Time operator from parametrization invariance and implications for cosmology

Motivated by the parametrization invariance of cosmological Lagrangians and their equivalence to systems describing the motion of particles in curved backgrounds, we identify the phase space analogue of the notion of proper time. We define the corresponding quantum operator, which results in being canonically conjugate to that of the vanishing Hamiltonian. In the context of particle dynamics, this leads to an uncertainty relation of the form $\Delta E_0 \,\Delta T \geq \hbar$, where $E_0$ is the rest energy of the particle. By studying the non-relativistic limit, we show that the action of the operator reduces to multiplication by the classical time coordinate. Finally, we derive the generic expression for the introduced time variable in the cosmological setting.