Rotations and e, $\nu$ Propagators, Part III

TL;DR

This paper investigates the conditions under which electron and neutrino propagators derived from rotation invariant projection operators maintain spacetime symmetry, revealing that only specific bases spanning four-dimensional space satisfy this symmetry.

Contribution

It demonstrates that only two specific bases from previous parts produce spacetime symmetric propagators, providing a geometric interpretation involving four-dimensional planes.

Findings

Only the bases considered in Parts I and II yield spacetime symmetric propagators.

Spacetime symmetry requires bases spanning four dimensions in Euclidean space.

The unfaithfulness of the spinor representation explains the four-component structure of spin 1/2 wave functions.

Abstract

In Parts I and II we showed that e, $\nu$ propagators can be derived from rotation invariant projection operators, thereby providing examples of how quantities with spacetime symmetry can be obtained by constraining rotationally symmetric objects. One constraint is the restriction of the basis; only two kinds of bases were considered, one for the electron and one for the neutrino. In this part, we find that, of a wide range of bases each consistent with the constraint process, only the two kinds of bases considered in Parts I and II give spacetime symmetric propagators. We interpret the result geometrically. The spinor representation is unfaithful in four dimensional Euclidean space which explains why spin 1/2 wave functions have four, not two, components. Then we show how a basis relates to two planes in four dimensional Euclidean space. A pair of planes spanning two or three dimensions does not allow spacetime symmetry. Spacetime symmetry requires two planes that span four dimensions. PACS: 11.30.-j, 11.30.Cp, and 03.65.Fd