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. 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 <ath>\log p</math>. That is,

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

with

$E_{p}=\log p$

The second-quantized version of this Hamiltonian converts states into particles, the primons. A multi-partcile state is denotedby 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 $E_{n}=\log n$.

The statistical mechanics partition function is given by the [[Riemann zeta function]]:

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

after identifying $s=\beta =1/k_{B}T$ where $k_{B}$ is Boltzmann's constant and T is the temperature.

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.

More complex models

The analogy can be somewhat further extended in topological field theory and K-theory, where prime ideals take the role of the prime numbers, the group representations take the role of integers, group characters taking the place the [[Dirichlet character]]s, and so on.

References

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