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 ${\displaystyle \psi }$

${\displaystyle D_{\mu }\psi =(\partial _{\mu }+A_{\mu })\psi }$,

in which

• ${\displaystyle A_{\mu }}$a matrix potential of the form is with ${\displaystyle A_{\mu }=-it^{a}A_{\mu }^{a}}$
  • Hermitian matrices and ${\displaystyle t^{a}}$
  • real functions of spacetime , ${\displaystyle A_{\mu }^{a}}$

so the field strength tensor results from this theory

${\displaystyle {\begin{aligned}F_{\mu \nu }&=D_{\mu }A_{\nu }-D_{\nu }A_{\mu }\\&=-it^{a}(\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+f^{abc}A_{\mu }^{b}A_{\nu }^{c}),\end{aligned}}}$

where the real numbers come from the commutator . ${\displaystyle f^{abc}}$ ${\displaystyle [t^{a},t^{b}]=if^{abc}t^{c}}$

The Lagrangian for the field can then be chosen as , this is the Yang-Mills Lagrangian. ${\displaystyle L}$ ${\displaystyle A_{\mu }}$ ${\displaystyle L\propto F_{\mu \nu }^{a}F^{a\mu \nu }}$

Electromagnetism

For quantum electrodynamics corresponds to the known vector potential . Since its components interchange, the form of the field strength tensor is simplified ${\displaystyle A_{\mu }}$

${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$

For its further properties, see Electromagnetic field strength tensor .

literature

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