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Concept
Non-abelian gauge transformation
Résumé
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.
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.
In particular, its generators ,
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.
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
Since any representation of a compact group is equivalent to a unitary representation, we take
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.
Source officielle
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Cours associés (1)
PHYS-741: Gauge Theories and the Standard Model
The goal of this course is to explain the conceptual and mathematical bases of the Standard Model of fundamental interactions and to illustrate in detail its phenomenological consequences.