Dirac Matrices and Feynman's Rest of the Universe

TL;DR

This paper explores the relationship between Dirac matrices, symmetries in quantum mechanics, and space-time, proposing a connection between the 'rest of the universe' and Lorentz group symmetries.

Contribution

It demonstrates how the symmetry group of coupled oscillators can extend to a larger group related to space-time symmetries, linking quantum mechanics and the universe's structure.

Findings

The fifteen Majorana matrices generate the SL(4,r) group.

The ten generators of Sp(4) relate to quantum mechanics.

The transition from Sp(4) to SL(4,r) involves phase space area changes.

Abstract

There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four $\gamma$ matrices. These fifteen matrices can also serve as the generators of the group $SL(4,r)$. The second set consists of ten generators of the $Sp(4)$ group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of $Sp(4)$ to that of $SL(4,r)$ if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman's rest of the universe makes this $Sp(4)$-to-$SL(4,r)$ transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups $SL(4,r)$ and $Sp(4)$ are locally isomorphic to the Lorentz groups O(3,3) and O(3,2) respectively. This allows us to interpret Feynman's rest of the universe in terms of space-time symmetry.