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{{Short description|Quantum field theories based on non-commuting gauge symmetry groups leading to self-interacting gauge fields}}

{{Quantum book backlink|Quantum field theory}}
'''Non-Abelian gauge theory''' is a class of quantum field theories in which the underlying gauge symmetry group is non-commutative, meaning that the order of symmetry transformations matters.<ref name="weinberg">Weinberg, S. ''The Quantum Theory of Fields'' (1995).</ref> These theories generalize Abelian gauge theories such as quantum electrodynamics and form the foundation of the strong and weak interactions.

<div style="float:right; border:1px solid #ccc; padding:4px; background:#fff8dc; margin:0 0 1em 1em; width:420px;">
[[File:Non_Abelian_gauge_theory_structure.jpg|400px]]
<div style="font-size:90%;">Non-Abelian gauge symmetry: interacting gauge fields arising from non-commuting symmetry generators</div>
</div>

== Abelian vs non-Abelian symmetry ==
In Abelian gauge theories, such as <math>U(1)</math>, the group elements commute:
<math>
T_a T_b = T_b T_a
</math>

In contrast, non-Abelian groups such as <math>SU(N)</math> satisfy:
<math>
[T_a, T_b] = i f_{abc} T_c
</math>

where <math>f_{abc}</math> are the structure constants of the group.<ref name="peskin">Peskin, M. E.; Schroeder, D. V. ''An Introduction to Quantum Field Theory'' (1995).</ref>

This non-commutativity leads to fundamentally new physical features.

== Gauge fields and covariant derivative ==
To maintain local gauge invariance, one introduces multiple gauge fields <math>A_\mu^a(x)</math>, one for each generator of the symmetry group.

The covariant derivative becomes:
<math>
D_\mu = \partial_\mu + i g A_\mu^a T_a
</math>

where:
* <math>T_a</math> are the generators
* <math>g</math> is the coupling constant

This structure ensures invariance under local transformations of the non-Abelian group.<ref name="schwartz">Schwartz, M. D. ''Quantum Field Theory and the Standard Model'' (2014).</ref>

== Field strength tensor ==
The field strength tensor generalizes to:
<math>
F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f_{abc} A_\mu^b A_\nu^c
</math>

The additional term:
<math>
g f_{abc} A_\mu^b A_\nu^c
</math>

arises from the non-commuting nature of the group and leads to self-interactions of the gauge fields.<ref name="weinberg"/>

== Self-interacting gauge fields ==
Unlike Abelian theories, non-Abelian gauge fields carry the charge associated with the symmetry.

This means that:

* gauge bosons can interact with each other
* the theory is inherently nonlinear

These self-interactions are essential for understanding the behavior of the strong and weak forces.

== Example: SU(3) and SU(2) ==
Important non-Abelian gauge groups include:

* <math>SU(3)</math> → quantum chromodynamics (QCD)
* <math>SU(2)</math> → weak interaction

These groups describe the internal symmetries of fundamental particles and determine how they interact.

== Yang–Mills theory ==
Non-Abelian gauge theories are often called Yang–Mills theories, after Yang and Mills who first formulated them.<ref name="yangmills">Yang, C. N.; Mills, R. L. (1954). Conservation of isotopic spin and isotopic gauge invariance.</ref>

The Yang–Mills Lagrangian is:
<math>
\mathcal{L} = -\frac{1}{4} F_{\mu\nu}^a F^{\mu\nu a}
</math>

This describes the dynamics of the gauge fields and their interactions.

== Physical consequences ==
Non-Abelian gauge theories exhibit rich physical phenomena:

* confinement in QCD
* asymptotic freedom at high energies
* spontaneous symmetry breaking (in extended models)

These features distinguish them from simpler Abelian theories.

== Conceptual importance ==
Non-Abelian gauge theories form the backbone of modern particle physics. They explain:

* the structure of strong and weak interactions
* the behavior of gauge bosons
* the organization of the Standard Model

They represent a profound generalization of the gauge principle.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum field theory (QFT) core|1}}

Physics:Quantum Non-Abelian gauge theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template