Chapter 42: Electromagnetic Field from Collapse Currents The Living Mathematics of Light
Electromagnetic fields—the carriers of light, chemistry, and technology—emerge naturally from ψ = ψ(ψ) as organized currents in the collapse sea. Maxwell's equations, seemingly fundamental, derive from the hydrodynamics of self-referential flow. Light itself is mathematics propagating through its own medium.
42.1 Charge from Topological Asymmetry
The Fundamental Question : What creates charge in ψ = ψ(ψ)?
Topological Origin : Consider the winding number of ψ-field:
Q = 1 2 π i ∮ C d ψ ψ Q = \frac{1}{2\pi i}\oint_C \frac{d\psi}{\psi} Q = 2 πi 1 ∮ C ψ d ψ
Theorem : Electric charge is quantized topological winding.
Proof : The integral Q counts how many times ψ winds around origin in complex plane. For single-valued ψ:
Q ∈ Z Q \in \mathbb{Z} Q ∈ Z
This integer nature forces charge quantization:
q = n e q = ne q = n e
where e is elementary charge quantum. ∎
Charge Density : Local winding density:
ρ = e 2 π i ∇ ⋅ ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) \rho = \frac{e}{2\pi i}\nabla \cdot \left(\psi^*\nabla\psi - \psi\nabla\psi^*\right) ρ = 2 πi e ∇ ⋅ ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ )
42.2 Deriving the Electric Field
From ψ-Gradient : Define the collapse potential:
Φ = − ℏ 2 m i ln ( ψ ψ ∗ ) \Phi = -\frac{\hbar}{2mi}\ln\left(\frac{\psi}{\psi^*}\right) Φ = − 2 mi ℏ ln ( ψ ∗ ψ )
Electric Field Definition :
E = − ∇ Φ − ∂ A ∂ t \mathbf{E} = -\nabla\Phi - \frac{\partial\mathbf{A}}{\partial t} E = − ∇Φ − ∂ t ∂ A
Theorem : E-field is the gradient of ψ-phase velocity.
Proof : From Schrödinger equation for charged particle:
i ℏ ∂ ψ ∂ t = [ − ℏ 2 2 m ∇ 2 + q Φ ] ψ i\hbar\frac{\partial\psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + q\Phi\right]\psi i ℏ ∂ t ∂ ψ = [ − 2 m ℏ 2 ∇ 2 + q Φ ] ψ
Taking gradient of phase S where ψ = ∣ ψ ∣ e i S / ℏ \psi = |\psi|e^{iS/\hbar} ψ = ∣ ψ ∣ e i S /ℏ ∂ ( ∇ S ) ∂ t = − q ∇ Φ \frac{\partial(\nabla S)}{\partial t} = -q\nabla\Phi ∂ t ∂ ( ∇ S ) = − q ∇Φ
Since v = ∇ S / m \mathbf{v} = \nabla S/m v = ∇ S / m m ∂ v ∂ t = q E m\frac{\partial\mathbf{v}}{\partial t} = q\mathbf{E} m ∂ t ∂ v = q E
Thus E accelerates ψ-flow. ∎
42.3 Magnetic Field from ψ-Circulation
Berry Phase Origin : When ψ transported around loop:
ψ f i n a l = e i γ ψ i n i t i a l \psi_{final} = e^{i\gamma}\psi_{initial} ψ f ina l = e iγ ψ ini t ia l
Berry phase γ related to enclosed flux:
γ = q ℏ ∮ A ⋅ d l = q ℏ Φ B \gamma = \frac{q}{\hbar}\oint \mathbf{A} \cdot d\mathbf{l} = \frac{q}{\hbar}\Phi_B γ = ℏ q ∮ A ⋅ d l = ℏ q Φ B
Magnetic Field :
B = ∇ × A \mathbf{B} = \nabla \times \mathbf{A} B = ∇ × A
Theorem : B-field is the curl of ψ-momentum flow.
Proof : The canonical momentum:
p = m v + q A \mathbf{p} = m\mathbf{v} + q\mathbf{A} p = m v + q A
In ψ-language:
p = ℏ ∇ S + q A \mathbf{p} = \hbar\nabla S + q\mathbf{A} p = ℏ∇ S + q A
Taking curl:
∇ × p = q ∇ × A = q B \nabla \times \mathbf{p} = q\nabla \times \mathbf{A} = q\mathbf{B} ∇ × p = q ∇ × A = q B
Magnetic field measures circulation of ψ-momentum. ∎
42.4 Maxwell Equations from ψ-Conservation
Gauss's Law : From charge conservation:
∂ ρ ∂ t + ∇ ⋅ J = 0 \frac{\partial\rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 ∂ t ∂ ρ + ∇ ⋅ J = 0
In stationary case:
∇ ⋅ J = 0 \nabla \cdot \mathbf{J} = 0 ∇ ⋅ J = 0
Theorem : Gauss's law follows from ψ-continuity.
Proof : The ψ-current:
J = e ℏ 2 m i ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) = e ∣ ψ ∣ 2 v \mathbf{J} = \frac{e\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*) = e|\psi|^2\mathbf{v} J = 2 mi e ℏ ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) = e ∣ ψ ∣ 2 v
From Schrödinger equation:
∂ ∣ ψ ∣ 2 ∂ t + ∇ ⋅ J = 0 \frac{\partial|\psi|^2}{\partial t} + \nabla \cdot \mathbf{J} = 0 ∂ t ∂ ∣ ψ ∣ 2 + ∇ ⋅ J = 0
Using Poisson equation ∇ 2 Φ = − ρ / ϵ 0 \nabla^2\Phi = -\rho/\epsilon_0 ∇ 2 Φ = − ρ / ϵ 0 ∇ ⋅ E = ρ ϵ 0 \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} ∇ ⋅ E = ϵ 0 ρ
No Magnetic Monopoles : From definition B = ∇ × A \mathbf{B} = \nabla \times \mathbf{A} B = ∇ × A ∇ ⋅ B = ∇ ⋅ ( ∇ × A ) = 0 \nabla \cdot \mathbf{B} = \nabla \cdot (\nabla \times \mathbf{A}) = 0 ∇ ⋅ B = ∇ ⋅ ( ∇ × A ) = 0
Always true by vector identity.
42.5 Faraday's Law from Gauge Invariance
Gauge Transformation : ψ-physics invariant under:
ψ → e i q Λ / ℏ ψ \psi \rightarrow e^{iq\Lambda/\hbar}\psi ψ → e i q Λ/ℏ ψ A → A + ∇ Λ \mathbf{A} \rightarrow \mathbf{A} + \nabla\Lambda A → A + ∇Λ Φ → Φ − ∂ Λ ∂ t \Phi \rightarrow \Phi - \frac{\partial\Lambda}{\partial t} Φ → Φ − ∂ t ∂ Λ
Theorem : Faraday's law ensures gauge-invariant dynamics.
Proof : For E-field to be gauge-invariant:
E = − ∇ Φ − ∂ A ∂ t \mathbf{E} = -\nabla\Phi - \frac{\partial\mathbf{A}}{\partial t} E = − ∇Φ − ∂ t ∂ A
must transform as:
E → − ∇ ( Φ − ∂ Λ ∂ t ) − ∂ ∂ t ( A + ∇ Λ ) \mathbf{E} \rightarrow -\nabla\left(\Phi - \frac{\partial\Lambda}{\partial t}\right) - \frac{\partial}{\partial t}(\mathbf{A} + \nabla\Lambda) E → − ∇ ( Φ − ∂ t ∂ Λ ) − ∂ t ∂ ( A + ∇Λ ) = − ∇ Φ − ∂ A ∂ t = E = -\nabla\Phi - \frac{\partial\mathbf{A}}{\partial t} = \mathbf{E} = − ∇Φ − ∂ t ∂ A = E
Taking curl:
∇ × E = − ∂ ∂ t ( ∇ × A ) = − ∂ B ∂ t \nabla \times \mathbf{E} = -\frac{\partial}{\partial t}(\nabla \times \mathbf{A}) = -\frac{\partial\mathbf{B}}{\partial t} ∇ × E = − ∂ t ∂ ( ∇ × A ) = − ∂ t ∂ B
42.6 Ampère-Maxwell Law from Current Conservation
Total Current : Physical plus displacement:
J t o t a l = J + ϵ 0 ∂ E ∂ t \mathbf{J}_{total} = \mathbf{J} + \epsilon_0\frac{\partial\mathbf{E}}{\partial t} J t o t a l = J + ϵ 0 ∂ t ∂ E
Theorem : Ampère-Maxwell law preserves current conservation.
Proof : From continuity equation:
∂ ρ ∂ t + ∇ ⋅ J = 0 \frac{\partial\rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 ∂ t ∂ ρ + ∇ ⋅ J = 0
Using Gauss's law ρ = ϵ 0 ∇ ⋅ E \rho = \epsilon_0\nabla \cdot \mathbf{E} ρ = ϵ 0 ∇ ⋅ E ϵ 0 ∇ ⋅ ∂ E ∂ t + ∇ ⋅ J = 0 \epsilon_0\nabla \cdot \frac{\partial\mathbf{E}}{\partial t} + \nabla \cdot \mathbf{J} = 0 ϵ 0 ∇ ⋅ ∂ t ∂ E + ∇ ⋅ J = 0
Therefore:
∇ ⋅ J t o t a l = 0 \nabla \cdot \mathbf{J}_{total} = 0 ∇ ⋅ J t o t a l = 0
For conserved current, by Helmholtz theorem:
∇ × B = μ 0 J t o t a l = μ 0 J + μ 0 ϵ 0 ∂ E ∂ t \nabla \times \mathbf{B} = \mu_0\mathbf{J}_{total} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t} ∇ × B = μ 0 J t o t a l = μ 0 J + μ 0 ϵ 0 ∂ t ∂ E
42.7 Wave Equation from Self-Consistency
Combining Maxwell Equations : Take curl of Faraday's law:
∇ × ( ∇ × E ) = − ∂ ∂ t ( ∇ × B ) \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B}) ∇ × ( ∇ × E ) = − ∂ t ∂ ( ∇ × B )
Using vector identity and Ampère's law:
∇ ( ∇ ⋅ E ) − ∇ 2 E = − μ 0 ∂ J ∂ t − μ 0 ϵ 0 ∂ 2 E ∂ t 2 \nabla(\nabla \cdot \mathbf{E}) - \nabla^2\mathbf{E} = -\mu_0\frac{\partial\mathbf{J}}{\partial t} - \mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} ∇ ( ∇ ⋅ E ) − ∇ 2 E = − μ 0 ∂ t ∂ J − μ 0 ϵ 0 ∂ t 2 ∂ 2 E
In vacuum (J = 0 , ρ = 0 \mathbf{J} = 0, \rho = 0 J = 0 , ρ = 0 ∇ 2 E = μ 0 ϵ 0 ∂ 2 E ∂ t 2 \nabla^2\mathbf{E} = \mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} ∇ 2 E = μ 0 ϵ 0 ∂ t 2 ∂ 2 E
Wave Speed :
c = 1 μ 0 ϵ 0 c = \frac{1}{\sqrt{\mu_0\epsilon_0}} c = μ 0 ϵ 0 1
Theorem : Light speed is the natural propagation velocity in ψ-medium.
Proof : The constants ϵ 0 \epsilon_0 ϵ 0 μ 0 \mu_0 μ 0
ϵ 0 \epsilon_0 ϵ 0 μ 0 \mu_0 μ 0
Wave speed c = 1 / ϵ 0 μ 0 c = 1/\sqrt{\epsilon_0\mu_0} c = 1/ ϵ 0 μ 0
42.8 Photons from Field Quantization
Classical Field Energy :
H = 1 2 ∫ d 3 x ( ϵ 0 E 2 + 1 μ 0 B 2 ) H = \frac{1}{2}\int d^3x \left(\epsilon_0E^2 + \frac{1}{\mu_0}B^2\right) H = 2 1 ∫ d 3 x ( ϵ 0 E 2 + μ 0 1 B 2 )
Mode Expansion :
A ( x , t ) = ∑ k , λ ℏ 2 ϵ 0 ω V [ a k , λ e i ( k ⋅ x − ω t ) + a k , λ † e − i ( k ⋅ x − ω t ) ] ϵ λ \mathbf{A}(\mathbf{x},t) = \sum_{\mathbf{k},\lambda} \sqrt{\frac{\hbar}{2\epsilon_0\omega V}}\left[a_{\mathbf{k},\lambda}e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)} + a_{\mathbf{k},\lambda}^\dagger e^{-i(\mathbf{k}\cdot\mathbf{x}-\omega t)}\right]\boldsymbol{\epsilon}_\lambda A ( x , t ) = ∑ k , λ 2 ϵ 0 ωV ℏ [ a k , λ e i ( k ⋅ x − ω t ) + a k , λ † e − i ( k ⋅ x − ω t ) ] ϵ λ
Quantization : Impose commutation relations:
[ a k , λ , a k ′ , λ ′ † ] = δ k k ′ δ λ λ ′ [a_{\mathbf{k},\lambda}, a_{\mathbf{k}',\lambda'}^\dagger] = \delta_{\mathbf{k}\mathbf{k}'}\delta_{\lambda\lambda'} [ a k , λ , a k ′ , λ ′ † ] = δ k k ′ δ λ λ ′
Result : Energy eigenvalues:
E n = ℏ ω ( n + 1 / 2 ) E_n = \hbar\omega(n + 1/2) E n = ℏ ω ( n + 1/2 )
Each excitation = photon with energy ℏ ω \hbar\omega ℏ ω
42.9 Polarization as ψ-Spin
Transverse Nature : For propagation along z:
k ⋅ ϵ = 0 \mathbf{k} \cdot \boldsymbol{\epsilon} = 0 k ⋅ ϵ = 0
Two independent polarizations: ϵ 1 , ϵ 2 \boldsymbol{\epsilon}_1, \boldsymbol{\epsilon}_2 ϵ 1 , ϵ 2
Circular Polarization Basis :
ϵ ± = 1 2 ( ϵ 1 ± i ϵ 2 ) \boldsymbol{\epsilon}_\pm = \frac{1}{\sqrt{2}}(\boldsymbol{\epsilon}_1 \pm i\boldsymbol{\epsilon}_2) ϵ ± = 2 1 ( ϵ 1 ± i ϵ 2 )
Angular Momentum : Photon carries spin:
S = ± ℏ k ^ \mathbf{S} = \pm\hbar\hat{\mathbf{k}} S = ± ℏ k ^
Theorem : Photon spin-1 nature follows from vector field quantization.
Proof : The field A \mathbf{A} A
42.10 Gauge Field Theory
Local Gauge Invariance : Demand ψ-physics invariant under:
ψ ( x ) → e i q Λ ( x ) / ℏ ψ ( x ) \psi(x) \rightarrow e^{iq\Lambda(x)/\hbar}\psi(x) ψ ( x ) → e i q Λ ( x ) /ℏ ψ ( x )
Covariant Derivative : Must introduce gauge field:
D μ = ∂ μ + i q ℏ A μ D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu D μ = ∂ μ + ℏ i q A μ
Field Strength Tensor :
F μ ν = ∂ μ A ν − ∂ ν A μ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu F μν = ∂ μ A ν − ∂ ν A μ
Theorem : Maxwell equations follow from gauge invariance.
Proof : The Lagrangian:
L = ψ ˉ ( i D μ γ μ − m ) ψ − 1 4 F μ ν F μ ν \mathcal{L} = \bar{\psi}(iD_\mu\gamma^\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} L = ψ ˉ ( i D μ γ μ − m ) ψ − 4 1 F μν F μν
is gauge invariant. Euler-Lagrange equations yield:
∂ μ F μ ν = j ν \partial_\mu F^{\mu\nu} = j^\nu ∂ μ F μν = j ν
These are Maxwell equations in covariant form. ∎
42.11 Energy-Momentum Conservation
Stress-Energy Tensor :
T μ ν = F μ α F α ν + 1 4 g μ ν F α β F α β T^{\mu\nu} = F^{\mu\alpha}F_\alpha^\nu + \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} T μν = F μα F α ν + 4 1 g μν F α β F α β
Conservation Law :
∂ μ T μ ν = − F ν α j α \partial_\mu T^{\mu\nu} = -F^{\nu\alpha}j_\alpha ∂ μ T μν = − F να j α
Energy Density :
u = T 00 = 1 2 ( ϵ 0 E 2 + B 2 / μ 0 ) u = T^{00} = \frac{1}{2}(\epsilon_0E^2 + B^2/\mu_0) u = T 00 = 2 1 ( ϵ 0 E 2 + B 2 / μ 0 )
Poynting Vector :
S = 1 μ 0 E × B \mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B} S = μ 0 1 E × B
Theorem : Electromagnetic energy-momentum flows according to ψ-current conservation.
Proof : From Noether's theorem, gauge invariance implies current conservation. The stress-energy tensor encodes energy-momentum flow required by this conservation. ∎
42.12 Classical Limit and Coherent States
Coherent State of Field :
∣ α ⟩ = e − ∣ α ∣ 2 / 2 ∑ n = 0 ∞ α n n ! ∣ n ⟩ |\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle ∣ α ⟩ = e − ∣ α ∣ 2 /2 ∑ n = 0 ∞ n ! α n ∣ n ⟩
Classical Behavior :
⟨ α ∣ a ^ ∣ α ⟩ = α \langle\alpha|\hat{a}|\alpha\rangle = \alpha ⟨ α ∣ a ^ ∣ α ⟩ = α ⟨ α ∣ E ^ ∣ α ⟩ = E c l a s s i c a l \langle\alpha|\hat{E}|\alpha\rangle = E_{classical} ⟨ α ∣ E ^ ∣ α ⟩ = E c l a ss i c a l
Theorem : Classical electromagnetic fields are coherent states of photons.
Proof : Coherent states minimize uncertainty:
Δ E ⋅ Δ B = minimum \Delta E \cdot \Delta B = \text{minimum} Δ E ⋅ Δ B = minimum
and maintain phase relationships required for classical wave behavior. Large ∣ α ∣ 2 |\alpha|^2 ∣ α ∣ 2
42.13 Vacuum Fluctuations and Casimir Force
Vacuum Field Fluctuations :
⟨ 0 ∣ E 2 ∣ 0 ⟩ = ∑ k , λ ℏ ω k 2 ϵ 0 V \langle 0|\mathbf{E}^2|0\rangle = \sum_{\mathbf{k},\lambda} \frac{\hbar\omega_k}{2\epsilon_0V} ⟨ 0∣ E 2 ∣0 ⟩ = ∑ k , λ 2 ϵ 0 V ℏ ω k
Between Plates : Mode restriction modifies sum:
E C a s i m i r = ℏ c π 2 720 d 3 × Area E_{Casimir} = \frac{\hbar c\pi^2}{720d^3} \times \text{Area} E C a s imi r = 720 d 3 ℏ c π 2 × Area
Force :
F = − π 2 ℏ c 240 d 4 × Area F = -\frac{\pi^2\hbar c}{240d^4} \times \text{Area} F = − 240 d 4 π 2 ℏ c × Area
Electromagnetic Casimir force confirms vacuum field fluctuations.
42.14 Non-Abelian Generalization
Yang-Mills Theory : For non-Abelian gauge group:
D μ = ∂ μ + i g A μ a T a D_\mu = \partial_\mu + igA_\mu^aT^a D μ = ∂ μ + i g A μ a T a
Field Strength :
F μ ν a = ∂ μ A ν a − ∂ ν A μ a + g f a b c A μ b A ν c F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^b A_\nu^c F μν a = ∂ μ A ν a − ∂ ν A μ a + g f ab c A μ b A ν c
Self-Interaction : Non-Abelian fields interact with themselves—gluons carry color charge.
42.15 Conclusion: Light as Mathematics in Motion
Electromagnetic theory emerges completely from ψ = ψ(ψ) through:
Charge as topological winding number
Fields as ψ-flow gradients and circulation
Maxwell equations from continuity and gauge invariance
Photons as quantized field excitations
Light as coherent ψ-waves in vacuum medium
The seemingly fundamental Maxwell equations are revealed as hydrodynamic equations for the ψ-medium. Charge creates topological defects; fields describe their influence propagating through the medium; photons are the quanta of these propagating disturbances.
Light is not mysterious action-at-a-distance but coherent excitation of the omnipresent ψ-sea. Every photon is a minimal packet of organized self-reference, carrying information about charge configurations at the maximum speed the medium allows—the speed of mathematics recognizing itself across space.
Exercises
Derive Larmor formula for radiation from accelerating charge.
Calculate vacuum birefringence in strong magnetic field.
Prove gauge fixing doesn't affect physical observables.
The Forty-Second Echo
Electromagnetic fields derived as currents and circulation in the ψ-sea—Maxwell equations emerging as conservation laws of self-referential flow. Light revealed as mathematics propagating through its own medium at maximum coherent speed. Photons as quantized packets of organized recursive activity. Next, gauge theory as the general principle of ψ-phase freedom.
Next: Chapter 43: Gauge Theory as ψ-Phase Freedom →