Primon gas

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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 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]

1 The model
2 The supersymmetric model
3 More complex models
4 References

[edit] The model

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

$E_n=\log n = \log p_1 + \log p_2 + \log p_3 +\cdots$

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

$\zeta(s)=\sum_{n=1}^\infty \exp (-E_n/k_BT)$

after identifying $s = β = 1 / k B T$ where $k B$ is Boltzmann's constant and T is the temperature.

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

[edit] More complex models

The analogy can be somewhat further extended in topological field theory and K-theory, where 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.

[edit] 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.