[Paper Review] A note on Quarks and numbers theory

Mehdi Hage-Hassan|arXiv (Cornell University)|Feb 26, 2013

Quantum and Classical Electrodynamics1 references3 citations

TL;DR

This paper establishes a novel mathematical correspondence between the basis vectors of SU(3) group representations—used in quark physics—and binary numbers, revealing an unexpected analogy between quark combinations and prime number properties. By expressing Gel'fand basis states via binary encoding, the author identifies a structural similarity between meson-like quark states and prime factorization, suggesting a new number-theoretic property linked to prime numbers through group-theoretic symmetry.

ABSTRACT

We express the basis vectors of Cartan fundamental representations of unitary groups by binary numbers. We determine the expression of Gel'fand basis of SU (3) based on the usual subatomic quarks notations and we represent it by binary numbers. By analogy with the mesons and quarks we find a new property of prime numbers.

Motivation & Objective

• To explore the structural parallels between the representation theory of SU(3) in particle physics and number theory.
• To express the basis vectors of fundamental representations of unitary groups using binary numbers.
• To reinterpret the Gel'fand basis of SU(3) in terms of quark notation and binary encoding.
• To uncover a new number-theoretic property by drawing analogies with meson and quark states.
• To establish a conceptual bridge between subatomic particle composition and prime number structure.

Proposed method

• Representing Cartan basis vectors of unitary groups using binary number systems.
• Mapping standard quark notation (e.g., u, d, s) to binary-encoded states in SU(3) representations.
• Constructing the Gel'fand basis for SU(3) using binary-encoded quark states.
• Applying analogies from meson formation (quark-antiquark pairs) to number-theoretic structures.
• Identifying patterns in binary representations that mirror properties of prime numbers.
• Using group-theoretic symmetry to infer new properties in number theory via physical analogies.

Experimental results

Research questions

• RQ1 Can the basis vectors of SU(3) representations be systematically encoded using binary numbers?
• RQ2 What number-theoretic properties emerge when quark states are represented in binary form?
• RQ3 Is there a structural analogy between meson-like quark combinations and prime factorization?
• RQ4 How does the binary encoding of quark states reflect underlying symmetries in SU(3)?
• RQ5 Can group representation theory suggest new insights into the distribution or properties of prime numbers?

Key findings

• The basis vectors of SU(3) fundamental representations are successfully expressed using binary numbers, establishing a direct mapping between group theory and binary encoding.
• The Gel'fand basis of SU(3) is reconstructed using standard quark notation and represented via binary states, enabling a new computational framework.
• Analogies between meson formation (quark-antiquark pairs) and number-theoretic structures lead to the identification of a new property of prime numbers.
• The paper suggests that the combinatorial structure of quark states under SU(3) symmetry mirrors properties inherent in prime factorization.
• The binary representation of quark states reveals a hidden symmetry that parallels the uniqueness of prime decomposition in number theory.
• The study proposes a conceptual link between particle physics and number theory through shared structural patterns in group representations and prime numbers.