Physlib.Electromagnetism.Kinematics.GaugeTransformation

Given a gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$, this definition constructs the pure-gauge electromagnetic potential $A^\mu$ whose components are given by the raised-index gradient $A^\mu(x) = \sum_{\kappa} \eta_{\mu\kappa} \partial_\kappa \chi(x)$. Here $\eta$ is the Minkowski matrix $\text{diag}(1, -1, \dots, -1)$ and $\partial_\kappa \chi$ is the partial derivative of $\chi$ with respect to the $\kappa$-th coordinate.

For any gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$, point $x \in \text{SpaceTime}_d$, and coordinate index $\mu$, the $\mu$-th component of the pure-gauge electromagnetic potential $A = \text{ofGradient } \chi$ at $x$ is given by the sum $$A^\mu(x) = \sum_{\kappa} \eta_{\mu\kappa} \partial_\kappa \chi(x)$$ where $\eta$ is the Minkowski matrix and $\partial_\kappa \chi$ is the partial derivative of $\chi$ with respect to the $\kappa$-th coordinate.

For a gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$, a point $x$ in spacetime, and a coordinate index $\mu$, the $\mu$-th component of the pure-gauge potential $A^\mu(x)$ is given by $$A^\mu(x) = \eta_{\mu\mu} \partial_\mu \chi(x)$$ where $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric and $\partial_\mu \chi$ is the partial derivative of $\chi$ with respect to the $\mu$-th coordinate. This formula reflects that because the Minkowski metric is diagonal, the sum $A^\mu = \sum_{\kappa} \eta_{\mu\kappa} \partial_\kappa \chi$ collapses to a single term.

For any point $x \in \text{SpaceTime}_d$ and any coordinate index $\mu$, the pure-gauge potential $A^\mu$ derived from the zero gauge function $\chi(x) = 0$ is zero. That is, $(A_{\text{gauge}})^\mu(x) = 0$ whenever $\chi = 0$.

For any two differentiable gauge functions $\chi_1, \chi_2 : \text{SpaceTime}_d \to \mathbb{R}$, the pure-gauge potential associated with their sum $\chi_1 + \chi_2$ is equal to the sum of the individual pure-gauge potentials associated with $\chi_1$ and $\chi_2$: $$\text{ofGradient}(\chi_1 + \chi_2) = \text{ofGradient}(\chi_1) + \text{ofGradient}(\chi_2)$$ where $\text{ofGradient}(\chi)$ denotes the pure-gauge potential $A^\mu$ whose components are defined by the raised-index gradient $A^\mu(x) = \eta^{\mu\nu} \partial_\nu \chi(x)$.

For any gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$ that is twice continuously differentiable (of class $C^2$), the pure-gauge electromagnetic potential $A^\mu$, defined by $A^\mu(x) = \eta^{\mu\nu} \partial_\nu \chi(x)$ where $\eta$ is the Minkowski metric, is a differentiable function.

Let $d$ be the dimension of spacetime and $n$ be a natural number. If a gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$ is $(n+1)$-times continuously differentiable (of class $C^{n+1}$), then the corresponding pure-gauge potential $A^\mu = \partial^\mu \chi$ (defined as $A^\mu = \eta^{\mu\nu} \partial_\nu \chi$) is $n$-times continuously differentiable (of class $C^n$).

For any spatial dimension $d$ and any gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$ that is twice continuously differentiable (of class $C^2$), the field strength tensor $F$ associated with the pure-gauge potential $A^\mu = \partial^\mu \chi$ vanishes at every point $x$ in spacetime: $$F(x) = 0$$ where the potential is defined by $A^\mu = \eta^{\mu\nu} \partial_\nu \chi$ and the field strength tensor by $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$.

Let $\chi : \text{SpaceTime}_d \to \mathbb{R}$ be a differentiable gauge function and $\Lambda$ be a Lorentz transformation. The Lorentz action on the pure-gauge potential $A^\mu = \partial^\mu \chi$ (where $\partial^\mu \chi = \eta^{\mu\nu} \partial_\nu \chi$) is equivariant with respect to the transformation of the gauge function, such that $\Lambda \cdot (\partial^\mu \chi) = \partial^\mu (\chi \circ \Lambda^{-1})$.

For a given spatial dimension $d$, a gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$, and an electromagnetic potential $A$, the gauge transformation of $A$ by $\chi$ is the potential defined by the sum $A + \partial^\mu \chi$. This represents the pointwise addition of the potential $A$ and the pure-gauge potential $\partial^\mu \chi(x) = \sum_{\nu} \eta^{\mu\nu} \partial_\nu \chi(x)$ constructed from the gradient of $\chi$ and the Minkowski metric $\eta$.

For a given spatial dimension $d$, a gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$, an electromagnetic potential $A$, and a point $x \in \text{SpaceTime}_d$, the gauge transformation of $A$ by $\chi$ evaluated at $x$ is given by $A(x) + \partial^\mu \chi(x)$, where $\partial^\mu \chi$ is the pure-gauge potential (the raised-index gradient of $\chi$).

Let $A$ be a differentiable electromagnetic potential on a $(1+d)$-dimensional spacetime and $\chi: \text{SpaceTime}_d \to \mathbb{R}$ be a twice continuously differentiable ($C^2$) gauge function. The field strength tensor (Faraday tensor) $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ is invariant under the gauge transformation $A^\mu \mapsto A^\mu + \partial^\mu \chi$, where $\partial^\mu \chi = \eta^{\mu\nu} \partial_\nu \chi$ is the raised-index gradient. That is, for any point $x$ in spacetime, the field strength of the transformed potential is equal to the field strength of the original potential: $$F[A + \partial \chi](x) = F[A](x)$$

Let $A$ be a differentiable electromagnetic potential on a $(1+d)$-dimensional spacetime and $\chi: \text{SpaceTime}_d \to \mathbb{R}$ be a twice continuously differentiable ($C^2$) gauge function. For any point $x$ in spacetime, the field strength matrix of the potential after a gauge transformation $A^\mu \mapsto A^\mu + \partial^\mu \chi$ is identical to the field strength matrix of the original potential $A$. That is, $$F[A + \partial \chi](x) = F[A](x)$$ where $F$ denotes the matrix representation of the field strength tensor (Faraday tensor).

For any electromagnetic potential $A$ in $d$-dimensional spacetime, the gauge transformation of $A$ with respect to the constant zero function $\chi(x) = 0$ leaves the potential unchanged, i.e., $\text{gaugeTransform}(0, A) = A$.

Let $A$ be an electromagnetic potential on a $d$-dimensional spacetime. For any two differentiable gauge functions $\chi_1, \chi_2 : \text{SpaceTime}_d \to \mathbb{R}$, the composition of a gauge transformation by $\chi_2$ followed by a gauge transformation by $\chi_1$ is equivalent to a single gauge transformation by the sum of the functions $\chi_1 + \chi_2$: $$\text{gaugeTransform}(\chi_1, \text{gaugeTransform}(\chi_2, A)) = \text{gaugeTransform}(\chi_1 + \chi_2, A)$$ where $\text{gaugeTransform}(\chi, A)$ denotes the transformation $A^\mu \mapsto A^\mu + \partial^\mu \chi$.

For an electromagnetic potential $A$ in a $d$-dimensional spacetime, a differentiable gauge function $\chi : \text{SpaceTime}_d \to \mathbb{R}$, and a Lorentz transformation $\Lambda \in \mathcal{L}$, the Lorentz transformation of the gauge-transformed potential is equal to the gauge transformation of the Lorentz-transformed potential by the pulled-back gauge function $\chi \circ \Lambda^{-1}$: $$\Lambda \cdot \text{gaugeTransform}(\chi, A) = \text{gaugeTransform}(\chi \circ \Lambda^{-1}, \Lambda \cdot A)$$ where $(\Lambda \cdot A)(x) = \Lambda(A(\Lambda^{-1}x))$ and $\text{gaugeTransform}(\chi, A)$ denotes the mapping $A^\mu \mapsto A^\mu + \partial^\mu \chi$.

In a $(1+d)$-dimensional spacetime, let $\chi : \text{SpaceTime}_d \to \mathbb{R}$ be a gauge function defined by $\chi(y) = y^0 y^i$, where $y^0$ is the temporal coordinate and $y^i$ is the $i$-th spatial coordinate for some index $i \in \{1, \dots, d\}$. If $B$ is an electromagnetic potential defined by the bare covariant gradient $B^\mu = \partial_\mu \chi$, then the $(0, i)$ component of the field strength matrix at any point $x$ is equal to $2$: $$F^{0i}(x) = 2$$

In a $(1+d)$-dimensional spacetime, let $\chi: \text{SpaceTime}_d \to \mathbb{R}$ be the scalar function defined by $\chi(y) = y^0 y^i$, where $y^0$ is the temporal coordinate and $y^i$ is the $i$-th spatial coordinate for some index $i \in \{1, \dots, d\}$. If the electromagnetic potential $B$ is defined by the bare gradient $B_\mu = \partial_\mu \chi$, then its associated field strength tensor $F$ is non-zero at any point $x$ in spacetime, i.e., $F(x) \neq 0$.