In theoretical physics

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, a non-abelian gauge transformation means a gauge transformation taking values in some group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 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

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

 was U(1), which is commutative.

For a non-abelian

Non-abelian

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, which is commutative.For a...

 Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 G, its elements do not commute, i.e. they do in general not satisfy
.

The quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s marked the introduction of non-abelian structures in mathematics.

In particular, its generators ,
which form a basis for the vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 of infinitesimal transformation

Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...

s (the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

), have a commutation rule:


The structure constants quantify the lack of commutativity, and do not then all vanish. We can deduce that
are antisymmetric in the first two indices and real.

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


Within this orthonormal basis

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

, the structure constants are then antisymmetric with respect to all three indices.

An element of the group can be expressed near the identity element

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

 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

Compact group

In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

 is equivalent to a unitary representation

Unitary representation

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...

, 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.

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