Ordinary classical mechanics of point particles may be regarded as the theory of action functionals on mapping space?s of maps from the real line to some space.
In classical field theory? one instead studies functionals on mapping space?s on higher dimensional domains.
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)
Important examples of classical field theories are
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 corresponginf quantum field theory. If we think of quantum field theory in terms of functorial quantum field theory, then the domains of the mapping spaces mentionmed 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).