nLab classical field theory

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

• electromagnetism

• gravity, supergravity

Quantization 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

Textbook accounts:

• Clifford Truesdell, Richard A. Toupin: The Classical Field Theories, in: Principles of Classical Mechanics and Field Theory, Encyclopedia of Physics, volume III/1, Springer (1960) 226-858 [doi:10.1007/978-3-642-45943-6_2]

• S. Flügge (ed.), Encyclopedia of Physics Volume III/I, Principles of Classical Mechanics and Field Theory, Springer 1960 [doi:10.1007/978-3-642-45943-6]

• Pierre Deligne, Daniel Freed, Classical field theory (1999) (pdf)

  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)

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, Advanced classical field theory, World Scientific (2009) [doi:10.1142/7189]

• Jean-Louis Basdevant: Variational Principles in Physics, Springer (2007) [doi:10.1007/978-3-031-21692-3]

Discussion of recursive solutions to classical field equations and their relation to the quantum perturbation theory:

• Robert Helling, Solving classical field equations (pdf)

On smooth sets as a convenient category for variational calculus of Lagrangian classical field theory:

• Grigorios Giotopoulos, Hisham Sati, Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]