On the Clifford-Algebraic Structure of the First-Order Klein-Gordon Factorisation

This work revisits the first–order Lorentz–covariant factorisation of the Klein–Gordon operator without assuming a priori that fields take values in a complex vector space. It is shown that the requirement of a first–order factorisation already induces the spacetime Clifford algebra as the natural algebraic framework.

Formulating the problem intrinsically leads to a carrier space admitting both left and right Clifford action. Within this setting, the classification of admissible factorisations, together with the requirement of physically propagating solutions, selects a class of equations equivalent to the Dirac equation up to algebraic realisation. In these cases, the complex unit arises geometrically as a spacelike bivector acting on the field.

The analysis clarifies the algebraic origin of the Dirac equation and its associated complex structure, showing that both emerge from the structure of the factorisation problem rather than being imposed from the outset.