What is classical field theory really about?

Classical field theory admits a comprehensive mathematical formulation in a very general setting of reducible degenerate Lagrangian systems on an arbitrary fiber bundle. However, we really observe only three classical fields. These are Dirac fermions, electromagnetic and gravitational fields. It may happen that a Higgs vacuum is a classical field, too. One also considers classical theory of non-Abelian gauge fields though these fields that we know (W and Z bosons and gluons) are quantum fields.

A problem is that contemporary quantum field theory mainly is developed as quantization of classical fields. Namely, the propagators in Feynman diagrams of perturbative quantum field theory are Green’s functions of classical field equations. The Feynman diagram technique is formalized in the terms of functional integrals which contain classical field Lagrangians. Thus, classical field theory is a necessary step towards quantum field theory of today. Note that there are many different hypothetic quantum fields in contemporary quantum filed models. To describe them, one therefore, one should be able to consider the classical counterparts of these fields. For instance, it may happen that ghosts, which play the role of parameters of gauge symmetries in BRST theory, are real fields.

However, there are classical field models which can not be quantized in principle. Quantization of a Lagrangian field theory essentially depends on its degeneracy characterized by a family of reducible Noether identities. Such a family is defined if and only if a certain condition holds. Fortunately, this is the case of irreducible Lagrangian systems, including all the above-mentioned field theories.

It is a general tendency that theoreticians develop their theory in a very general form though only its particular variant is realistic. Then we understand its peculiarity. For instance, gravitation theory on an arbitrary manifold is formulated, but a real space-time apparently is globally hyperbolic and parallelizable. Hamiltonian mechanics on an arbitrary symplectic manifold is considered, but a momentum phase space of a mechanical system usually is the cotangent bundle over its coordinate space, and so on.

References:

G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, 2009)