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## Finite noncommutative probability, the Born rule, and wave function collapse

The previous post on noncommutative probability was too long to leave much room for examples of random algebras. In this post we will describe all finite-dimensional random algebras with faithful states and all states on them. This will lead, in particular, to a derivation of the Born rule from statistical mechanics. We will then give a mathematical description of wave function collapse as taking a conditional expectation.

## A little more about zeta functions and statistical mechanics

In the previous post we described the following result characterizing the zeta distribution.

Theorem: Let $a_n = \mathbb{P}(X = n)$ be a probability distribution on $\mathbb{N}$. Suppose that the exponents in the prime factorization of $n$ are chosen independently and according to a geometric distribution, and further suppose that $a_n$ is monotonically decreasing. Then $a_n = \frac{1}{\zeta(s)} \left( \frac{1}{n^s} \right)$ for some real $s > 1$.

I have been thinking about the first condition, and I no longer like it. At least, I don’t like how I arrived at it. Here is a better way to conceptualize it: given that $n | X$, the probability distribution on $\frac{X}{n}$ should be the same as the original distribution on $X$. By Bayes’ theorem, this is equivalent to the condition that

$\displaystyle \frac{a_{mn}}{a_n + a_{2n} + a_{3n} + ...} = \frac{a_m}{a_1 + a_2 + ...}$

which in turn is equivalent to the condition that

$\displaystyle \frac{a_{mn}}{a_m} = \frac{a_n + a_{2n} + a_{3n} + ...}{a_1 + a_2 + a_3 + ...}$.

(I am adopting the natural assumption that $a_n > 0$ for all $n$. No sense in excluding a positive integer from any reasonable probability distribution on $\mathbb{N}$.) In other words, $\frac{a_{mn}}{a_m}$ is independent of $m$, from which it follows that $a_{mn} = c a_m a_n$ for some constant $c$. From here it already follows that $a_n$ is determined by $a_p$ for $p$ prime and that the exponents in the prime factorization are chosen geometrically. And now the condition that $a_n$ is monotonically decreasing gives the zeta distribution as before. So I think we should use the following characterization theorem instead.

Theorem: Let $a_n = \mathbb{P}(X = n)$ be a probability distribution on $\mathbb{N}$. Suppose that $a_{nm} = c a_n a_m$ for all $n, m \ge 1$ and some $c$, and further suppose that $a_n$ is monotonically decreasing. Then $a_n = \frac{1}{\zeta(s)} \left( \frac{1}{n^s} \right)$ for some real $s > 1$.

More generally, the following situation covers all the examples we have used so far. Let $M$ be a free commutative monoid on generators $p_1, p_2, ...$, and let $\phi : M \to \mathbb{R}$ be a homomorphism. Let $a_m = \mathbb{P}(X = m)$ be a probability distribution on $M$. Suppose that $a_{nm} = c a_n a_m$ for all $n, m \in M$ and some $c$, and further suppose that if $\phi(n) \ge \phi(m)$ then $a_n \le a_m$. Then $a_m = \frac{1}{\zeta_M(s)} e^{-\phi(m) s}$ for some $s$ such that the zeta function

$\displaystyle \zeta_M(s) = \sum_{m \in M} e^{-\phi(m) s}$

converges. Moreover, $\zeta_M(s)$ has the Euler product

$\displaystyle \zeta_M(s) = \prod_{i=1}^{\infty} \frac{1}{1 - e^{- \phi(p_i) s}}$.

Recall that in the statistical-mechanical interpretation, we are looking at a system whose states are finite collections of particles of types $p_1, p_2, ...$ and whose energies are given by $\phi(p_i)$; then the above is just the partition function. In the special case of the zeta function of a Dedekind abstract number ring, $M = M_R$ is the commutative monoid of nonzero ideals of $R$ under multiplication, which is free on the prime ideals by unique factorization, and $\phi(I) = \log N(I)$. In the special case of the dynamical zeta function of an invertible map $f : X \to X$, $M = M_X$ is the free commutative monoid on orbits of $f$ (equivalently, the invariant submonoid of the free commutative monoid on $X$), and $\phi(P) = \log |P|$, where $|P|$ is the number of points in $P$.

An interesting result that demonstrates, among other things, the ubiquity of $\pi$ in mathematics is that the probability that two random positive integers are relatively prime is $\frac{6}{\pi^2}$. A more revealing way to write this number is $\frac{1}{\zeta(2)}$, where
$\displaystyle \zeta(s) = \sum_{n \ge 1} \frac{1}{n^s}$
is the Riemann zeta function. A few weeks ago this result came up on math.SE in the following form: if you are standing at the origin in $\mathbb{R}^2$ and there is an infinitely thin tree placed at every integer lattice point, then $\frac{6}{\pi^2}$ is the proportion of the lattice points that you can see. In this post I’d like to explain why this “should” be true. This will give me a chance to blog about some material from another math.SE answer of mine which I’ve been meaning to get to, and along the way we’ll reach several other interesting destinations.