By exploring the correspondence between the Buckingham-Pi theorem and Unique Factorization Domains using Koopman Operators, we find that the Prime Numbers have an emergent Spectral Geometry.

By exploring the correspondence between the Buckingham-Pi theorem and Unique Factorisation Domains, we derive the notion of a Buckingham-Pi Space where each UFD has an associated Buckingham-Pi Space. Furthermore, by using Koopman Operators to explore dynamics on Buckingham-Pi Spaces we find that the prime numbers have an emergent Spectral Geometry. Hence, the Prime Numbers satisfy the metrological requirements of a fundamental physical theory.

This analysis lays a theoretical foundation for profound correspondences between number theory and physics using methods from representation theory. Before beginning our analysis, we shall introduce essential notions from both algebraic number theory and physics.

A ring is a set \(S\) with two binary operations, \((+,\cdot)\) satisfying the following conditions:

- Additive associativity: \(\forall a,b,c \in S, (a+b)+c = a+(b+c)\)
- Additive commutativity: \(\forall a,b,c \in S, a+b = b+a\)
- Additive identity: \(\exists 0 \in S \forall a \in S, 0+a=a+0=a\)
- Additive inverse: \(\forall a \in S \exists -a \in S, a+(-a)=(-a)+a= 0\)
- Left and right distributivity: \(\forall a,b,c \in S, a \cdot (b+c)=a\cdot b+a\cdot c=(b+c)\cdot a=(b\cdot a)+(c \cdot a)\)
- Multiplicative associativity: \(\forall a,b,c \in S, (a \cdot b)\cdot c = a \cdot (b \cdot c)\)

Commutative rings satisfy all of the above criteria as well as:

- Multiplicative commutativity: \(\forall a,b,c \in S, a \cdot b = b \cdot a\)

A field satisfies the criteria of commutative rings as well as:

Multiplicative identity: \(\exists e \in S \forall a \in S \setminus \{0\}, e \cdot a = a \cdot e = a\)

Multiplicative inverse: \(\forall a \in S \setminus \{0\} \exists a^{-1} \in S, a \cdot a^{-1}=a^{-1} \cdot a = e\)

*For specific examples of rings, we may consider the integers
which don’t have multiplicative inverses and real 2x2 matrices which
don’t satisfy the criterion of multiplicative commutativity.*

A ring that is commutative under multiplication, has a multiplicative identity element, and has no divisors of \(0\).

For an arbitrary ring \((R,+,\cdot)\), let \((R,+)\) be its additive group. \(I\) is an ideal if it satisfies two conditions:

- \((I,+)\) is a subgroup of \((R,+)\).
- \(\forall x \in R \forall y \in I, x \cdot y = y \cdot x \in I\)

For concreteness, the set of even integers is an ideal in the ring of integers \(\mathbb{Z}\).

An ideal \(P\) of a commutative ring \(R\) is prime if it satisfies the following:

- If \(a,b \in R\) such that \(a \cdot b \in P\) then \(a \in P\) or \(b \in P\).
- \(P\) is not the whole ring \(R\).

The notion of a Prime Ideal generalizes Euclid’s lemma: if \(p\) is a prime number, \(a,b \in \mathbb{N}\) and \(p\) divides \(a \cdot b\) then \(a \bmod p = 0\) or \(b \bmod p = 0\). In this sense, Prime Ideals are generated by prime elements.

\(u \in R\) is a unit if \(\exists v \in R\),

\[\begin{equation} v \cdot u = u \cdot v = 1 \end{equation}\]

where the set of units of \(R\) naturally forms a group under multiplication, known as the group of units.

\(p \in R\) is prime if it is not the zero element or a unit and whenever \(p\) divides \(a \cdot b\) for \(a,b \in R\) then \(a \bmod p = 0\) or \(b \bmod p = 0\).

A Unique Factorisation Domain is any integral domain where every non-zero and non-invertible element has a unique factorisation i.e. an essentially unique decomposition as the product of prime elements.

If we know the appropriate physical units for the observable \(\Omega\) then we have made significant progress in understanding its behavior. In particular, the Buckingham-Pi theorem tells us how many physical units we need and what we can do with free parameters if we happen to have more physical measurements than necessary to model the behavior of \(\Omega\).

Let’s suppose we have succeeded in identifying an equation that describes the evolution of \(\Omega\) as a function of \(N\) physical units. Then we have:

\[\begin{equation} \exists \alpha_i \in \mathbb{Z}, \Omega = \prod_{i=1}^N w_i^{\alpha_i} \tag{1} \end{equation}\]

where \(w_i\) are our physical units, and the reason why all exponents are of integer order is that integer-order differential equations describe the physics of the Observable Universe.

Now, we also know that each unit of \(\Omega\) may be expressed in terms of the fundamental units \(U = \{u_i\}_{i=1}^k\) so we have:

\[\begin{equation} \exists \beta_i \in \mathbb{Z}, \Omega = \prod_{i=1}^k u_i^{\beta_i} \tag{2} \end{equation}\]

and for each physical unit,

\[\begin{equation} \exists \lambda_{i,j} \in \mathbb{Z}, w_j^{\alpha_j} = \prod_{i=1}^k u_i^{\lambda_{i,j} \cdot \alpha_j} \tag{3} \end{equation}\]

This allows a representation in terms of the system of equations:

\[\begin{equation} \sum_{j=1}^N \lambda_{i,j} \cdot \alpha_j = \beta_i \tag{4} \end{equation}\]

so we have:

\[\begin{equation} \Lambda \cdot \vec{\alpha} = \vec{\beta} \tag{5} \end{equation}\]

In order to translate between different civilisations which might have different metrological standards, we may use the last relation to model the problem using dimensionless parameters. This amounts to finding \(\vec{\Delta \alpha}\) such that:

\[\begin{equation} \Lambda \cdot (\vec{\alpha}+\vec{\Delta \alpha}) = \Lambda \cdot \vec{\alpha}= \vec{\beta} \tag{6} \end{equation}\]

where the set of all possible \(\vec{\Delta \alpha}\) defines the null-space of \(\Lambda\).

From the rank-nullity theorem, we may deduce that:

\[\begin{equation} \text{dim}(\text{Null}(\Lambda)) = N - k \tag{7} \end{equation}\]

is the number of dimensionless parameters with which we may describe our physical system.

A system of units of measure that satisfies the criteria of the Buckingham-Pi theorem implicitly defines a vector space \(\mathcal{B}\) that satisfies the following criteria:

\(\mathcal{B}\) is equipped with a system of fundamental units \(\{p_i\}_{i=1}^N\) which are co-prime.

For any two measurements \(\hat{a}\) and \(\hat{b}\) that are not dimensionless, and hence non-zero, their types are given by:

\[\begin{equation} \exists \alpha_i \in \mathbb{Z}, \text{TypeOf}(\hat{a}) = \prod_{i=1}^N p_i^{\alpha_i} \tag{8} \end{equation}\]

\[\begin{equation} \exists \beta_i \in \mathbb{Z}, \text{TypeOf}(\hat{b}) = \prod_{i=1}^N p_i^{\beta_i} \tag{9} \end{equation}\]

A type hierarchy naturally emerges from the observation that \(\text{Type}(\hat{a})\) is a sub-type of \(\text{Type}(\hat{b})\) if \(\text{Type}(\hat{a})\) divides \(\text{Type}(\hat{b})\).

The Buckingham-Pi Space \(\mathcal{B}\) is the set of equivalence relations over typed units:

\[\begin{equation} \text{Type}(\hat{a}) \sim \text{Type}(\hat{b}) \iff \frac{\hat{a}}{\hat{b}} \in \mathbb{R} \tag{10} \end{equation}\]

where \(\hat{a}\) and \(\hat{b}\) are not dimensionless measurements.

- Any element \(\varphi \in \mathcal{B}\) generates an additive group which satisfies the criteria of a one-dimensional vector space:

\[\begin{equation} \hat{\varphi} = \{\lambda \cdot \varphi: \forall \lambda \in \mathbb{R} \exists! \varphi \in \mathcal{B}\} \tag{11} \end{equation}\]

In agreement with (5), higher-dimensional vector spaces may be derived from \(\{\varphi_{i=1}\}_{i=1}^n\) if \(\varphi_i\) and \(\varphi_{j \neq i}\) are co-prime.

Based on this presentation of the Buckingham-Pi theorem and Unique Factorisation Domains, we may make two related observations.

First, when the objects to be measured are rational numbers then the prime numbers satisfy the criteria of fundamental units in the Buckingham-Pi theorem. Hence, the rational numbers define a Buckingham-Pi Space.

Second, the reader may verify that each Unique Factorisation Domain has an associated Buckingham-Pi Space. This leads us to the Spectral Geometry of \(\mathbb{P}\) via the Koopman Operator(aka the Composition Operator).

Let \(g: S \rightarrow \mathbb{R}\) be a real-valued observable of a dynamical system:

\[\begin{equation} \forall k \in \mathbb{N}, x_{k+1} = T \circ x_k \tag{12} \end{equation}\]

where \(x_k \in S \subset \mathbb{R}^n\) and \(T: S \rightarrow S\) is a dynamic map. As a corollary of the Buckingham-Pi theorem, the collection of all such observables forms a vector space.

The Koopman Operator \(U\) is a linear transform on this vector space given by:

\[\begin{equation} U g(x_k) = g \circ T(x_k) = g(x_{k+1}) \tag{13} \end{equation}\]

in a discrete setting, where the linearity of the Koopman Operator follows from the linearity of the composition operator:

\[\begin{equation} U \circ (g_1 + g_2) (x) = g_1 \circ T(x) + g_2 \circ T(x) = U g_1(x) + U g_2(x) \tag{14} \end{equation}\]

so we may think of the Koopman Operator as lifting dynamics from the space of states to the space of observables.

These considerations of the Koopman Operator naturally lead to the emergence of the Spectral Geometry of Unique Factorisation Domains. For concreteness, let’s suppose \(\vec{x} \in M \subset \mathbb{R}^n\) contains all the information concerning the flow field at a particular time, so \(g(x)\) may be a vector of any quantity of interest such as velocity measurements.

If we let \(\varphi_j(x): M \rightarrow R\) denote eigenfunctions and \(\lambda_j \in \mathbb{C}\) denote eigenvalues of \(U\), we have:

\[\begin{equation} \forall j \in \mathbb{N}, U \circ \varphi_j (x) = \lambda_j \cdot \varphi_j(x) \tag{15} \end{equation}\]

If each of the components of \(\vec{g}\) lie within the span of the eigenfunctions \(\varphi_j\) then we may expand the vector-valued \(\vec{g}\) in terms of the eigenfunctions:

\[\begin{equation} g(\vec{x}) = \sum_{j=1}^{\infty} \varphi_j(x) \cdot \vec{v_j} \tag{16} \end{equation}\]

so we may think of \(\vec{g}\) as a linear combination of the eigenfunctions \(\varphi_j\) of \(U\) where \(\vec{v}_j\) are the vector coefficients in the expansion.

By induction, we find the iterates of \(\vec{x_0}\) are given by:

\[\begin{equation} g(\vec{x_k}) = \sum_{j=1}^\infty U^k \circ \varphi_j(\vec{x_0}) \cdot \vec{v_j}=\sum_{j=1}^{\infty} \lambda_j^k \cdot \varphi_j(\vec{x_0}) \cdot \vec{v_j} \tag{17} \end{equation}\]

so we may think of the Koopman Operator \(U\) as an infinite-dimensional operator that acts on vectors from a Hilbert Space.

Kac, Mark. “Can One Hear the Shape of a Drum?”. American Mathematical Monthly. 1966.

Lapidus, Michel L.; Pomerance, Carl. “The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums”, Proc. London Math. Soc. 1993.

Koopman, B. O. “Hamiltonian Systems and Transformation in Hilbert Space”. Proceedings of the National Academy of Sciences. 1931.

Steven L. Brunton. Notes on Koopman Operator Theory. 2019.

Joseph Bakarji, Jared Callaham, Steven L. Brunton, J. Nathan Kutz. Dimensionally Consistent Learning with Buckingham Pi. 2022.

For attribution, please cite this work as

Rocke (2022, April 17). Kepler Lounge: The Spectral Geometry of the Prime Numbers. Retrieved from keplerlounge.com

BibTeX citation

@misc{rocke2022the, author = {Rocke, Aidan}, title = {Kepler Lounge: The Spectral Geometry of the Prime Numbers}, url = {keplerlounge.com}, year = {2022} }