nLab
classical field theory

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

Ordinary classical mechanics of point particles may be regarded as the theory of action functionals on mapping spaces of maps from the real line to some space.

In classical field theory one instead studies functionals on mapping spaces on higher dimensional domains.

Classical gauge theory

Of particular interest are classical field theories that are gauge theories. A powerful formalism for handling these is provided by BV theory, which effectively realizes spaces of classical fields as ∞-Lie algebroids. BV-formalism can be understood as a means to capture a classical gauge field theory in such a way that it lends itself to quantization. (See below)

Examples

Important examples of classical field theories are

Quantizaton of classical field theory

When it was realized that fundamental physics is governed by quantum field theory it became clear that classical field theory of fundamental fields can only be an approximation to the corresponding quantum field theory. If we think of quantum field theory in terms of functorial quantum field theory, then the domains of the mapping spaces mentioned above are the cobordisms that this FQFT is a functor on. The quantization of classical field theories to quantum field theories is a major issue in theoretical and mathematical physics (see also renormalization and geometric quantization).

References

Texbook references include

  • S. Flügge (ed.), Encyclopedia of Physics Volume III/I, Principles of Classical Mechanics and Field Theory, Springer 1960

For more see the references at multisymplectic geometry.

A discussion of recursive solutions to classical field equations and their relation to the quantum perturbation theory is in

  • Robert Helling?, Solving classical field equations (pdf)

Revised on September 22, 2013 18:41:18 by Urs Schreiber (89.204.139.76)