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Finite canonical partition is invariant under static electromagnetic gauge changes

Physical scope

Finite one-particle canonical model on a bounded region with static smooth potentials and a fixed action reference; only time-independent gauge changes are covered, with no claim of field dynamics or empirical realization.

Statement

Let $Ω \subset R^{3}$ be nonempty bounded open with compact closure, $m > 0$ , and $q_{p} \in R$ the signed particle charge in coulombs, as defined in Static electromagnetic coupling and typed canonical momentum and distinct from its exact elementary-charge constant $e$ . Let $a_{0} > 0$ be an action reference and $β > 0$ have units J $^{- 1}$ . Let $A, φ$ satisfy the smoothness assumptions of Static electromagnetic coupling and typed canonical momentum, with $φ$ bounded. Adopt the canonical preparation Canonical thermal preparation and the single-particle reference convention Classical particle counting and physical observables relative to $d x d p / a_{0}^{3}$ . Then $Z_{β} = \frac{1}{a_{0}^{3}} \int_{Ω} \int_{R^{3}} e^{- β H (x, p)} d p d x = \frac{1}{a_{0}^{3}} {(\frac{2 π m}{β})}^{3 / 2} \int_{Ω} e^{- β q_{p} φ (x)} d x$ is finite and positive. If $χ \in C^{3}$ is time-independent and $A_{χ} = A + \nabla χ$ , $φ_{χ} = φ$ , then the paired canonical coordinate map $T_{χ} (x, p) = (x, p + q_{p} \nabla χ (x))$ satisfies $Z_{β, χ} = Z_{β} .$ It carries the canonical probability to the transformed canonical probability and leaves the expectation of every integrable measurable observable of $(x, π)$ unchanged, where $π = p - q_{p} A (x)$ . The result is a conditional prediction of the adopted static model, not an experimental observation.

Facts & Assumptions

Given: The static external-potential model, smooth time-independent gauge function, fixed action reference and canonical preparation.

[A1]

$Ω$ has finite positive volume, $φ$ is bounded, $m > 0$ , $β > 0$ , and the canonical density is the one adopted in Canonical thermal preparation relative to the stated phase reference (Classical configuration, cotangent phase and reference measures) and single-particle counting scale (Classical particle counting and physical observables).

[A2]

$A, φ$ obey the static coupling definition and gauge signs (Static electromagnetic coupling and typed canonical momentum, Potential representations, gauge transformations and the Lorenz and Coulomb conditions); $χ$ is time-independent $C^{3}$ .

[F1]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ proves $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ . Tonelli gives $\int_{R^{3}} e^{- ∣ x ∣^{2}} d^{3} x = π^{3 / 2}$ . For $a = β / (2 m) > 0$ , the linear map $u \mapsto \sqrt{a} u$ has Jacobian $a^{3 / 2}$ , so the local scaling calculation gives $\int_{R^{3}} e^{- β ∣ u ∣^{2} / (2 m)} d^{3} u = a^{- 3 / 2} π^{3 / 2} = (2 π m / β)^{3 / 2}$ .

[F2]

Tonelli permits the nonnegative position-momentum integration (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F3]

The nonnegative change-of-variables theorem applies to a $C^{1}$ diffeomorphism with its absolute Jacobian under Countable Choice (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F4]

The fixed action reference $a_{0}$ has units of action (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

Proof

Given: [A1]–[A2] and the Gaussian, product-integration, unit and change-of-variables interfaces [F1]–[F4].

1.1A1A2F1F2F4algebra

Put $u = p - q_{p} A (x)$ in the inner integral. Translation preserves Lebesgue measure and $H = ∣ u ∣^{2} / (2 m) + q_{p} φ (x)$ . By [F1]–[F2], the displayed formula for $Z_{β}$ follows. Since $e^{- β q_{p} φ}$ is bounded above and below by positive constants on finite positive-volume $Ω$ , the normalizer is finite and strictly positive.

2.1A2F3step 1.1algebra∎

The map $T_{χ}$ has inverse $T_{- χ}$ and derivative $D T_{χ} = (\begin{array}{cc} I & 0 \\ q_{p} D^{2} χ & I \end{array}), \det D T_{χ} = 1.$ By the paired gauge transformation, $H_{χ} (T_{χ} (x, p)) = ∣ p + q_{p} \nabla χ - q_{p} (A + \nabla χ) ∣^{2} / (2 m) + q_{p} φ = H (x, p)$ . Apply [F3] to the nonnegative integrand and this diffeomorphism; its unit Jacobian gives $Z_{β, χ} = Z_{β}$ . The same substitution carries normalized densities into one another. Moreover $(p + q_{p} \nabla χ) - q_{p} (A + \nabla χ) = p - q_{p} A = π$ , so any integrable function of $(x, π)$ has identical expectations in the paired states.

Finite canonical partition is invariant under static electromagnetic gauge changes

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources