Primon gas

Consider a Hilbert space H with an orthonormal basis of states ${\displaystyle |p\rangle }$ labelled by the prime numbers p. Second quantization gives a new Hilbert space K, the bosonic Fock space on H, where states describe collections of primes - which we can call primons if we think of them as analogous to particles in quantum field theory. This Fock space has an orthonormal basis given by finite multisets of primes. In other words, to specify one of these basis elements we can list the number ${\displaystyle k_{p}=0,1,2,\dots }$ of primons for each prime ${\displaystyle p}$:

${\displaystyle |k_{2},k_{3},k_{5},k_{7},k_{11},\ldots ,k_{p},\ldots \rangle }$

where the total ${\displaystyle \sum _{p}k_{p}}$ is finite. Since any positive natural number ${\displaystyle n}$ has a unique factorization into primes:

${\displaystyle n=2^{k_{2}}\cdot 3^{k_{3}}\cdot 5^{k_{5}}\cdot 7^{k_{7}}\cdot 11^{k_{11}}\cdots p^{k_{p}}\cdots }$

we can also denote the basis elements of the Fock space as simply ${\displaystyle |n\rangle }$ where ${\displaystyle n=1,2,3,\dots .}$

In short, the Fock space for primons has an orthonormal basis given by the positive natural numbers, but we think of each such number ${\displaystyle n}$ as a collection of primons: its prime factors, counted with multiplicity.

Given the state ${\displaystyle x_{n}=n}$, we may use the Koopman operator[7] ${\displaystyle \Phi }$ to lift dynamics from the space of states to the space of observables:

${\displaystyle \Phi \circ {\textbf {log}}\circ x_{n}={\textbf {log}}\circ F\circ x_{n}={\textbf {log}}\circ x_{n+1}}$

where ${\displaystyle {\textbf {log}}}$ is an algorithm for integer factorisation, analogous to the discrete logarithm, and ${\displaystyle F}$ is the successor function. Thus, we have:

${\displaystyle {\textbf {log}}\circ x_{n}=\bigoplus _{k}a_{k}\cdot \ln p_{k}}$

A precise motivation for defining the Koopman operator ${\displaystyle \Phi }$ is that it represents a global linearisation of ${\displaystyle F}$, which views linear combinations of eigenstates as integer partitions. In fact, the reader may easily check that the successor function is not a linear function:

${\displaystyle \forall n\in \mathbb {N} ,F(n)=n+1\implies \forall x,y\in \mathbb {N} ^{*},F(x+y)\neq F(x)+F(y)}$

Hence, ${\displaystyle \Phi }$ is canonical.

If we take a simple quantum Hamiltonian H to have eigenvalues proportional to log p, that is,

${\displaystyle H|p\rangle =E_{p}|p\rangle }$

with

${\displaystyle E_{p}=E_{0}\log p\,,}$

we are naturally led to

${\displaystyle E_{n}=\sum _{p}k_{p}E_{p}=E_{0}\cdot \sum _{p}k_{p}\log p=E_{0}\log n}$

Let's suppose we would like to know the average time, suitably-normalised, that the Riemann gas spends in a particular subspace. How might this frequency be related to the dimension of this subspace?

If we characterize distinct linear subspaces as Erdős-Kac data which have the form of sparse binary vectors, using the Erdős-Kac theorem we may actually demonstrate that this frequency depends upon nothing more than the dimension of the subspace. In fact, if ${\displaystyle \omega (n)}$ counts the number of unique prime divisors of ${\displaystyle n\in \mathbb {N} }$ then the Erdős-Kac law tells us that for large ${\displaystyle n}$:

${\displaystyle {\frac {\omega (n)-\ln \ln n}{\sqrt {\ln \ln n}}}\sim {\mathcal {N}}(0,1)}$

has the standard normal distribution.

What is even more remarkable is that although the Erdős-Kac theorem has the form of a statistical observation, it could not have been discovered using statistical methods.[8] Indeed, for ${\displaystyle X\sim U([1,N])}$ the normal order of ${\displaystyle \omega (X)}$ only begins to emerge for ${\displaystyle N\geq 10^{100}}$.

The partition function Z is given by the Riemann zeta function:

${\displaystyle Z(T):=\sum _{n=1}^{\infty }\exp \left({\frac {-E_{n}}{k_{\text{B}}T}}\right)=\sum _{n=1}^{\infty }\exp \left({\frac {-E_{0}\log n}{k_{\text{B}}T}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s)}$

with s = E₀/k_BT where k_B is the Boltzmann constant and T is the absolute temperature.

The divergence of the zeta function at s = 1 corresponds to the divergence of the partition function at a Hagedorn temperature of T_H = E₀/k_B.