In this limit, con¬nement results from the kinetic energy [100] of the compact

link variables. The potential energy generated by the magnetic ¬eld, which has

been the crucial ingredient in the construction of the Nielsen“Olesen Vortex and

the ™t Hooft“Polyakov monopole, is negligible in this limit. It is no accident that,

as we have seen, Polyakov-loop variables, which as group elements are compact,

also exhibit con¬nement-like behavior.

Apart from instantons as the genuine topological objects, Yang“Mills theo-

ries exhibit non-trivial topological properties related to the center of the gauge

group. The center symmetry as a residual gauge symmetry o¬ers the possibility

to formulate con¬nement as a symmetry property and to characterize con¬ned

and decon¬ned phases. The role of the center vortices (gauge transformations

which are singular on a two dimensional space-time sheet) remains to be clari¬ed.

The existence of obstructions in imposing gauge conditions is another non-trivial

property of non-abelian gauge theories which might be related to con¬nement.

I have described the appearance of monopoles as the results of such obstruc-

tions in so-called diagonalization or abelian gauges. These singular ¬elds can

be characterized by topological methods and, on a formal level, are akin to

the ™t Hooft“Polyakov monopole. I have described the di¬culties in developing

a viable framework for formulating their dynamics which is supposed to yield

con¬nement via a dual Meissner e¬ect.

Acknowledgment

I thank M. Thies, L.v. Smekal, and J. Pawlowski for discussions on the vari-

ous subjects of these notes. I™m indebted to J. J¨ckel and F. Ste¬en for their

a

meticulous reading of the manuscript and for their many valuable suggestions

for improvement.

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Aspects of BRST Quantization

J.W. van Holten

National Institute for Nuclear and High-Energy Physics (NIKHEF) P.O. Box 41882,

1009 DB Amsterdam, The Netherlands, and Department of Physics and Astronomy,

Faculty of Science, Vrije Universiteit Amsterdam

Abstract. BRST-methods provide elegant and powerful tools for the construction

and analysis of constrained systems, including models of particles, strings and ¬elds.

These lectures provide an elementary introduction to the ideas, illustrated with some

important physical applications.

1 Symmetries and Constraints

The time evolution of physical systems is described mathematically by di¬er-

ential equations of various degree of complexity, such as Newton™s equation

in classical mechanics, Maxwell™s equations for the electro-magnetic ¬eld, or

Schr¨dinger™s equation in quantum theory. In most cases these equations have

o

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 straight-

forward. 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