Newton laws and Schroedinger's equation from continuous energy flow
2026-01-26
One-Sentence Summary. In source-free Maxwell theory, the same local continuity structure yields (i) global conservation laws and Newton-like motion for localized energy knots, and (ii) the Schrödinger equation as the narrow-band envelope limit of a toroidal standing mode, with \(m\) and \(\hbar\) emerging from the fundamental mode.
Abstract. We start from source-free Maxwell equations and derive the wave equation, then derive exact local continuity laws for energy, momentum, and angular momentum using the Poynting theorem and the Maxwell stress tensor. Integrating these identities yields global conservation statements. When electromagnetic energy is localized into a persistent circulating knot, its center-of-energy motion obeys Newton-like inertia and momentum-balance relations as flux bookkeeping, not as postulates. We then study a self-confined toroidal standing mode and isolate its forward-time narrow-band envelope via an analytic-signal projection. Keeping derivative terms exactly gives an envelope equation with a controlled remainder of order \((\Delta\omega/\omega_{11})^2\); discarding only that bounded term yields the Schrödinger equation. In this construction, \(m=E_{11}/c^2\) and \(\hbar=E_{11}/\omega_{11}\) are geometric properties of the fundamental toroidal mode.
Keywords. Maxwell theory, source-free electromagnetism, continuity equation, Poynting theorem, Maxwell stress tensor, momentum conservation, angular momentum conservation, electromagnetic knots, toroidal standing modes, analytic signal, narrow-band limit, emergent inertia, emergent Planck constant, emergent quantum mechanics
We want a document that does not assume:
We assume only:
We show that: - Newton-like mechanics for a localized object is integrated continuity bookkeeping, - Schrödinger dynamics is the narrow-band envelope limit of Maxwell waves on a toroidal mode.
In a source-free region:
\[ \nabla \cdot \mathbf{E} = 0, \qquad \nabla \cdot \mathbf{B} = 0, \]
\[ \nabla \times \mathbf{E} = -\partial_t \mathbf{B}, \qquad \nabla \times \mathbf{B} = \mu_0\epsilon_0\,\partial_t \mathbf{E}. \]
From this system we derive wave propagation. Taking the curl of Faraday’s law and substituting the curl equation for \(\mathbf{B}\) yields
\[ \nabla^2\mathbf{E}-\mu_0\epsilon_0\,\partial_t^2\mathbf{E}=0. \]
An identical equation follows for \(\mathbf{B}\).
From the coefficient of the time-derivative term, the wave speed is
\[ c=\frac{1}{\sqrt{\mu_0\epsilon_0}}. \]
\[ u = \frac{\epsilon_0}{2}\mathbf{E}^2 + \frac{1}{2\mu_0}\mathbf{B}^2, \qquad \mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}. \]
\[ \mathbf{g} = \frac{\mathbf{S}}{c^2} = \epsilon_0\,\mathbf{E}\times\mathbf{B}. \]
\[ T_{ij} = \epsilon_0\left(E_iE_j - \frac{1}{2}\delta_{ij}\mathbf{E}^2\right) + \frac{1}{\mu_0}\left(B_iB_j - \frac{1}{2}\delta_{ij}\mathbf{B}^2\right). \]
\(T_{ij}\) is the local momentum-flux bookkeeping of the field.
Start from:
\[ \nabla \times \mathbf{E} = -\partial_t \mathbf{B}, \qquad \nabla \times \mathbf{B} = \mu_0\epsilon_0\,\partial_t \mathbf{E}. \]
Use:
\[ \nabla\cdot(\mathbf{E}\times\mathbf{B}) = \mathbf{B}\cdot(\nabla\times\mathbf{E}) - \mathbf{E}\cdot(\nabla\times\mathbf{B}). \]
Substitute Maxwell and rewrite as derivatives:
\[ \nabla\cdot\mathbf{S}=-\partial_t u, \qquad\Rightarrow\qquad \partial_t u + \nabla\cdot\mathbf{S} = 0. \]
Integrate over a fixed volume \(V\) with boundary \(\partial V\):
\[ \frac{d}{dt}\int_V u\,d^3x + \int_{\partial V}\mathbf{S}\cdot d\mathbf{A}=0. \]
Define:
\[ U_V=\int_V u\,d^3x, \qquad \Phi_V=\int_{\partial V}\mathbf{S}\cdot d\mathbf{A}, \]
so:
\[ \frac{d}{dt}U_V=-\Phi_V. \]
If \(\Phi_V=0\), then \(U_V\) is constant.
Start from:
\[ \mathbf{g}=\epsilon_0\,\mathbf{E}\times\mathbf{B}. \]
Differentiate and substitute Maxwell:
\[ \partial_t\mathbf{g} = \frac{1}{\mu_0}(\nabla\times\mathbf{B})\times\mathbf{B} - \epsilon_0\,\mathbf{E}\times(\nabla\times\mathbf{E}). \]
Use:
\[ (\nabla\times\mathbf{A})\times\mathbf{A} = (\mathbf{A}\cdot\nabla)\mathbf{A} - \frac{1}{2}\nabla(\mathbf{A}^2) + \mathbf{A}(\nabla\cdot\mathbf{A}), \]
and \(\nabla\cdot\mathbf{E}=0\), \(\nabla\cdot\mathbf{B}=0\), giving
\[ (\nabla\times\mathbf{A})\times\mathbf{A} = (\mathbf{A}\cdot\nabla)\mathbf{A} - \frac{1}{2}\nabla(\mathbf{A}^2). \]
So:
\[ \partial_t\mathbf{g} = \frac{1}{\mu_0}\left((\mathbf{B}\cdot\nabla)\mathbf{B}-\frac{1}{2}\nabla(\mathbf{B}^2)\right) - \epsilon_0\left((\mathbf{E}\cdot\nabla)\mathbf{E}-\frac{1}{2}\nabla(\mathbf{E}^2)\right). \]
In components, using \(\partial_j(B_iB_j)=(\mathbf{B}\cdot\nabla)B_i\) and \(\partial_j(E_iE_j)=(\mathbf{E}\cdot\nabla)E_i\), this becomes:
\[ \partial_t g_i = -\partial_j\left[ \epsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}\mathbf{E}^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}\mathbf{B}^2\right) \right]. \]
With the definition of \(T_{ij}\):
\[ \partial_t g_i + \partial_j T_{ij} = 0, \qquad\text{equivalently}\qquad \partial_t\mathbf{g}+\nabla\cdot\mathbf{T}=0. \]
Integrate over \(V\):
\[ \frac{d}{dt}\int_V g_i\,d^3x + \int_{\partial V}T_{ij}n_j\,dA=0. \]
Define:
\[ P_i(V)=\int_V g_i\,d^3x, \qquad F_i^{(\text{boundary})}(V)=\int_{\partial V}T_{ij}n_j\,dA, \]
so:
\[ \frac{d}{dt}P_i(V)=-F_i^{(\text{boundary})}(V). \]
\[ \boldsymbol{\ell}=\mathbf{x}\times\mathbf{g}, \qquad \mathbf{L}(V)=\int_V \mathbf{x}\times\mathbf{g}\,d^3x. \]
Using momentum continuity, angular momentum changes only by torque flux:
\[ \frac{d}{dt}\mathbf{L}(V) = -\int_{\partial V}(\mathbf{x}\times(\mathbf{T}\cdot\mathbf{n}))\,dA. \]
Let \(K(t)\) be a moving region such that energy is concentrated inside it and boundary flux is small.
\[ E_K = \int_{K(t)} u\,d^3x, \qquad \mathbf{P}_K = \int_{K(t)} \mathbf{g}\,d^3x. \]
\[ \mathbf{X}_K = \frac{1}{E_K}\int_{K(t)} \mathbf{x}\,u\,d^3x. \]
When boundary terms are negligible:
\[ E_K\,\dot{\mathbf{X}}_K \approx \int_K \mathbf{S}\,d^3x, \qquad \mathbf{P}_K \approx \frac{E_K}{c^2}\dot{\mathbf{X}}_K. \]
Define:
\[ m_K:=\frac{E_K}{c^2}. \]
\[ \frac{d}{dt}\mathbf{P}_K = -\int_{\partial K}\mathbf{T}\cdot\mathbf{n}\,dA =:\mathbf{F}_K. \]
If \(E_K\) is roughly constant:
\[ m_K\,\ddot{\mathbf{X}}_K \approx \mathbf{F}_K. \]
For any Cartesian component \(F(\mathbf{r},t)\) of \(\mathbf{E}\) or \(\mathbf{B}\):
\[ \left(\nabla^{2}-\frac{1}{c^{2}}\partial_t^{2}\right)F(\mathbf{r},t)=0. \]
Take a toroidal topology with radii \(R\) and \(r\). Integer windings \((n_1,n_2)\) give
\[ k_1=\frac{n_1}{R}, \qquad k_2=\frac{n_2}{r}, \qquad k^{2}=k_1^{2}+k_2^{2}, \qquad \omega_{n_1n_2}=ck. \]
Define the fundamental mode \((1,1)\) with \((E_{11},\omega_{11})\) and
\[ \hbar=\frac{E_{11}}{\omega_{11}}, \qquad m=\frac{E_{11}}{c^{2}}. \]
Define the analytic (positive-frequency) signal:
\[ F^{(+)}(\mathbf{r},t)=\int_{0}^{\infty}\tilde F(\mathbf{r},\omega)\,e^{-i\omega t}\,d\omega. \]
Extract the carrier at \(\omega_{11}\):
\[ \psi(\mathbf{r},t)=e^{i\omega_{11}t}\,F^{(+)}(\mathbf{r},t). \]
Substitution into the wave equation yields:
\[ \nabla^{2}\psi-\frac{1}{c^{2}}\partial_t^{2}\psi +\frac{2i\omega_{11}}{c^{2}}\partial_t\psi +\frac{\omega_{11}^{2}}{c^{2}}\psi=0. \]
Rearrange:
\[ i\partial_t\psi=-\frac{c^{2}}{2\omega_{11}}\nabla^{2}\psi +\frac{1}{2\omega_{11}c^{2}}\partial_t^{2}\psi. \]
If the envelope has RMS bandwidth \(\Delta\omega\) with \(\epsilon=\Delta\omega/\omega_{11}\ll1\), then the last term is bounded by \(O(\epsilon^2)\) in norm.
Dropping only this controlled term gives:
\[ i\partial_t\psi=-\frac{c^{2}}{2\omega_{11}}\nabla^{2}\psi+O(\epsilon^{2}). \]
Using \(\hbar=E_{11}/\omega_{11}\) and \(m=E_{11}/c^2\) turns the coefficient into \(\hbar/(2m)\), yielding:
\[ i\hbar\,\partial_t\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+O(\epsilon^{2}). \]
These are separate questions.
In a source-free Maxwell universe, continuity laws are identities, not postulates. When energy localizes into a persistent knot, its coarse motion follows from flux balance and looks Newtonian. When a toroidal mode is narrow-band, its envelope obeys Schrödinger dynamics up to a controlled \(O(\epsilon^2)\) correction.