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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Primon ： ウィキペディア英語版 Primon gas 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〔D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathematical Physics 127 (1990) pp. 239–252.〕 and Bernard Julia.〔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.〕 Later works by Bakas and Bowick〔I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881〕 and Spector 〔D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927〕 explored the connection of such systems to string theory. ==The model== Consider a simple quantum Hamiltonian ''H'' having eigenstates $|p\backslash rangle$ labelled by the prime numbers ''p'', and having energies proportional to log ''p''. That is, :$H|p\backslash rangle\; =\; E\_p\; |p\backslash rangle$ with :$E\_p=E\_0\; \backslash log\; p\; \backslash ,$ 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 $p$: :$|n\backslash rangle\; =\; |k\_2,\; k\_3,\; k\_5,\; k\_7,\; k\_,\; \backslash ldots,\; k\_p,\; \backslash ldots\backslash rangle$ This corresponds to the factorization of $n$ into primes: :$n\; =\; 2^\; \backslash cdot\; 3^\; \backslash cdot\; 5^\; \backslash cdot\; 7^\; \backslash cdot\; 11^\; \backslash cdots$ 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 :$E(n)\; =\; \backslash sum\_p\; k\_p\; E\_p\; =\; E\_0\; \backslash cdot\; \backslash sum\_p\; k\_p\; \backslash log\; p\; =\; E\_0\; \backslash log\; n$ The statistical mechanics partition function ''Z'' is given by the Riemann zeta function: :$Z(T)\; :=\; \backslash sum\_^\backslash infty\; \backslash exp\; \backslash left(\backslash frac\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash exp\; \backslash left(\backslash frac\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash zeta\; (s)$ with ''s'' = ''E''0/''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''0/''k''B. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Primon gas」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.