A new perspective on spacetime 4D rotations and the SO(4) transformation group

Abstract

In quantum field theory, every conservation law is enforced by adequate symmetries. The Lorentz group, which represents two continuous symmetries: rotations in 3D Euclidean space and Lorentz boosts, generates conservation of the angular momentum tensor. This article shows that both Lorentz boosts and 3D rotations can be replaced by spatio-temporal rotations in Minkowski spacetime. Noether’s theorem certifies that the consequences of 4D rotations are identical to the Lorentz transformation group SO(4). I demonstrate how to build a spatio-temporal rotation matrix that preserves the spacetime interval, and that the resulting matrix represents the special orthogonal group of order 4 (SO(4)). Keywords: 4D rotations, Lorentz transformation, Noether’s theorem