Abstract
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.