Statistical field theory

Statistical field theory is the language we use when the microscopic description of a many-body system contains far more information than is needed to understand its large-scale behaviour. The basic move is to replace the individual degrees of freedom by coarse-grained fields - an order parameter, a density, a phase, a current - and to ask which effective theory governs their fluctuations. Near a continuous phase transition this leads naturally to universality: microscopically different systems can share the same long-distance physics because the renormalisation group removes details that are irrelevant at large scales. Fixed points, scaling dimensions and relevant deformations then organise the problem, while correlation functions encode the observable fluctuations. Statistical field theory is therefore not simply quantum field theory with the word statistical added to it. It is a general framework for turning the hierarchy of scales in a many-body problem into a controlled description of collective phenomena, whether the system is at equilibrium, close to a critical point, or driven away from equilibrium.

The modern subject has become both more precise and substantially broader. At criticality, the conformal bootstrap can determine scaling dimensions and operator-product data without relying on a weak-coupling expansion, while conformal perturbation theory provides a systematic way of using this information away from the fixed point. Renormalisation-group methods remain central, but they are now routinely combined with lattice calculations, Monte Carlo methods and high-precision numerical information. At the same time, statistical field theory has moved decisively into non-equilibrium physics. Martin-Siggia-Rose and Schwinger-Keldysh formulations allow noise, response and dissipation to be treated directly at the level of an action, while active matter forces us to confront field theories in which detailed balance is absent from the outset. Large-deviation methods and stochastic effective theories add a complementary description of rare fluctuations. The important point is that these are not disconnected developments: they are different ways of asking which parts of microscopic dynamics survive coarse graining and which constraints continue to control the infrared theory.

Our contributions sit on both sides of this equilibrium and non-equilibrium divide. We have developed and applied conformal perturbation theory to extract off-critical observables from critical data and confronted those predictions with lattice results, while related work has studied confinement and effective strings in three-dimensional gauge theories. We have also used duality and topological field theory to understand phases and infrared equivalences that are not captured by ordinary Landau symmetry breaking. On the non-equilibrium side, our work on flocking matter has identified thermodynamic constraints and exact scaling information, and our broader programme uses statistical field theory to connect active matter, fluctuating hydrodynamics and stochastic effective descriptions. Across these projects the aim is the same: not to write down the most general field theory that symmetry permits, but to determine which structures are actually forced by universality, probability, thermodynamics or the underlying microscopic dynamics, and then to test those structures against analytical, numerical or lattice information.