action gauge symmetry group U(1) gauge boson symmetry Lagrangian photon function field local symmetry
where α(x) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory. Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations 




e

i
α
(
x
)




{\displaystyle e^{i\alpha (x)}}

 and 




e

i

α
′

(
x
)




{\displaystyle e^{i\alpha '(x)}}

 is yet another symmetry transformation 




e

i
[
α
(
x
)
+

α
′

(
x
)
]




{\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}

. For any α(x), 




e

i
α
(
x
)




{\displaystyle e^{i\alpha (x)}}

 is an element of the U(1) group, thus QED is said to have U(1) gauge symmetry. The photon field Aμ may be referred to as the U(1) gauge boson.
