Sunday, July 11, 2010
DANGER, Will Robinson! It DOES NOT Commute!
As I begin a detailed study of the Mathematics of Physics (known Physics) I shall begin with William Rowan Hamilton and the study of Quaternions, and Group Theory beginning with Galois Theory.
Each is dip and rich and old and ancient, but way important.
Consider the following equation, which is beautifully simple yet very key. It is the quaternion equation Hamilton came up with in an inspirational flash, and scratched on a bridge in Ireland:
i^2 = j^2 = k^2 = ijk = -1
Sweet. Vectors do NOT commute, not when multiplied together, meaning if the order is taken into account, you will get different results based on which vector you multiply first. Vectors do commute when added, but not when multiplied.
From that simple fact, all good things follow.