Unbounded vector potential and effective Pauli model

Use SI units with mass $m>0$ in kilograms, charge $q\in \mathbb{R}$ in coulombs, action constant $\mathrm{\hslash }>0$ in joule-seconds, vector-potential components $A_j$ in tesla-metres, and scalar potential-energy function $V$ in joules. For each integer $d\ge 1$, let $A\in C^{\mathrm{\infty }}({\mathbb{R}}^d;{\mathbb{R}}^d)$ be real and let $V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d;\mathbb{R})$ be bounded below. Adopt the general-dimensional minimal-coupling assignment whose core form is $q_{A,V}[u]=\frac{1}{2m}{\sum }_{j=1}^d\mathrm{\parallel }(-i\mathrm{\hslash }{\mathrm{\partial }}_j-q{A}_j)u{\mathrm{\parallel }}_2^2+{\int }_{{\mathbb{R}}^d}V\mathrm{\mid }u{\mathrm{\mid }}^2,u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})\mathrm{.}$ Use its closed form closure and the unique self-adjoint representing Hamiltonian. For dimensional form norms use the fixed reference energy $E_0=1\mathrm{J}$: apply the closed-form representation to $q_{A,V}\mathrm{/}{E}_0$ and multiply the representing operator by $E_0$.

The magnetic-field notation is dimension-specific. In $d=2$, with coordinates $(x,y)$ and $A=(A_x,A_y)$, define the scalar perpendicular field $B_{\perp }(x,y)={\mathrm{\partial }}_x{A}_y(x,y)-{\mathrm{\partial }}_y{A}_x(x,y),$ in teslas. Thus $A=(0,Bx)$ gives $B_{\perp }=B$ for a constant signed scalar $B$. In $d=3$, for coordinates $(x,y,z)$ and $A=(A_x,A_y,A_z)$, define the field vector $\mathbf{B}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ in teslas by $\mathbf{B}=({\mathrm{\partial }}_y{A}_z-{\mathrm{\partial }}_z{A}_y,{\mathrm{\partial }}_z{A}_x-{\mathrm{\partial }}_x{A}_z,{\mathrm{\partial }}_x{A}_y-{\mathrm{\partial }}_y{A}_x)=\mathrm{\nabla }\times A\mathrm{.}$ No magnetic-field object is assigned by this notation in $d=1$.

For an optional spin-half degree of freedom in three dimensions, define ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),S_j=\frac{\mathrm{\hslash }}{2}{\sigma }_j\mathrm{.}$ For a core spinor $u=(u_1,u_2)\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{C}}^2)$, define the spin-independent form $q_{A,V}^{(2)}[u]=q_{A,V}[u_1]+q_{A,V}[u_2]\mathrm{.}$ For a declared dimensionless real coupling $g$, adopt the effective Pauli core form $q_{A,V,g}^{\mathrm{P}}[u]=q_{A,V}^{(2)}[u]-\frac{gq}{2m}{\int }_{{\mathbb{R}}^3}u(x{)}^{\ast }(S\cdot \mathbf{B}(x))u(x)dx,$ where $u^{\ast }$ is conjugate transpose and $(S\cdot \mathbf{B})(x)={\sum }_{j=1}^3{S}_j{B}_j(x)$ for the three-dimensional field vector just defined. Include this term only in model instances where the resulting form is closable and bounded below, and use its closed form closure and representing Hamiltonian. No universal semiboundedness for arbitrary $A,V,g$ is asserted.

David Tong, Applications of Quantum Mechanics, §§1.1.1 and 1.5

V. P. Nair, Notes on Quantum Mechanics, §§11.2–11.3

Aram Harrow, MIT 8.06 Lecture Notes, Chapter 4, §3.1

The core form data and the closability argument are given in Magnetic quadratic-form data and Closability of smooth magnetic forms with locally integrable potentials. The form-representation interface Semibounded closed-form representation and perturbation specifies the unique self-adjoint operator associated with the adopted closed form. The generic closability supplier assumes Countable Choice and Dependent Choice; the representation supplier assumes the Axiom of Choice and Countable Choice. These are the mathematical hypotheses of the construction, not additional physical postulates.

The mathematical identity in Pauli square identity and the formal g=2 coupling is a formal-core identity. This postulate does not identify $g$ with an electron measurement, derive the anomalous magnetic moment, or use the Dirac or quantum-electrodynamic theories. The linear Landau gauge is covered by the local closed-form construction; no bounded-coefficient $H^2$ hypothesis is imposed.