  The emergence of classical behaviour in quantum theory is often ascribed to
the interaction of a quantum system with its environment, which can be
interpreted as environmental monitoring of the system. As a result,
off-diagonal elements of the density matrix of the system are damped in the
basis of a preferred observable, often taken to be the position, leading to the
phenomenon of decoherence. This effect can be modelled dynamically in terms of
a Lindblad equation driven by the position operator. Here the question of
decoherence resulting from a monitoring of position and momentum, i.e. a
phase-space measurement, by the environment is addressed. There is no standard
quantum observable corresponding to the detection of phase-space points, which
is forbidden by Heisenberg's uncertainty principle. This issue is addressed by
use of a coherent-state-based positive operator-valued measure (POVM) for
modelling phase-space monitoring by the environment. In this scheme,
decoherence in phase space implies the diagonalisation of the density matrix in
both position and momentum representations. This is shown to be linked to a
Lindblad dynamics where position and momentum appear as two independent
Lindblad operators.
