Quantum Symmetries and the Robustness of Dynamics

A privileged role in the description of a quantum system is played by its symmetries, namely the operators that remain constant in time in the Heisenberg picture. It is well known that these correspond to operators that commute with the Hamiltonian of the system. When the Hamiltonian is slightly perturbed, however, symmetries do not all behave in the same way. Some remain nearly conserved throughout the evolution and are therefore called robust symmetries. Others instead drift significantly from their initial values over sufficiently long times, and are known as fragile symmetries. This thesis provides a precise characterization of both robust and fragile symmetries, together with quantitative bounds on their deviation under perturbations.