[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/primon-gas-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/primon-gas-wikipedia\/","headline":"Primon gas – Wikipedia","name":"Primon gas – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Model from mathematical physics In mathematical physics, the primon gas or free Riemann","datePublished":"2017-10-10","dateModified":"2017-10-10","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4aa152d3a6e11e9759ca10ae564c371e2609337f","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4aa152d3a6e11e9759ca10ae564c371e2609337f","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/primon-gas-wikipedia\/","wordCount":3464,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Model from mathematical physicsIn 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[1] and Bernard Julia.[2] Later works by Bakas and Bowick[3] and Spector [4] explored the connection of such systems tostring theory.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsThe model[edit]State space[edit]Energies[edit]Statistical mechanics[edit]The supersymmetric model[edit]More complex models[edit]References[edit]External links[edit]The model[edit]State space[edit]Consider a Hilbert space H with an orthonormal basis of states |p\u27e9{displaystyle |prangle } 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         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4kp=0,1,2,\u2026{displaystyle k_{p}=0,1,2,dots } of primons for each prime p{displaystyle p}:|k2,k3,k5,k7,k11,\u2026,kp,\u2026\u27e9{displaystyle |k_{2},k_{3},k_{5},k_{7},k_{11},ldots ,k_{p},ldots rangle }where the total \u2211pkp{displaystyle sum _{p}k_{p}} is finite.  Since any positive natural number n{displaystyle n} has a unique factorization into primes:n=2k2\u22c53k3\u22c55k5\u22c57k7\u22c511k11\u22efpkp\u22ef{displaystyle 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 }we can also denote the basis elements of the Fock space as simply |n\u27e9{displaystyle |nrangle } where n=1,2,3,\u2026.{displaystyle n=1,2,3,dots .}In short, the Fock space for primons has an orthonormal basis given by the positive natural numbers, but we think of each such number n{displaystyle n} as a collection of primons: its prime factors, counted with multiplicity.Energies[edit]If we take a simple quantum Hamiltonian H to have eigenvalues proportional to\u00a0log\u00a0p, that is,H|p\u27e9=Ep|p\u27e9{displaystyle H|prangle =E_{p}|prangle }withEp=E0log\u2061p,{displaystyle E_{p}=E_{0}log p,,}we are naturally led toEn=\u2211pkpEp=E0\u22c5\u2211pkplog\u2061p=E0log\u2061n{displaystyle E_{n}=sum _{p}k_{p}E_{p}=E_{0}cdot sum _{p}k_{p}log p=E_{0}log n}Statistical mechanics[edit]The partition function Z is given by the Riemann zeta function:Z(T):=\u2211n=1\u221eexp\u2061(\u2212EnkBT)=\u2211n=1\u221eexp\u2061(\u2212E0log\u2061nkBT)=\u2211n=1\u221e1ns=\u03b6(s){displaystyle Z(T):=sum _{n=1}^{infty }exp left({frac {-E_{n}}{k_{text{B}}T}}right)=sum _{n=1}^{infty }exp left({frac {-E_{0}log n}{k_{text{B}}T}}right)=sum _{n=1}^{infty }{frac {1}{n^{s}}}=zeta (s)}with s\u00a0=\u00a0E0\/kBT where kB is the Boltzmann constant and T is the absolute temperature.The divergence of the zeta function at s\u00a0=\u00a01 corresponds to the divergence of the partition function at a Hagedorn temperature of\u00a0TH\u00a0=\u00a0E0\/kB.The supersymmetric model[edit]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\u2013statistics theorem, field states with an even number of particles are bosons, while those with an odd number of particles are fermions. The fermion operator (\u22121)F has a very concrete realization in this model as the M\u00f6bius function \u03bc(n){displaystyle mu (n)}, in that the M\u00f6bius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.More complex models[edit]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[edit]^ D. Spector, Supersymmetry and the M\u00f6bius Inversion Function, Communications in Mathematical Physics 127 (1990) pp. 239\u2013252.^ 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\u2013293.^ 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\u20131927External links[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/primon-gas-wikipedia\/#breadcrumbitem","name":"Primon gas – Wikipedia"}}]}]