θ (x, y)$ fixes the transverse phase gradient
$\nabla_{⊥} ϕ = (\partial_{x} θ, \partial_{y} θ)$ .
Different $θ$ choices give the standard polarisation families.

2 Linear polarisation

Choose

θ (x, y) = α x ⟹ \nabla_{⊥} ϕ = α \hat{x} .

The phase surfaces are parallel planes tilted in $x$ ; the coherence
gradient oscillates purely along $\hat{x}$ – the PBG picture of linear
polarisation.

3 Circular polarisation

Take

θ (x, y) = m \arctan (y / x),

so that

\nabla_{⊥} ϕ = \frac{m}{x^{2} + y^{2}} (- y \hat{x} + x \hat{y}),

a spiral around the propagation axis.

$m = + 1$ → left-handed, counter-clockwise phase advance
$m = - 1$ → right-handed

Interpretation – photon “spin” $\pm 1$ is the handedness of this
internal phase rotation; no intrinsic angular-momentum operator is needed.

4 Elliptical & arbitrary states

Any linear combination

θ (x, y) = a x + b y

or more complex $θ (x, y)$ that keeps the anchoring cost density

C_{γ} = α | \nabla_{⊥} ϕ |^{2}

finite is a valid, stable polarisation mode.
The usual Poincaré sphere of polarisations is realised as the set of all
allowed $θ$ .

5 Implications & predictions

Conventional view	PBG reinterpretation
Polariser projects $E$ -field	Aligns phase gradient $\nabla_{⊥} ϕ$
Spin-1 operator $S_{z} = \pm 1$	Integer winding $m = \pm 1$ in $θ (x, y)$
Entangled polarisations	Locked transverse phase fields of two modes

All textbook results (Malus law, Stokes parameters, Bell tests) follow
directly from the geometry of $θ (x, y)$ and coherence-cost coupling.

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Appendix N - Neutrino Mass | [Index](./Appendix Master) | Appendix P - Orbital Motion