Physics:Quantum Non-Abelian gauge theory

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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.[1] These theories generalize Abelian gauge theories such as quantum electrodynamics and form the foundation of the strong and weak interactions.

Non-Abelian gauge symmetry: interacting gauge fields arising from non-commuting symmetry generators

Abelian vs non-Abelian symmetry

In Abelian gauge theories, such as U(1), the group elements commute: TaTb=TbTa

In contrast, non-Abelian groups such as SU(N) satisfy: [Ta,Tb]=ifabcTc

where fabc are the structure constants of the group.[2]

This non-commutativity leads to fundamentally new physical features.

Gauge fields and covariant derivative

To maintain local gauge invariance, one introduces multiple gauge fields Aμa(x), one for each generator of the symmetry group.

The covariant derivative becomes: Dμ=∂μ+igAμaTa

where:

  • Ta are the generators
  • g is the coupling constant

This structure ensures invariance under local transformations of the non-Abelian group.[3]

Field strength tensor

The field strength tensor generalizes to: Fμνa=∂μAνa−∂νAμa+gfabcAμbAνc

The additional term: gfabcAμbAνc

arises from the non-commuting nature of the group and leads to self-interactions of the gauge fields.[1]

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:

  • SU(3) → quantum chromodynamics (QCD)
  • SU(2) → 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.[4]

The Yang–Mills Lagrangian is: ℒ=−14FμνaFμνa

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

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ 1.0 1.1 Weinberg, S. The Quantum Theory of Fields (1995).
  2. ↑ Peskin, M. E.; Schroeder, D. V. An Introduction to Quantum Field Theory (1995).
  3. ↑ Schwartz, M. D. Quantum Field Theory and the Standard Model (2014).
  4. ↑ Yang, C. N.; Mills, R. L. (1954). Conservation of isotopic spin and isotopic gauge invariance.
Author: Harold Foppele

Source attribution: Quantum field theory (QFT) core

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