Manual Noncovariant Gauges in Canonical Formalism

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Though the path integral method is very convenient for the proof of unitarity and renormalizability of gauge theories, the canonical formalism is eventually necessary to expose the issues in a self-consistent way. These notes are written as an introduction to postgraduate students, lecturers and researchers in the field and assume prior knowledge of quantum field theory. Though the path integral method is very convenient for the proof of unitarity and renormalizability of gauge theories, the canonical formalism is eventually necessary to to expose the issues in a self-consistent way.

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Read more Read less. K; ed. Review From the reviews: "This book develops a consistent formulation for the handling of noncovariant gauges in the quantization process. No customer reviews. The most frequently used noncovariant gauges are the Coulomb and axial gauges.

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In Quantum Electrodynamics, the Coulomb gauge, which is very useful in the search of classical solutions, involves only physical degrees of freedom but the theory is nonlocal and covariance is cumbersome. Things are even worse in nonabelian theories where, in addition to necessary Faddeev-Popov ghosts, many inconsistencies like Gribov [7] ambiguities, operator ordering troubles,. In general, the discussion of noncovariant gauges begins with the use of the propagator they generate and without any reference to the basic field theory formalism.

There is no hope of giving a definite answer to the difficulties they generate if they are not considered from their very beginning. The aim of this book is to build a consistent formulation for the handling of noncovariant gauges in the quantization process. They will be set on the same level of consistency as covariant ones. Although path integral methods are very useful in the proof of unitarity and renormalizability of the theory, the canonical formalism is necessary to set the problem in a consistent way.

From it, two points are obvious and their neglect is generating most of the troubles encountered in the current literature.

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  8. The usual gauge theories involve two first class constraints acting therefore on two degrees of freedom which are unphysical. Gauge fixing, as understood in the quantization procedure, involves three different mechanisms according to the number of unphysical degrees of freedom left in the theory. Class I gauges involve only the physical degrees of freedom.

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    Class II gauges involve, in addition to the physical degrees of freedom, an unphysical one. Two unphysical degrees of freedom and their associated Faddeev-Popov ghosts are met in class III gauges. In noncovariant gauges, all the coordinate frames are not equivalent. Some of them involve a lesser number of degrees of freedom than those needed in general frames. They are called singular and their use in quantization procedure will imply ambiguities. This book is not a general treatise on field theory. For instance, the fermionic fields are not taken into account because the troubles with quantization and related problems are not generated by them but are directly related to the gauge fields themselves.

    In the same way, questions which are not directly related to gauge fields or noncovariant gauges will not be taken into account here, except when needed for the understanding. For instance, an introduction to the quantization of constrained systems is completely developed in order to show the necessity of distinguishing three classes of gauges. Our tackling of noncovariant gauges rests on an extension of the results from covariant to noncovariant gauges. Such tensor cannot be completely arbitrary because the Cauchy problem associated with field equations must be well defined.

    Moreover the gauge condition gives the time evolution of the A0 -field. Otherwise, as in the Coulomb gauge, the quantum theory is nonlocal and therefore inconsistent. Extension of the results from covariant to noncovariant gauges becomes then almost straightforward when the gauge condition is class III. The generalization to such class II gauges is also considered here. The theory can be quantized but trouble occurs with the nonphysical degree of freedom.

    It is not governed by a second-order differential equation. This fact generates two problems: 1. The usual interpretation of antiparticle as a particle moving backward in space and time is no longer possible.

    The Hamiltonian is not bounded from below so that a vacuum cannot be defined as the lowest energy state. These problems are such that the propagator cannot be defined in the usual way. It is left open. In the framework of perturbative theory, the usual methods of handling ultraviolet divergent integrals through dimensional regularization are also extended from covariant to noncovariant class III gauges.

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    These singularities are generated by a lack of isotropic power counting. In particular, the ghost loops become infinite even when dimensionally regularized. Regularization of these singularities can be made by interpolating between the gauge with singular C-matrix and relativistic gauges. Singularities appear as poles at the critical value of the interpolating parameter.

    Such poles cancel out in the expression of the full S-matrix. For instance, in the gluon self-energy, the ghost-loop poles are cancelled by gluon-loop poles. In order to make clearer the appearance of singularities generated by singular C-matrices, the problem is also discussed in the framework of an approach which is free of ultra-violet divergences. The main result is that no class III gauge with decoupling Faddeev-Popov ghosts exists, in opposition to a widespread belief.

    It is hoped that this book, which is the result of many years of work, will be of help in the future study of noncovariant gauges. Acknowledgements Long discussions and a fruitful collaboration during many years with Dr. Hubert Caprasse are at the basis of this work.

    A careful reading and critical comments of the manuscript by Dr. References 1. Barnich, G. Stora, R. B52, v Becchi, C. Acta 23, v Faddeev, L. B, 1 v viii Preface 8. Gupta, S. London A63, v 9.

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    Itzykson, C. Peskin, M. Tyutin, I. Let us begin with the canonical quantization of a simple mechanical system characterized by its Lagrangian L.

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    Here and in the following, except when the opposite is explicitly stated, the upper or lower position of the indices respects the usual rules of tensor calculus. The Hamiltonian is a function of the variables qi and pi given by the Legendre transform 1. In a first time, one assumes that the velocities can be expressed univoquely in terms of canonical momenta so that the Hamiltonian is univoquely defined by 1.

    The point x is then a continuous extension of the discrete index i and the summation extends to an integration.

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    It is often abusively called the Lagrangian. Locality of the theory is also assumed. This means that L x,t is a function, not a functional, of the fields.