Mass is conventionally introduced into physical theories as a passive parameter, m₀. As such, it plays no dynamical role in the theory, nor can it change. But in practice, particles decay and recombine, changing their mass. They also acquire binding energies, changing their mass, and may also have an energy uncertainty, and so also a mass uncertainty. Similarly, the proper time of a particle is described along its trajectory. But quantum mechanically, trajectories can be split and recombined, or they may not be well-defined at all. So the proper time also has a dynamical role to play. We also show that there is a natural extension to the equivalence principle that is needed to include unstable particles. Both proper time and mass should be treated as quantum-mechanical operators, whose values are determined by measurement. The Hamiltonian formalism has a natural extension to include them as an extra coordinate and conjugate momentum, allowing one to construct both a classical and quantum theory of particles that can decay, have binding energies and obey the uncertainty principle.