Physics notation $\langle X \rangle$ for expected value

Question

What's the motivation behind the notation $\langle X \rangle$ often used in physics for the expected value of the random variable $X$? The angled brackets tend to point me towards quantum mechanics and are reminiscent of $\langle \hat A \rangle_\psi = \langle \psi | \hat A | \psi \rangle$, but as far as I can tell, the analogy ends there. Are there other (more compelling) motivations?

Answer 1

See this blog post for an extended answer. Here is a brief summary:

Let's say that a random C*-algebra is a pair consisting of a C*-algebra $A$ and a state on $A$, which is a linear functional $\langle - \rangle : A \to \mathbb{C}$ such that

1. $\langle a^{\dagger} \rangle = \overline{ \langle a \rangle }$
2. $\langle 1 \rangle = 1$, and
3. $\langle a^{\dagger} a \rangle \ge 0$ for all $a \in A$.

The Riesz-Markov theorem implies that if $A$ is commutative, then not only (by Gelfand-Naimark) is $A = C(X)$ the algebra of functions on a compact Hausdorff space, but $\langle - \rangle$ is given by taking expected values with respect to a Radon probability measure on $X$.

If $A$ is noncommutative then this is a setup for doing "noncommutative" or "quantum" probability. A typical example here is that $A = B(H)$ is the C*-algebra of bounded operators on a Hilbert space $H$, in which case we can take $\langle A \rangle = \langle \psi, A \psi \rangle$ for any unit vector $\psi \in H$. Conversely, given a random C*-algebra $A$ we can write down a (semi)-inner product on $A$ given by

$$\langle a, b \rangle = \langle a^{\dagger} b \rangle$$

and this begins the GNS construction, which in the end produces a Hilbert space on which $A$ acts and a vector in this Hilbert space reproducing the state on $A$.