Relational Analysis of Dirac Equation in Momentum Representation

TL;DR

This paper derives the Dirac equation in momentum space from a relational approach using binary complex relations, showing that the 4D pseudo-Euclidean space naturally emerges without prior assumption.

Contribution

It introduces a relational framework to derive the Dirac equation, emphasizing the emergence of momentum space from fundamental relations rather than geometric assumptions.

Findings

Dirac equation derived from binary relation systems

Momentum space emerges naturally from relational considerations

Constructed bispinor wave functions for free fermions

Abstract

In terms of the relational approach to space-time geometry and physical interactions, we show that the Dirac equation for a free fermion in the momentum representation can be obtained starting from a binary system of complex relations (BSCR) between elements of two abstract sets. With the derivation performed we show that the 4-dimensional pseudo-Euclidean momentum space is not needed a priori but naturally emerges from considerations of rather general character (2-spinor algebra). A bispinor wave function is constructed for a fermion with positive energy and an arbitrary distribution of momenta. Special attention is paid to physical assumptions that should be made to enable the construction.