In QED, photons do not interact with each other at tree level. In a non-abelian gauge theory, gauge bosons do interact with each other. What is the mathematical origin of this difference?
ANon-abelian gauge bosons are massive, and massive particles always interact
BThe field strength tensor F^a_{mu nu} for a non-abelian group contains a term g f^{abc} A^b_mu A^c_nu that is quadratic in the gauge field — when substituted into the kinetic term F^2, this produces cubic and quartic self-interaction vertices that have no analog in QED
CNon-abelian gauge bosons have spin-2 instead of spin-1
DThe non-abelian gauge group has more generators, requiring more interaction terms
In QED, F_{mu nu} = partial_mu A_nu - partial_nu A_mu is linear in A, so F^2 contains only quadratic terms (free propagation) — no photon self-interactions. In a non-abelian theory, F^a_{mu nu} = partial_mu A^a_nu - partial_nu A^a_mu + g f^{abc} A^b_mu A^c_nu, where f^{abc} are the structure constants of the gauge group. The extra term is quadratic in A, so F^2 contains cubic (three-gluon vertex) and quartic (four-gluon vertex) interaction terms. These self-interactions exist because non-abelian gauge bosons carry charge under their own gauge group — gluons carry color charge, while photons carry no electric charge.
Question 2 Multiple Choice
The number of gauge bosons in a Yang-Mills theory equals the number of generators of the gauge group. SU(N) has N^2 - 1 generators. How many gluons does QCD (SU(3) gauge theory) have?
A3
B6
C8
D9
SU(3) has 3^2 - 1 = 8 generators, so QCD has 8 gluons. Each gluon carries a color-anticolor combination (but not all 9 combinations — the color-singlet combination is removed, leaving 8). For comparison, SU(2) has 3 generators (the W+, W-, and Z bosons before electroweak symmetry breaking), and U(1) has 1 generator (the photon).
Question 3 True / False
Quantizing a non-abelian gauge theory requires introducing Faddeev-Popov ghost fields, which are scalar fields that obey Fermi statistics. Why are ghosts necessary, and why don't they appear in QED?
TTrue
FFalse
Answer: True
Ghost fields are necessary in non-abelian theories to maintain unitarity (conservation of probability) in covariant gauges. The gauge-fixing procedure overcounts gauge-equivalent field configurations, and ghosts cancel the contributions of unphysical longitudinal and timelike gauge boson polarizations in loops. In QED, the ghost fields decouple (they do not interact with photons because the abelian structure constants vanish) and can be ignored. In non-abelian theories, ghosts couple to gluons through the structure constants f^{abc} and must be included in all loop calculations. Ghosts are not physical particles — they are computational tools that appear as internal lines in Feynman diagrams but never as external states.
Question 4 Short Answer
Explain why the self-interaction of non-abelian gauge bosons leads to qualitatively different physics from QED, giving at least two specific physical consequences.
Think about your answer, then reveal below.
Model answer: First, gluon self-interaction produces asymptotic freedom: the QCD beta function is negative (beta = -11 N_c/(48 pi^2) g^3 + ... for SU(N_c) with no fermions), meaning the coupling decreases at high energies. This is opposite to QED (where vacuum polarization from charged fermions makes beta positive) and occurs because gluon loops contribute to the vacuum polarization with the opposite sign to fermion loops, and dominate when the number of fermion flavors is not too large. Second, the strong coupling at low energies leads to confinement: quarks and gluons cannot exist as free particles but are permanently bound into color-neutral hadrons. This has no analog in QED, where the coupling is weak at all accessible energies.
Additional consequences include the existence of glueballs (bound states of pure glue, with no quarks), the rich spectrum of hadrons, jet production in high-energy collisions (reflecting the underlying quark-gluon dynamics), and the QCD phase transition at high temperature (deconfinement). All trace back to the non-abelian self-interaction.