In theoretical physics, a non-abelian gauge transformation[1] 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.
For a non-abelian Lie group G, its elements do not commute, i.e. they in general do not satisfy The quaternions marked the introduction of non-abelian structures in mathematics.
, which form a basis for the vector space of infinitesimal transformations (the Lie algebra), have a commutation rule: 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.
of the group can be expressed near the identity element in the form where
be a field that transforms covariantly in a given representation
This means that under a transformation we get Since any representation of a compact group is equivalent to a unitary representation, we take to be a unitary matrix without loss of generality.
By the unitarity of the representation, scalar products like
are invariant under global transformation of the non-abelian group.
Any Lagrangian constructed out of such scalar products is globally invariant: