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

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?

• Similar question/answer here. Of which it says it's probably a coincidence. But maybe a year later there's some more answers? Sep 30, 2016 at 14:38
• But also as an added note, $\langle v \rangle$ was/is used for average or mean speed in statistical mechanics. That would also be an expected value but I'm not sure what notation Boltzmann would've been using for example. Sep 30, 2016 at 14:46

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 $A$.