# 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