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Minimal Dirac spin and orbital coefficient comparison

Physical scope

A uniform-field evaluation of the minimal Dirac spin coefficient and its comparison with a separately defined ideal orbital cyclotron coefficient. It is not a full Landau spectrum, a radiative anomaly calculation, or an experimental prediction.

Statement

Take $q = - e$ with $e > 0$ , $m > 0$ , and a constant field $B = B_{0} \hat{z}$ with $B_{0} > 0$ . It is realized by the smooth SI potential $A (x, y, z) = \frac{B_{0}}{2} (- y, x, 0), ϕ = 0,$ for which $\nabla \times A = B_{0} \hat{z}$ , where $\hat{z} = (0, 0, 1)$ . Under the Countable Choice and adopted-action hypotheses of Minimal Dirac magnetic benchmark from the adopted action, the formal spin term is $- (q / m) S \cdot B = (e / m) S_{z} B_{0}$ . Since $Σ_{z} = diag (1, - 1, 1, - 1)$ , its two spin eigenvalue coefficients are $Δ E_{↑, ↓} = \pm \frac{e ℏ B_{0}}{2 m}, Δ E_{g a p} = \frac{e ℏ B_{0}}{m} .$ Define the comparison orbital coefficient by $ω_{c, o r b} : = e B_{0} / m$ and $ν_{c, o r b} : = ω_{c, o r b} / (2 π)$ . The cyclic coefficient frequency of the spin gap is $ν_{s, c o e f f} : = \frac{Δ E_{g a p}}{2 π ℏ} = \frac{e B_{0}}{2 π m} = ν_{c, o r b} .$ This is a comparison of formal minimal coefficients. It is not the full Landau spectrum, a radiative correction, a prediction of zero measured anomaly, or a laboratory measurement.

Facts & Assumptions

Given: $q = - e \neq 0$ , $m > 0$ , $B_{0} > 0$ , the displayed smooth pure-magnetic potential, and the theorem's Countable Choice and adopted-action hypotheses and conditional minimal Dirac coefficient in Minimal Dirac magnetic benchmark from the adopted action.

[F1]

Minimal Dirac magnetic benchmark from the adopted action: the spin term is $- (q / m) S \cdot B$ with $S = ℏ Σ / 2$ and its matching convention gives $g_{s} = 2$ at the minimal coefficient level.

[F2]

QFT example units, metric, gauge, and pairing conventions: $e$ is in coulombs, $ℏ$ in $J s$ , $m$ in kilograms, and $B_{0}$ in tesla.

[F3]

Example observable types and scoped meanings: an adopted coefficient is not a source-reported empirical estimate; empirical uncertainty and conditions would need to accompany a measurement claim.

[A1]

The comparison coefficient is defined as $ω_{c, o r b} : = e B_{0} / m$ and $ν_{c, o r b} : = ω_{c, o r b} / (2 π)$ . This conventional orbital coefficient is supplied for comparison; no orbit equation is derived here.

Example

The uniform field makes the spin eigenvalues and the formal coefficient comparison explicit.

Verification

1.1givenF1algebra

Direct differentiation gives $\nabla \times \frac{B_{0}}{2} (- y, x, 0) = B_{0} \hat{z}$ , so the potential is smooth and satisfies the field hypothesis of [F1]. From [F1] and $q = - e$ , the spin coefficient is $+ (e / m) S_{z} B_{0}$ . The explicit diagonal matrix $Σ_{z}$ has eigenvalues $+ 1$ and $- 1$ , so $S_{z}$ has eigenvalues $+ ℏ / 2$ and $- ℏ / 2$ . This gives the two stated shifts and their difference $e ℏ B_{0} / m$ .

2.1step 1.1F2F3A1algebra∎

Dividing that positive gap by $2 π ℏ$ yields $ν_{s, c o e f f} = e B_{0} / (2 π m)$ , equal to $ν_{c, o r b}$ by [A1]. The unit of $e B_{0} / m$ is $s^{- 1}$ because $[T] = k g / (C s)$ ; the energy gap has unit joules. The equality is among model coefficients and uses no observation or statistical inference.

Minimal Dirac spin and orbital coefficient comparison

Statement

Facts & Assumptions

Example

Verification

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