Field strength tensor

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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 .


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


in which

  • a matrix potential of the form is with
    • Hermitian matrices and
    • real functions of spacetime ,

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.


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 .


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