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In theoretical physics
Theoretical physics

Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain experimental data taken of the natural world....
, 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 Binary operation that combines any two of its element to form a third element....
 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 the physics of the electromagnetic field, a field which exerts a force on Elementary particles with the property of electric charge and which is reciprocally affected by the presence and motion of such particles....
 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....
 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 Differential structure....
 G, its elements do not commute, i.e. they do in general not satisfy

.

The quaternion
Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions 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

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 of infinitesimal transformation
Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limit form of small transformation . For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space....
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....
), have a commutation rule:

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






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In theoretical physics
Theoretical physics

Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain experimental data taken of the natural world....
, 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 Binary operation that combines any two of its element to form a third element....
 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 the physics of the electromagnetic field, a field which exerts a force on Elementary particles with the property of electric charge and which is reciprocally affected by the presence and motion of such particles....
 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....
 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 Differential structure....
 G, its elements do not commute, i.e. they do in general not satisfy

.

The quaternion
Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions 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

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 of infinitesimal transformation
Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limit form of small transformation . For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space....
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....
), 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
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
) as

Within this orthonormal basis
Orthonormal basis

In mathematics, an orthonormal basis of an inner product space V , is a set of mutually orthogonality vectors of magnitude 1 that span the space when infinite linear combinations are allowed....
, 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 p of G on a complex Hilbert space V such that p is a unitary operator for every g ? G....
, we take

to be a unitary matrix
Unitary matrix

In mathematics, a unitary matrix is an n by n complex number matrix U satisfying the condition where is the identity matrix and is the conjugate transpose of U....
 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: