Quaternion commutators

In Eq. (5), the product q q is quaternion-valued and non-commutative, but not antisymmetric in the indices p and v. The B<3> held and structure of 0(3) electrodynamics must be found from a special case of Eq. (5) showing that 0(3) electrodynamics is a Yang-Mills theory and also a theory of general relativity [1]. The important conclusion reached is that Yang-Mills theories can be derived from the irreducible representations of the Einstein group. This result is consistent with the fact that all theories of physics must be theories of general relativity in principle. From Eq. (1), it is possible to write four-valued, generally covariant, components such as... 

In general, all the off-diagonal elements of the quaternion-valued commutator term [the fifth term in Sachs Eq. (4.19)] exist, and in this appendix, it is shown, by a choice of metric, that one of these components is the Ba> field discussed in the text. The B<3) field is the fundamental signature of 0(3) electrodynamics discussed in Vol. 114, part 2. In this appendix, we also give the most general form of the vector potential in curved spacetime, a form that also has longitudinal and transverse components under all conditions, including the vacuum. In the Maxwell-Heaviside theory, on the other hand, the vector... 

Thus the multiplication of quaternion units is non-commutative. In eq. (1) q is to be interpreted as a compound symbol that stands for two different objects the real quaternion,... 

The product of two rotations is a rotation. Obtain an expression for the Cayley-Klein parameters of the product as a function of the parameters of its factors. Is the product commutative The SU 2) matrices may also be identified as normalized quaternions. 

In the 4D equation, space and time coordinates are inextricably entangled. Its mathematical solutions are hypercomplex functions, or quaternions, without a commutative algebra. Quaternions are used to describe what is known as spherical rotation, also called the spin function, and the complex rotation known as the Lorentz transformation of special relativity. 

The appearance of non-commuting quantum variables can now also be traced back to the non-commutative algebra of 4D hypercomplex functions. On projection into 3D by the separation of space and time variables, the quaternion variables are reduced to complex functions that characterize orbital angular momentum, but the commutation properties remain. Not appreciating the essence of complex wave... 

The final result depends on the order in which the operations are applied, because of the fact that the quaternions t] and do not commute. The quantity Q is called the tensor (stretcher), and the exponential is called the versor (turner) of the operator. 

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