Physlib.Relativity.LorentzGroup.Basic

For a given natural number $d$, the Lorentz group is the set of real matrices $\Lambda$ of dimension $(1+d) \times (1+d)$ that satisfy the condition $\Lambda \Lambda^* = I$, where $I$ is the identity matrix and $\Lambda^* = \eta \Lambda^T \eta$ is the Minkowski dual of $\Lambda$, defined using the Minkowski metric $\eta = \text{diag}(1, -1, \dots, -1)$.

The symbol $\mathcal{L}$ is defined as the mathematical notation for the Lorentz group, which consists of real matrices of size $(1+d) \times (1+d)$ that preserve the Minkowski metric.

For a given natural number $d$ and a real matrix $\Lambda$ of dimension $(1+d) \times (1+d)$, $\Lambda$ belongs to the Lorentz group $\mathcal{L}$ if and only if $\Lambda \Lambda^* = I$, where $\Lambda^*$ is the Minkowski dual of $\Lambda$ defined as $\eta \Lambda^T \eta$, and $I$ is the identity matrix.

For a given natural number $d$ and a real matrix $\Lambda$ of dimension $(1+d) \times (1+d)$, $\Lambda$ belongs to the Lorentz group $\mathcal{L}$ if and only if $\Lambda^* \Lambda = I$, where $\Lambda^*$ is the Minkowski dual of $\Lambda$ (defined as $\eta \Lambda^T \eta$ using the Minkowski metric $\eta$) and $I$ is the identity matrix.

For any natural number $d$ and any real matrix $\Lambda$ of dimension $(1+d) \times (1+d)$, $\Lambda$ belongs to the Lorentz group $\mathcal{L}$ if and only if its transpose $\Lambda^T$ belongs to the Lorentz group.

For any real matrix $\Lambda$ of dimension $(1+d) \times (1+d)$, $\Lambda$ belongs to the Lorentz group $\mathcal{L}$ if and only if its negation $-\Lambda$ belongs to the Lorentz group.

For any two matrices $\Lambda$ and $\Lambda'$ that are members of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, their matrix product $\Lambda \Lambda'$ is also a member of the Lorentz group. The Lorentz group is defined as the set of real $(1+d) \times (1+d)$ matrices $\Lambda$ satisfying $\Lambda \Lambda^* = I$, where $\Lambda^* = \eta \Lambda^T \eta$ is the Minkowski dual and $\eta$ is the Minkowski metric $\text{diag}(1, -1, \dots, -1)$.

For a given natural number $d$, the identity matrix $I$ of dimension $(1+d) \times (1+d)$ is an element of the Lorentz group $\mathcal{L}$.

For any matrix $\Lambda$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, its Minkowski dual $\Lambda^* = \eta \Lambda^T \eta$ is also an element of the Lorentz group $\mathcal{L}$.

For a given natural number $d$ representing spatial dimensions, a real matrix $\Lambda$ of dimension $(1+d) \times (1+d)$ belongs to the Lorentz group $\mathcal{L}$ if and only if it satisfies the condition $$\Lambda^T \eta \Lambda = \eta$$ where $\eta$ is the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$ and $\Lambda^T$ denotes the transpose of $\Lambda$.

For a given natural number $d$ representing spatial dimensions, the Lorentz group $\mathcal{L}$ (the set of real matrices $\Lambda$ of dimension $(1+d) \times (1+d)$ satisfying $\Lambda^T \eta \Lambda = \eta$) forms a group under matrix multiplication. The identity element of the group is the identity matrix $I$, and the inverse of an element $\Lambda \in \mathcal{L}$ is given by its Minkowski dual $\Lambda^* = \eta \Lambda^T \eta$, where $\eta = \text{diag}(1, -1, \dots, -1)$.

The Lorentz group $\mathcal{L}$ for $d$ spatial dimensions is equipped with a topological space structure. This topology is the subspace topology (subtype topology) inherited from the space of all $(1+d) \times (1+d)$ real matrices $\text{M}_{1+d}(\mathbb{R})$.

For any element $\Lambda$ of the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the group inverse $\Lambda^{-1}$ is equal to its Minkowski dual $\Lambda^*$, which is defined as $\Lambda^* = \eta \Lambda^T \eta$, where $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric.

For any element $\Lambda$ of the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the underlying matrix of the group inverse $\Lambda^{-1}$ is equal to the matrix inverse of $\Lambda$.

For any element $\Lambda$ of the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the underlying $(1+d) \times (1+d)$ real matrix is invertible.

For any element $\Lambda$ of the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the product of its matrix inverse $\Lambda^{-1}$ and the matrix $\Lambda$ is the identity matrix $I$, that is: $$\Lambda^{-1} \Lambda = I$$

For any element $\Lambda$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, the product of its matrix representation and its inverse matrix $\Lambda^{-1}$ is the identity matrix $I$, i.e., $\Lambda \Lambda^{-1} = I$.

Let $d$ be a natural number and let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions. Then the matrix representation of $\Lambda$ satisfies the identity $$\Lambda \eta \Lambda^T = \eta$$ where $\eta$ is the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$ and $\Lambda^T$ denotes the transpose of $\Lambda$.

Let $d$ be a natural number and let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions. Then the matrix representation of $\Lambda$ satisfies the identity $$\Lambda^T \eta \Lambda = \eta$$ where $\eta$ is the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$ and $\Lambda^T$ denotes the transpose of $\Lambda$.

For an element $\Lambda$ in the Lorentz group $\text{LorentzGroup } d$ (the set of real $(1+d) \times (1+d)$ matrices preserving the Minkowski metric), this function returns its matrix transpose $\Lambda^T$, which is also an element of the Lorentz group.

Let $I$ be the identity element of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions. The transpose of the identity element is equal to the identity element itself, i.e., $I^T = I$.

For any two Lorentz transformations $\Lambda$ and $\Lambda'$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, the transpose of their product is equal to the product of their transposes in reverse order: $$(\Lambda \Lambda')^T = \Lambda'^T \Lambda^T$$ where the product $\Lambda \Lambda'$ is the group multiplication (matrix multiplication) and $T$ denotes the matrix transpose.

For any element $\Lambda$ of the Lorentz group for $d$ spatial dimensions, the underlying matrix of the transpose of $\Lambda$ is equal to the transpose of the matrix representation of $\Lambda$. That is, $(\Lambda^T)_{ij} = \Lambda_{ji}$.

For any element $\Lambda$ of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, the group inverse of its transpose is equal to the transpose of its group inverse: $$(\Lambda^T)^{-1} = (\Lambda^{-1})^T$$ where $\Lambda^T$ denotes the matrix transpose and $\Lambda^{-1}$ denotes the group inverse within the Lorentz group.

For any element $\Lambda$ of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, let $\eta$ be the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$. Then the matrix representation of $\Lambda$ satisfies the identity: $$\Lambda \eta = \eta (\Lambda^{-1})^T$$ where $\Lambda^{-1}$ denotes the group inverse of $\Lambda$ and $(\cdot)^T$ denotes the matrix transpose.

Let $d$ be a natural number and let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions. Let $\eta$ be the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$. Then the following identity holds: $$\eta \Lambda = (\Lambda^{-1})^T \eta$$ where $\Lambda$ denotes the matrix representation of the Lorentz transformation, $\Lambda^{-1}$ is its group inverse, and $(\Lambda^{-1})^T$ is the transpose of that inverse.

For any element $\Lambda$ in the Lorentz group $\mathcal{L}$ of $d$ spatial dimensions, its negation $-\Lambda$ is defined as the entry-wise negation of the matrix $\Lambda$.

Let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions. The matrix representation of the negation $-\Lambda$ is equal to the entry-wise negation of the matrix representation of $\Lambda$.

Let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions. The group inverse of the negation of $\Lambda$ is equal to the negation of the group inverse of $\Lambda$, i.e., $(-\Lambda)^{-1} = -\Lambda^{-1}$.

For a given natural number $d$ representing the number of spatial dimensions, this is the group homomorphism that maps an element $\Lambda$ of the Lorentz group $\mathcal{L}$ to the general linear group $GL(1+d, \mathbb{R})$. It identifies each Lorentz matrix as an invertible real matrix of dimension $(1+d) \times (1+d)$.

For a given natural number $d$ representing the number of spatial dimensions, the group homomorphism $\text{toGL} : \mathcal{L} \to GL(1+d, \mathbb{R})$, which maps an element of the Lorentz group to its corresponding matrix in the general linear group, is injective.

For a given natural number $d$ representing the number of spatial dimensions, this is the group homomorphism from the Lorentz group $\mathcal{L}$ to the product of the monoid of $(1+d) \times (1+d)$ real matrices and its opposite monoid, denoted $M_{1+d}(\mathbb{R}) \times M_{1+d}(\mathbb{R})^{\text{op}}$. For any Lorentz transformation $\Lambda \in \mathcal{L}$, the map is defined as $\Lambda \mapsto (\Lambda, \Lambda^*)$, where $\Lambda^* = \eta \Lambda^T \eta$ is the Minkowski dual (which coincides with the inverse $\Lambda^{-1}$ for elements of the Lorentz group) and $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric.

Let $\mathcal{L}$ be the Lorentz group for $d$ spatial dimensions, consisting of real matrices of dimension $(1+d) \times (1+d)$. For any Lorentz transformation $\Lambda \in \mathcal{L}$, the value of the group homomorphism $\text{toProd} : \mathcal{L} \to M_{1+d}(\mathbb{R}) \times M_{1+d}(\mathbb{R})^{\text{op}}$ is given by the pair $(\Lambda, \Lambda^*)$, where $\Lambda^*$ is the Minkowski dual of $\Lambda$ defined by $\Lambda^* = \eta \Lambda^T \eta$ (and viewed as an element of the opposite monoid $M_{1+d}(\mathbb{R})^{\text{op}}$), where $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric.

For the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the group homomorphism $\text{toProd} : \mathcal{L} \to M_{1+d}(\mathbb{R}) \times M_{1+d}(\mathbb{R})^{\text{op}}$ defined by $\Lambda \mapsto (\Lambda, \Lambda^*)$ is injective, where $\Lambda^* = \eta \Lambda^T \eta$ is the Minkowski dual and $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric.

For the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the map $\Lambda \mapsto (\Lambda, \Lambda^*)$ from $\mathcal{L}$ to the product space $M_{1+d}(\mathbb{R}) \times M_{1+d}(\mathbb{R})^{\text{op}}$ is continuous, where $\Lambda^* = \eta \Lambda^T \eta$ is the Minkowski dual and $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric.

For the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions, the map $\Lambda \mapsto (\Lambda, \Lambda^*)$ from $\mathcal{L}$ to the product space $M_{1+d}(\mathbb{R}) \times M_{1+d}(\mathbb{R})^{\text{op}}$ is a topological embedding. Here, $M_{1+d}(\mathbb{R})$ is the space of $(1+d) \times (1+d)$ real matrices, $M_{1+d}(\mathbb{R})^{\text{op}}$ is its opposite monoid, $\Lambda^* = \eta \Lambda^T \eta$ is the Minkowski dual, and $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric.

For a given natural number $d$ representing the number of spatial dimensions, the natural map $\text{toGL} : \mathcal{L} \to GL(1+d, \mathbb{R})$, which identifies each element of the Lorentz group $\mathcal{L}$ as an invertible $(1+d) \times (1+d)$ real matrix, is a topological embedding. Here, the Lorentz group $\mathcal{L}$ is equipped with the subspace topology inherited from the space of all real matrices $M_{1+d}(\mathbb{R})$.

For a given natural number $d$ representing the number of spatial dimensions, the Lorentz group $\mathcal{L}$ (the group of $(1+d) \times (1+d)$ real matrices that preserve the Minkowski metric $\eta$) is a topological group. This structure is defined using the subspace topology inherited from the space of all real matrices $M_{1+d}(\mathbb{R})$, under which the group operations of matrix multiplication and inversion are continuous.

For a given natural number $d$ representing spatial dimensions, this definition provides a monoid homomorphism from the Lorentz group $\mathcal{L}$ to the set of $(1+d) \times (1+d)$ complex matrices. Specifically, it maps each real matrix $\Lambda \in \mathcal{L}$ to a complex matrix by applying the canonical embedding of real numbers into complex numbers, $\mathbb{R} \hookrightarrow \mathbb{C}$, to each entry of the matrix.

For any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, the $(1+d) \times (1+d)$ complex matrix obtained by mapping the real entries of $\Lambda$ to the complex field $\mathbb{C}$ is invertible. The inverse of this matrix is the image of the group inverse $\Lambda^{-1}$ under the same mapping.

For any element $\Lambda$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, the matrix inverse of its complex representation is equal to the complex representation of its group inverse. That is, $$(\text{toComplex } \Lambda)^{-1} = \text{toComplex } (\Lambda^{-1})$$ where $\text{toComplex}$ is the monoid homomorphism that embeds real matrices into the space of complex matrices.

Let $d$ be a natural number representing the number of spatial dimensions. For any element $\Lambda$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, its representation as a complex matrix satisfies the identity $$\Lambda \eta \Lambda^T = \eta$$ where $\eta$ is the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$ with entries embedded into the complex numbers $\mathbb{C}$, and $\Lambda^T$ denotes the transpose of the matrix.

For a given natural number $d$ and an element $\Lambda$ of the Lorentz group $\mathcal{L}$ (the group of real $(1+d) \times (1+d)$ matrices preserving the Minkowski metric), let $\Lambda_\mathbb{C}$ be the complex matrix obtained by the canonical embedding of the real entries of $\Lambda$ into $\mathbb{C}$. Then the following identity holds: $$\Lambda_\mathbb{C}^T \eta_\mathbb{C} \Lambda_\mathbb{C} = \eta_\mathbb{C}$$ where $\eta_\mathbb{C}$ is the $(1+d) \times (1+d)$ Minkowski matrix $\text{diag}(1, -1, \dots, -1)$ with entries considered as complex numbers, and $\Lambda_\mathbb{C}^T$ denotes the transpose of $\Lambda_\mathbb{C}$.

For any natural number $d$ representing the number of spatial dimensions, any element $\Lambda$ of the Lorentz group $\mathcal{L}$, and any real vector $v \in \mathbb{R}^{1+d}$, the following identity holds: $$\Lambda_{\mathbb{C}} v_{\mathbb{C}} = (\Lambda v)_{\mathbb{C}}$$ where $\Lambda_{\mathbb{C}}$ denotes the complex matrix obtained by embedding the real entries of $\Lambda$ into $\mathbb{C}$, and $v_{\mathbb{C}}$ denotes the complex vector obtained by embedding the real entries of $v$ into $\mathbb{C}$.

The parity transformation is the element of the Lorentz group $\mathcal{L}$ in $d$ spatial dimensions represented by the Minkowski matrix $\eta = \text{diag}(1, -1, \dots, -1)$. It acts on spacetime coordinates by preserving the time component and inverting the spatial components.