Standard model physics from an algebra?

Abstract: This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Using nothing more than $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ acting on itself, we set out to find standard model particle representations. From the complex quaternionic portion of the algebra, we find generalized ideals, and show that they describe concisely all of the Lorentz representations of the standard model. From the complex octonionic portion of the algebra, we find minimal left ideals, and show that they mirror the behaviour of a generation of quarks and leptons under $su(3)c$ and $u(1){em}$. We then demonstrate a rudimentary electroweak model which yields a straightforward explanation as to why $SU(2)_L$ acts only on left-handed states. This holds in the case of leptons. Finally, we demonstrate how $\mathbb{C}\otimes\mathbb{O}$ can generate a 64-$\mathbb{C}$-dimensional algebra, wherein we find the $SU(3)_c$ irreducible representations corresponding to three generations of quarks and leptons. We then conclude by showing how to arrive at all 48 electric charges.

• The paper introduces an innovative algebraic framework that derives Lorentz representations from complex quaternions and octonions.
• It models quarks and leptons via minimal left ideals and explains electric charge as a number operator with quantized eigenvalues.
• The work reveals that gauge symmetries arise as algebraic automorphisms, offering insights into chirality and a three-generation model.

An Insightful Overview of "Standard Model Physics from an Algebra" by C. Furey

C. Furey's dissertation presents a novel approach to particle physics by attempting to derive aspects of the Standard Model (SM) purely from algebraic structures, specifically leveraging the complex quaternions ($\mathbb{C} \otimes \mathbb{H}$) and the complex octonions ($\mathbb{C} \otimes \mathbb{O}$). This work integrates ideas from non-associative and associative algebras to propose representations of particles and forces, a venture that attempts to unify aspects of the SM purely algebraically without introducing extraneous physical assumptions.

Summary of Methodology and Results

1. Algebraic Structures for Lorentz Representations:
   • The dissertation provides a comprehensive method using $\mathbb{C} \otimes \mathbb{H}$ to produce all necessary Lorentz representations for the SM, including scalar fields, spinors, and the field strength tensor. These representations are derived through generalized ideals based on complex, Hermitian, and quaternionic invariant actions.
2. Quarks and Leptons from $\mathbb{C} \otimes \mathbb{O}$:
   • By extending the work of G\"{u}naydin and G\"{u}rsey, Furey exploits $\mathbb{C} \otimes \mathbb{O}$ to describe both quark and lepton structures through a series of elegant mathematical constructs known as minimal left ideals. These ideals mimic quantum states in a Fock space, with ladder operators facilitating transitions akin to those in particle physics.
3. Electric Charge as a Number Operator:
   • Within the octonionic framework, electric charge emerges as a number operator with quantized eigenvalues corresponding to the charges of the SM fermions. This suggests a natural origin for charge quantization.
4. Gauge Symmetries as Algebraic Automorphisms:
   • Furey identifies the unbroken symmetries of the SM—$SU(3)_c$ and $U(1)_{em}$—as intrinsic symmetries of these algebraic structures. These symmetries are shown to govern the transformations of the minimal left ideals representing particle states.
5. The Chirality Puzzle:
   • A critical insight from this work is the explanation of parity in weak interactions. The mathematical framework inherently favors left-handed particles for $SU(2)_L$ interactions while excluding right-handed particles, offering a purely algebraic perspective on what is traditionally an empirically observed asymmetry in particle physics.
6. Three-Generation Model from $\mathbb{C} \otimes \overleftarrow{\mathbb{O}}$:
   • The thesis discovers representations within the complex octonionic chain algebra that correspond to three generations of SM particles under $SU(3)_c$. This extends beyond the usual single-generation models typical in unification theories and suggests a structured pathway to extend the SM to encompass all observed fermion families.

Implications and Future Directions

The implications of Furey's work are profound. By deriving particle physics relationships from algebraic structures, this approach could bridge gaps between quantum mechanics and quantum field theory by reducing reliance on manifold-based representations of space-time. Furey's use of algebra shows promise in providing a mathematically rigorous foundation for phenomena traditionally explained through complex Lagrangian dynamics.

Moreover, the integration of non-associative algebras into the domain of unified theories could shed light on unexplored connections between geometry and quantum field phenomena, offering a potential avenue for addressing the unification of gravity with quantum mechanics.

Conclusion

While this seminal dissertation does not produce a complete and tested theory of everything, it sets the stage for innovative exploration of algebraic unification in particle physics. Furey’s work not only challenges existing paradigms but also fosters a unique methodological perspective that inspires theoretical and mathematical developments beyond the boundaries traditionally defined by the SM. Future research inspired by these findings could focus on further mathematical rigor, computational modeling, and experimental prediction mechanisms stemming from algebraic foundations.