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[citation needed] 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
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.
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.
Examples are as well as the required structure of an orientation, needed for a notion of integration over all of the manifold
, generated by the Killing vector fields.
In this case the fields of the theory should transform in a representation of
For example, for Minkowski space, the symmetries are the Poincaré group
describing the (continuous) symmetries of internal degrees of freedom.
In field theory this connection is also viewed as a covariant derivative
whose action on various fields is defined later.
-valued 1-form on P satisfying technical conditions of 'projection' and 'right-equivariance': details found in the principal connection article.
Under a trivialization this can be written as a local gauge field
It is this local form of the connection which is identified with gauge fields in physics.
The collection of these, together with gauge fields, is the matter content of the theory.
This completes the mathematical prerequisites for a large number of interesting theories, including those given in the examples section above.
is flat, that is, (Pseudo-)Euclidean space, there are many useful simplifications that make theories less conceptually difficult to deal with.
The simplifications come from the observation that flat spacetime is contractible: it is then a theorem in algebraic topology that any fibre bundle over flat
, and therefore identify the connection globally as a gauge field
, and then we need not view fields as sections but simply as functions
In other words, vector bundles at different points are comparable.
Then the spacetime covariant derivative on tensor or spin-tensor fields is simply the partial derivative in flat coordinates.
However the gauge covariant derivative may require a non-trivial connection
which is considered to be the gauge field of the theory.
In weak gravitational curvature, flat spacetime often serves as a good approximation to weakly curved spacetime.
The Standard Model is defined on flat spacetime, and has produced the most accurate precision tests of physics to date.