In mathematical physics, the primon gas or Riemann gas[1] discovered by Bernard Julia[2] 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.
[3] Later works by Ioannis Bakas and Mark Bowick,[4] and Spector[5] explored the connection of such systems to string theory.
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
has a unique factorization into primes: we can also denote the basis elements of the Fock space as simply
as a collection of primons: its prime factors, counted with multiplicity.
is an algorithm for integer factorisation, analogous to the discrete logarithm, and
Thus, we have: A precise motivation for defining the Koopman operator
, which views linear combinations of eigenstates as integer partitions.
If we take a simple quantum Hamiltonian H to have eigenvalues proportional to log p, that is, with for some positive constant
, we are naturally led to Let's suppose we would like to know the average time, suitably-normalised, that the Riemann gas spends in a particular 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.
The above second-quantized model takes the particles to be bosons.
If the particles are taken to be fermions, then the Pauli exclusion principle prohibits multi-particle states which include squares of primes.
The fermion operator (−1)F has a very concrete realization in this model as the Möbius function
, in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.
The connections between number theory and quantum field theory can be somewhat further extended into connections between topological field theory and K-theory, where, corresponding to the example above, the spectrum of a ring takes the role of the spectrum of energy eigenvalues, the prime ideals take the role of the prime numbers, the group representations take the role of integers, group characters taking the place the Dirichlet characters, and so on.