https://cds.cern.ch/record/406087/files/CM-P00061709.pdf

Quantization of relativistic systems with constraints

The purpose of this review article is to explain and illustrate in detail the technique of dimensional regularization, which is a major mathematical tool in the renormalization program of gauge theories. The most important single feature of the new technique is the concept of analytic continuation in the number of space-time dimensions $2\ensuremath{\omega}$, where the regulating parameter $\ensuremath{\omega}$ is complex in general, and $\ensuremath{\omega}=2$ corresponds to four-dimensional space-time. The technique of dimensional regularization preserves the local gauge symmetry of the underlying Lagrangian and thereby permits a consistent gauge-invariant treatment of divergent Feynman integrals to all orders in perturbation theory. The method can thus be applied---as demonstrated in this article---not only to Abelian gauge models, but more importantly to non-Abelian theories such as Yang-Mills fields and quantum gravity, to which the majority of conventional regularization procedures is inapplicable. We illustrate both the advantages and the limitation of dimensional regularization, as well as its extension to massless particles.

Introduction to the Technique of Dimensional Regularization

The most important single attribute of noncovariant gauges is their ghost-free nature. Although noncovariant gauges have been an integral part of quantum field theory for many decades, their effectiveness in the quantization of non-Abelian theories and their broad range of applicability have only recently been appreciated by theorists at large. The purpose of this review is to explain and illustrate the essential characteristics of some typical noncovariant gauges, such as the axial gauge, the planar gauge, the light-cone gauge, and the temporal gauge. The author's aim is to acquaint the reader not only with the basic properties of these ghost-free gauges, but also with their deficiencies and advantages over covariant gauges, their computational idiosyncrasies, and their dominant areas of application.

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Introduction to noncovariant gauges

We discuss renormalization for a class of Lagrangians containing an explicit mass term for the vector field and a four-vertex for the Faddeev-Popov ghost field. The action is shown to be invariant under nonlinear gauge transformations, which are a generalization of those introduced by Becchi, Rouet and Stora. This invariance guarantees the validity of the Slavnov-Taylor identities, which are then used to discuss the renormalization of the theory. The problem of unitarity is not discussed here. Canonical and Lorentz-covariant formalism is employed.

On a class of Lagrangian models for massive and massless Yang-mills fields

Conventional particle theories such as the Standard Model have a number of freely adjustable coupling constants and mass parameters, depending on the symmetry algebra of the local gauge group and the representations chosen for the spinor and scalar fields There seems to be no physical principle to determine these parameters as long as they stay within certain domains dictated by the renormalization group Here however, reasons are given to demand that, when gravity is coupled to the system, local conformal invariance should be a spontaneously broken exact symmetry The argument has to do with the requirement that black holes obey a complementarity principle relating ingoing observers to outside observers, or equivalently, initial states to final states This condition fixes all parameters, including masses and the cosmological constant We suspect that only examples can be found where these are all of order one in Planck units, but the values depend on the algebra chosen This paper combines findings reported in two previous preprints (G ’t Hooft in arXiv:10090669
 [gr-qc], 2010; arXiv:10110061
 [gr-qc], 2010) and puts these in a clearer perspective by shifting the emphasis towards the implications for particle models

/pdf/a-class-of-elementary-particle-models-without-any-adjustable-f6e9dvaaan.pdf

A Class of Elementary Particle Models Without Any Adjustable Real Parameters

A method is suggested (and applied to the Yang-Mills and gravitational fields) for the construction of the generating functional ($S$ matrix) for fields possessing an invariance group. The unitarity and gauge independence of the $S$ matrix on the mass shell are seen explicitly.

I. V. Tyutin

Papers

S Matrix for Yang-Mills and Gravitational Fields