Minimal Dirac magnetic square and formal Pauli benchmark

Let $m>0$ be a particle mass in kilograms, $q\in \mathbb{R}$ its charge in coulombs, and $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ a smooth spatial vector potential with components in tesla-metres. Use standard Cartesian coordinates $x_i$ measured in metres and write ${\mathrm{\partial }}_i:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}_i$. Let ${ϵ}_{ijk}$ be the alternating symbol with ${ϵ}_{123}=1$, and sum repeated spatial indices from $1$ to $3$. Define $B_k(x)={ϵ}_{kij}{\mathrm{\partial }}_i{A}_j(x)$ and $\mathbf{B}=(B_1,B_2,B_3):{\mathbb{R}}^3\to {\mathbb{R}}^3$; its smooth components are in tesla.

Let $\mathcal{C}=C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{C}}^4)$ be the common invariant test domain. In this item, “common core” refers only to this shared domain of the displayed differential expressions; no operator-theoretic core property for a closure is asserted. Define the dimensionless finite matrices ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$ $\beta =\left(\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}\right),{\alpha }_i=\left(\begin{array}{cc}0& {\sigma }_i\\ {\sigma }_i& 0\end{array}\right),{\mathrm{\Sigma }}_i=\mathrm{diag}({\sigma }_i,{\sigma }_i),$ and use $\mathrm{\hslash },c$ of QFT example units, metric, gauge, and pairing conventions. On $\mathcal{C}$, define ${\pi }_i\psi =-i\mathrm{\hslash }{\mathrm{\partial }}_i\psi -q{A}_i\psi ,H_D\psi =c{\sum }_{i=1}^3{\alpha }_i{\pi }_i\psi +\beta m{c}^2\psi \mathrm{.}$ These maps preserve $\mathcal{C}$. The components ${\pi }_i$ have units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$ and $H_D$ has units joules. Define the operator expressions ${\mathit{\pi }}^2\psi :={\sum }_{i=1}^3{\pi }_i({\pi }_i\psi ),(\mathbf{\Sigma }\cdot \mathbf{B})\psi :={\sum }_{i=1}^3{\mathrm{\Sigma }}_i{B}_i\psi \mathrm{.}$ Here $[{\pi }_i,{\pi }_j]:={\pi }_i\circ {\pi }_j-{\pi }_j\circ {\pi }_i$, and $H_D^2$ denotes composition on $\mathcal{C}$. On $\mathcal{C}$, the identities are $[{\pi }_i,{\pi }_j]=iq\mathrm{\hslash }{ϵ}_{ijk}{B}_k,H_D^2=m^2{c}^4{I}_4+c^2({\mathit{\pi }}^2-q\mathrm{\hslash }\mathbf{\Sigma }\cdot \mathbf{B})\mathrm{.}$ The operator $K:={\mathit{\pi }}^2-q\mathrm{\hslash }\mathbf{\Sigma }\cdot \mathbf{B}$ maps $\mathcal{C}$ to itself and has units $(\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}{)}^2$.

For commuting formal indeterminates $t,X$ over $\mathbb{R}$, define the first-order square-root coefficient $a$ by $(m{c}^2+taX{)}^2\equiv m^2{c}^4+t{c}^{2}X(\mathrm{m}\mathrm{o}\mathrm{d}{t}^2)\mathrm{.}$ Comparing the coefficient of $tX$ gives $a=1\mathrm{/}(2m)$; substituting $K$ for $X$ gives the formal first-degree correction $K\mathrm{/}(2m)$, with no operator square-root assertion. Its spin term is $-(q\mathrm{/}m)\mathbf{S}\cdot \mathbf{B}$, where $S_i=\mathrm{\hslash }{\mathrm{\Sigma }}_i\mathrm{/}2$, $\mathbf{S}=(S_1,S_2,S_3)$, and $(\mathbf{S}\cdot \mathbf{B})\psi :={\sum }_{i=1}^3{S}_i{B}_i\psi$. The magnetic-moment dot product below uses the same componentwise sum convention.

Under the coefficient convention $H_{\mathrm{m}\mathrm{a}\mathrm{g}}=-{\mathit{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{g}}\cdot \mathbf{B}$ and ${\mathit{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{g}}=g_s(q\mathrm{/}(2m))\mathbf{S}$, equality of the formal coefficients as a linear function of $q$ gives $g_s=2$ for the minimal tree-level term. Here $H_{\mathrm{m}\mathrm{a}\mathrm{g}}$ has units joules and ${\mathit{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{g}}$ has units $\mathrm{A}{\mathrm{m}}^2$. No magnetic self-adjoint closure, full spectrum, operator square-root theorem, radiative correction, or empirical prediction is asserted.