nLab Tomonaga-Schwinger equation



The generalization of the Schrödinger equation from quantum mechanics to quantum field theory:

from Torre-Varadarajan 98 p.2:

The idea of evolving a quantum field from any Cauchy surface to any other seems to have originated in the mid 1940’s with the work of Tomonaga [1] and Schwinger [2] on relativisticquantum field theory. Tomonaga and Schwinger wanted an invariant generalization of the Schrödinger equation, which describes time evolution of the state of a quantum field relative to a fixed inertial reference frame. By allowing for all possible Cauchy surfaces in the description of dynamical evolution one easily accommodates all possible notions of time for all possible inertial observers. Thus a dynamical formalism incorporating arbitrary Cauchy surfaces does allow for an invariant generalization of the Schrödinger equation. Since, the space of Cauchy surfaces is infinite-dimensional, it is impossible to describe time evolution along arbitrary surfaces by using a single time parameter. In essence, one needs a distinct time parameter for every possible foliation of spacet ime. As shown by Tomonaga and Schwinger, if one formulates dynamics in terms of general Cauchy surfaces, the resulting dynamical evolution equation is, formally, a functional differential equation, which is usually called the “Tomonaga-Schwinger equation”.


Named after Shin'ichirō Tomonaga and Julian Schwinger.

Last revised on September 8, 2017 at 09:27:21. See the history of this page for a list of all contributions to it.