Non-abelian gauge theories

Classical action

• Write the general expressions for the covariant derivative of a set of matter fields, for the transformation properties of the gauge fields and for the field-strength tensor. Compare these with the corresponding expression in QED.
• Why a simple non-compact Lie group is not suitable for a physical gauge theory?

Path integral formulation

• Write the action of a non-abelian gauge theory after gauge fixing (in the Lorentz gauge). Comment the differences with QED. What are the ghosts? Why should we take the ghosts as anticommuting fields? Why for ghosts the spin-statystics connection is violated? Why should we not care about ghosts in QED? What is the ghost number?
• What is the BRS symmetry? Write the action of the BRS transformations on all the fields of the theory. What do we mean by nilpotency of the BRS transformations? Write the gauge-fixed action for a gauge theory in a general gauge.
• What is the physical relevance of the BRS symmetry? In the operator formalism, how do we define the Hilbert space of physical state starting from the Fock space generated by all oscillators? Does this Hilbert space contain ghosts?

Functional Identities

• Write the Slavnov-Taylor identities satisfied by the generating functional of the 1PI Green functions.
• Give a short account of the use of Slavnov-Taylor identities in discussing the one-loop renormalization of a non-abelian gauge theory.
  What is the difference between a theory renormalizable by power counting and a strictly renormalizable one?