We first review the basic theory of a general class of symmetric spaces with canonical reflections, midpoints, and displacement groups. We introduce a notion of gyrogroups established by A. A. Ungar and define gyrovector spaces slightly different from Ungar's setting. We see the categorical equivalence of symmetric spaces and gyrovector spaces with respect to their corresponding operations. In a smooth manifold with spray we define weighted means using the exponential map and develop the Lie-Trotter formula with respect to midpoint operation. Via the idea that we associate a spray with a Loos symmetric space, we construct an analytic scalar multiplication on a smooth gyrocommutative gyrogroup with unique square roots. We furthermore develop the concepts of parallel transport and parallelogram. Later we see that the exponential map associated with spray in the Finsler gyrovector space with seminegative curvature gives us a length minimizing geodesic. Analogous to define a partial order on a vector space, we construct the partial order on the gyrovector space and investigate its properties related with what we call the gyrolines and the cogyrolines. Finally we apply the concept of gyrogroup structure to the setting of density matrices, especially qubits generated by Bloch vectors, and show the equivalence between the set of Bloch vectors and the set of Lorentz boosts.
Networked Digital Library of Theses and Dissertations