Primon gas
In mathematical physics, the Primon gas or Riemann gas[1] discovered by Bernard Julia[2] is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee-Yang theorem. 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[3]. Later works by Bakas and Bowick[4] and Spector [5] explored the connection of such systems to string theory.
The model
State space
Consider a Hilbert space H with an orthonormal basis of states labelled by the prime numbers p. Second quantization gives a new Hilbert space K, the bosonic Fock space on H, where states describe collections of primes - which we can call primons if we think of them as analogous to particles in quantum field theory. This Fock space has an orthonormal basis given by finite multisets of primes. In other words, to specify one of these basis elements we can list the number of primons for each prime :
where the total is finite. Since any positive natural number has a unique factorization into primes:
we can also denote the basis elements of the Fock space as simply where
In short, the Fock space for primons has an orthonormal basis given by the positive natural numbers, but we think of each such number as a collection of primons: its prime factors, counted with multiplicity.
Energies
If we take a simple quantum Hamiltonian H to have eigenvalues proportional to log p, that is,
with
we are naturally led to
Statistical mechanics
The partition function Z is given by the Riemann zeta function:
with s = E0/kBT where kB is the Boltzmann 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 TH = E0/kB.
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 , 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
- ^ D. J. G. Dueñas and N. F. Svaiter. Thermodynamics of the Bosonic Randomized Riemann Gas. arXiv preprint arXiv:1401.8190.
- ^ 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.
- ^ D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathematical Physics 127 (1990) pp. 239–252.
- ^ I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881
- ^ D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927