Physlib.SpaceAndTime.SpaceTime.SpaceTimeAlgebra.Star

The star operation (involution) on the spacetime algebra $\text{SpaceTimeAlgebra} \cong \mathbb{C}[[x^\mu]]$ is defined by applying complex conjugation to each coefficient of the formal power series. For a power series $f = \sum a_n x^n$, its conjugate $\text{star } f$ (or $f^*$) is given by $\sum \bar{a}_n x^n$, where $\bar{a}_n$ denotes the complex conjugate of the coefficient $a_n$.

In the spacetime algebra $\mathbb{C}[[x^\mu]]$, for any formal power series $f$ and any multi-index $n \in \mathbb{N}^4$ (representing the exponents of the variables $x^\mu$), the coefficient of the monomial $x^n$ in the conjugate series $f^*$ is equal to the complex conjugate of the coefficient of $x^n$ in $f$. That is, $\text{coeff}_n(f^*) = \overline{\text{coeff}_n(f)}$.

For any formal power series $f$ in the spacetime algebra $\text{SpaceTimeAlgebra} \cong \mathbb{C}[[x^\mu]]$, the constant coefficient of the conjugate series $\text{star } f$ is equal to the complex conjugate (star) of the constant coefficient of $f$. That is, $\text{constantCoeff}(\text{star } f) = \text{star}(\text{constantCoeff } f)$.

The spacetime algebra $\text{SpaceTimeAlgebra} \cong \mathbb{C}[[x^\mu]]$ is a star ring where the star operation $f \mapsto f^*$ is defined by applying complex conjugation to each coefficient of the formal power series. This structure satisfies the following properties for all $f, g \in \text{SpaceTimeAlgebra}$: 1. Additivity: $(f + g)^* = f^* + g^*$ 2. Conjugate-multiplicativity: $(fg)^* = g^* f^*$ 3. Involution: $(f^*)^* = f$

In the spacetime algebra, let $C: \mathbb{C} \to \text{SpaceTimeAlgebra}$ be the map that identifies a complex number $a$ with a constant formal power series. For any complex number $a \in \mathbb{C}$, the star operation (complex conjugation of coefficients) applied to the constant series $C(a)$ is equal to the constant series formed by the complex conjugate of $a$. That is, $\text{star}(C(a)) = C(\text{star}(a))$, where $\text{star}(a)$ denotes the complex conjugate $\bar{a}$.

In the spacetime algebra, for any coordinate index $\mu \in \text{Fin } 1 \oplus \text{Fin } 3$, the star operation (complex conjugation) fixes the coordinate variable $x^\mu$. That is, $(x^\mu)^* = x^\mu$.

The spacetime algebra $\mathbb{C}[[x^\mu]]$ (the ring of formal power series in four variables with complex coefficients) is a star module over the real numbers $\mathbb{R}$. This means that the star operation (defined as coefficient-wise complex conjugation) commutes with multiplication by real scalars: for any $r \in \mathbb{R}$ and any $f \in \mathbb{C}[[x^\mu]]$, the conjugate of the scaled series is equal to the scalar times the conjugate of the series, i.e., $(r \cdot f)^* = r \cdot f^*$.

For any complex scalar $c \in \mathbb{C}$ and any element $f$ of the spacetime algebra (represented as a formal power series $f \in \mathbb{C}[[x^\mu]]$), the star operation (complex conjugation) satisfies the identity $(c \cdot f)^* = \bar{c} \cdot f^*$, where $\bar{c}$ denotes the complex conjugate of $c$ and $f^*$ is the series obtained by conjugating each coefficient of $f$.

In the spacetime algebra $\mathbb{C}[[x^\mu]]$, let $f^*$ denote the complex conjugate of a formal power series $f$ (obtained by conjugating each of its coefficients). For any coordinate index $\mu \in \text{Fin } 1 \oplus \text{Fin } 3$, the formal partial derivative with respect to $x^\mu$ commutes with the star operation: $$\frac{\partial f^*}{\partial x^\mu} = \left( \frac{\partial f}{\partial x^\mu} \right)^*$$

In the spacetime algebra $\text{SpaceTimeAlgebra} \cong \mathbb{C}[[x^\mu]]$, let $f^*$ denote the complex conjugate of a formal power series $f$ (the series obtained by conjugating each of its coefficients). For any multiset $s$ of coordinate indices $\mu \in \text{Fin } 1 \oplus \text{Fin } 3$, the iterated formal partial derivative $\partial^s$ commutes with the star operation: $$\partial^s (f^*) = (\partial^s f)^*$$