Uncoupled theories

• 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.

Coupled theories

• 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.

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.

Gauge, principal bundles and connections

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

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

${\displaystyle P\xrightarrow {\pi } M}$

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

Associated vector bundles and matter content

An associated vector bundle ${\displaystyle E\xrightarrow {\pi } M}$ 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 ${\displaystyle E'\xrightarrow {\pi } M}$, the Lagrangian is a function ${\displaystyle L:E'\rightarrow \mathbb {R} }$.

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 ${\displaystyle E'}$ to be a bundle where the fibre at ${\displaystyle p}$ is ${\displaystyle V\otimes T_{p}^{*}M}$. 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.

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.