Massless Majorana and Weyl fermions cannot be distinguished

In this note, I will demonstrate that a theory of a massless neutrino (in 3+1-dimensional quantum field theory) can be described either as a theory of a massless Weyl fermion or as a theory of a massless four-component Majorana fermion. The two formulations are indistinguishable, as they arise from exactly the same Lagrangian when expressed in terms of two-component fermions. I will also exhibit the equivalence of one massive Dirac fermion with a theory of two mass-degenerate Majorana fermions. Finally, I will argue that no discontinuities arise when taking the m → 0 limit of a theory of one Majorana fermion of mass m, or when taking the m1 → m2 limit of a theory of two Majorana fermions of mass m1 and m2, respectively. First, some facts about two-component anti-commuting spinors. Introduce undotted spinors ξα with α = 1, 2 and dotted spinors η α̇ ≡ η. Indices are raised and lowered with ǫ = iσ (where σ is the usual Pauli matrix) and ǫαβ = −iσ (ǫ-tensors with dotted indices are defined similarly). We introduce σ = (I;σ) and σ = (I;−σ), where I is the 2×2 identity matrix. Explicitly, σ and σ possess the following spinor index structure: σ αα̇ and σ . Finally, spinor products are defined as