The time evolution of physical systems is described mathematically by differential equations of various degree of complexity, such as Newton’s equation in classical mechanics, Maxwell’s equations for the electromagnetic field, or Schrödinger’s equation in quantum theory. In most cases these equations have to be supplemented with additional constraints, like initial conditions and/or boundary conditions, which select only one – or sometimes a restricted subset – of the solutions as relevant to the physical system of interest. Quite often the preferred dynamical equations of a physical system are not formulated directly in terms of observable degrees of freedom, but in terms of more primitive quantities, such as potentials, from which the physical observables are to be constructed in a second separate step of the analysis. As a result, the interpretation of the solutions of the evolution equation is not always straightforward. In some cases certain solutions have to be excluded, as they do not describe physically realizable situations; or it may happen that certain classes of apparently different solutions are physically indistinguishable and describe the same actual history of the system. The BRST-formalism [1,2] has been developed specifically to deal with such situations. The roots of this approach to constrained dynamical systems are found in attempts to quantize General Relativity [3,4] and Yang–Mills theories . Out of these roots has grown an elegant and powerful framework for dealing with quite general classes of constrained systems using ideas borrowed from algebraic geometry. 1
Aspects of BRST Quantization J.W. van Holten
Only in the late 1980s, when QFT was reformulated in fiber bundle language for application to problems in the topology of low-dimensional manifolds (topological quantum field theory), did it become apparent that the BRST “transformation” is fundamentally geometrical in character. In this light, “BRST quantization” becomes more than an alternate way to arrive at anomaly-cancelling ghosts. It is a different perspective on what the ghost fields represent, why the Faddeev–Popov method works, and how it is related to the use of Hamiltonian mechanics to construct a perturbative framework. The relationship between gauge invariance and “BRST invariance” forces the choice of a Hamiltonian system whose states are composed of “particles” according to the rules familiar from the canonical quantization formalism. This esoteric consistency condition therefore comes quite close to explaining how quanta and fermions arise in physics to begin with.