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We present a new approach for investigating quantum effects in laser-driven plasma. Unlike the modelling strategies underpinning particle-in-cell codes that include the effects of quantum electrodynamics, our new field theory incorporates multi-particle effects from the outset. Our approach is based on the path-integral quantisation of a classical bi-scalar field theory describing the behaviour of a laser pulse propagating through an underdense plasma. Results established in the context of quantum field theory on curved spacetime are used to derive a non-linear, non-local, effective field theory that describes the evolution of the laser-driven plasma due to quantum fluctuations. As the first application of our new theory, we explore the behaviour of perturbations to fields describing a uniform, monochromatic, laser beam propagating through a uniform plasma. Our results suggest that quantum fluctuations could play a significant role in the evolution of an underdense plasma, with plasma frequency $10\,{\rm THz}$, driven by a laser pulse with wavelength $\lesssim 150\,{\rm nm}$, width $30\,{\mu\rm m}$ and duration $>100\,{\rm fs}$. Such parameters should be realisable in an experiment using a facility such as the European XFEL.