# Primon gas

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Primon_gas
> Markdown URL: https://mediated.wiki/source/Primon_gas.md
> Source: https://en.wikipedia.org/wiki/Primon_gas
> Source revision: 1350335533
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Model from mathematical physics}}
In [mathematical physics](/source/mathematical_physics), the '''primon gas''' or '''Riemann gas'''<ref>D. J. G. Dueñas and N. F. Svaiter. Thermodynamics of the Bosonic Randomized Riemann Gas. arXiv preprint arXiv:1401.8190.</ref> discovered by [Bernard Julia](/source/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>  is a [model](/source/model) illustrating correspondences between [number theory](/source/number_theory) and methods in [quantum field theory](/source/quantum_field_theory), [statistical mechanics](/source/statistical_mechanics) and [dynamical systems](/source/dynamical_systems) such as the [Lee–Yang theorem](/source/Lee%E2%80%93Yang_theorem). It is a quantum field theory of a set of non-interacting particles, the '''primons'''; it is called a [gas](/source/gas) or a ''free model'' because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald Spector.<ref>D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathematical Physics 127 (1990) pp. 239–252.</ref> Later works by Ioannis Bakas and [Mark Bowick](/source/Mark_Bowick),<ref>I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881</ref> 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 [string theory](/source/string_theory).

== The model==
=== State space ===
Consider a [Hilbert space](/source/Hilbert_space) '''H''' with an orthonormal basis of states <math>|p\rangle</math> labelled by the [prime number](/source/prime_number)s ''p''. [Second quantization](/source/Second_quantization) gives a new Hilbert space '''K''', the [bosonic Fock space](/source/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](/source/multisets) of primes.  In other words, to specify one of these basis elements we can list the number  <math>k_p = 0 , 1, 2, \dots</math> of primons for each prime <math>p</math>:

:<math>|k_2, k_3, k_5, k_7, k_{11}, \ldots, k_p, \ldots\rangle</math>

where the total <math>\sum_p k_p</math> is finite.  Since any positive natural number <math>n</math> has a unique factorization into primes:

:<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>

we can also denote the basis elements of the Fock space as simply <math>|n\rangle</math> where <math>n = 1,2,3, \dots. </math>

In short, the Fock space for primons has an orthonormal basis given by the positive natural numbers, but we think of each such number <math>n</math> as a collection of primons: its prime factors, counted with multiplicity.

=== Identifying the Hamiltonian via the Koopman operator ===

Given the state <math>x_n = n</math>, we may use the Koopman operator<ref>Steven L. Brunton. Notes on Koopman Operator Theory. Cambridge University Press. 2019.</ref> <math>\Phi</math> to lift dynamics from the space of states to the space of observables: 

:<math>\Phi \circ \textbf{log} \circ x_n = \textbf{log} \circ F \circ x_n = \textbf{log} \circ x_{n+1} </math>

where <math>\textbf{log}</math> is an algorithm for integer factorisation, analogous to the discrete logarithm, and <math>F</math> is the successor function. Thus, we have: 

:<math>\textbf{log} \circ x_n = \bigoplus_k a_k \cdot \ln p_k </math>

A precise motivation for defining the Koopman operator <math>\Phi</math> is that it represents a global linearisation of <math>F</math>, which views linear combinations of eigenstates as 
integer partitions. In fact, the reader may easily check that the successor function is 
not a linear function: 

:<math>\forall n \in \mathbb{N}, F(n) = n+1 \implies \forall x,y \in \mathbb{N}^*, F(x+y) \neq F(x)+F(y)</math>

Hence, <math>\Phi</math> is canonical. 

=== Energies ===
If we take a simple [quantum Hamiltonian](/source/quantum_Hamiltonian) ''H'' to have eigenvalues proportional to&nbsp;log&nbsp;''p'', that is,

:<math>H|p\rangle = E_p |p\rangle</math>

with

:<math>E_p=E \log p </math>

for some positive constant <math>E</math>, we are naturally led to

:<math>E_n = \sum_p k_p E_p = E \cdot \sum_p k_p \log p = E \log n</math>

=== Statistics of the phase-space dimension ===

Let's suppose we would like to know the average time, suitably-normalised, that the Riemann gas spends in a particular subspace. How might this frequency be related to the dimension of this subspace? 

If we characterize distinct linear subspaces as Erdős-Kac data which have the form of sparse binary vectors, using the [Erdős–Kac theorem](/source/Erd%C5%91s%E2%80%93Kac_theorem) we may actually demonstrate that this frequency depends upon nothing more than the dimension of the subspace. In fact, if <math>\omega(n)</math> counts the number of unique prime divisors of <math>n \in \mathbb{N}</math> then the Erdős–Kac law tells us that for large <math>n</math>: 

:<math> \frac{\omega(n)-\ln \ln n}{\sqrt{\ln \ln n}} \sim \mathcal{N}(0,1)
</math>

has the standard normal distribution. 

What is even more remarkable is that although the Erdős-Kac theorem has the form of a statistical observation, it could not have been discovered using statistical methods.<ref>BubbleZ (https://mathoverflow.net/users/470546/bubblez), Theorems that are essentially impossible to guess by empirical observation, URL (version: 2021-12-29): https://mathoverflow.net/q/412762

</ref> Indeed, for <math>X \sim U([1,N])</math> the normal order of <math>\omega(X)</math> only begins to emerge 
for <math>N \geq 10^{100}</math>. 

=== Statistical mechanics ===
The [partition function](/source/partition_function_(mathematics)) ''Z'' of the primon gas is given by the [Riemann zeta function](/source/Riemann_zeta_function):

:<math>Z(T) := \sum_{n=1}^\infty \exp \left(\frac{-E_n}{k_\text{B} T}\right) = \sum_{n=1}^\infty \exp \left(\frac{-E \log n}{k_\text{B} T}\right) = \sum_{n=1}^\infty \frac{1}{n^s} = \zeta (s) </math>

with ''s''&nbsp;=&nbsp;''E''/''k''<sub>B</sub>''T'' where ''k''<sub>B</sub> is the [Boltzmann constant](/source/Boltzmann_constant) and ''T'' is the absolute [temperature](/source/temperature). 

The divergence of the zeta function at ''s''&nbsp;=&nbsp;1 corresponds to the divergence of the partition function at a [Hagedorn temperature](/source/Hagedorn_temperature) of&nbsp;''T''<sub>H</sub>&nbsp;=&nbsp;''E''/''k''<sub>B</sub>.

==Supersymmetric model==
The above second-quantized model takes the particles to be [boson](/source/boson)s. If the particles are taken to be [fermion](/source/fermion)s, then the [Pauli exclusion principle](/source/Pauli_exclusion_principle) prohibits multi-particle states which include squares of primes. By the [spin–statistics theorem](/source/spin%E2%80%93statistics_theorem), field states with an even number of particles are bosons, while those with an odd number of particles are fermions. The fermion operator [(&minus;1)<sup>F</sup>](/source/(-1)%5EF) has a very concrete realization in this model as the [Möbius function](/source/M%C3%B6bius_function) <math>\mu(n)</math>, 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](/source/topological_field_theory) and [K-theory](/source/K-theory), where, corresponding to the example above, the [spectrum of a ring](/source/spectrum_of_a_ring) takes the role of the spectrum of energy eigenvalues, the [prime ideal](/source/prime_ideal)s take the role of the prime numbers, the [group representation](/source/group_representation)s take the role of integers, [group character](/source/group_character)s taking the place the [Dirichlet character](/source/Dirichlet_character)s, and so on.

==References==
<references/>

==External links==
* [John Baez](/source/John_Baez), [http://math.ucr.edu/home/baez/week199.html This Week's Finds in Mathematical Physics, Week 199]

Category:Number theory
Category:Quantum field theory
Category:statistical mechanics

---
Adapted from the Wikipedia article [Primon gas](https://en.wikipedia.org/wiki/Primon_gas) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Primon_gas?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.