PhyslibAlpha.AlgebraicFramework.HilbertSpace.Dynamics.Automorphism

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ (denoted in the formal text as $H \to L[\mathbb{C}] H$) be the algebra of bounded linear operators on $H$. Every $\ast$-automorphism of $\mathcal{B}(H)$ is implemented by unitary conjugation. That is, for any $\ast$-automorphism $\Phi : \mathcal{B}(H) \to \mathcal{B}(H)$, there exists a unitary operator $U$ on $H$ such that $\Phi(A) = U A U^*$ for all $A \in \mathcal{B}(H)$.

Let $H$ be a complex Hilbert space and $B(H)$ be the algebra of bounded linear operators on $H$. For any two unitary operators $u, v \in \mathcal{U}(H)$, the $\ast$-automorphisms of $B(H)$ induced by conjugation (i.e., $A \mapsto uAu^\ast$ and $A \mapsto vAv^\ast$) are equal if and only if there exists a scalar phase $c \in \mathbb{C}$ with $|c| = 1$ such that $u = cv$.

For a complex Hilbert space $H$, let $U(H)$ be the group of unitary bounded linear operators on $H$. The projective unitary group $PU(H)$ is the quotient group $U(H) / \ker(\text{Ad})$, where $\text{Ad}$ is the homomorphism mapping a unitary $U$ to the $\star$-automorphism of the algebra of bounded operators $B(H)$ given by $A \mapsto U A U^*$. This quotient effectively removes the scalar phases from the unitary group.

Let $H$ be a complex Hilbert space. The projective unitary group $PU(H)$ is equipped with a group structure, which is defined as the quotient group of the group of unitary bounded linear operators $\mathcal{U}(H)$ by the subgroup of scalar phases (the kernel of the conjugation action on bounded operators).

For a complex Hilbert space $H$, let $\mathcal{B}(H)$ (denoted as $H \to L[\mathbb{C}] H$) be the algebra of bounded linear operators. The projective unitary group $PU(H)$ is isomorphic as a group to the group of $\star$-automorphisms of $\mathcal{B}(H)$, denoted as $\text{Aut}_{\star}(\mathcal{B}(H))$. This isomorphism realizes the fact that every $\star$-automorphism of the algebra of bounded operators is implemented by conjugation by a unitary operator, and two unitaries implement the same automorphism if and only if they differ by a scalar phase.

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ (denoted `H →L[ℂ] H`) be the algebra of bounded linear operators on $H$. Given a unitary one-parameter group $U(t)$ on $H$, this definition constructs the induced one-parameter group of $\star$-automorphisms of $\mathcal{B}(H)$. For each $t \in \mathbb{R}$, the automorphism maps an operator $a \in \mathcal{B}(H)$ to $U(t) a U(t)^*$, representing the reversible dynamics induced by $U$.

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ be the algebra of bounded linear operators on $H$. For a unitary one-parameter group $U(t)$ and its induced one-parameter group of $\star$-automorphisms $\alpha_t$, the action of the automorphism at time $t \in \mathbb{R}$ on an operator $a \in \mathcal{B}(H)$ is given by $$\alpha_t(a) = U(t) a U(t)^*,$$ where $U(t)^*$ denotes the adjoint (star) of the unitary operator $U(t)$.

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ be the algebra of bounded linear operators on $H$. For a norm-continuous unitary one-parameter group $U(t)$ with generator $G$, let $\alpha_t: \mathcal{B}(H) \to \mathcal{B}(H)$ be the induced one-parameter group of $\star$-automorphisms defined by $\alpha_t(a) = U(t) a U(t)^*$. Then the derivative of the map $t \mapsto \alpha_t(a)$ at any time $t \in \mathbb{R}$ is given by the Heisenberg-type equation: $$\frac{d}{dt} \alpha_t(a) = [\alpha_t(a), i G],$$ where $[A, B] = AB - BA$ denotes the standard commutator of operators.

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ be the algebra of bounded linear operators on $H$. Let $U: \mathbb{R} \to \mathcal{B}(H)$ be a norm-continuous unitary one-parameter group with generator $G \in \mathcal{B}(H)$ (defined such that $U(t) = e^{-itG}$). Suppose $\alpha: \mathbb{R} \to (\mathcal{B}(H) \to \mathcal{B}(H))$ is a family of maps satisfying the initial condition $\alpha_0 = \text{id}_{\mathcal{B}(H)}$ and the differential equation (Heisenberg initial-value problem) $$\frac{d}{dt} \alpha_t(a) = [\alpha_t(a), iG]$$ for all $a \in \mathcal{B}(H)$ and $t \in \mathbb{R}$, where $[x, y] = xy - yx$ denotes the ring commutator. Then $\alpha$ is the unique conjugation flow defined by $$\alpha_t(a) = U(t) a U(t)^*$$ for all $t \in \mathbb{R}$ and $a \in \mathcal{B}(H)$.