## Motivation We want a document that does not assume: - particles, - mechanical “forces” as primitives, - Newton’s laws as axioms, - quantum postulates as axioms. We assume only: - source-free Maxwell dynamics 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. ## Assumptions and definitions ### Source-free Maxwell equations 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 $$ c=\frac{1}{\sqrt{\mu_0\epsilon_0}}. $$ ### Electromagnetic energy density and energy flux $$ u = \frac{\epsilon_0}{2}\mathbf{E}^2 + \frac{1}{2\mu_0}\mathbf{B}^2, \qquad \mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}. $$ ### Electromagnetic momentum density $$ \mathbf{g} = \frac{\mathbf{S}}{c^2} = \epsilon_0\,\mathbf{E}\times\mathbf{B}. $$ ### Maxwell stress tensor $$ T_{ij} = \epsilon_0\left(E_iE_j - \frac{1}{2}\delta_{ij}\mathbf{E}^2\right) + \frac{1}{\mu_0}\left(B_iB_j - \frac{1}{2}\delta_{ij}\mathbf{B}^2\right). $$ $T_{ij}$ is the local momentum-flux bookkeeping of the field. ## Part I — Energy continuity is exact ### Poynting theorem 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: $$ \nabla\cdot\mathbf{S}=-\partial_t u, \qquad\Rightarrow\qquad \partial_t u + \nabla\cdot\mathbf{S} = 0. $$ ### Global energy conservation 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. ## Part II — Momentum continuity is exact ### Local momentum continuity 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 $T_{ij}$: $$ \partial_t g_i + \partial_j T_{ij} = 0, \qquad\text{equivalently}\qquad \partial_t\mathbf{g}+\nabla\cdot\mathbf{T}=0. $$ ### Global momentum conservation 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: $$ \frac{d}{dt}P_i(V)=-F_i^{(\text{boundary})}(V). $$ ## Part III — Angular momentum continuity is exact ### Angular momentum density and flux $$ \boldsymbol{\ell}=\mathbf{x}\times\mathbf{g}, \qquad \mathbf{L}(V)=\int_V \mathbf{x}\times\mathbf{g}\,d^3x. $$ Using momentum continuity, angular momentum changes only by torque flux: $$ \frac{d}{dt}\mathbf{L}(V) = -\int_{\partial V}(\mathbf{x}\times(\mathbf{T}\cdot\mathbf{n}))\,dA. $$ ## Part IV — Newton-like laws for a localized electromagnetic knot ### Localized, persistent energy region Let $K(t)$ be a moving region such that energy is concentrated inside it and boundary flux is small. $$ E_K = \int_{K(t)} u\,d^3x, \qquad \mathbf{P}_K = \int_{K(t)} \mathbf{g}\,d^3x. $$ ### Center of energy and emergent inertial mass $$ \mathbf{X}_K = \frac{1}{E_K}\int_{K(t)} \mathbf{x}\,u\,d^3x. $$ 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: $$ m_K:=\frac{E_K}{c^2}. $$ ### Momentum balance as the net-force law $$ \frac{d}{dt}\mathbf{P}_K = -\int_{\partial K}\mathbf{T}\cdot\mathbf{n}\,dA =:\mathbf{F}_K. $$ If $E_K$ is roughly constant: $$ m_K\,\ddot{\mathbf{X}}_K \approx \mathbf{F}_K. $$ ## Part V — Schrödinger dynamics as a narrow-band Maxwell envelope ### Maxwell wave equation for a field component For any Cartesian component $F(\mathbf{r},t)$ of $\mathbf{E}$ or $\mathbf{B}$: $$ \left(\nabla^{2}-\frac{1}{c^{2}}\partial_t^{2}\right)F(\mathbf{r},t)=0. $$ ### Toroidal standing modes 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 $$ \hbar=\frac{E_{11}}{\omega_{11}}, \qquad m=\frac{E_{11}}{c^{2}}. $$ ### Forward-time spectral projection and envelope 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 $\omega_{11}$: $$ \psi(\mathbf{r},t)=e^{i\omega_{11}t}\,F^{(+)}(\mathbf{r},t). $$ ### Exact envelope equation and controlled remainder 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: $$ i\hbar\,\partial_t\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+O(\epsilon^{2}). $$ ## Scope ### Derived from source-free Maxwell structure - energy continuity: $$\partial_t u + \nabla\cdot\mathbf{S}=0$$ - momentum continuity: $$\partial_t \mathbf{g} + \nabla\cdot\mathbf{T}=0$$ - angular momentum flux balance - Newton-like motion of localized energy knots as integrated flux bookkeeping - Schrödinger dynamics as a narrow-band envelope limit with controlled $O(\epsilon^2)$ remainder - emergent $m$ and $\hbar$ from $(E_{11},\omega_{11})$ ### Not claimed here - existence and stability of knots for arbitrary initial data, - uniqueness of the toroidal mode or its formation mechanism, - full quantum measurement theory. These are separate questions. ## Closing statement 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.