Summary: The reaction-diffusion master equation (RDME) is a model for chemical systems in which both noise in the chemical reaction process and the diffusion of molecules are important. It extends the chemical master equation for well-mixed chemical reactions by discretizing space into a collection of voxels. In this work, we show how the RDME may be rewritten as an equivalent ’particle tracking’ model following the motion and interaction of individual molecules on a lattice. This new representation can be interpreted as a discrete version of the spatially-continuous ‘probability distribution function’ stochastic reaction-diffusion model studied by M. Doi [J. Phys. A, Math. Gen. 9, 1465–1477 (1976)]. We show how this new representation can be mapped to a quantum field theory, complementing the existing work by Peliti mapping the RDME, and Doi mapping his spatially continuous model, to quantum field theories. The formal continuum limit, as the voxel size approaches zero, of the ‘particle tracking’ representation is studied to consider the question of whether the RDME approximates any spatially continuous model.