Time Optimal Quantum Control of Spinor States

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.