Physlib.QuantumMechanics.OperatorAlgebra.Observables.Lie

Let $A$ be a complex $C^*$-algebra. For any two observables $a, b$ (self-adjoint elements in $A$), the Lie bracket $\unicode{x2045}a, b\unicode{x2046}$ is defined as the imaginary part of their product in the algebra, $\text{Im}(ab)$. This is equivalent to the expression $\unicode{x2045}a, b\unicode{x2046} = -\frac{i}{2}(ab - ba)$, where $i$ is the imaginary unit.

Let $A$ be a complex $C^*$-algebra. For any two observables $a, b \in A$ (i.e., self-adjoint elements such that $a^* = a$ and $b^* = b$), the Lie bracket $\unicode{x2045}a, b\unicode{x2046}$ is equal to $-\frac{i}{2}(ab - ba)$, where $i$ is the imaginary unit.

Let $A$ be a complex $C^*$-algebra. For any observables (self-adjoint elements) $a, b, c \in A$, the Lie bracket $\unicode{x2045}\cdot, \cdot\unicode{x2046}$, defined as $\unicode{x2045}a, b\unicode{x2046} = \text{Im}(ab) = -\frac{i}{2}(ab - ba)$, satisfies the left-distributive property: $\unicode{x2045}a + b, c\unicode{x2046} = \unicode{x2045}a, c\unicode{x2046} + \unicode{x2045}b, c\unicode{x2046}$.

Let $A$ be a complex $C^*$-algebra. For any observables $a, b, c \in A$ (self-adjoint elements), the Lie bracket satisfies $\unicode{x2045}a, b + c\unicode{x2046} = \unicode{x2045}a, b\unicode{x2046} + \unicode{x2045}a, c\unicode{x2046}$. The bracket is defined as $\unicode{x2045}x, y\unicode{x2046} = -\frac{i}{2}(xy - yx)$, which corresponds to the imaginary part of the product of two observables.

Let $A$ be a complex $C^*$-algebra. For any observable $a \in A$ (a self-adjoint element such that $a^* = a$), the Lie bracket of the observable with itself is zero, i.e., $\unicode{x2045}a, a\unicode{x2046} = 0$. Here, the Lie bracket is defined as the imaginary part of the product, $\unicode{x2045}a, b\unicode{x2046} = -\frac{i}{2}(ab - ba)$.

Let $A$ be a complex $C^*$-algebra. For any observables $a, b, c \in A$ (self-adjoint elements), the Lie bracket $\unicode{x2045} \cdot, \cdot \unicode{x2046}$, defined as $\unicode{x2045}a, b\unicode{x2046} = -\frac{i}{2}(ab - ba)$, satisfies the Leibniz identity: \[ \unicode{x2045}a, \unicode{x2045}b, c\unicode{x2046}\unicode{x2046} = \unicode{x2045}\unicode{x2045}a, b\unicode{x2046}, c\unicode{x2046} + \unicode{x2045}b, \unicode{x2045}a, c\unicode{x2046}\unicode{x2046} \]

Let $A$ be a complex $C^*$-algebra. The set of observables in $A$, defined as the self-adjoint elements $a \in A$ such that $a^* = a$, forms a Lie ring under the Lie bracket defined by $\unicode{x2045}a, b\unicode{x2046} = -\frac{i}{2}(ab - ba)$. This bracket corresponds to the imaginary part of the product $ab$ and satisfies the following properties for all observables $a, b, c$: 1. Left and right distributivity over addition: $\unicode{x2045}a + b, c\unicode{x2046} = \unicode{x2045}a, c\unicode{x2046} + \unicode{x2045}b, c\unicode{x2046}$ and $\unicode{x2045}a, b + c\unicode{x2046} = \unicode{x2045}a, b\unicode{x2046} + \unicode{x2045}a, c\unicode{x2046}$. 2. The alternating property: $\unicode{x2045}a, a\unicode{x2046} = 0$. 3. The Leibniz identity: $\unicode{x2045}a, \unicode{x2045}b, c\unicode{x2046}\unicode{x2046} = \unicode{x2045}\unicode{x2045}a, b\unicode{x2046}, c\unicode{x2046} + \unicode{x2045}b, \unicode{x2045}a, c\unicode{x2046}\unicode{x2046}$.

Let $A$ be a complex $C^*$-algebra. For any real number $t \in \mathbb{R}$ and any observables $a, b \in A$ (where an observable is a self-adjoint element), the Lie bracket $\unicode{x2045}a, b\unicode{x2046} = -\frac{i}{2}(ab - ba)$ satisfies the identity $\unicode{x2045}a, t \cdot b\unicode{x2046} = t \cdot \unicode{x2045}a, b\unicode{x2046}$.

Let $A$ be a complex $C^*$-algebra. The space of observables in $A$, consisting of the self-adjoint elements $\{a \in A \mid a^* = a\}$, forms a Lie algebra over the real numbers $\mathbb{R}$. The Lie bracket is defined by $[a, b] = -\frac{i}{2}(ab - ba)$, which corresponds to the skew-symmetric (imaginary) part of the product of two observables.

Let $A$ be a unital complex $C^*$-algebra. For any observable $a \in A$ (i.e., a self-adjoint element $a = a^*$), the Lie bracket of $a$ with the identity element $1$ is zero, expressed as $\unicode{x2045}a, 1\unicode{x2046} = 0$.

Let $A$ be a complex $C^*$-algebra and let $1$ be its identity element. For any observable $a \in A$ (a self-adjoint element), the Lie bracket of $1$ and $a$ is zero, i.e., $\unicode{x2045}1, a\unicode{x2046} = 0$, where the bracket is defined as $\unicode{x2045}a, b\unicode{x2046} = -\frac{i}{2}(ab - ba)$.