This thesis develops a theory for approximate quantum time-correlation functions, Matsubara dynamics, that rigorously describes how to combine quantum statistics with classical dynamics. Matsubara dynamics is based on Feynman's path integral formulation of quantum mechanics and is expected to describe the physics of any system that satisfies quantum Boltzmann statistics and exhibits rapid quantum decoherence, e.g. liquid water at room temperature.

Having derived the Matsubara dynamics theory and explored the symmetry properties that it shares with the quantum Kubo time-correlation function, we demonstrate that two heuristic computational methods, Centroid Molecular Dynamics and Ring Polymer Molecular Dynamics, are based on quantifiable approximations to the Matsubara dynamics time-correlation function. This provides these methods with a stronger theoretical foundation and helps to explain their strengths and shortcomings. We then apply the Matsubara dynamics theory to a recently developed computational method of Poulsen $et\; al$. called the planetary model. We show that the planetary model is based on a harmonic approximation to Matsubara dynamics that is engineered to maintain the conservation of the quantum Boltzmann distribution, so quantum statistics and classical dynamics remain harmonised.

By making practical modifications to the planetary model, we were able to calculate infrared absorption spectra for a point charge model of condensed-phase water over a range of thermodynamic conditions. We find that this harmonic approximation to Matsubara dynamics provides a good description of bending and vibrational motions and is expected to be a useful tool for future spectroscopic studies of more complex, polarisable models of water.