Covariant classical field theory

Requisite mathematical structures

In order to formulate a classical field theory, the following structures are needed:

Spacetime

A smooth manifold ${\displaystyle M}$.

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.

Structures on spacetime

The spacetime often comes with additional structure. Examples are

• Metric: a (pseudo-)Riemannian metric ${\displaystyle \mathbf{g}}$ on ${\displaystyle M}$.
• Metric up to conformal equivalence

as well as the required structure of an orientation, needed for a notion of integration over all of the manifold ${\displaystyle M}$.

Symmetries of spacetime

The spacetime ${\displaystyle M}$ may admit symmetries. For example, if it is equipped with a metric ${\displaystyle \mathbf{g}}$, these are the isometries of ${\displaystyle M}$, generated by the Killing vector fields. The symmetries form a group ${\displaystyle \text{Aut}(M)}$, the automorphisms of spacetime. In this case the fields of the theory should transform in a representation of ${\displaystyle \text{Aut}(M)}$.

For example, for Minkowski space, the symmetries are the Poincaré group ${\displaystyle \text{Iso}(1,3)}$.

Gauge, principal bundles and connections

A Lie group ${\displaystyle G}$ describing the (continuous) symmetries of internal degrees of freedom. This is referred to as the gauge group. The corresponding Lie algebra through the Lie group–Lie algebra correspondence is denoted ${\displaystyle \mathfrak{g}}$.

A principal ${\displaystyle G}$-bundle ${\displaystyle P}$, otherwise known as a ${\displaystyle G}$-torsor. This is sometimes written as

${\displaystyle P\stackrel{\pi }{\to }M}$

where ${\displaystyle \pi }$ is the canonical projection map on ${\displaystyle P}$ and ${\displaystyle M}$ is the base manifold.

Connections and gauge fields

Here we take the view of the connection as a principal connection. In field theory this connection is also viewed as a covariant derivative ${\displaystyle \mathrm{\nabla }}$ whose action on various fields is defined later.

A principal connection denoted ${\displaystyle \mathcal{A}}$ is a ${\displaystyle \mathfrak{g}}$-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 ${\displaystyle A_{\mu }(x)}$, a ${\displaystyle \mathfrak{g}}$-valued 1-form on a trivialization patch ${\displaystyle U\subset M}$. It is this local form of the connection which is identified with gauge fields in physics. When the base manifold ${\displaystyle M}$ is flat, there are simplifications which remove this subtlety.

Associated vector bundles and matter content

An associated vector bundle ${\displaystyle E\stackrel{\pi }{\to }M}$ associated to the principal bundle ${\displaystyle P}$ through a representation ${\displaystyle \rho .}$

For completeness, given a representation ${\displaystyle (V,G,\rho )}$, the fiber of ${\displaystyle E}$ is ${\displaystyle V}$.

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.

Lagrangian

A Lagrangian ${\displaystyle L}$: given a fiber bundle ${\displaystyle E^{\prime }\stackrel{\pi }{\to }M}$, the Lagrangian is a function ${\displaystyle L:E^{\prime }\to \mathbb{R}}$.

Suppose that the matter content is given by sections of ${\displaystyle E}$ with fibre ${\displaystyle V}$ from above. Then for example, more concretely we may consider ${\displaystyle E^{\prime }}$ to be a bundle where the fibre at ${\displaystyle p}$ is ${\displaystyle V\otimes T_p^{\ast }M}$. This then allows ${\displaystyle L}$ to be viewed as a functional of a field.

This completes the mathematical prerequisites for a large number of interesting theories, including those given in the examples section above.

Theories on flat spacetime

When the base manifold ${\displaystyle M}$ 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 ${\displaystyle M}$ is trivial.

In particular, this allows us to pick a global trivialization of ${\displaystyle P}$, and therefore identify the connection globally as a gauge field ${\displaystyle A_{\mu }.}$

Furthermore, there is a trivial connection ${\displaystyle A_{0,\mu }}$ which allows us to identify associated vector bundles as ${\displaystyle E=M\times V}$, and then we need not view fields as sections but simply as functions ${\displaystyle M\to V}$. In other words, vector bundles at different points are comparable. In addition, for flat spacetime the Levi-Civita connection is the trivial connection on the frame bundle.

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 ${\displaystyle A_{\mu }}$ which is considered to be the gauge field of the theory.

Accuracy as a physical model

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.