The rotations of four-dimensional Euclidean space

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Lie groups and Lie algebras: rotations, SU(2) and the double cover

Problem

The rotations of four-dimensional Euclidean space, which keep lengths and orientation, form

SO(4)

, the real

4\times4

orthogonal matrices

\mathsf{R}

with

\mathsf{R}^{\mathsf T}\mathsf{R}=\mathsf{1}

and

\det \mathsf{R}=1

. ((a)) Count the dimension of

SO(4)

from the orthogonality condition on a general real

4\times4

matrix. ((b)) State how many independent Hermitian generators a one-parameter-subgroup construction requires, and identify them with rotations in the coordinate planes of the four axes.

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