[Paper Review] Varieties of Dirac equation and flavors of leptons and quarks

Ruo Peng Wang|ArXiv.org|Jul 17, 2001

International Science and Diplomacy1 references4 citations

TL;DR

This paper proposes that there are twelve distinct varieties of the Dirac equation for spin-1/2 fermions, classified by their spin operator, Lorentz multiplier, and parity operator. These varieties naturally group into six pairs that correspond to the three generations of quarks and leptons, with charged-current weak interactions only connecting fermions from the same pair—explaining why quarks and leptons transform only within their generation.

ABSTRACT

I show that there exist twelve independent Dirac equations for spin 1/2 fermions. The Dirac fields that satisfy these equations can be grouped into six pairs according to the way they transform under continuous space-time transformations. These six pairs of Dirac equations correspond to the three quark generations and the three lepton generations. The charged V-A currents can be formed only from fields of the same pair. This property of the Dirac fields implies that a quark or lepton may be transformed only into its partner of the same generation via the charged-current weak interaction. According to the properties of the charged-current weak interaction, I conclude that different elementary fermion fields must satisfy different Dirac equations, and there may not be more than twelve flavors of elementary fermions that are already known.

Motivation & Objective

• To resolve the lack of natural explanation in the Standard Model for why quarks and leptons only transform within their generation via charged-current weak interactions.
• To identify a fundamental field-theoretic distinction between elementary fermions beyond mass differences.
• To show that the observed three generations of quarks and leptons arise from a complete classification of Dirac equation varieties.
• To establish that only twelve distinct fermion types are possible, corresponding to the twelve independent Dirac equations.
• To demonstrate that V-A currents can only form from fermion fields satisfying the same type of Dirac equation, enforcing generation-specific transitions.

Proposed method

• Identify all 16 linearly independent 4×4 matrices and classify them into two sets: Σ^I_j and Σ^II_j, each defining a distinct spin operator s^I_j and s^II_j.
• Derive the Dirac equation using matrices α and β that satisfy the standard Dirac algebra and transformation properties under Lorentz transformations.
• Define the Lorentz multiplier λ and parity operator β such that each combination (λ, β, s) defines a unique Dirac equation variety.
• Enumerate all combinations: two spin operators, six Lorentz multipliers (λ = Σ^I_j or Σ^II_j), and six parity operators (β = Σ^I_k or Σ^II_k with j≠k), yielding 12 distinct equations.
• Use unitary transformations (e.g., U, (1±iΣ^A_j)/√2) to show that all 12 equations are related but not equivalent, preserving physical distinction.
• Analyze the transformation properties of the V-A current under space-time symmetries to show that only fields with identical λ and s can form such currents.

Experimental results

Research questions

• RQ1 Why do quarks and leptons only undergo charged-current weak interactions within their respective generations?
• RQ2 What fundamental field-theoretic distinction separates different elementary fermions beyond mass?
• RQ3 How many distinct types of Dirac equations exist for spin-1/2 massive fermions?
• RQ4 Can the structure of the V-A current be derived from the transformation properties of Dirac fields under space-time symmetries?
• RQ5 Is there a natural explanation for the existence of exactly three generations of quarks and leptons?

Key findings

• There exist exactly twelve independent Dirac equations for spin-1/2 fermions, classified by their spin operator, Lorentz multiplier λ, and parity operator β.
• These twelve equations form six pairs, each pair sharing the same Lorentz multiplier and spin operator, corresponding to the three quark generations and three lepton generations.
• Charged V-A weak currents can only be formed from fermion fields satisfying Dirac equations of the same pair, explaining generation-specific transitions.
• Fermions of different generations must satisfy different Dirac equations, implying no more than twelve distinct elementary fermion flavors exist.
• The unitary transformation between equations preserves the algebraic structure but not the physical equivalence, as the Lorentz multiplier and spin operator define distinct physical fields.
• The observed three-generation structure of quarks and leptons is a direct consequence of the classification of Dirac equation varieties, with no room for additional generations beyond the known twelve fermion types.