  We show how continuous matrix product states of quantum field theories can be
described in terms of the dissipative non-equilibrium dynamics of a
lower-dimensional auxiliary boundary field theory. We demonstrate that the
spatial correlation functions of the bulk field can be brought into one-to-one
correspondence with the temporal statistics of the quantum jumps of the
boundary field. This equivalence: (1) illustrates an intimate connection
between the theory of continuous quantum measurement and quantum field theory;
(2) gives an explicit construction of the boundary field theory allowing the
extension of real-space renormalization group methods to arbitrary dimensional
quantum field theories without the introduction of a lattice parameter; and (3)
yields a novel interpretation of recent cavity QED experiments in terms of
quantum field theory, and hence paves the way toward observing genuine quantum
phase transitions in such zero-dimensional driven quantum systems.
