U(N) Based Transformations in N-Squared Dimensions

TL;DR

This paper introduces a method to construct transformations in N-squared dimensional manifolds based on the properties of the unitary group U(N), generalizing spacetime symmetries beyond four dimensions.

Contribution

It provides a novel framework for deriving transformations in higher-dimensional spaces using U(N), extending familiar spacetime concepts to arbitrary N.

Findings

For N=2, recovers familiar spacetime transformations

Identifies invariant 3+1 dimensional subspaces within higher-dimensional manifolds

Shows boosts do not preserve intervals for N>2

Abstract

Consider the example of the relationship of the group O(3) of rotations in 3-space to the special unitary group SU(2). Given other unitary groups, what transformations can we find? In this paper we describe a method of constructing transformations in an N-squared dimensional manifold based on the properties of the unitary group U(N). With N = 2, the method gives the familiar rotations, boosts, translations and scale transformations of four dimensional spacetime. In the language of spacetime now applied for any N, we show that rotations preserve distances and times. There is a scale-changing transformation. For N more than 2, boosts do not preserve generalized spacetime intervals, but each N-squared dimensional manifold contains a 3+1 dimensional subspace with invariant spacetime intervals. That each manifold has a four dimensional spacetime subspace correlates with the property of U(N) that U(2) is a subgroup.