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.
In order to formulate a classical field theory, the following structures are needed:
A smooth manifold
M
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
g
M
as well as the required structure of an orientation, needed for a notion of integration over all of the manifold
M
The spacetime
M
g
M
Aut(M)
Aut(M)
Iso(1,3)
G
ak{g}
G
P
G
P\xrightarrow{\pi}M
\pi
P
M
\nabla
A principal connection denoted
l{A}
ak{g}
Under a trivialization this can be written as a local gauge field
A\mu(x)
ak{g}
U\subsetM
M
E\xrightarrow{\pi}M
P
\rho.
For completeness, given a representation
(V,G,\rho)
E
V
A field or matter field is a section of an associated vector bundle. The collection of these, together with gauge fields, is the matter content of the theory.
A Lagrangian
L
E'\xrightarrow{\pi}M
L:E' → R
Suppose that the matter content is given by sections of
E
V
E'
p
V ⊗
| *M | |
| T | |
| p |
L
This completes the mathematical prerequisites for a large number of interesting theories, including those given in the examples section above.
When the base manifold
M
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
M
In particular, this allows us to pick a global trivialization of
P
A\mu.
Furthermore, there is a trivial connection
A0,\mu
E=M x V
M → V
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
A\mu
In weak gravitational curvature, flat spacetime often serves as a good approximation to weakly curved spacetime. For experiment, this approximation is good. The Standard Model is defined on flat spacetime, and has produced the most accurate precision tests of physics to date.