Positive-energy spin–size inequality

Fix $c>0$ and a future unit observer $n$ in an affine Minkowski spacetime as in Oriented time-oriented affine Minkowski space (the spacetime primitives) and Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. Identify the positive-definite rest space $n^{\perp }$ isometrically with ${\mathbb{R}}^3$, with its Euclidean norm $\mathrm{\mid }\cdot \mathrm{\mid }$ and Lebesgue measure $d^{3}x$. Define the closed Euclidean ball by ${\bar{B}}_{\mathrm{\ell }}(0):=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }\le \mathrm{\ell }\}\mathrm{.}$ Let $e:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ be measurable with $e\in L^1({\mathbb{R}}^3)$, and let $j:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be measurable with each component $j_i\in L^1({\mathbb{R}}^3)$ for $i=1,2,3$. Thus $j$ is vector-valued integrable and every vector integral below is the componentwise Lebesgue integral. Assume $e$ and $j$ are both zero almost everywhere outside ${\bar{B}}_{\mathrm{\ell }}(0)$ for some $\mathrm{\ell }>0$, and that $\mathrm{\mid }j(x)\mathrm{\mid }\le e(x)\mathrm{/}c$ almost everywhere. Define $E_0:={\int }_{{\mathbb{R}}^3}e(x)d^{3}x>0,M:=E_0\mathrm{/}{c}^2,{\mathbf{s}}_{\ast }:={\int }_{{\mathbb{R}}^3}x\times j(x)d^{3}x\mathrm{.}$ Then ${\mathbf{s}}_{\ast }$ is absolutely integrable and $\mathrm{\mid }{\mathbf{s}}_{\ast }\mathrm{\mid }\le \mathrm{\ell }Mc,\text{equivalently}\mathrm{\ell }\ge \frac{\mathrm{\mid }{\mathbf{s}}_{\ast }\mathrm{\mid }}{Mc}\mathrm{.}$ The hypotheses are the full mathematical content of this estimate; it is not a bound for a quantum point particle or for a source whose energy density or momentum density fails them.

This is a conditional material inequality. Positivity of $e$ and the pointwise energy-current bound $\mathrm{\mid }j\mathrm{\mid }\le e\mathrm{/}c$ are load-bearing hypotheses, not consequences of the definition of a quantum or point particle.