Physlib.SpaceAndTime.TimeAndSpace.EuclideanGroup.SchwartzAction

The continuous representation of the Euclidean group $E(d)$ on the space of Schwartz maps $\mathcal{S}(\text{Time} \times \text{Space } d, F)$ over the real numbers $\mathbb{R}$. For an element $g \in E(d)$ and a Schwartz function $\eta$, the action is defined by the pullback: $$(g \cdot \eta)(tx) = \eta(g^{-1} \cdot tx)$$ where $tx \in \text{Time} \times \text{Space } d$ represents a point in the space-time manifold. The group action $g^{-1} \cdot tx$ acts on the spatial component of the coordinate while leaving the time component invariant. This definition characterizes the action as a continuous linear representation $\rho : E(d) \to \text{End}(\mathcal{S}(\text{Time} \times \text{Space } d, F))$, where each $\rho(g)$ is a continuous linear operator on the Schwartz space.

For any dimension $d \in \mathbb{N}$, let $g$ be an element of the Euclidean group $E(d)$, $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, F)$ be a Schwartz map, and $tx$ be a point in the space-time manifold $\text{Time} \times \text{Space } d$. The value of the continuous representation of the Euclidean group action $\text{schwartzEuclideanAction}$ on the map $\eta$ at the point $tx$ is given by the pointwise formula: $$(\text{schwartzEuclideanAction}(g, \eta))(tx) = \eta(g^{-1} \cdot tx)$$ where $g^{-1} \cdot tx$ represents the action of the inverse element $g^{-1}$ on the space-time coordinate $tx$ (which leaves the time component invariant and transforms the spatial component).

The Euclidean group $E(d)$ acts on the space of Schwartz maps $\mathcal{S}(\text{Time} \times \text{Space } d, F)$ (where $F$ is a normed space) via a pullback action. For any group element $g \in E(d)$ and any Schwartz map $\eta$, the action results in a new Schwartz map defined by $(g \cdot \eta)(tx) = \eta(g^{-1} \cdot tx)$ for all $tx \in \text{Time} \times \text{Space } d$. The group transformation $g^{-1}$ acts on the spatial components of the spacetime coordinates while the time component remains invariant.

For any dimension $d \in \mathbb{N}$, let $g$ be an element of the Euclidean group $E(d)$, $\eta$ be a Schwartz map in $\mathcal{S}(\text{Time} \times \text{Space } d, F)$, and $tx$ be a point in the spacetime $\text{Time} \times \text{Space } d$. The evaluation of the group action of $g$ on $\eta$ at the point $tx$ is given by the pointwise formula: $$(g \cdot \eta)(tx) = \eta(g^{-1} \cdot tx)$$ where $g^{-1} \cdot tx$ denotes the action of the inverse element $g^{-1}$ on the spacetime coordinate $tx$ (which transforms the spatial part and leaves the time part invariant).

For any dimension $d \in \mathbb{N}$, given elements $g$ and $h$ of the Euclidean group $E(d)$ and a Schwartz map $\eta \in \mathcal{S}(\text{TimeAndSpace } d, F)$, applying the Euclidean action of $g$ and then the Euclidean action of $h$ is equivalent to applying the pullback action of the product $h \cdot g$: $$\text{schwartzEuclideanAction}(h, \text{schwartzEuclideanAction}(g, \eta)) = \text{schwartzEuclideanAction}(h \cdot g, \eta)$$ This reflects that the continuous representation of the Euclidean group on the space of Schwartz maps follows the group multiplication law.

Let $E(d)$ denote the Euclidean group of dimension $d$. For any element $g \in E(d)$, the pullback action on the space of Schwartz maps $\mathcal{S}(\text{Time} \times \text{Space } d, F)$, defined by $(g \cdot \eta)(tx) = \eta(g^{-1} \cdot tx)$, is injective. That is, for any $g \in E(d)$, the map $\eta \mapsto g \cdot \eta$ is an injective function from $\mathcal{S}(\text{Time} \times \text{Space } d, F)$ to itself.

Let $d$ be a natural number and $g$ be an element of the Euclidean group $E(d)$. For the space of Schwartz maps $\mathcal{S}(\text{Time} \times \text{Space } d, F)$ mapping from space-time to a normed space $F$, the pullback action of $g$ on a Schwartz map $\eta$, defined by $(g \cdot \eta)(tx) = \eta(g^{-1} \cdot tx)$, is surjective.