Complex systems are composed of many particles or agents that move and interact with one another. In most real-world applications, these systems involve a varying number of particles or agents that change owing to interactions with the environment or their internal dynamics. The underlying mathematical framework to model these systems must incorporate the spatial transport of particles or agents and their interactions, as well as changes to their copy numbers, all of which can be formulated in terms of stochastic reaction-diffusion processes. However, the standard probabilistic representation of these processes can be overly complex because of the combinatorial aspects arising from the nonlinear interactions and varying particle numbers. This review addresses the main field theory representations of stochastic reaction-diffusion systems, which handle these issues “under the hood.” To begin, the focus is on bringing techniques familiar to theoretical physicists—such as second quantization, Fock space, path integrals, and quantum field theory—back into the classical domain of reaction-diffusion systems. It is demonstrated how various field theory representations, which have evolved historically, can be unified under a single basis-independent representation. Existing quantum-based methods and notation are then extended to work directly on the level of the unifying representation, and it is illustrated how they can be used to consistently obtain previous known results in a more straightforward manner, such as through numerical discretizations and relations between model parameters at multiple scales. Throughout, the review contextualizes how these representations mirror well-known models of chemical physics depending on their spatial resolution, as well as the corresponding macroscopic (large copy-number) limits. The framework presented here may find applications in a diverse set of scientific fields, including physical chemistry, theoretical ecology, epidemiology, game theory, and socioeconomic models of complex systems, specifically in the modeling and multiscale simulation of complex systems with varying numbers of particles or agents. The presentation is done in a self-contained educational and unifying manner such that it can be followed by researchers across several fields.

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• Chemical Physics & Physical Chemistry
• Chemical reactions
• Complex systems
• Stochastic processes
• Path-integral methods