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Nonzero pure-gauge potentials representing the zero field pair

Physical scope

An example of gauge representations of the zero field pair: the full linear-generator family includes the zero potential pair at zero parameters and nonzero constant pairs at nonzero parameter tuples. An explicit nonzero witness demonstrates noninjectivity of the potential-to-field map. It asserts no source model, Maxwell equation, empirical situation or observable potential value.

Example

Let $Ω \subseteq R^{3}$ be open and let $I \subseteq R$ be an interval (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length); for a possibly non-open $I$ , the joint classes below are read in the 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). Let $χ : Ω \times I \to R$ be of class $C^{2}$ jointly. If $I$ is a singleton, choose one ambient $C^{2}$ extension as part of the data and define the potential pair and all its derivatives by the same ambient formulas before restriction; singleton values of $χ$ alone do not determine these time derivatives. For an empty product all identities below are vacuous.

Then the pair

$(φ, A) : = T_{χ} (0, 0) = (- \partial_{t} χ, \nabla χ)$

is a potential pair of class $C^{1}$ jointly on $Ω \times I$ whose field pair is the zero field pair:

$E (φ, A) = E (0, 0) = 0, B (φ, A) = curl \nabla χ = 0 on Ω \times I$

(Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression), the electric identity by the gauge invariance of Gauge invariance of the electric and magnetic fields applied to the zero pair, whose field pair is $(0, 0)$ , and the magnetic identity also directly by The curl of the gradient of a $C^{2}$ function vanishes for the $C^{2}$ function $χ$ .

All displayed quantities may be read as numerical SI components: $x$ in m, $t$ in s, $χ$ in V s, $a$ in T m and $β$ in V, as in Potential representations, gauge transformations and the Lorenz and Coulomb conditions; in particular $e_{1}$ below specifies a vector-potential component of one T m.

Explicit instance: for $a \in R^{3}$ , $β \in R$ and $χ (x, t) = a \cdot x - β t$ on $R^{3} \times R$ , the pair is the constant pair $(φ, A) = (β, a)$ , with $E = - \nabla β - \partial_{t} a = 0$ and $B = curl a = 0$ by Divergence and curl of a $C^{1}$ vector field and the vanishing of all first derivatives of a constant (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ). This pair is nonzero if and only if $(β, a) \neq (0, 0)$ ; for $β = 0$ and $a = 0$ it is exactly the zero pair. To exhibit noninjectivity explicitly, take $β = 0$ and $a = e_{1} = (1, 0, 0)$ on $R^{3} \times R$ . Then $(φ, A) = (0, e_{1})$ and $(0, 0)$ are distinct potential pairs with the same zero field pair. Thus the map from potential pairs to field pairs of The electric and magnetic fields as fixed-time spatial vector fields is not injective: nonzero potentials may represent the zero field pair (pure-gauge fields), and no nonzero value of $φ$ or $A$ alone is evidence for a nonzero field, consistently with Gauge invariance of the electric and magnetic fields and Potential representations, gauge transformations and the Lorenz and Coulomb conditions. The example is a statement about the representation of the zero field configuration; it asserts no Maxwell equation, no source configuration and no empirical situation.

Facts & Assumptions

Given: An open set $Ω \subseteq R^{3}$ , an interval $I \subseteq R$ , a $C^{2}$ function $χ$ on $Ω \times I$ in the restriction convention, with a chosen ambient extension when $I$ is a singleton, the pair $(φ, A) = T_{χ} (0, 0)$ , the parameters $a \in R^{3}$ , $β \in R$ with $χ (x, t) = a \cdot x - β t$ on $R^{3} \times R$ , and the vector $e_{1} = (1, 0, 0)$ .

[F1]

Potential-pair vocabulary and gauge transform: the zero pair $(0, 0)$ is a potential pair of class $C^{m}$ for every $m$ ; for a class $C^{m}$ pair and a class $C^{m}$ gauge function with $m \geq 2$ the transform is $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ and is a potential pair of class $C^{m - 1}$ ; the field pair of a class $C^{1}$ pair $(φ, A)$ is $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ ; the values of the potentials and of $χ$ are auxiliary and no value is an observable (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, Potential representations, gauge transformations and the Lorenz and Coulomb conditions).

[F2]

Gauge invariance: on $Ω \times I$ with $Ω \subseteq R^{3}$ open and $I$ an interval with at least two elements, in the stated restriction convention, if $(φ, A)$ is a potential pair of class $C^{k}$ jointly and $χ$ a gauge function of class $C^{k}$ with $k \geq 2$ , then $T_{χ} (φ, A)$ has the same field pair as $(φ, A)$ (Gauge invariance of the electric and magnetic fields).

[F3]

The curl of a gradient of a $C^{2}$ scalar function vanishes pointwise (The curl of the gradient of a $C^{2}$ function vanishes).

[F4]

Calculus of sums, constants and linear functions: partial derivatives are one-variable derivatives along coordinate lines, sums and scalar multiples of differentiable maps are differentiable with the corresponding derivatives, and every first partial derivative of a constant function or field is zero; for $χ (x, t) = a \cdot x - β t$ the coordinate partial derivatives are $\partial_{i} χ = a_{i}$ and $\partial_{t} χ = - β$ (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Divergence and curl of a $C^{1}$ vector field).

[F5]

Framework vocabulary: the electric and magnetic fields are the maps $Ω \times I \to R^{3}$ read in the frame basis with the stated regularity, and the potential-to-field assignment is that of the cited definitions (The electric and magnetic fields as fixed-time spatial vector fields).

[F6]

Restriction convention for a possibly non-open interval, and the classes it yields (Joint regularity of the radial vector and scalar potentials on a star-shaped domain).

Verification

technique · direct; identify the transformed pair as the gauge transform of the zero pair, use gauge invariance for the zero field pair, recompute the constant instance, and exhibit the nonzero witness

1.1givenF1F2F3F4F6

The general pure-gauge pair. The zero pair $(0, 0)$ is a potential pair of class $C^{2}$ by [F1], and $χ$ is a gauge function of class $C^{2}$ , so the transform $T_{χ} (0, 0) = (- \partial_{t} χ, \nabla χ)$ is a potential pair of class $C^{1}$ by [F1]; its field pair is defined and, for a nondegenerate $I$ , by [F2] with $k = 2$ equals the field pair of $(0, 0)$ . For singleton $I$ , apply [F2] on open spatial-time boxes in the chosen extension neighbourhood and then restrict the ambient pair and its jets to $Ω \times I$ ; these local computations cover the singleton slice. For an empty product the identities are vacuous. That field pair is $E (0, 0) = - \nabla 0 - \partial_{t} 0 = 0$ and $B (0, 0) = curl 0 = 0$ by [F1] and [F4], because all first derivatives of the constant zero pair vanish. Hence $E (φ, A) = 0$ ; separately, $B (φ, A) = curl \nabla χ = 0$ by [F3] applied to the $C^{2}$ function $χ$ . Thus every $C^{2}$ generator produces a pure-gauge pair, and the resulting field pair is the zero field pair of [F5].

2.1step 1.1F1F4

The constant instance. For $χ (x, t) = a \cdot x - β t$ on $R^{3} \times R$ , [F4] gives $\nabla χ = a$ and $\partial_{t} χ = - β$ , so $(φ, A) = T_{χ} (0, 0) = (β, a)$ as constant maps on $R^{3} \times R$ . For this constant pair, $E = - \nabla β - \partial_{t} a = 0$ and $B = curl a = 0$ by [F1] and [F4]: the gradient of the constant scalar $β$ and the time derivative of the constant field $a$ vanish, and the curl of the constant field $a$ vanishes because each of its first partial derivatives is zero.

3.1step 1.1step 2.1F1F4

The zero-parameter branch and the nonzero witness. The constant pair $(β, a)$ equals the zero pair $(0, 0)$ if and only if $β = 0$ and $a = 0$ , equality of maps on the nonempty set $R^{3} \times R$ being componentwise equality of the constants; at $β = 0$ , $a = 0$ the generator is $χ = 0$ and $T_{0}$ is the identity by [F1], so the zero pair is recovered. Taking instead $β = 0$ and $a = e_{1}$ gives a pair $(φ, A) = (0, e_{1})$ that is not the zero pair, while by steps 1.1 and 2.1 its field pair is $(0, 0)$ : explicitly $φ = 0$ , $A = (1, 0, 0)$ are constant, so $E = - \nabla 0 - \partial_{t} e_{1} = 0$ and $B = curl e_{1} = 0$ by [F4]. Two distinct potential pairs therefore have the same field pair, and the map from potential pairs to field pairs of [F5] is not injective.

4.1step 1.1step 3.1F1F5∎

Conclusion and scope. Every $C^{2}$ gauge generator gives a potential pair with the zero field pair (step 1.1); the linear family $χ (x, t) = a \cdot x - β t$ realizes the constant pairs $(β, a)$ , with the zero pair exactly at the zero parameters and the witness $(0, e_{1})$ distinct from $(0, 0)$ while still representing the zero field pair (steps 2.1 and 3.1). No Maxwell equation, source configuration, choice principle or empirical situation is used, and no nonzero value of $φ$ or $A$ is asserted to be evidence for a nonzero field.

Remarks

What the example shows about representations. The construction sends every $C^{2}$ generator to a pure-gauge pair with the zero field pair, so the potential-to-field map is not injective; the witness $(0, e_{1})$ uses the simplest constant family, and nothing beyond the displayed pure-gauge family is asserted about representatives of other field pairs.

Nonzero pure-gauge potentials representing the zero field pair

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

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