In mathematical physics, the primon gas or Riemann gas discovered by Bernard Julia is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee–Yang theorem. It is a quantum field theory of a set of non-interacting particles, the primons; it is called a gas or a free model because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald Spector. Later works by Ioannis Bakas and Mark Bowick, and Spector explored the connection of such systems to string theory.
The model
State space Consider a Hilbert space H with an orthonormal basis of states | p ⟩ {\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 k p = 0 , 1 , 2 , … {\displaystyle k_{p}=0,1,2,\dots } of primons for each prime p {\displaystyle p} :
| k 2 , k 3 , k 5 , k 7 , k 11 , … , k p , … ⟩ {\displaystyle |k_{2},k_{3},k_{5},k_{7},k_{11},\ldots ,k_{p},\ldots \rangle }
where the total ∑ p k p {\displaystyle \sum _{p}k_{p}} is finite. Since any positive natural number n {\displaystyle n} has a unique factorization into primes:
n = 2 k 2 ⋅ 3 k 3 ⋅ 5 k 5 ⋅ 7 k 7 ⋅ 11 k 11 ⋯ p k p ⋯ {\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 | n ⟩ {\displaystyle |n\rangle } where n = 1 , 2 , 3 , … . {\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 n {\displaystyle n} as a collection of primons: its prime factors, counted with multiplicity.
Identifying the Hamiltonian via the Koopman operator Given the state x n = n {\displaystyle x_{n}=n} , we may use the Koopman operator Φ {\displaystyle \Phi } to lift dynamics from the space of states to the space of observables:
Φ ∘ log ∘ x n = log ∘ F ∘ x n = log ∘ x n + 1 {\displaystyle \Phi \circ {\textbf {log}}\circ x_{n}={\textbf {log}}\circ F\circ x_{n}={\textbf {log}}\circ x_{n+1}}
where log {\displaystyle {\textbf {log}}} is an algorithm for integer factorisation, analogous to the discrete logarithm, and F {\displaystyle F} is the successor function. Thus, we have:
log ∘ x n = ⨁ k a k ⋅ ln p k {\displaystyle {\textbf {log}}\circ x_{n}=\bigoplus _{k}a_{k}\cdot \ln p_{k}}
A precise motivation for defining the Koopman operator Φ {\displaystyle \Phi } is that it represents a global linearisation of F {\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:
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