  We present a theoretical investigation of the properties of quantum
correlation functions in a dissipative multimode system. We define a total mth
order equal-time correlation function, summed over all modes, which is shown to
be conserved if the Hamiltonian possesses U(1) symmetry, provided any
dissipation processes are linear in the system operators. As examples, we
demonstrate this conservation using numerical simulations of a coupled cavity
system and the Jaynes-Cummings model.
