Using the mapping from the Fokker-Planck description of classical stochastic dynamics into a quantum Hamiltonian, we argue that a dynamical glass transition can be given a precise definition in terms of a static (equilibrium) quantum phase transition. This transition is characterized by a massive collapse of the excited states, leading to a divergent static quantum susceptibility throughout the glassy phase, which directly relates to a non-vanishing Edwards-Anderson order parameter. The quantum mechanical language allows to search for off-diagonal order parameters to detect the classical ‘dynamic order’ using transverse bases – an approach that corresponds to non-static observables in the original classical language. Even in the absence of a local order parameter, the transition can be detected via quantum fidelity measures on the ground state wavefunction, which we show to translate directly into a singularity in the heat capacity of the classical system.