Quantum field theory is a framework in which physical fields are treated as quantum systems. Mathematically, a field assigns an operator or generalized function to each point of spacetime, while states belong to a Hilbert space. Observables localized in spacetime regions are represented by operators, and their correlation functions encode the predictions of the theory.
Two complementary mathematical formulations are widely used. In the operator approach, fields are operator-valued distributions subject to locality and symmetry axioms. In the functional integral approach, correlation functions are expressed formally as integrals over a space of fields. After analytic continuation to Euclidean space, the latter leads to Euclidean quantum field theory, whose axioms can reconstruct a Lorentzian theory.
Quantum field theory originated in physics, but its rigorous formulations and connections with geometry, topology, representation theory, and functional analysis make it an active mathematical subject.
