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 α ( x ) and e i α α ′ ( x ) is yet another symmetry transformation e i [ α α ( x ) + α α ′ ( x ) ] . For any α (x), e i α α ( x ) is an element of the U(1) group, thus QED is said to have U(1) gauge symmetry. The photon field A μ may be referred to as the U(1) gauge boson.