Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator

Adopt as a physical assumption of the framework, for the declared massive spin-$s$ particle model of The massive spin-s particle model of the framework: state space, operators, units and frame readings and only within the scope stated below, the following effective magnetic-moment coupling (Dimensions, SI unit conventions and quantity types of the framework, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations).

(i) Gyromagnetic parameter and magnetic moment. For the declared spin label $s$ assign a real gyromagnetic parameter ${\gamma }_s$, with unit ${\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, and define the magnetic moment of the model as the internal-factor operator $\mathit{\mu }:={\gamma }_s\mathbf{S}=({\gamma }_s{S}_1,{\gamma }_s{S}_2,{\gamma }_s{S}_3),$ with unit $\mathrm{J}{\mathrm{T}}^{-1}$. Here $\mathbf{S}=(S_1,S_2,S_3)$ are the spin operators of Finite spin matrices read on the internal factor as in The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c). The parameter ${\gamma }_s$ is an empirical model parameter of the species: it is part of the declared model data, with uncertainty, and it is not derived from $s$, from the mass parameter $m$ or from the charge of the particle; no relation between ${\gamma }_s$ and any other species parameter is asserted by this postulate.

(ii) Prescribed field and Zeeman energy operator. Let $\mathbf{B}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be a real bounded measurable vector field, the prescribed magnetic field profile, with components $B_i$ in tesla, entering the model as externally prescribed data (The state space of the framework: preparations, pure rays and density operators, Componentwise position and momentum operators on the component-valued wave-function space); in the declared inertial frame the model adopts the Zeeman energy operator $H_Z:=H_0+M_W,H_0:=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2,M_{W}f:=-{\sum }_{i=1}^3{\gamma }_s{B}_i(\cdot )\left(S_{i}f\right)(\cdot ),$ on the componentwise domain $D(H_Z):=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r)$, the domain of $H_0$ (componentwise, Componentwise position and momentum operators on the component-valued wave-function space; ${\sum }_k{P}_k^2$ is the componentwise free operator with that domain by Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) applied to each component). The added term $M_W$ is bounded with $\mathrm{\parallel }{M}_W\mathrm{\parallel }\le \mathrm{\mid }{\gamma }_s\mathrm{\mid }{\sum }_i\mathrm{\parallel }{B}_i{\mathrm{\parallel }}_{\mathrm{\infty }}{\max }_i\mathrm{\parallel }{S}_i\mathrm{\parallel }$; its matrix value is Hermitian almost everywhere since $B_i$ and ${\gamma }_s$ are real and each finite spin matrix is self-adjoint. Integrating the fibrewise inner-product identity gives $M_W^{\ast }=M_W$ on $H_s$. For completeness, if $y\in D((H_0+M_W{)}^{\ast })$ and $z=(H_0+M_W{)}^{\ast }y$, then $⟨H_{0}x,y⟩=⟨x,z-M_{W}y⟩$ for every $x\in D(H_0)$. Self-adjointness of $H_0$ gives $y\in D(H_0)$ and $H_{0}y=z-M_{W}y$; the reverse adjoint inclusion follows from symmetry. Thus $H_Z=H_Z^{\ast }$ on the unchanged domain (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators), with unit joule, and the assignment $t\mapsto e^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}$ is a strongly continuous one-parameter unitary group read in the autonomous-dynamics postulate P4 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, State and composition postulate P1: Hilbert spaces, preparations and composite systems, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned sharp observables and the outcome law of the framework). The profile $\mathbf{B}$ is a model input; the items of this page assert neither the Maxwell equations nor a source or boundary realization of $\mathbf{B}$, and the boundedness of $\mathbf{B}$ is an explicit hypothesis, not a general property of magnetic fields.

(iii) What is assumed and what is not. (a) The coupling $-\mathit{\mu }\cdot \mathbf{B}(Q)$ is an effective interaction: it is adopted, not derived from the spin representation P6, from the minimal-coupling postulate P5a of the framework or from classical electromagnetism. (b) No value of ${\gamma }_s$ and no comparison with measured gyromagnetic ratios or $g$-factors is made here. (c) No spin-orbit coupling, no fine or hyperfine structure, no relativistic correction and no radiation reaction is included. (d) A time-dependent field profile requires its own domain and evolution hypotheses and is outside this postulate. (e) The internal factor is not a spatial current distribution and $\mathit{\mu }$ is not asserted to arise from a rotating charge distribution.

(iv) Status. This postulate is an adopted model assumption with the physical scope above; it is motivated by and formulated after the cited sources, which treat the coupling as a model input, and no deduction of it from the framework's other postulates is claimed.

• Why a bounded profile. The domain statement is that of the free operator on $W^{2,2}$; it does not cover linearly growing fields, point dipoles or fields whose components leave $L^{\mathrm{\infty }}$, each of which would require a separate form-domain or resolvent analysis.
• Units check. ${\gamma }_s$ in ${\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$ times $S_i$ in $\mathrm{J}\mathrm{s}$ gives $\mathrm{J}{\mathrm{T}}^{-1}$ for ${\mu }_i$, and ${\mu }_i{B}_i$ in $\mathrm{J}{\mathrm{T}}^{-1}\cdot \mathrm{T}=\mathrm{J}$; the phase $H_{Z}t\mathrm{/}\mathrm{\hslash }$ is dimensionless. The parameter ${\gamma }_s$ is read in the conventions of Dimensions, SI unit conventions and quantity types of the framework; no numerical value is asserted and, in particular, no exact value of any $g$-factor is assumed.