Quantum Symmetry: Projective representations of relativistic and symplectic groups; 2nd Edition [Print Replica] Kindle Edition

This book is intended for graduate students and researchers interested in the general mathematical framework of projective representations and its application to groups that are of physical interest in the study of physical quantum symmetries. Projective representations are required for quantum symmetries as physical transition probabilities between physical states in quantum theory are given by the square of the modulus of the states. This allows a phase degree of freedom, the quantum phase, that manifests in the symmetries through projective representations.Part I of the book is a primer of the mathematical theory required for the study of symmetries. Lie groups (and in particular, matrix groups) are reviewed. The general properties of their representations are studied. Finally, the theory of projective representations is developed for connected Lie groups.Part II applies this to the inhomogeneous Lorentz group to describe the inertial states of special relativistic quantum mechanics and then is also applied to the Galilean 'nonrelativistic' limit. Part III studies the Weyl-Heisenberg group and its origin in the projective representations of the inhomogeneous symplectic group, and its orthogonal and unitary subgroups, that are basic symmetries of the Hamilton formulation of mechanics. The projective representations are directly computed using the central extension of the inhomogeneous symplectic group. This shows how the Heisenberg commutation relations are a direct result of the transition probabilities being given by the square of the modulus of states and the resulting quantum phase. Read more