Five-dimensional spinor bilinears

Explicit spatial Clifford algebra

Use the Hermitian Pauli matrices $X,Y,Z$ and define

$$\gamma_1=X\otimes I_2,\quad \gamma_2=Y\otimes I_2,\quad \gamma_3=Z\otimes X,\quad \gamma_4=Z\otimes Y,\quad \gamma_5=Z\otimes Z. \tag{1}$$

They obey $\{\gamma_a,\gamma_b\}=2\delta_{ab}I_4$. These matrices describe spinor components in five ordinary spatial tangent directions. Matrix rank is not spatial dimension. Set

$$C=\gamma_2\gamma_4.$$

Direct multiplication gives

$$C^T=-C,\qquad C^*=C,\qquad C^2=-I_4,\qquad C^\dagger C=I_4, \qquad C\gamma_aC^{-1}=\gamma_a^T. \tag{2}$$

The transpose-intertwining sign in (2) is positive. In odd dimensions the admissible sign matters. The convention is the $d\bmod8=5$ case of Van Proeyen, Tools for supersymmetry, §3.1, equation (3.11) and Table 1. The equations here provide an explicit verification independently of the classification table.

Rotation invariance of the bilinear

The infinitesimal spatial spin generators are

$$\Sigma_{ab}=\frac14[\gamma_a,\gamma_b].$$

Taking the transpose reverses the commutator, so (2) gives

$$\Sigma_{ab}^T C+C\Sigma_{ab}=0. \tag{3}$$

Exponentiating any real combination of these generators yields $S^TCS=C$. Therefore

$$B(\psi,\chi)=\psi^T C\chi \tag{4}$$

is a nondegenerate complex bilinear invariant under $\operatorname{Spin}(5)$. On a Riemannian five-dimensional spin manifold it patches consistently under the spin-frame transition maps and is preserved by the spin connection. Additional twisting bundles or sector representations must supply their own invariant contractions.

Quaternionic reality and field statistics

Let $\mathcal K$ denote componentwise complex conjugation. The antilinear map $J=C\mathcal K$ satisfies

$$J\gamma_a=\gamma_aJ,\qquad J^2=-I_4. \tag{5}$$

It supplies a quaternionic structure. In particular, $J\psi=\psi$ has only the zero solution, because applying $J$ again gives $-\psi=\psi$. This is a different statement from the existence of the invariant bilinear (4).

Statistics and the other field indices now matter. For commuting components, antisymmetry gives $\psi^T C\psi=0$, while $\psi^T C\chi$ can be nonzero for independent component vectors. For Grassmann-odd components,

$$\psi^T C\psi=2\sum_{i<j}C_{ij}\psi_i\psi_j \tag{6}$$

need not vanish. A multiplet also carries whatever symmetry conditions apply to its additional labels; those conditions cannot be read from the spatial Clifford matrices alone.

Under a common phase $\psi\mapsto e^{i\alpha}\psi$, a bilinear with two such factors has weight two. An action required to conserve that phase charge cannot contain this term with a neutral fixed coefficient. Establishing the relevant conserved charge and its full interaction content is an action-level calculation.

Use in the coupling investigation

Equations (2)–(6) supply concrete algebra for a five-dimensional sector calculation. They do not identify the physical neutrino field, impose its statistics, fix its mass, or settle lepton-number conservation. A physical Majorana or Dirac conclusion additionally requires the spacetime spinor structure, the occupied bundle, the complete allowed interactions, and the associated symmetries. The neutrino channel control states the separate neutral and nonzero-response conditions to test.

Verification

geometric_channel_checks.py checks all Clifford anticommutators, the transpose and reality relations, all infinitesimal bilinear identities, finite spin rotations, and explicit commuting and Grassmann coefficient controls. A passing matrix calculation does not establish an all-orders physical selection rule.