Examination of Dirac, Majorana, and Weyl Fermions
This paper, authored by Palash B. Pal, serves as a pedagogical exploration into the fundamental distinctions and relationships between Dirac, Majorana, and Weyl fermions. While covering the historical context and theoretical formulations, it addresses the underpinning of fermion classifications in terms of the proper Lorentz group and diverges from conventional relationships with discrete symmetries such as charge conjugation and CP. The analysis highlights Majorana fermions and derives useful insights regarding their interactions and representations in field theory.
Historical Context and Theoretical Framework
The groundwork for the paper rests on Dirac's formulation of relativistic electrons and Weyl’s subsequent simplifications for massless fermions. Pauli's introduction of neutrinos further unfolded as a critical development. Majorana’s unique contribution established a novel categorization of fermions as self-conjugate particles, thus setting a stage for contemporary research inquiries regarding the nature of neutrino masses and supersymmetric theories.
A foundational element is the differentiation between massless (Weyl) and massive (Dirac and Majorana) fermions in relation to chirality and helicity. The detailed mathematical exposition includes Klein-Gordon and Dirac equations, underscoring representations and reality conditions essential for understanding fermions in various frameworks, including electroweak interactions and beyond.
Majorana and Weyl Fermions: Essential Distinctions
The representation and reality of Majorana fermions emerge as central tenets, specifically their character as esserially simpler, real-valued entities compared to complex Dirac fermions. Majorana fermions require explicit representation distinctions, with focus on transformations via matrix manipulations such as the Majorana representation, through which the unique real nature is established.
Weyl fermions, notably, are described via 2-component spinors, useful for massless particle representations. Their role as irreducible representations of the Lorentz group provides building blocks for general fermionic fields where the intricacies of chirality and helicity may be delineated without ambiguities in the massless limit.
Practical and Theoretical Implications
The paper ventures into practical applications reflected in Feynman rules and interaction processes. The treatment of Majorana fermions yields expanded Feynman diagram considerations due to their ability to act as their own antiparticles, thereby adding complexity to computation in particle physics processes such as annihilation and creation interactions.
Majorana and Weyl fermions in field theory reveal essential paradigms for model building, especially in symmetric theories, providing potential avenues for understanding neutrinos beyond the Standard Model given non-zero mass implications. This pedagogical insight extends to implications in CPT symmetry considerations and broader applications in theoretical physics.
Conclusion
Palash B. Pal’s discussion enriches the understanding of fermion types essential for advanced theoretical pursuits in particle physics. The conceptual clarity established through representation-independent manipulations and symmetry considerations serves as a foundation for further exploration of neutrino properties and potential advancements in supersymmetric theories.
