Physlib.SpaceAndTime.SpaceTime.LorentzAction

For a given spatial dimension $d$, the Lorentz group $\mathcal{L}$ acts on the space of real-valued Schwartz functions $\mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ through a group homomorphism into the space of continuous linear maps. For any Lorentz transformation $\Lambda \in \mathcal{L}$ and any Schwartz function $\eta$, the action is defined by $(\Lambda \cdot \eta)(x) = \eta(\Lambda^{-1} x)$ for all $x \in \text{SpaceTime}_d$.

For any spatial dimension $d$, let $\Lambda_1, \Lambda_2$ be elements of the Lorentz group $\mathcal{L}$ and let $\eta \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ be a real-valued Schwartz function. The action of the Lorentz group on Schwartz functions satisfies the property that applying $\Lambda_2$ to the result of $\Lambda_1$ acting on $\eta$ is equal to the product $\Lambda_2 \Lambda_1$ acting on $\eta$: $$\Lambda_2 \cdot (\Lambda_1 \cdot \eta) = (\Lambda_2 \Lambda_1) \cdot \eta$$

For any spatial dimension $d$, let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ and let $\eta \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ be a real-valued Schwartz function. For any point $x \in \text{SpaceTime}_d$, the action of $\Lambda$ on $\eta$ is defined by the pointwise evaluation: $$(\Lambda \cdot \eta)(x) = \eta(\Lambda^{-1} x)$$ where $\Lambda^{-1}$ is the inverse of the Lorentz transformation $\Lambda$.

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, the action of $\Lambda$ on the space of real-valued Schwartz functions $\mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$, denoted as $\text{schwartzAction}(\Lambda)$, is an injective map. That is, if $\eta_1, \eta_2 \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ such that $\text{schwartzAction}(\Lambda)(\eta_1) = \text{schwartzAction}(\Lambda)(\eta_2)$, then $\eta_1 = \eta_2$.

For any spatial dimension $d$ and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, the action of $\Lambda$ on the space of real-valued Schwartz functions $\mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ is a surjective map. That is, for every Schwartz function $\zeta \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$, there exists a Schwartz function $\eta \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ such that $\Lambda \cdot \eta = \zeta$, where the action is defined by $(\Lambda \cdot \eta)(x) = \eta(\Lambda^{-1} x)$.

This definition establishes a group action (scalar multiplication) of the Lorentz group $\mathcal{L}$ on the space of distributions $f: \text{SpaceTime}_d \to_d[\mathbb{R}] M$, where $M$ is a space equipped with a tensorial structure. For any Lorentz transformation $\Lambda \in \mathcal{L}$ and distribution $f$, the action $\Lambda \cdot f$ is defined as the composition of continuous linear maps $\rho(\Lambda) \circ f \circ \pi(\Lambda^{-1})$. In this composition, $\pi(\Lambda^{-1})$ is the action of the inverse Lorentz transformation on the Schwartz space $\mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$, and $\rho(\Lambda)$ is the continuous linear map representing the Lorentz action on the target space $M$.

For a given natural number $d$ representing spatial dimensions, let $\mathcal{L}$ be the Lorentz group and $M$ be a normed space equipped with a tensorial structure. Let $f: \text{SpaceTime}_d \to_d[\mathbb{R}] M$ be an $M$-valued distribution (viewed as a continuous linear map from the Schwartz space $\mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$ to $M$). For any Lorentz transformation $\Lambda \in \mathcal{L}$ and any Schwartz function $\eta \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$, the induced action of $\Lambda$ on the distribution $f$ satisfies: $$(\Lambda \cdot f)(\eta) = \Lambda \cdot (f(\Lambda^{-1} \cdot \eta))$$ where $\Lambda^{-1} \cdot \eta$ is the action of the inverse Lorentz transformation on the Schwartz function, defined by $(\Lambda^{-1} \cdot \eta)(x) = \eta(\Lambda x)$, and the $\Lambda \cdot (\dots)$ on the right-hand side denotes the Lorentz action on the target space $M$.

For a natural number $d$ representing spatial dimensions and a normed real vector space $M$ equipped with a tensorial structure, the Lorentz group $\mathcal{L}$ acts distributively on the space of $M$-valued distributions on $\text{SpaceTime}_d$, denoted by $\text{SpaceTime}_d \to_d[\mathbb{R}] M$. This structure ensures that for any Lorentz transformations $\Lambda, \Lambda_1, \Lambda_2 \in \mathcal{L}$ and distributions $f, f_1, f_2$, the action satisfies: 1. $I \cdot f = f$, where $I$ is the identity transformation. 2. $(\Lambda_1 \Lambda_2) \cdot f = \Lambda_1 \cdot (\Lambda_2 \cdot f)$. 3. $\Lambda \cdot (f_1 + f_2) = \Lambda \cdot f_1 + \Lambda \cdot f_2$. 4. $\Lambda \cdot 0 = 0$.

For a given spatial dimension $d$, let $\mathcal{L}$ be the Lorentz group and let $M$ be a real normed space with a tensorial structure. For any scalar $a \in \mathbb{R}$, Lorentz transformation $\Lambda \in \mathcal{L}$, and $M$-valued distribution $f : \text{SpaceTime}_d \to_d[\mathbb{R}] M$, the action of the Lorentz group commutes with the real scalar multiplication, such that $a \cdot (\Lambda \cdot f) = \Lambda \cdot (a \cdot f)$.

For a Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$ of $d$ spatial dimensions, this definition provides the linear map over $\mathbb{R}$ that acts on the space of distributions $f: \text{SpaceTime}_d \to_d[\mathbb{R}] M$, where $M$ is a normed vector space with a tensorial structure. This linear map sends each distribution $f$ to its transformed version $\Lambda \cdot f$ under the group action of the Lorentz group on the space of distributions.