Primon gas
In mathematical physics, the primon gas or free Riemann gas is a toy model illustrating in a simple way some correspondences between number theory and ideas in quantum field theory and dynamical systems. 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 is attributed to Bernard Julia [1]
The model
Consider a simple quantum Hamiltonian H having eigenstates labelled by the prime numbers p, and having energies proportional to . That is,
with
The second-quantized version of this Hamiltonian converts states into particles, the primons. A multi-particle state is denoted by a natural number n as
The labelling by the integer n is unique, since every prime number has a unique factorization into primes. The energy of such a multi-particle state is clearly
The statistical mechanics partition function is given by the Riemann zeta function:
with where is Boltzmann's constant and T is the absolute temperature. The divergence of the zeta function at corresponds to the divergence of the partition function at a Hagedorn temperature of .
The supersymmetric model
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. By the spin-statistics theorem, field states with an even number of particles are bosons, while those with an odd number of particles are fermions. The fermion operator (−1)F has a very concrete realization in this model as the Mobius function , in that the Mobius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.
References
- ^ Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in Physics, Vol. 47, Springer-Verlag, Berlin, 1990, pp. 276-293.