Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== We start from the Lagrangian density we've built: ===

L=12c2(∂ϕ∂t)2−12(∂ϕ∂r)2−V(ϕ)\mathcal{L} = \frac{1}{2c^2} \left( \frac{\partial \phi}{\partial t} \right)^2 - \frac{1}{2} \left( \frac{\partial \phi}{\partial r} \right)^2 - V(\phi)L=2c21(∂t∂ϕ)2−21(∂r∂ϕ)2−V(ϕ)
Now let’s construct the Hamiltonian, which represents the total energy in terms of:
* Kinetic energy of the wave over time
* Gradient energy over space (spreading out)
* Potential energy from symmetry breaking

===== From the Lagrangian, the canonical momentum is: =====

π(r,t)=∂L∂(∂tϕ)=1c2∂ϕ∂t\pi(r,t) = \frac{\partial \mathcal{L}}{\partial (\partial_t \phi)} = \frac{1}{c^2} \frac{\partial \phi}{\partial t}π(r,t)=∂(∂tϕ)∂L=c21∂t∂ϕ

===== The Hamiltonian density is: =====

H=π∂ϕ∂t−L\mathcal{H} = \pi \frac{\partial \phi}{\partial t} - \mathcal{L}H=π∂t∂ϕ−L
Substitute in π\piπ and L\mathcal{L}L:

H=12c2(∂ϕ∂t)2+12(∂ϕ∂r)2+V(ϕ)\mathcal{H} = \frac{1}{2c^2} \left( \frac{\partial \phi}{\partial t} \right)^2 + \frac{1}{2} \left( \frac{\partial \phi}{\partial r} \right)^2 + V(\phi)H=2c21(∂t∂ϕ)2+21(∂r∂ϕ)2+V(ϕ)
✅ This matches physical intuition:
* First term = kinetic energy
* Second term = gradient/spread energy
* Third term = potential (due to symmetry breaking)

===== Now integrate over the spherical surface of radius rrr: =====

E=∫H⋅dA=∫02π∫0πH(r,t)⋅r2sin⁡θ dθdϕE = \int \mathcal{H} \cdot dA = \int_0^{2\pi} \int_0^\pi \mathcal{H}(r,t) \cdot r^2 \sin\theta \, d\theta d\phiE=∫H⋅dA=∫02π∫0πH(r,t)⋅r2sinθdθdϕ
E=4πr2⋅H(r,t)E = 4\pi r^2 \cdot \mathcal{H}(r,t)E=4πr2⋅H(r,t)
: 

So the total field energy over the sphere becomes:

E(t)=4πr2[12c2(∂ϕ∂t)2+12(∂ϕ∂r)2+V(ϕ)]E(t) = 4\pi r^2 \left[ \frac{1}{2c^2} \left( \frac{\partial \phi}{\partial t} \right)^2 + \frac{1}{2} \left( \frac{\partial \phi}{\partial r} \right)^2 + V(\phi) \right]E(t)=4πr2[2c21(∂t∂ϕ)2+21(∂r∂ϕ)2+V(ϕ)]

===== Let’s now link each part to your geometrical vision: =====

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==== We’ve successfully: ====
* Defined a QAT Lagrangian using h/2πh/2\pih/2π and spherical geometry
* Derived the spherical wave equation
* Added a symmetry-breaking potential to describe measurement
* Built the Hamiltonian → total energy on the sphere

This gives us the foundation to model wavefunction evolution, quantum collapse, and even charge movement as the geometry of the field changes.

==== We now have two powerful directions available: ====
# Quantize this Hamiltonian → Start building a true Quantum Field Theory over spherical geometry
# Introduce coupling to charge → Show how breaking spherical symmetry causes electron motion, i.e., charge flow, EM fields, or even current

Would you like to go forward with quantization, or start modeling charge emergence from broken symmetry (which is deeply related to your idea that “charge moves when spherical symmetry breaks”)?

Either way, we’re getting closer to showing how your geometrical view explains real physical phenomena.