Non-abelian gauge transformation

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. The original choice of G in the physics of electromagnetism was U(1), which is commutative.

For a non-abelian Lie group G, its elements do not commute, i.e. they do in general not satisfy

$a*b=b*a \,$.

The quaternions marked the introduction of non-abelian structures in mathematics.

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

$\left[t^a,t^b\right] = t^a t^b - t^b t^a = C^{abc} t^c.$

The structure constants quantify the lack of commutativity, and do not then all vanish. We can deduce that

C^abc are antisymmetric in the first two indices and real.

The normalization is usually chosen (using the Kronecker delta) as

$Tr(t^at^b) = \frac{1}{2}\delta^{ab}.$

Within this orthonormal basis, the structure constants are then antisymmetric with respect to all three indices.

An element ω of the group can be expressed near the identity element in the form

ω = exp(θ^at^a),

where θ^a are the parameters of the transformation.

Let $\varphi(x)$ be a field that transforms covariantly in a given representation T(ω). This means that under a transformation we get

$\varphi(x) \to \varphi'(x) = T(\omega)\varphi(x).$

Since any representation of a compact group is equivalent to a unitary representation, we take

T(ω)

to be a unitary matrix without loss of generality. We assume that the Lagrangian $\mathcal{L}$ depends only on the field $\varphi(x)$ and the derivative $\partial_\mu\varphi(x)$:

$\mathcal{L} = \mathcal{L}\big(\varphi(x),\partial_\mu\varphi(x)\big).$

If the group element ω is independent of the spacetime coordinates (global symmetry), the derivative of the transformed field is equivalent to the transformation of the field derivatives:

$\partial_\mu T(\omega)\varphi(x) = T(\omega)\partial_\mu\varphi(x).$

Thus the field φ and its derivative transform in the same way. By the unitarity of the representation, scalar products like $(\varphi,\varphi)$, $(\partial_\mu\varphi,\partial_\mu\varphi)$ or $(\varphi,\partial_\mu\varphi)$ are invariant under global transformation of the non-Abelian group.

Any Lagrangian constructed out of such scalar products is globally invariant:

$\mathcal{L}\big(\varphi(x),\partial_\mu\varphi(x)\big) = \mathcal{L}\big(T(\omega)\varphi,T(\omega)\partial_\mu \varphi\big).$

References