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Minimal Dirac magnetic benchmark from the adopted action

Physical scope

A conditional compact-variation Dirac equation and formal minimal magnetic coefficient for a smooth static pure magnetic background. The result is the minimal/tree Pauli coefficient only: it proves no self-adjoint magnetic closure, full spectrum, Landau-level construction, radiative correction, or empirical equality.

Statement

Assume Countable Choice, ${A C}_{ω}$ The Axiom of Countable Choice ( ${A C}_{ω}$ ), as in the mathematical Dirac-matrix supplier chain. Also assume the adopted local Maxwell–Dirac action of Explicit charged Dirac/Maxwell local coupling model with electron charge $q = - e$ , $e > 0$ , and rest mass $m > 0$ , with $q, e$ in coulombs and $m$ in kilograms. Use the oriented, time-oriented affine Minkowski spacetime and inertial chart $x = (c t, x)$ with $η = diag (- 1, 1, 1, 1)$ of Quantum field quantities and compatible SI normalizations. On $R^{4}$ , the static lower potential is $A_{S I, a} (x^{0}, x) = (0, A (x))$ : it is independent of $x^{0}$ , with $ϕ = 0$ and $A \in C^{\infty} (R^{3}; R^{3})$ in SI units $T m$ . Define $B = \nabla \times A$ in tesla, $π_{i} = - i ℏ \partial_{i} - q A_{i}, H_{D} = c \sum_{i = 1}^{3} α_{i} π_{i} + β m c^{2},$ as spatial differential expressions on $C^{\infty} (R^{3}; C^{4})$ . Let $D = C_{c}^{\infty} (R^{3}; C^{4})$ be their common test domain; smooth coefficients make $π_{i}$ and $H_{D}$ map $D$ into itself. Use the Dirac matrices and adjoint of Dirac Clifford matrices and energy projections and $Σ_{i} = diag (σ_{i}, σ_{i}), S_{i} = \frac{ℏ}{2} Σ_{i} .$ The matrices $α_{i}, β, Σ_{i}$ are dimensionless, $S_{i}$ has units $J s$ , $π_{i}$ has units $k g m s^{- 1}$ , and $H_{D}$ has units joules. Let $ψ : R^{4} \to C^{4}$ be a smooth spacetime spinor field. For a compactly supported direction $ξ \in C_{c}^{\infty} (R^{4}; C^{4})$ , holding $A$ fixed and using real $ε$ , define $δ_{ξ} S_{c l} : = {\frac{d}{d ε} ∣}_{ε = 0} ℏ \int_{R^{4}} (L_{n a t} [ψ + ε ξ, A] - L_{n a t} [ψ, A]) d^{4} x,$ where $L_{n a t}$ is the adopted density in Explicit charged Dirac/Maxwell local coupling model. The integrand has compact support, so this variation is finite. Assume $δ_{ξ} S_{c l} = 0$ for every such $ξ$ . Then the compactly supported spinor-variation Euler–Lagrange equation of the adopted action holds pointwise as a differential equation. In the natural action normalization of Quantum field quantities and compatible SI normalizations, $ψ$ has engineering length degree $- 3 / 2$ , and this equation is $i ℏ \partial_{t} ψ = H_{D} ψ$ for this pure magnetic specialization. Independently of this condition on $ψ$ , on $D$ the spatial differential expression satisfies $H_{D}^{2} = m^{2} c^{4} I_{4} + c^{2} (π^{2} - q ℏ Σ \cdot B) .$ Writing the formal first-order square-root coefficient as $m c^{2} + K / (2 m)$ , where $K = π^{2} - q ℏ Σ \cdot B$ , the spin term is $- (q / m) S \cdot B$ . Under the explicitly chosen magnetic-energy convention $H_{m a g} = - μ_{m a g} \cdot B, μ_{m a g} = g_{s} \frac{q}{2 m} S,$ where $H_{m a g}$ is an energy-valued matrix coefficient in joules, $μ_{m a g}$ is a $4 \times 4$ matrix-valued magnetic moment coefficient with units $A m^{2}$ , and $g_{s}$ is dimensionless. Comparing the coefficient of $S \cdot B$ as a linear-form coefficient gives $g_{s} = 2$ for this minimal tree coefficient. The Dirac equation is asserted only for fields satisfying the stated stationarity condition. The square and coefficient are formal on the stated test domain; no magnetic self-adjoint closure, full spectrum, radiative correction, or empirical measurement is asserted.

Facts & Assumptions

Given: Countable Choice; a smooth static spatial potential on the fixed Minkowski chart, $ϕ = 0$ , $q = - e \neq 0$ , $m > 0$ , the test domain $D$ , the stated compactly supported spinor-variation stationarity condition, and the explicit SI/natural, metric, and Dirac-matrix conventions below.

[F1]

Explicit charged Dirac/Maxwell local coupling model: the adopted natural action is $L_{n a t} = - \frac{1}{4} f_{a b} f^{a b} + \frac{i}{2} (\overset{ˉ}{ψ} γ^{a} D_{a} ψ - (D_{a} \overset{ˉ}{ψ}) γ^{a} ψ) - μ \overset{ˉ}{ψ} ψ,$ with $D_{a} = \partial_{a} - i q_{n} a_{a}$ , $x^{0} = c t$ , $μ = m c / ℏ$ , and compactly supported local variations. Its own text leaves the Euler–Lagrange consequence for a consuming proof.

[F2]

Charged gauge data and SI-natural conventions and Charge, canonical photon field and observer units: $η = (-, +, +, +)$ , $A_{0} = - ϕ / c$ , $a_{a} = A_{a} / \sqrt{μ_{0} ℏ c}$ , and $q_{n} a_{a} = q_{S I} A_{a} / ℏ$ ; $γ^{0} = β$ , $γ^{i} = β α_{i}$ , $β^{2} = I_{4}$ , and ${γ^{a}, γ^{b}} = - 2 η^{a b} I_{4}$ .

[F3]

Minimal Dirac magnetic square and formal Pauli benchmark: for smooth $A$ , $B = \nabla \times A$ , real $q$ , $m > 0$ , and the common core $D$ , it proves $[π_{i}, π_{j}] = i q ℏ ϵ_{i j k} B_{k}$ and the displayed square identity; it derives the formal $K / (2 m)$ coefficient and its spin term without assuming self-adjointness.

[F4]

Dirac Clifford matrices and energy projections: the fixed finite matrices have $β = diag (I_{2}, - I_{2})$ , $α_{i} = (\begin{array}{cc} 0 & σ_{i} \\ σ_{i} & 0 \end{array})$ , $γ^{0} = β$ , $γ^{i} = β α_{i}$ , and $\overset{ˉ}{ψ} = ψ^{†} β$ . The present calculation uses only these explicit finite matrices under the stated Countable Choice hypothesis, not the supplier's infinite-space operator conclusions.

[F5]

QFT example units, metric, gauge, and pairing conventions and Quantum field quantities and compatible SI normalizations: $x^{0} = c t$ , $A_{i}$ is in tesla-metres, $q$ is in coulombs, $m$ is in kilograms, $ℏ$ is in joule-seconds, and $ℏ c$ converts inverse-length natural coordinates to SI energy units.

[F6]

Example observable types and scoped meanings: a formal coefficient and a source-reported empirical estimate are different types; the latter carries its source conditions and uncertainty and is not a theorem premise here.

[A1]

Explicit charged Dirac/Maxwell local coupling model is an adopted classical local model with commuting smooth spinor variables. Its physical interpretation does not quantize the interacting action or supply a continuum QED solution.

Proof

1.1givenF1F2A1algebra

Let $ξ$ be a compactly supported smooth complex spinor variation and write $δ \overset{ˉ}{ψ} = \overset{ˉ}{ξ}$ . Vary the symmetrized spinor action in [F1] while holding $a_{a}$ fixed. Since $γ^{a}$ is constant and $D_{a} \overset{ˉ}{ξ} = \partial_{a} \overset{ˉ}{ξ} + i q_{n} a_{a} \overset{ˉ}{ξ}$ , compact integration by parts gives $\int (D_{a} \overset{ˉ}{ξ}) γ^{a} ψ d^{4} x = - \int \overset{ˉ}{ξ} γ^{a} D_{a} ψ d^{4} x$ : the derivative term changes sign, and the connection term becomes $+ i q_{n} a_{a}$ on the right, matching the conjugate-charge term on the left. Thus $δ (S_{c l} / ℏ) = 2 Re \int ξ^{†} β (i γ^{a} D_{a} - μ) ψ d^{4} x$ . No field product outside the smooth classical action is used.

2.1givenstep 1.1F2algebra

Put $R = (i γ^{a} D_{a} - μ) ψ$ . By the stated stationarity condition, choose $ξ = ρ β R$ for any nonnegative compactly supported smooth real bump $ρ$ . Since $β^{2} = I$ and $β = β^{†}$ , the first variation in 1.1 gives $0 = 2 \int ρ R^{†} R d^{4} x$ . If $R$ were nonzero at a point, continuity would make $R^{†} R > 0$ on a neighborhood, and a bump supported there would make the integral positive; hence $R = 0$ pointwise. Conversely, if $R = 0$ , the first variation in 1.1 vanishes. Therefore the compact spinor-variation Euler–Lagrange equation is $(i γ^{a} D_{a} - μ) ψ = 0$ .

3.1givenstep 2.1F2F4F5algebra

With $A_{0} = 0$ , the conversion in [F2] gives $D_{0} = c^{- 1} \partial_{t}, D_{i} = \partial_{i} - i q A_{i} / ℏ .$ Use $γ^{0} = β$ and $γ^{i} = β α_{i}$ in the Euler–Lagrange equation just obtained, multiply on the left by $β$ , and use $μ = m c / ℏ$ . After multiplication by $ℏ c$ , the result is $i ℏ \partial_{t} ψ = [c α \cdot (- i ℏ \nabla - q A) + β m c^{2}] ψ = H_{D} ψ .$ This is an equality of the stated differential expressions. It establishes the QX12 Hamiltonian convention on $D$ and assumes no operator closure.

4.1givenstep 3.1F3F4algebra

The smooth multiplication and derivative maps preserve $D$ . By [F3], their commutator is $[π_{i}, π_{j}] = i q ℏ ϵ_{i j k} B_{k}$ there, and the Pauli-block identity $α_{i} α_{j} = δ_{i j} I + i ϵ_{i j k} Σ_{k}$ contracts the antisymmetric part to $\frac{i}{2} ϵ_{i j k} Σ_{k} [π_{i}, π_{j}] = - q ℏ Σ \cdot B .$ The cross terms with $β m c^{2}$ cancel because $α_{i} β + β α_{i} = 0$ , while $β^{2} = I$ . Hence the exact core identity in the Statement follows, with every product acting on $D$ .

5.1givenstep 4.1F3F6algebra∎

In a formal symbol $t$ , square $m c^{2} + t a K + O (t^{2})$ and equate it to $m^{2} c^{4} + t c^{2} K + O (t^{2})$ . The scalar coefficient is defined before substitution by $(m c^{2} + t a X)^{2} \equiv m^{2} c^{4} + t c^{2} X (m o d t^{2})$ in $R [X, t] / (t^{2})$ , as in [F3]. Comparing $t X$ gives $2 m c^{2} a = c^{2}$ , hence $a = 1 / (2 m)$ since $m > 0$ , and substituting $X = K$ gives the displayed correction. The spin part of $K / (2 m)$ is $- \frac{q ℏ}{2 m} Σ \cdot B = - \frac{q}{m} S \cdot B .$ Equating this with $- g_{s} (q / (2 m)) S \cdot B$ and using $q = - e \neq 0$ and the nonzero Pauli spin matrices gives $g_{s} = 2$ . This is a formal coefficient comparison only, not a source-reported empirical estimate under the type distinction [F6].

Minimal Dirac magnetic benchmark from the adopted action

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources