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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 was independently discovered by
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| Donald Spector<ref>D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathemtical Physics 127 (1990) pp. 239–252.</ref> and [[Bernard Julia]].<ref>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.</ref> Later works by Bakas and
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| Bowick<ref>I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881</ref>
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| and Spector <ref>D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927</ref> explored the connection of such systems to
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| string theory.
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| ==The model==
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| Consider a simple [[quantum Hamiltonian]] ''H'' having [[eigenstate]]s <math>|p\rangle</math> labelled by the [[prime number]]s ''p'', and having energies proportional to log ''p''. That is,
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| :<math>H|p\rangle = E_p |p\rangle</math>
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| with
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| :<math>E_p=E_0 \log p \, </math>
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| The [[second quantization|second-quantized]] version of this Hamiltonian converts states into particles, the '''primons'''. A multi-particle state is given by the numbers <math>k_p</math> of primons in the single-particle states <math>p</math>:
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| :<math>|n\rangle = |k_2, k_3, k_5, k_7, k_{11}, \ldots, k_p, \ldots\rangle</math> | |
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| This corresponds to the factorization of <math>n</math> into primes:
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| :<math>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</math>
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| The labelling by the integer ''n'' is unique, since every number has a unique factorization into primes.
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| The energy of such a multi-particle state is clearly
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| :<math>E(n) = \sum_p k_p E_p = E_0 \cdot \sum_p k_p \log p = E_0 \log n</math>
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| The statistical mechanics [[partition function (mathematics)|partition function]] ''Z'' is given by the [[Riemann zeta function]]:
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| :<math>Z(T) := \sum_{n=1}^\infty \exp \left(\frac{-E(n)}{k_B T}\right) = \sum_{n=1}^\infty \exp \left(\frac{-E_0 \log n}{k_B T}\right) = \sum_{n=1}^\infty \frac{1}{n^s} = \zeta (s) </math>
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| with ''s'' = ''E''<sub>0</sub>/''k''<sub>B</sub>''T'' where ''k''<sub>B</sub> 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''<sub>H</sub> = ''E''<sub>0</sub>/''k''<sub>B</sub>.
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| ==The supersymmetric model==
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| The above second-quantized model takes the particles to be [[boson]]s. If the particles are taken to be [[fermion]]s, 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|(−1)<sup>F</sup>]] has a very concrete realization in this model as the [[Möbius function]] <math>\mu(n)</math>, in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.
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| ==More complex models==
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| 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 ideal]]s take the role of the prime numbers, the [[group representation]]s take the role of integers, [[group character]]s taking the place the [[Dirichlet character]]s, and so on.
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| ==References==
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| <references/>
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| * [[John Baez]], [http://math.ucr.edu/home/baez/week199.html This Week's Finds in Mathematical Physics, Week 199]
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| [[Category:Number theory]]
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| [[Category:Quantum field theory]]
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