Editing Non-abelian gauge transformation

{{Short description|Transformation rule for non-abelian gauge groups}}
{{technical|date=June 2012}}
{{onesource|date=April 2016}}

In [[theoretical physics]], a '''non-abelian gauge transformation'''<ref>{{Cite journal |last=Lahiri |first=Amitabha |date=2002-08-20 |title=GAUGE TRANSFORMATIONS OF THE NON-ABELIAN TWO-FORM |url=https://www.worldscientific.com/doi/abs/10.1142/S0217732302007752 |journal=Modern Physics Letters A |language=en |volume=17 |issue=25 |pages=1643–1650 |doi=10.1142/S0217732302007752 |issn=0217-7323|arxiv=hep-th/0107104 }}</ref> means a [[gauge transformation]] taking values in some [[group (mathematics)|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 group|non-abelian]] [[Lie group]] ''G'', its elements do not commute, i.e. they in general do ''not'' satisfy 

:<math>a*b=b*a \,</math>. 

The [[quaternion]]s marked the introduction of non-abelian structures in mathematics.

In particular, its generators <math>t^a</math>,
which form a basis for the [[vector space]] of [[infinitesimal transformation]]s (the [[Lie algebra]]), have a 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> 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 

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

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

An element <math>\omega</math> of the group can be expressed near the [[identity element]] 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 a [[unitary matrix]] 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 derivative of the transformed 
field is equivalent to the transformation of the field derivatives:

:<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 product]]s 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(x),T(\omega)\partial_\mu \varphi(x)\big).</math>

==References==
{{reflist}}

{{DEFAULTSORT:Non-Abelian Gauge Transformation}}
[[Category:Gauge theories]]