  An analysis of the motion of a relativistic electron under a linear
constraint in four dimensions is presented. Interesting results are given that
show that the state of the electron is well defined under the formalism of time
optimal quantum state control. We establish compact mechanisms for achieving
time dependent unitary evolution, and present new calculation methods for
time-ordered exponential operators. A powerful modification of the
brachistochrone technique is presented that allows solution of a class of
problems via matrix decomposition of the Hamiltonian. These techniques allow us
to arrive at a series of solutions for quantum systems that have readily
accessible physical realisations. We contrast the output of the theory when
constrained to a relativistic space-time to that of other physical systems of
lower dimensionality. Some comment is given regarding hypercomplex numbers and
their application in quantum mechanics.
