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Gauge invariance of the electric and magnetic fields

Physical scope

The gauge invariance of the field pair of the classical vacuum electromagnetic framework on a laboratory region: gauge related potential representations of the framework's fields, in the conventions of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, define the same electric and magnetic fields. The statement is conditional on the joint regularity of the pair and the generator; it contains no claim about the existence of generators meeting prescribed data, no empirical claim, and no assertion that a particular gauge is convenient or available.

Statement

Let $c > 0$ , let $Ω \subseteq R^{3}$ be open, let $I \subseteq R$ be an interval with at least two elements (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), let $k \geq 2$ be an integer, let $(φ, A)$ be a potential pair of class $C^{k}$ jointly on $Ω \times I$ and let $χ$ be a gauge function of class $C^{k}$ jointly on $Ω \times I$ , all in the conventions of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression and Potential representations, gauge transformations and the Lorenz and Coulomb conditions: $E (φ, A) = - \nabla φ - \partial_{t} A$ , $B (φ, A) = curl A$ and $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ ; for a non-open $I$ the joint $C^{k}$ classes are read in the open-neighbourhood restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain (for open $I$ this is the ordinary class of $C^{k}$ Euclidean maps and diffeomorphisms).

Then $T_{χ} (φ, A)$ is a potential pair of class $C^{k - 1}$ jointly on $Ω \times I$ and its field pair is the field pair of $(φ, A)$ :

$E (T_{χ} (φ, A)) = E (φ, A), B (T_{χ} (φ, A)) = B (φ, A) on Ω \times I,$

the common values being maps $Ω \times I \to R^{3}$ of class $C^{k - 1}$ jointly, that is, fields of the framework in the sense of The electric and magnetic fields as fixed-time spatial vector fields. In particular gauge related potential representations of one field pair have equal field pairs: the electric and magnetic fields are gauge invariants of the framework, and no value of $φ$ , of $A$ or of $χ$ is asserted to be observable.

The proof uses only the defining formulas, the vanishing of $curl \nabla χ$ for the $C^{2}$ function $χ$ and the equality of the time-space mixed partials $\partial_{t} \partial_{i} χ = \partial_{i} \partial_{t} χ$ ; no Maxwell equation, no hypothesis on $Ω$ beyond openness, and no choice principle is used.

Facts & Assumptions

Given: An open set $Ω \subseteq R^{3}$ , an interval $I$ with at least two elements, $c > 0$ , an integer $k \geq 2$ , a jointly $C^{k}$ potential pair $(φ, A)$ and a jointly $C^{k}$ gauge function $χ$ on $Ω \times I$ .

[F1]

Potential-pair formulas: for a pair $(φ, A)$ of class $C^{m}$ , $m \geq 1$ , $E (φ, A) = - \nabla φ - \partial_{t} A$ and $B (φ, A) = curl A$ are of class $C^{m - 1}$ ; for $m \geq 2$ , the transform of a $C^{m}$ pair by a $C^{m}$ gauge function is $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ and is of class $C^{m - 1}$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression); for a non-open $I$ the joint classes and the componentwise computations are read in the open-neighbourhood restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain, whose relative time derivatives at included endpoints reproduce the two identities below pointwise on $Ω \times I$ .

[F2]

Framework vocabulary: a potential representation of a field pair is a potential pair with that field pair, a gauge function of class $C^{m}$ has $m \geq 2$ , two pairs are gauge related when one is the gauge transform of the other, and the field pair of a representation is its physical content (Potential representations, gauge transformations and the Lorenz and Coulomb conditions).

[F3]

Linearity of the curl: for $C^{1}$ fields $F, G$ and reals $α, β$ one has $curl (α F + β G) = α curl F + β curl G$ (Divergence and curl are linear and satisfy the scalar product rules).

[F4]

For a $C^{2}$ scalar function $χ$ the field $\nabla χ$ is $C^{1}$ and $curl \nabla χ = 0$ (The curl of the gradient of a $C^{2}$ function vanishes).

[F5]

For a $C^{2}$ function on an open subset of $R^{m}$ the mixed coordinate partial derivatives commute, so $\partial_{t} \partial_{i} χ = \partial_{i} \partial_{t} χ$ for the jointly $C^{2}$ function $χ$ (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F6]

Differentiability and class bookkeeping: sums of totally differentiable maps are totally differentiable with the sum of the derivatives and a map is of class $C^{m}$ when all its componentwise iterated partial derivatives up to order $m$ exist and are continuous, partial derivatives being one-variable derivatives along coordinate lines (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives, $C^{k}$ Euclidean maps and diffeomorphisms, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F7]

The electric and magnetic fields of the framework are maps $Ω \times I \to R^{3}$ at fixed time, read in the frame basis with the stated regularity and polar/axial conventions (The electric and magnetic fields as fixed-time spatial vector fields).

Proof

technique · direct; compute the two field formulas of the transformed pair and use the vanishing curl of a gradient and the commutativity of mixed partials

1.1givenF1F2F6

The transformed pair and its class. The partial derivatives $\partial_{t} χ$ and $\nabla χ$ of the jointly $C^{k}$ function $χ$ are jointly $C^{k - 1}$ , and the pair members $φ$ and $A$ are jointly $C^{k}$ , hence jointly $C^{k - 1}$ ; by the closure of the class under sums and the componentwise reading of [F6], the maps $φ - \partial_{t} χ$ and $A + \nabla χ$ are jointly $C^{k - 1}$ , so $T_{χ} (φ, A)$ is a potential pair of class $C^{k - 1}$ in the sense of [F1] and [F2]. Since $k - 1 \geq 1$ , its field pair is defined by [F1].

1.2givenF1F3F4

Magnetic field. $curl (A + \nabla χ) = curl A + curl \nabla χ$ by the linearity of the curl [F3]; the function $χ$ is jointly $C^{k}$ with $k \geq 2$ , hence jointly $C^{2}$ , so $curl \nabla χ = 0$ by [F4]; therefore $B (T_{χ} (φ, A)) = curl (A + \nabla χ) = curl A = B (φ, A)$ on $Ω \times I$ .

1.3givenF1F5F6

Electric field. Expanding with the definitions of [F1] and the componentwise linearity of the total derivative for sums [F6], $E (T_{χ} (φ, A)) = - \nabla (φ - \partial_{t} χ) - \partial_{t} (A + \nabla χ) = - \nabla φ - \partial_{t} A + \nabla \partial_{t} χ - \partial_{t} \nabla χ = E (φ, A) + (\nabla \partial_{t} χ - \partial_{t} \nabla χ)$ . The bracket vanishes componentwise: its $i$ -th component is $\partial_{i} \partial_{t} χ - \partial_{t} \partial_{i} χ$ , and the jointly $C^{2}$ function $χ$ has commuting mixed partials by [F5]. Hence $E (T_{χ} (φ, A)) = E (φ, A)$ on $Ω \times I$ .

2.1step 1.1step 1.2step 1.3F1F2F7∎

Conclusion. Step 1.1 shows that the transform is a potential pair of class $C^{k - 1}$ ; steps 1.2 and 1.3 show that its magnetic and electric fields equal those of $(φ, A)$ , and by [F1] both are jointly $C^{k - 1}$ , hence framework fields of the shape recorded in [F7]. Gauge relatedness of two pairs means exactly that one is the transform of the other by some gauge function [F2], so gauge related representations have equal field pairs, and the only physical content attached to a representation in [F2] is its field pair. No Maxwell equation, domain hypothesis beyond openness, or choice principle was used.

Remarks

Why $k \geq 2$ and not $k \geq 1$ . The scalar identity uses $curl \nabla χ = 0$ , which needs $χ$ to be $C^{2}$ , and the electric identity uses the symmetry of the second mixed partial of $χ$ . For a merely $C^{1}$ generator the two field expressions are still formally equal, but the intermediate quantities $curl \nabla χ$ and the mixed partials are then not defined.
No converse. The theorem says gauge transformations do not change the fields. Which potential pairs represent the same field pair, and which gauge conditions are reachable, are separate questions treated elsewhere on this page.

Gauge invariance of the electric and magnetic fields

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources