Non-abelian gauge transformation

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Consider a non-abelian Lie group. Its elements do not commute, i.e. they do in general not satisfy <math>a*b=b*a \,</math>. Also see Quaternion for a better explanation for non-abelian in mathematics.

In particular, its generators <math>t^a</math>, which form a basis for the vector space of infinitesimal transformations (called the Lie algebra), have the following commutation rule:

<math>\left[t^a,t^b\right] = t^a t^b - t^b t^a = C^{abc} t^c.</math>

The structure constants <math>C^{abc}</math> are antisymmetric in the first two indices and real. The normalization is usually chosen as

<math>Tr(t^at^b) = \frac{1}{2}\delta^{ab}.</math>

Within this orthonormal basis, the structure constants are antisymmetric with respect to all three indices. An element <math>\omega</math> of the group can be expressed near unity in the form <math>\omega = exp(\theta^at^a)</math>, where <math>\theta^a</math> are the parameters of the transformation.

Let <math>\varphi(x)</math> be a field that transforms covariantly in a given representation <math>T(\omega)</math>. This means that under a transformation we get

<math>\varphi(x) \to \varphi'(x) = T(\omega)\varphi(x).</math>

Since any representation of a compact group is equivalent to a unitary representation, we take <math>T(\omega)</math> to be unitary without loss of generality. We assume that the Lagrangian <math>\mathcal{L}</math> depends only on the field <math>\varphi(x)</math> and the derivative <math>\partial_\mu\varphi(x)</math>:

<math>\mathcal{L} = \mathcal{L}\big(\varphi(x),\partial_\mu\varphi(x)\big).</math>

If the group element <math>\omega</math> is independent of the spacetime coordinates (global symmetry), the derivation of the transformed field is equivalent to the transformation of the derived field:

<math>\partial_\mu T(\omega)\varphi(x) = T(\omega)\partial_\mu\varphi(x).</math>

Thus the field <math>\varphi</math> and its derivative transform in the same way. By the unitarity of the representation, scalar products like <math>(\varphi,\varphi)</math>, <math>(\partial_\mu\varphi,\partial_\mu\varphi)</math> or <math>(\varphi,\partial_\mu\varphi)</math> are invariant under global transformation of the non-Abelian group. Any Lagrangian constructed out of such scalar products is globally invariant:

<math>\mathcal{L}\big(\varphi(x),\partial_\mu\varphi(x)\big) = \mathcal{L}\big(T(\omega)\varphi,T(\omega)\partial_\mu \varphi\big).</math>