Covariant classical field theory

In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics is formulated as covariant classical field theory on fiber bundles over the time axis R. Many important examples of classical field theories which are of interest in quantum field theory are given below. In particular, these are the theories which make up the Standard model of particle physics. These examples will be used in the discussion of the general mathematical formulation of classical field theory. Scalar field theory Klein−Gordon theory Spinor theories Dirac theory Weyl theory Majorana theory Gauge theories Maxwell theory Yang–Mills theory. This is the only theory in the uncoupled theory list which contains interactions: Yang–Mills contains self-interactions. Yukawa coupling: coupling of scalar and spinor fields. Scalar electrodynamics/chromodynamics: coupling of scalar and gauge fields. Quantum electrodynamics/chromodynamics: coupling of spinor and gauge fields. Despite these being named quantum theories, the Lagrangians can be considered as those of a classical field theory. In order to formulate a classical field theory, the following structures are needed: A smooth manifold . This is variously known as the world manifold (for emphasizing the manifold without additional structures such as a metric), spacetime (when equipped with a Lorentzian metric), or the base manifold for a more geometrical viewpoint. The spacetime often comes with additional structure. Examples are Metric: a (pseudo-)Riemannian metric on .