We develop the path integral formalism for studying cosmological
perturbations in multi-field inflation, which is particularly well suited to
study quantum theories with gauge symmetries such as diffeomorphism invariance.
We formulate the gauge fixing conditions based on the Poisson brackets of the
constraints, from which we derive two convenient gauges that are appropriate
for multi-field inflation. We then adopt the in-in formalism to derive the most
general expression for the power spectrum of the curvature perturbation
including the corrections from the interactions of the curvature mode with
other light degrees of freedom. We also discuss the contributions of the
interactions to the bispectrum.