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 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]

1 The model
2 The supersymmetric model
3 See Also
4 References

[edit] 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

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 $T H = E o / k B$ .

[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] See Also

Godel Number

[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.