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Coulomb-gauge canonical and kinetic integrals

Physical scope

Conditional classical vacuum electromagnetism in the fixed SI laboratory chart, restricted to the real Schwartz Fourier-packet fields and the particular Coulomb-gauge potential constructed on the preceding A page. The item defines global angular-momentum functionals relative to the fixed spatial origin. It does not assert that the separate terms are local, gauge-independent, universally measurable, or defined for non-decaying plane waves.

Definition

Let $A, E, B : R^{3} \to R^{3}$ be any fixed-time fields from the compactly supported transverse Fourier construction: $A, E, B$ are real Schwartz fields, $div A = 0$ , $B = curl A$ , and $div E = 0$ . Use the positively oriented laboratory coordinates $x = (x^{1}, x^{2}, x^{3})$ and the fixed origin $0$ . Every $R^{3}$ -valued integrand below is integrated componentwise: on $Q_{R} = [- R, R]^{3}$ , integrate each of its three real components by the ordinary real Riemann integral. Its whole-space integral is the finite componentwise cube-exhaustion limit specified in the preceding Fourier-field Definition; the estimates in the Verification establish absolute integrability in that definition's sense. For each scalar component $A_{i}$ , define $(x \times \nabla) A_{i}$ to have components $\sum_{m, j = 1}^{3} ϵ_{ℓ m j} x_{m} \partial_{j} A_{i}$ , where $ϵ_{ℓ m j}$ is the right-handed Levi-Civita symbol in this chart. Define the total kinetic angular-momentum functional and its Coulomb-gauge canonical orbital and spin functionals by

$J_{k i n} : = ε_{0} \int_{R^{3}} x \times (E \times B) d^{3} x, L_{c a n} : = ε_{0} \sum_{i = 1}^{3} \int_{R^{3}} E_{i} (x \times \nabla) A_{i} d^{3} x, S_{c a n} : = ε_{0} \int_{R^{3}} E \times A d^{3} x .$

Each is a finite vector with SI units $J s$ . The words “kinetic,” “canonical orbital,” and “spin” are names for these functionals on this fixed potential class; they do not identify three separately gauge-invariant local densities. A generic gauge transform is outside this definition's domain.

Definition references

Classical transverse Fourier amplitudes

Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Axis-parallel rectangles in $R^{m}$ and their volume

The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$

Every continuous function on a closed nondegenerate rectangle in $R^{m}$ 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 $G^{'} = f$ 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 $R^{m}$

Complete ordered field (least-upper-bound property)

Facts & Assumptions

Given: The real Schwartz fields $A, E, B$ in the Definition, the fixed laboratory origin and orientation, and the SI vacuum constants.

[F1]

The Fourier-packet construction gives $A, E, B \in S (R^{3})$ , $div E = div A = 0$ , and $B = curl A$ . Its componentwise cube-exhaustion convention proves that a finite limit of the absolute component-sum integrals implies finite signed component-integral limits: integer-radius Cauchy convergence and the uniform absolute-tail estimate also control all real radii. (Classical transverse Fourier amplitudes)

[F2]

The momentum density and orbital angular-momentum density are $g = ε_{0} E \times B$ and $ℓ = x \times g$ , relative to the fixed laboratory origin. (Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum)

[F3]

$ε_{0} = (μ_{0} c^{2})^{- 1} > 0$ , $[ε_{0}] = F m^{- 1}$ , $[E] = V m^{- 1}$ , and $[A] = T m = V s m^{- 1}$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

[F4]

Continuous real functions on closed nondegenerate rectangles have Riemann integrals, and iterated integrals on product rectangles equal the multiple integral. (Axis-parallel rectangles in $R^{m}$ and their volume, The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable, Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections)

[F5]

The one-dimensional Riemann integral of a continuous derivative on a closed interval is the endpoint increment. (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ )

[F6]

Riemann integrals are additive under finite coordinate cuts, monotone for nonnegative integrands, and satisfy the absolute-value estimate. (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ )

[F7]

Every nonempty bounded-above subset of the Cauchy-sequence real field has a real least upper bound, as established by The Cauchy-sequence reals have the least-upper-bound property. This is the real field used here; Complete ordered field (least-upper-bound property) supplies the terminology rather than proving that property for $R$ .

Verification

1.1givenF1F4F5F6F7algebra

For every Schwartz component $f$ and every integer $N$ , the finite expansion of $\prod_{j = 1}^{3} (1 + ∣ x_{j} ∣)^{N}$ is controlled term-by-term by the seminorms in [F1], so $∣ f (x) ∣ \leq C_{N} \prod_{j} (1 + ∣ x_{j} ∣)^{- N}$ ; the same holds for its derivatives. Choose $N = 4$ . Each component of each vector integrand defining $J_{k i n}$ , $L_{c a n}$ , and $S_{c a n}$ is a sum of products of two Schwartz components, with at most one coordinate factor. Therefore, for each such vector integrand $V$ , $\sum_{j = 1}^{3} ∣ V_{j} (x) ∣ \leq C_{V} G (x)$ for some $C_{V} \geq 0$ , where $G (x) : = \prod_{j = 1}^{3} (1 + ∣ x_{j} ∣)^{- 7}$ . By [F5], for $R > 0$ the one-dimensional factor satisfies $\int_{- R}^{R} (1 + ∣ s ∣)^{- 7} d s = \frac{1}{3} (1 - (1 + R)^{- 6}) \leq \frac{1}{3}$ and tends to $1 / 3$ . Riemann-Fubini [F4] gives $\int_{Q_{R}} G d^{3} x = {(\frac{1}{3} (1 - (1 + R)^{- 6}))}^{3} \leq 1 / 27$ . Thus the nonnegative cube integrals $A_{R} (V) : = \int_{Q_{R}} \sum_{j = 1}^{3} ∣ V_{j} (x) ∣ d^{3} x$ are bounded above by $C_{V} / 27$ and are nondecreasing in $R$ by finite-slice additivity and monotonicity [F6]. The set ${A_{R} (V) : R > 0}$ is nonempty (take $R = 1$ ) and bounded above, so its finite real supremum $A (V)$ exists by [F7]. For every $ε > 0$ , choose $R_{0}$ with $A_{R_{0}} (V) > A (V) - ε$ ; then $A (V) - ε < A_{R} (V) \leq A (V)$ for all $R \geq R_{0}$ , proving that $A_{R} (V)$ has a finite limit. Hence each $V$ is absolutely integrable in the preceding Definition's sense, and its signed component integrals have finite cube-exhaustion limits by the proved absolute-tail implication in [F1]. This proves that all three functionals are well-defined finite vectors.

2.1F2F3algebra∎

The integrand of $J_{k i n}$ is $x$ (metres) crossed with the momentum density $g$ of [F2]. Since $g$ has units $k g m^{- 2} s^{- 1}$ , its integral after multiplication by $x$ has units $k g m^{2} s^{- 1} = J s$ . For each canonical term, $ε_{0} E A$ has units $ε_{0} E^{2} s$ because $A$ has the units of $E$ times time by [F3]. From $[ε_{0}] = F m^{- 1} = C V^{- 1} m^{- 1}$ and $1 C V = 1 J$ , $ε_{0} E^{2}$ has units $J m^{- 3}$ ; integration over volume again gives $J s$ . These are the units asserted in the Definition.

Coulomb-gauge canonical and kinetic integrals

Definition

Definition references

Facts & Assumptions

Verification

Depends on

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