  We recently obtained a quantum-Boltzmann-conserving classical dynamics by
making a single change to the derivation of the `Classical Wigner'
approximation. Here, we show that the further approximation of this `Matsubara
dynamics' gives rise to two popular heuristic methods for treating quantum
Boltzmann time-correlation functions: centroid molecular dynamics (CMD) and
ring-polymer molecular dynamics (RPMD). We show that CMD is a mean-field
approximation to Matsubara dynamics, obtained by discarding (classical)
fluctuations around the centroid, and that RPMD is the result of discarding a
term in the Matsubara Liouvillian which shifts the frequencies of these
fluctuations. These findings are consistent with previous numerical results,
and give explicit formulae for the terms that CMD and RPMD leave out.
