# Can the strong force be explained by quaternions?

I wonder if the strong force can be explained by quaternions without the use of matrices.

I heard that it could, but the source was not reliable.

• Probably they are relating something like pion model (isospin model to strong force), which has an algebraic strucuture equal to unit quaternions, but not the SU(3) QCD algebra. – Hydro Guy Dec 19 '14 at 22:27
• Quaternions can be represented by matrices, so that would be mathematically equivalent, anyway. – CuriousOne Dec 19 '14 at 23:08

## 1 Answer

Your source probably misunderstood.

In Gauge Theory, the U(1) group describes electromagnetism, SU(2) describes weak interaction, and SU(3) describes strong interaction.

Mathematically, U(1) is isomorphic to the unitary complex numbers (or the unit circle), SU(2) to unit quaternions (or the unit 3-sphere), and SU(3) to a more complicated thing.

Furthermore, as CuriousOne comments, elements of an n-dimensional field can be represented as nx1 matrices (n=4 for quaternion). The weak force can be explained with quaternions instead of matrices, but underneath it's really just the same math in different clothing.

Describing the strong force requires at least 8 independent terms, but quaternions only have 4. Maybe your source meant octonions.