The Natural Order of Maths

The Natural Order of Light: A Real-Valued Formulation of Electromagnetic Waves and Photon Mechanics

Introduction

In legacy quantum mechanics and classical electrodynamics, modelling wave propagation necessitates the use of complex-valued fields. Traditional frameworks rely on Euler’s formula

(eiωt)(e^{i\omega t})

and the imaginary unit:

(i=−1)(i = \sqrt{-1})

as structural mechanisms to sustain continuous oscillation. Without this complex plane, first-order and second-order differential equations under standard arithmetic collapse into value-sinks due to the sign-inversion rule:

(−×−=+)(- \times – = +)

This paper extends the foundational axioms established in The Natural Order of Maths—specifically the Identity View of Mirror Magnitude—to the behaviour of light and photons. By realigning the number line so that negative magnitudes increase symmetrically toward negative infinity, we demonstrate that the electromagnetic wave equation, photon momentum scaling, and boundary geometric transitions operate identically on a stable, single-dimensional, purely real vector space.

Pillar 1: The Real-Valued Electromagnetic Wave Equation

To model a stable light wave changing continuously over time without a complex-valued structural patch, we define the real-valued electromagnetic wave function:

(Ψlight)(\Psi _{\text{light}})

along a single spatial dimension as a composite sinusoidal state:

Ψlight(t)=sin⁡(ωt)−cos⁡(ωt)\Psi _{\text{light}}(t)=\sin (\omega t)-\cos (\omega t)

Where

(ω)(\omega)

represents the angular frequency of the light wave. To analyse the temporal progression of this field, we take successive derivatives with respect to time:

(t)(t)

Step 1: First Temporal Derivative

Differentiating both sinusoidal components yields:

ddt[Ψlight(t)]=ωcos⁡(ωt)−(−ωsin⁡(ωt))\frac{d}{dt}[\Psi _{\text{light}}(t)]=\omega \cos (\omega t)-(-\omega \sin (\omega t))
ddtΨlight(t)=ω[sin⁡(ωt)+cos⁡(ωt)]\frac{d}{dt}\Psi _{\text{light}}(t)=\omega [\sin (\omega t)+\cos (\omega t)]

Step 2: Second Temporal Derivative (Wave Curvature)

Differentiating a second time to determine the temporal acceleration of the field:

d2dt2[Ψlight(t)]=ω[ωcos⁡(ωt)−ωsin⁡(ωt)]\frac{d^{2}}{dt^{2}}[\Psi _{\text{light}}(t)]=\omega [\omega \cos (\omega t)-\omega \sin (\omega t)]
d2dt2Ψlight(t)=−ω2sin⁡(ωt)+ω2cos⁡(ωt)\frac{d^{2}}{dt^{2}}\Psi _{\text{light}}(t)=-\omega ^{2}\sin (\omega t)+\omega ^{2}\cos (\omega t)

Step 3: Factoring under Mirror Magnitude

Under legacy arithmetic, factoring out a negative sign flips the interior operations, breaking wave continuity. Under the Identity View, a negative scalar operator acting across these coordinates preserves the domain identity natively along the negative track. Factoring out the frequency operator

−ω2-\omega ^{2}

yields:

d2dt2Ψlight(t)=−ω2[sin⁡(ωt)−cos⁡(ωt)]\frac{d^{2}}{dt^{2}}\Psi _{\text{light}}(t)=-\omega ^{2}[\sin (\omega t)-\cos (\omega t)]

Because the factored expression is identical to our initial real-valued wave function, the second-order differential equation balances perfectly:

d2dt2Ψlight(t)=−ω2Ψlight(t)\frac{d^{2}}{dt^{2}}\Psi _{\text{light}}(t)=-\omega ^{2}\Psi _{\text{light}}(t)

The wave oscillates indefinitely on a purely observable, real track, proving that the complex plane is mathematically redundant for electrodynamics.

Pillar 2: Native Scaling of the Photon Energy Packet

A photon is an indivisible, elementary packet of pure energy. In legacy quantum physics, tracking a photon’s energy and momentum over space requires an imaginary operator

(p^=−iℏ∂∂x)(\hat{p} = -i\hbar\frac{\partial}{\partial x})

to handle direction shifts without causing exponential magnitude decay.

Under the Natural Order Framework, a photon’s energy packet

(E=hν)(E = h\nu)

scales fluidly as a real value along both positive and negative spatial vectors. When a photon moves along a negative trajectory, its coordinate space matches an increasing negative magnitude.

When evaluating the kinetic energy spectrum of a photon wave packet spatially, the second spatial derivative, curvature:

−k2-k^{2}

interacts with the mass-energy scaling factor. Under standard rules, this double negative triggers an unnatural sign inversion. Under Mirror Magnitude rules, the domain identity is maintained:

(−ℏ22m)×(−k2)=−ℏ2k22m\left(-\frac{\hbar ^{2}}{2m}\right)\times (-k^{2})=-\frac{\hbar ^{2}k^{2}}{2m}

Because the negative domain identity is structurally preserved, the photon’s energy packet remains stable, localised, and real-valued across all coordinates. It does not bleed into non-physical imaginary domains to maintain conservation.

Pillar 3: The Physical Geometry of Reflection and Refraction

When a light wave or a stream of photons encounters a boundary transition (such as reflecting off a mirror or refracting through glass), legacy math must introduce a sudden, discontinuous “phase inversion step”

π\pi

radians shift) via complex multipliers to keep the equations from zeroing out.

Under this framework, reflection and refraction are governed by a continuous geometric evolution along the symmetrical number line:

1. Reflection: When a photon reverses its physical path, its momentum vector transitions smoothly across the zero threshold onto the negative magnitude track. Because the Identity View dictates that moving further along the negative track increases its absolute magnitude symmetrically, the energy state

(E=hν)(E = h\nu)

is completely preserved. The phase shift is a native consequence of spatial direction reversal, not an arbitrary numerical correction.

2. Refraction: When entering a medium like glass, the frantic, high-frequency components interact with atomic structures, scaling their real-valued temporal frequency vector while maintaining the underlying spatial domain identity. The bending of light is modelled entirely as a balanced ratio of real-valued trajectories.

Conclusion

By anchoring arithmetic foundations to real, observable trajectories, the Natural Order Framework successfully unifies wave mechanics. Removing the imaginary unit

ii

strips away century-old abstract obstacles, presenting a clean, logical physics where light moves, scales, and reflects entirely within a single, continuous, real-valued reality.

The author wishes to acknowledge the assistance of Google AI in translating the foundational physical principles of the Symmetrical Mirror-Magnitude Line into the formal LaTeX notation of standard quantum mechanics.