BRST-Quantization of Gauge Field Theories

Over the last few decades, gauge field theories have gained an increasing interest in theoretical particle physics. This is a result of the fact that from the single concept of gauge invariance, important dynamical consequences can be derived, based on formal symmetry arguments only.
Several methods have been developed in order to quantize field theories. Probably the most powerful one is Feynman’s path integral approach, which is based on the intuitive idea of summing the amplitudes associated with the individual paths connecting a given initial and final state, to obtain the transition amplitude. Again, one reason for being so attractive is that in this way, all symmetries of the theory are manifestly preserved. Although mathematically not rigourously defined, the functional integral can be used to generate, in an unambiguous way, a perturbative expansion around some classical field configuration in terms of so-called Feynman diagrams.
This Feynman diagrammatic representation, however, can only be straightforwardly defined for a non-gauge theory, because it crucially relies on a regularity condition that excludes the possibility of gauge invariance. Nevertheless, there exists a way out of this dilemma. A first ansatz to the solution was given by L.D. Faddeev and V.N. Popov in 1967. It turned out to be possible to fix the gauge (and thus to turn the theory into a non-gauge theory) and yet to annihilate the influence of the gauge degrees of freedom, by introducing “ghost” fields of opposite statistics to these gauge variables. In this way, the local symmetry of the gauge theory is replaced by a global supersymmetry in an extended, graded configuration space, called BRST-symmetry. This BRST-symmetry is responsible for a physically correct (i.e. gauge independent, unitary) transition amplitude. Although the Faddeev-Popov method is only appropriate for gauge theories which are characterized by a closed gauge algebra, there exists an approach, directly based on this BRST-symmetry, which seems to be of a quite general applicability.
The purpose of this work is to explain the BRST-quantization procedure, both in the canonical (or Hamiltonian) and in the covariant (or Lagrangian) formalism. The organization of this work is as follows. Chapter 2 reviews the Faddeev-Popov method, which may be regarded as the starting point for the BRST-approach, revealing also its limitations. In Chapter 3, we will see that the Faddeev-Popov action is indeed, within its domain of validity, BRST-invariant. Historically, the canonical BRST-quantization for a general gauge theory had been established before the covariant Faddeev-Popov method had been generalized. Chapter 4 deals with this general, canonical method. In Chapter 5, we return to the covariant approach, generalizing it to deal with a large class of so-called open gauge theories. Although the two formalisms have been developed rather independently of each other, it was generally believed, by virtue of some explicit examples, that they were equivalent in general. Only very recently, this believe has received a firm basis. This general connection is demonstrated in Chapter 6. In all of these approaches, a restriction has been made to “irreducible” gauge theories. Chapter 7 is meant to sketch, in the light of a simple example, that this is in fact not a necessary restriction for the BRST-approach. Two appendices are added to review some aspects concerning Grassmann numbers and path integrals.