In
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, the primon gas or free Riemann gas is a
toy model
In the modeling of physics, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model.
* In "toy" mathematical models, ...
illustrating in a simple way some correspondences between
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
and ideas in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
and
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
. It is a quantum field theory of a set of non-interacting particles, the primons; it is called a
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
or a ''free model'' because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald Spector and
Bernard Julia. Later works by Bakas and Bowick 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
State space
Consider a Hilbert space H with an orthonormal basis of states
labelled by the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s ''p''.
Second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
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
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
:
:
with ''s'' = ''E''
0/''k''
B''T'' where ''k''
B is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and ''T'' is the absolute
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
.
The divergence of the zeta function at ''s'' = 1 corresponds to the divergence of the partition function at a
Hagedorn temperature
The Hagedorn temperature, ''T''H, is the temperature in theoretical physics where hadronic matter (i.e. ordinary matter) is no longer stable, and must either "evaporate" or convert into quark matter; as such, it can be thought of as the "boiling p ...
of ''T''
H = ''E''
0/''k''
B.
The supersymmetric model
The above second-quantized model takes the particles to be
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s. If the particles are taken to be
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s, then the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
prohibits multi-particle states which include squares of primes. By the
spin–statistics theorem
In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
, 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
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
, 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
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
, where, corresponding to the example above, the
spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
takes the role of the spectrum of energy eigenvalues, the
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s take the role of the prime numbers, the
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
s take the role of integers,
group character
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...
s taking the place the
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \chi ...
s, and so on.
References
External links
*
John Baez
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
This Week's Finds in Mathematical Physics, Week 199
Number theory
Quantum field theory
statistical mechanics