# Field strength tensor

A **field strength tensor** describes the fields in gauge theories . The best-known example is the electromagnetic field strength tensor for the calibration theory of electrodynamics , which describes the electric and magnetic field . Field strength tensors are mainly used in quantum field theories .

The field strength tensor is *not a * tensor in the actual mathematical sense, since its components are not real numbers , but elements of the Lie algebra belonging to the calibration group .

## General

If the covariant derivative of a field is defined in a gauge theory as

- ,

in which

so the field strength tensor results from this theory

where the real numbers come from the commutator .

The Lagrangian for the field can then be chosen as , this is the Yang-Mills Lagrangian.

## Electromagnetism

For quantum electrodynamics corresponds to the known vector potential . Since its components interchange, the form of the field strength tensor is simplified

For its further properties, see Electromagnetic field strength tensor .

## literature

- V. Parameswaran Nair:
*Quantum Field Theory - A Modern Perspective*, Springer 2005 - Chapter 10.1