PhyslibAlpha.QuantumMechanics.OperatorAlgebra.Measurement.SpectralMeasure

Given a $C^*$-algebra $A$, a state $\omega$ on $A$, and an observable $a \in A$ (a self-adjoint element), the spectral functional is the positive linear map from the space of continuous real-valued functions on the spectrum $\sigma(a)$, denoted $C(\sigma(a), \mathbb{R})$, to $\mathbb{R}$. For a function $f \in C(\sigma(a), \mathbb{R})$, the functional returns the real part of the state applied to the element $f(a)$ obtained via the continuous functional calculus: $f \mapsto \text{Re}(\omega(f(a)))$. Since $f$ is real-valued and $a$ is self-adjoint, $f(a)$ is self-adjoint, making the result real-valued.

Given a $C^*$-algebra $A$, a state $\omega$ on $A$, and an observable $a \in A$ (a self-adjoint element), this functional is the positive linear map from the space of continuous real-valued functions with compact support on the spectrum of $a$, denoted $C_c(\sigma(a), \mathbb{R})$, to $\mathbb{R}$. For a function $f \in C_c(\sigma(a), \mathbb{R})$, the functional maps $f$ to $\text{Re}(\omega(f(a)))$, where $f(a)$ is the element obtained via the continuous functional calculus. Since the spectrum $\sigma(a)$ is compact, this is the restriction of the spectral functional to the space of compactly supported functions, providing the necessary input for the Riesz–Markov–Kakutani representation theorem to produce a spectral measure.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element). The spectral measure associated with $\omega$ and $a$ is the unique Borel measure $\mu$ on the spectrum $\sigma(a) \subseteq \mathbb{R}$ such that for any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, the integral of $f$ with respect to $\mu$ satisfies: $$\int_{\sigma(a)} f \, d\mu = \text{Re}(\omega(f(a)))$$ where $f(a)$ is defined via the continuous functional calculus. This measure is obtained by applying the Riesz–Markov–Kakutani representation theorem to the positive linear functional $f \mapsto \text{Re}(\omega(f(a)))$ on the space of continuous functions with compact support $C_c(\sigma(a), \mathbb{R})$.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element). The spectral measure $\mu_{\omega, a}$ associated with $\omega$ and $a$—which is the unique Borel measure on the spectrum $\sigma(a) \subseteq \mathbb{R}$ such that $\int_{\sigma(a)} f \, d\mu_{\omega, a} = \text{Re}(\omega(f(a)))$ for any continuous real-valued function $f$—is a finite measure.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element). Let the spectral functional associated with $\omega$ and $a$ be the linear map $L: C(\sigma(a), \mathbb{R}) \to \mathbb{R}$ defined by $L(f) = \text{Re}(\omega(f(a)))$, where $f(a)$ is defined by the continuous functional calculus. Then $L(1) = 1$, where $1$ denotes the constant function on the spectrum $\sigma(a)$ that maps every element to 1.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element of $A$). The spectral measure $\mu_{\omega, a}$ associated with $\omega$ and $a$, which is a Borel measure defined on the spectrum $\sigma(a) \subseteq \mathbb{R}$, is a probability measure. That is, its total mass is 1, which corresponds to the normalization property of the state $\omega(1) = 1$.

Let $A$ be a unital $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element of $A$). Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$, and let $\mu_{\omega, a}$ be the spectral measure on $\sigma(a)$ associated with $\omega$ and $a$. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, the state applied to the element $f(a)$ (defined via the continuous functional calculus) satisfies: $$\omega(f(a)) = \int_{\sigma(a)} f(x) \, d\mu_{\omega, a}(x)$$ where the real-valued integral on the right is identified with its corresponding value in $\mathbb{C}$.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element). The **real spectral measure** $\mu_{\omega, a}$ is the probability measure on the real line $\mathbb{R}$ obtained by pushing forward the spectral measure (initially defined on the spectrum $\sigma(a) \subseteq \mathbb{R}$) along the inclusion map $\iota: \sigma(a) \hookrightarrow \mathbb{R}$. This measure is the unique probability measure on $\mathbb{R}$ supported on $\sigma(a)$ such that for any continuous real-valued function $f$, the integral of $f$ with respect to $\mu_{\omega, a}$ gives the expectation value: $$\int_{\mathbb{R}} f(x) \, d\mu_{\omega, a}(x) = \text{Re}(\omega(f(a)))$$ where $f(a)$ is determined by the continuous functional calculus.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a \in A$ an observable (a self-adjoint element). The real spectral measure $\mu_{\omega, a}$ associated with the state $\omega$ and observable $a$ is a probability measure on $\mathbb{R}$.

Let $A$ be a $C^*$-algebra, $\omega$ be a state on $A$, and $a$ be an observable. Let $\mu_{\omega, a}$ denote the real spectral measure on $\mathbb{R}$ and $\sigma(a) \subseteq \mathbb{R}$ be the spectrum of $a$. Then the measure of the complement of the spectrum $\sigma(a)$ with respect to $\mu_{\omega, a}$ is zero: $$\mu_{\omega, a}(\sigma(a)^c) = 0$$ In other words, the probability measure $\mu_{\omega, a}$ is concentrated on the spectrum of $a$.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable (a self-adjoint element of $A$). Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$ and $\mu_{\omega, a}$ denote the real spectral measure associated with the state $\omega$ and observable $a$. For any function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on the spectrum $\sigma(a)$, the expectation value of $f(a)$ (determined by the continuous functional calculus) is given by the integral of $f$ with respect to $\mu_{\omega, a}$: $$\omega(f(a)) = \int_{\mathbb{R}} f(x) \, d\mu_{\omega, a}(x)$$ where the integral on the right-hand side is a real number embedded into the complex numbers.

Let $a$ be an observable (a self-adjoint element) in a $C^*$-algebra $A$, and let $\sigma(a) \subseteq \mathbb{R}$ denote its spectrum. For any continuous function $g: \sigma(a) \to \mathbb{R}$ defined on the spectrum, there exists a function $f: \mathbb{R} \to \mathbb{R}$ such that $f$ is continuous on $\sigma(a)$ and $f(x) = g(x)$ for all $x \in \sigma(a)$.

Let $A$ be a $C^*$-algebra, $\omega$ a state on $A$, and $a$ an observable. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$ and let $\iota: \sigma(a) \to \mathbb{R}$ be the inclusion map. Suppose $\mu$ is a measure on $\mathbb{R}$ such that: 1. For every function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on $\sigma(a)$, the expectation of the observable $f(a)$ (defined via continuous functional calculus) satisfies $\omega(f(a)) = \int_{\mathbb{R}} f \, d\mu$. 2. The measure $\mu$ is the pushforward of its pullback to the spectrum, i.e., $\mu = \iota_*(\iota^* \mu)$, which implies $\mu$ is supported on $\sigma(a)$. Then, for any continuous function $g \in C(\sigma(a), \mathbb{R})$, the integral of $g$ with respect to the pulled-back measure $\iota^* \mu$ is equal to the integral of $g$ with respect to the spectral measure $\mu_{\omega, a}$: $$\int_{\sigma(a)} g \, d(\iota^* \mu) = \int_{\sigma(a)} g \, d\mu_{\omega, a}$$

Let $A$ be a $C^*$-algebra, $\omega$ be a state on $A$, and $a \in A$ be an observable (a self-adjoint element). Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. Suppose $\mu$ is a probability measure on $\mathbb{R}$ such that: 1. $\mu$ is concentrated on the spectrum of $a$, i.e., $\mu(\mathbb{R} \setminus \sigma(a)) = 0$. 2. For every function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on $\sigma(a)$, the identity $\omega(f(a)) = \int_{\mathbb{R}} f(x) \, d\mu(x)$ holds, where $f(a)$ is defined via the continuous functional calculus. Then $\mu$ is equal to the real spectral measure $\mu_{\omega, a}$.