Quaternion conjugate

A check can be made on the interpretation of the quaternion-valued metric if we take the quaternion conjugate ... 

The group Sp(n) consists of n x n matrices with quaternion entries such that the inverse is the conjugate transpose... 

In evaluating the PF ineq. (37) note that the conjugation operationgkgtgkT1, or kik l, is to be regarded as a single operation. The quaternion parameters X, A for the product of... 

Since there is no reflection symmetry in the quaternion formulation, the reflected quaternion q 1 must be distinguishable from. The conjugate differential fine element to ds is ds = q dxVL. The product of the quaternion and conjugate quaternion line elements is then the real-number-valued element that corresponds to the squared differential element of the Riemannian geometry ... 

In a similar fashion, application of the preceding analysis to the conjugated quaternion fields yields the accompanying equation to (40) ... 

Multiplying (40) on the right with the conjugated quaternion q and (41) on the left with the quaternion qy, then adding the resulting equations and using the identity... 

If we signify by Lm the part of the Lagrangian density that yields the matter variables on variation with respect to the quaternion variables, then the total Lagrangian density is L Lh I LM. Its variation with respect to the conjugated quaternion variables then yields the field equations [25] ... 

Variation with respect to the quaternion variables yields the conjugated quaternion field equation ... 

Note that the source term Tp is a quaternion and T p is a conjugated quaternion.)... 

Thus we have seen that the factorized quaternion field equations (46a) (or their conjugated equations (46b)) -the irreducible form of electromagnetism... 

As transposition of a product of matrices, conjugation of a quaternion product reverses the order of its factors ... 

In this section we will revisit KS regularization and present a new, elegant derivation of it, using quaternion algebra and an unconventional conjugate u referred to as the star conjugate of the quaternion u = uq + iu +j U2 + ku3 ... 

---