Introduction to entropy

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 was independently discovered by Donald Spector and Bernard Julia. Later works by Bakas and Bowick and Spector explored the connection of such systems to string theory.

The model

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

$H|p⟩=E_p|p⟩$

with

$E_p=E_0\log p$

The second-quantized version of this Hamiltonian converts states into particles, the primons. A multi-particle state is given by the numbers $k_p$ of primons in the single-particle states $p$:

$|n⟩=|k_2,k_3,k_5,k_7,k_{11},\dots ,k_p,\dots ⟩$

This corresponds to the factorization of $n$ into primes:

$n=2^{k_2}\cdot 3^{k_3}\cdot 5^{k_5}\cdot 7^{k_7}\cdot 11^{k_{11}}\cdots p^{k_p}\cdots$

The labelling by the integer n is unique, since every number has a unique factorization into primes.

The energy of such a multi-particle state is clearly

$E(n)=\sum _p{k}_p{E}_p=E_0\cdot \sum _p{k}_p\log p=E_0\log n$

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

$Z(T):=\sum _{n=1}^{\mathrm{\infty }}\exp \left(\frac{-E(n)}{k_{B}T}\right)=\sum _{n=1}^{\mathrm{\infty }}\exp \left(\frac{-E_0\log n}{k_{B}T}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}=\zeta (s)$

with s = E₀/k_BT 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 = E₀/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 Möbius function $\mu (n)$, in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.

More complex models

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.

References

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