In quantum theory, observables – things that are presumed to be measurable – are represented by linear operators on a Hilbert space, and states can be represented by elements of the Hilbert space. Quantum field theory (QFT) is a refinement of quantum theory in which each region of spacetime has an associated set of observables. The path integral formulation of QFT captures that association, and it also blurs the distinction between operators and states. An operator associated with a spacetime region R can be implemented by modifying the integrand of the path integral in a way that involves only the integration variables (field variables) in R. Evaluating the integrals over just those variables leaves a path integral with a state defined on the boundary of the now-excised region R. This leads to a way of thinking about QFT as a device that relates Hilbert spaces on different boundaries of spacetime to each other, sometimes called functorial QFT. This article introduces that way of thinking about QFT. The conventional concept of time evolution is a special case in which the path integral relates the Hilbert space associated with the initial time to the Hilbert space associated with the final time.
