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Types of quantum field thories
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.
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 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).
Textbook references include
Pierre Deligne, Daniel Freed, Classical field theory (1999) (pdf)
this is a chapter in
P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
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
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