This manuscript explores the connection between the fast relaxation processes in an undeformed polymer glass and essential trends in the mechanical toughness, a nonlinear mechanical property that is of practical interest for engineering polymers with high impact strength. We quantify the time scale of the molecular relaxations in the subnanosecond regime for a quiescent polycarbonate glass using inelastic and quasi-elastic neutron scattering and then correlate these processes with the macroscopic brittle-to-ductile transition (BDT), which demarcates a change in the dominant mechanism of failure and a marked increase in the material toughness. We show that the macroscopic phenomenon of the BDT corresponds to a change in the dominant dynamical process at the nanoscale. The brittle regime is characterized by collective vibrational modes (the so-called Boson peak) with a characteristic time scale tau approximate to 0.5-0.8 ps, while slower collective relaxations with tau approximate to 3 ps become dominant above the BDT. We further establish that the onset of ductility coincides with the appearance of anharmonicity in the mean-square atomic displacement < u(2)>. on a picosecond time scale, emphasizing that fast anharmonic molecular motions are important for energy dissipation. This builds upon our previous report correlating toughness with the amplitude of these anharmonic fluctuations across a wide range of polycarbonate glasses. Brillouin light scattering measurements are used to characterize the bulk and shear moduli of the material, revealing a concomitant upturn in Poisson's ratio in the region of the BDT, a phenomenon that has been reported in metallic and oxide glasses. The ratio of transverse acoustic mode velocity and the Boson peak frequency is used to estimate the length scale for these processes, indicating that the dynamic heterogeneities are collective across 100-1000s of atoms. These length scales are strikingly similar to the activation volume of yield derived from mechanical measurements, suggesting that these fast and collective relaxation processes may be related to the mechanisms of yield.