Physics:Quantum Field Theory Gauge symmetry

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Gauge symmetry in quantum field theory is a fundamental principle stating that certain transformations of the fields leave the physical predictions of a theory unchanged.[1] These symmetries determine the form of interactions and require the existence of gauge fields that mediate forces.

Gauge symmetry: local transformations of fields requiring compensating gauge fields to preserve invariance

Global and local symmetry

A symmetry transformation changes the fields without affecting observable quantities.

A global symmetry uses the same transformation everywhere: ψ(x)→eiαψ(x)

A local symmetry allows the transformation parameter to vary with space-time: ψ(x)→eiα(x)ψ(x)

Local symmetry is much more restrictive and leads directly to interactions.[2]

Emergence of gauge fields

A naive local transformation introduces extra terms in derivatives: ∂μψ(x)→(∂μ+i∂μα(x))ψ(x)

To restore invariance, a gauge field Aμ(x) is introduced and the derivative is replaced by the covariant derivative: Dμ=∂μ+igAμ

This ensures that the theory remains invariant under local transformations.[3]

Gauge invariance

The gauge field transforms simultaneously: Aμ(x)→Aμ(x)−1g∂μα(x)

This compensates the change in the matter field, preserving the symmetry of the Lagrangian.

Gauge invariance is therefore not just a mathematical property but a principle that determines the structure of interactions.

Example: Quantum electrodynamics

In QED, the symmetry group is U(1). The requirement of local gauge invariance leads directly to the electromagnetic interaction.

The interaction term: ψ¯γμAμψ

arises naturally from imposing gauge symmetry.[1]

Non-Abelian gauge theories

More complex gauge symmetries involve non-commuting groups such as:

  • SU(2)
  • SU(3)

These lead to non-Abelian gauge theories, where the gauge fields themselves interact.

This structure underlies:

  • the weak interaction
  • the strong interaction

and forms the basis of the Standard Model.[2]

Field strength tensor

The dynamics of gauge fields are described by the field strength tensor: Fμν=∂μAν−∂νAμ

In non-Abelian theories, additional terms appear due to field self-interactions.

The Lagrangian includes: −14FμνFμν

which governs the propagation of gauge fields.[3]

Physical interpretation

Gauge symmetry implies that certain degrees of freedom are not physically observable, but instead reflect redundancy in the mathematical description.

Observable quantities depend only on gauge-invariant combinations of fields.

Conceptual importance

Gauge symmetry is one of the central organizing principles of modern physics. It explains:

  • the existence of force-carrying particles
  • the structure of interactions
  • the unification of fundamental forces

All known fundamental interactions (except gravity in its classical form) are described by gauge theories.

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. ↑ 2.0 2.1 Peskin, M. E.; Schroeder, D. V. An Introduction to Quantum Field Theory (1995).
  3. ↑ 3.0 3.1 Schwartz, M. D. Quantum Field Theory and the Standard Model (2014).
Author: Harold Foppele

Source attribution: Quantum field theory (QFT) core

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