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

The model

Consider a simple quantum Hamiltonian H having eigenstates $|p\rangle$ labelled by the prime numbers p, and having energies proportional to $\log \,p$. That is,

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

with

$E_{p}=E_{o}\log \,p$

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

$|n\rangle =|p_{1},p_{2},p_{3},.\cdots \rangle =|p_{1}\rangle |p_{2}\rangle |p_{3}\rangle \cdots$

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

$E_{n}/E_{o}=\log n=\log p_{1}+\log p_{2}+\log p_{3}+\cdots$

The statistical mechanics partition function $Z$ is given by the Riemann zeta function:

$Z(T):=\sum _{n=1}^{\infty }\exp(-E_{n}/k_{B}T)=\sum _{n=1}^{\infty }\exp(-E_{o}\log n/k_{B}T)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s)$

with $s=E_{o}/k_{B}T$ where $k_{B}$ is Boltzmann's 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 $T_{H}=E_{o}/k_{B}$.

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 $\mu (n)$, in that the Mobius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.

References

• John Baez, This Week's Finds in Mathematical Physics, Week 199