Transverse fields, helicity and angular momentum — Examples

Given: The positive dimensional scales $\kappa$ and $E_0$, the coordinate unit vectors, $k=\kappa e_3$, $\omega =c\kappa$, $e=E_0{e}_1$, $b=(E_0\mathrm{/}c)e_2$, and the corresponding real plane wave.

The plane-wave encoding is $E=\mathrm{Re}(e\exp (i(k\cdot x-\omega t)))$ and $B=\mathrm{Re}(b\exp (i(k\cdot x-\omega t)))$; its fields are identically zero exactly when both amplitudes vanish. (Real plane-wave fields encoded by complex amplitudes)

For these transverse vacuum modes, the exact period average is $⟨u⟩={\epsilon }_0\mathrm{\mid }e{\mathrm{\mid }}^2\mathrm{/}2$, and every nonzero amplitude has infinite all-space box-exhaustion energy. (Period-averaged energy density, Poynting vector and momentum flux of a plane wave)

With $\omega =c\mathrm{\mid }k\mathrm{\mid }$, $k\cdot e=0$, and $b=k\times e\mathrm{/}\omega$, these plane waves satisfy the smooth source-free Maxwell equations; the plane-wave amplitudes determine the fields, and the pair is nonzero when $e\mathrm{\ne }0$. (Dispersion, transversality and the two polarization conditions for plane-wave solutions) The SI conventions give $c>0$ and ${\epsilon }_0>0$. (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

The angular-momentum functionals in question are defined on the real Schwartz vacuum packet fields of the Coulomb-gauge construction; their integrands are globally integrable on that domain, and every field in the Fourier-packet class has finite total energy. (Coulomb-gauge canonical and kinetic integrals, Classical transverse Fourier amplitudes)

Given: The positive length and SI amplitude scales, the cutoff $\chi$, the fields and sources defined above, and the adopted smooth Maxwell branch.

The source-free packet domain and its functionals are defined in Coulomb-gauge canonical and kinetic integrals. On each cube, the exact component identity is $J_R=L_R+S_R+{\epsilon }_0{\int }_{Q_R}(\mathrm{div}E)(x\times A)-{\epsilon }_0{\int }_{\mathrm{\partial }{Q}_R}(n\cdot E)(x\times A)$. (Equality of total angular momentum integrals)

Adopt the smooth branch of the vacuum Maxwell postulate: $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$, and $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$ for smooth fields and continuous sources. (The microscopic vacuum Maxwell equations: smooth and weak branches)

The fields are maps $E,B:{\mathbb{R}}^3\times \mathbb{R}\to {\mathbb{R}}^3$; the laboratory chart is positively oriented, and the source fields $\rho ,J$ are continuous maps to $\mathbb{R}$ and ${\mathbb{R}}^3$ with units $\mathrm{C}{\mathrm{m}}^{-3}$ and $\mathrm{A}{\mathrm{m}}^{-2}$. The divergence and curl use their coordinate partial-derivative formulas; for a $C^2$ vector field, divergence of curl vanishes, and continuous second partial derivatives commute. (The electric and magnetic fields as fixed-time spatial vector fields, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, Divergence and curl of a $C^1$ vector field, The divergence of the curl of a $C^2$ field vanishes, Clairaut--Schwarz theorem for continuous second partial derivatives)

The SI constants satisfy ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$, and $[E_0]=\mathrm{V}{\mathrm{m}}^{-1}$, $[A_0]=\mathrm{T}\mathrm{m}$, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$. (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

A potential pair satisfies $E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$ and $B=\mathrm{curl}A$. (Potential representations, gauge transformations and the Lorenz and Coulomb conditions)

There exists a smooth cutoff $\chi :{\mathbb{R}}^3\to [0,1]$ equal to $1$ on the unit ball and $0$ outside the radius-two ball. (Explicit compactly supported smooth cutoffs)

The coordinate cross product satisfies $x\times (0,g,0)=(-x_{3}g,0,x_{1}g)$ and is bilinear. (The cross product in ${\mathbb{R}}^3$)

Continuous functions on cubes and their faces are Riemann integrable, iterated integrals on product rectangles agree, the integral of a derivative is the endpoint increment, and the integral is monotone under pointwise inequalities. (Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume, The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$, Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable, Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ${\mathbb{R}}^m$)