Classical BMT polarization model

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure, and let $\mathrm{\nabla }$ be the metric-selected Levi–Civita connection of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $U$ and the prescribed external two-form $F$ be as in Charged and spinning test-body quantities, and let the translational motion obey the adopted action law External-field charged-particle action. For a $C^1$ spin or polarization vector $s$ along the worldline with $g(s,U)=0$, define $G:=g_s\mathrm{/}2$ and $K:=F(s,U)=g(s,F^{\mathrm{♯}}(U))$. Adopt the transport law

${\mathrm{\nabla }}_{U}s=\frac{q}{m}[G{F}^{\mathrm{♯}}(s)-(G-1)\frac{U}{c^2}K]\mathrm{.}$

Here ${\mathrm{\nabla }}_{U}s$ denotes the covariant derivative of the section $s$ along the specified curve, rather than differentiation of an unspecified ambient extension. More generally, for any $C^1$ vector field $V$ along this $C^2$ worldline $z$, define

$(D_{\tau }V{)}^a:=\frac{d{V}^a}{d\tau }+{\mathrm{\Gamma }}^a{}_{bc}(z(\tau ))U^b{V}^c,{\mathrm{\nabla }}_{U}s:=D_{\tau }s\mathrm{.}$

The components of $V$ are $C^1$ functions of $\tau$ in each smooth chart, and the displayed derivative is continuous. This is the pullback-connection derivative on $z^{\ast }TM$. To verify that the component definition is independent of the chart, let $y=y(x)$ and $J^{\alpha }{}_a:=\mathrm{\partial }{y}^{\alpha }\mathrm{/}\mathrm{\partial }{x}^a$. The connection transformation law gives

${\mathrm{\Gamma }}^{\mathrm{\prime }\alpha }{}_{\beta \gamma }{J}^{\beta }{}_b{J}^{\gamma }{}_c=J^{\alpha }{}_a{\mathrm{\Gamma }}^a{}_{bc}-{\mathrm{\partial }}_b{J}^{\alpha }{}_c\mathrm{.}$

Since $V^{\mathrm{\prime }\alpha }=J^{\alpha }{}_a{V}^a$, the ordinary chain and product rules give ${\dot{V}}^{\mathrm{\prime }\alpha }=J^{\alpha }{}_a{\dot{V}}^a+({\mathrm{\partial }}_b{J}^{\alpha }{}_c)U^b{V}^c$. The two transition-derivative terms cancel, leaving $(D_{\tau }V{)}^{\mathrm{\prime }\alpha }=J^{\alpha }{}_a(D_{\tau }V{)}^a$. Thus no smoothness of the curve beyond $C^2$, or of the section beyond $C^1$, is required. Metric compatibility also gives $\frac{d}{d\tau }g(V,W)=g(D_{\tau }V,W)+g(V,D_{\tau }W)$ for $C^1$ sections $V,W$: insert the component formulas and ${\mathrm{\partial }}_b{g}_{ac}={\mathrm{\Gamma }}^d{}_{ba}{g}_{dc}+{\mathrm{\Gamma }}^d{}_{bc}{g}_{ad}$. These conventions specify the mathematical derivative in the adopted transport law; they add no further physical motion assumption.

This is an ideal classical external-field dipole model with constant $q,m,g_s$. Its flat homogeneous-field specialization is the BMT polarization law in the stated RPM convention. The formula on a curved background, and its use for a spatially varying prescribed field, are separately adopted model assumptions; neither follows from the flat BMT source or from general covariance alone.

The model excludes an electric dipole moment, field-gradient or Stern–Gerlach translational forces, spin backreaction, radiation reaction, collisions and higher multipoles. The parameter $g_s$ is not fixed to $2$ and no measured value is asserted. Using this classical vector to represent a real quantum particle's polarization is a correspondence assumption; the postulate does not construct quantum spin or claim empirical validation.