  A wave function exposed to measurements undergoes pure state dynamics, with
deterministic unitary and probabilistic measurement induced state updates,
defining a quantum trajectory. For many-particle systems, the competition of
these different elements of dynamics can give rise to a scenario similar to
quantum phase transitions. To access it despite the randomness of single
quantum trajectories, we construct an $n$-replica Keldysh field theory for the
ensemble average of the $n$-th moment of the trajectory projector. A key
finding is that this field theory decouples into one set of degrees of freedom
that heats up indefinitely, while $n-1$ others can be cast into the form of
pure state evolutions generated by an effective non-Hermitian Hamiltonian. This
decoupling is exact for free theories, and useful for interacting ones. In
particular, we study locally measured Dirac fermions in $(1+1)$ dimensions,
which can be bosonized to a monitored interacting Luttinger liquid at long
wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a
quantum Sine-Gordon model with complex coefficients. A renormalization group
analysis reveals a gapless critical phase with logarithmic entanglement entropy
growth, and a gapped area law phase, separated by a
Berezinskii-Kosterlitz-Thouless transition. The physical picture emerging here
is a pinning of the trajectory wave function into eigenstates of the
measurement operators upon increasing the monitoring rate.
