  Caldeira and Leggett (CL) in a seminal paper derived a master equation
describing Markovian Quantum Brownian motion. Such an equation suffered of not
being completely positive, and many efforts have been made to solve this issue.
We show that, when a careful mathematical analysis is performed, the model
considered by CL leads to a non dissipative master equation. We argue that the
correct way to understand the master equation derived in the CL regime is to
consider it non-Markovian. Moreover, we show that if one wants to provide a
microscopic description of Quantum Brownian motion with the CL model, one
always needs to consider a non-Markovian dynamics. We conclude that dissipation
is a genuinely non-Markovian feature.
