Non-abelian gauge transformation
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In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group G, the elements of which do not obey the commutative law when they are multiplied. By contrast, the original choice of gauge group in the physics of electromagnetism had been U(1), which is commutative.
The quaternions marked the introduction of non-abelian structures in mathematics.
The structure constants quantify the lack of commutativity, and do not vanish. We can deduce that the structure constants are antisymmetric in the first two indices and real. The normalization is usually chosen (using the Kronecker delta) as
Within this orthonormal basis, the structure constants are then antisymmetric with respect to all three indices.
An element of the group can be expressed near the identity element in the form
where are the parameters of the transformation.
Let be a field that transforms covariantly in a given representation . This means that under a transformation we get
to be a unitary matrix without loss of generality. We assume that the Lagrangian depends only on the field and the derivative :
If the group element is independent of the spacetime coordinates (global symmetry), the derivative of the transformed field is equivalent to the transformation of the field derivatives:
Thus the field and its derivative transform in the same way. By the unitarity of the representation, scalar products like , or are invariant under global transformation of the non-Abelian group.
Any Lagrangian constructed out of such scalar products is globally invariant: