Gauge Theories

Gauge theories describe interactions using fields with redundant local descriptions. Their physical content depends on the gauge group, matter representations, couplings, and global choices. These profiles compare four-dimensional QED, pure Yang–Mills theory, and QCD, emphasizing how their ingredients change the questions that can be answered perturbatively.

Enter this chapter

Begin with Quantum Electrodynamics to connect a charged fermion to an Abelian gauge field. Read Yang–Mills Theory next to see why a non-Abelian field interacts with itself. Readers interested in hadrons can enter through QCD and use those two profiles to separate its ingredients.

Topics

1. Quantum Electrodynamics — Specify charge and photon coupling, test a gauge transformation, and interpret perturbative observables and infrared limits.
2. Yang–Mills Theory — Derive gauge-field self-interactions and use one-loop running to identify a dynamically generated scale.
3. Quantum Chromodynamics — Add quark flavors to the gluon sector and distinguish short-distance predictions from hadronic physics.

How the ideas fit together

The Abelian and non-Abelian field strengths differ by a commutator of gauge potentials. In Yang–Mills theory this term supplies gauge-field self-interactions, with observable consequences for running couplings. Quark fluctuations then modify that running in QCD. Comparing the coefficients is meaningful only after specifying the matter content and a common coupling normalization; see Srednicki 2007, §§69–73.

Short-distance asymptotic freedom gives a controlled expansion when all scales relevant to the observable permit it. The same calculation does not determine the long-distance spectrum: extrapolating a perturbative running coupling into its strong-coupling region cannot establish a mass gap or derive a hadron mass.

Gauge invariance also changes what should be compared. A gauge-fixed propagator is a useful computational ingredient, while a physical cross section, hadron correlation function, or suitably specified Wilson loop asks a different question. The profiles explain which quantities they calculate rather than identifying every field component with an observable particle.

Where to go next

Continue to Lower-Dimensional Models for more tractable settings in which mass generation and screening can be examined directly. Follow Chiral Effective Field Theory for a low-momentum description of QCD with different degrees of freedom.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§69–73. DOI.