  Free classical particles have well-defined momentum and position, while free
quantum particles have well-defined momentum but a position fully delocalized
over the sample volume. We develop a many-body formalism based on wave-packet
operators that connects these two limits, the thermal energy being distributed
between the state spatial extension and its thermal excitation. The
corresponding `mixed quantum-classical' states, which render the Boltzmann
operator diagonal, are the physically relevant states when the temperature is
finite. The formulation of many-body Hamiltonians in terms of these thermally
excited wave-packets and the resulting effective scatterings is provided.
