Diagrammar (CERN): G. 't Hooft, M. Veltman

Description

With the advent of gauge theories it became necessary to reconsider many well-established ideas in quantum field theories. The canonical formalism, formerly regarded as the most conventional and rigorous approach, has now been abandoned by many authors. The path-integral concept cannot replace the canonical formalism in defining a theory, since path integrals in four dimensions are meaningless without additional and rather ad hoc renormalization prescriptions.

Whatever approach is used, the result is always that the S-matrix is expressed in terms of a certain set of Feynman diagrams. Few physicists object nowadays to the idea that diagrams contain more truth thai the underlying formalism, and it seems only rational to take the final step and abandon operator formalism and path integrals as instruments of analysis.

Yet it would be very shortsighted to turn away completely from these methods. Many useful relations have been derived, and many more may be in the future. What must be done is to put them on a solid footing. The situation must be reversed: diagrams form the basis from which everything must be derived. They define the operational rules, and tell us when to worry about Schwinger terms, subtractions, and whatever other mythological objects need to be introduced.

The development of gauge theories owes much to path integrals and it is tempting to attach more than a heuristic value to path integral derivations. Although we do not rely on path integrals in this paper, one may think of expanding the exponent of the interaction Lagrangian in a Taylor series, so that the algebra of the Gaussian integrals becomes exactly identical to the scheme of manipulations with Feynman diagrams. That would leave us with the problems of giving the correct ie prescription in the propagators, and to find a decent renormalization scheme.

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