Noncovariant Gauges in Canonical Formalism (Lecture Notes in Physics 761)

Lecture Notes in Physics Founding Editors: W. Beiglb¨ock, J. Ehlers, K. Hepp, H. Weidenm¨uller Editorial Board R. Beig, Vienna, Austria W. Beiglb¨ock, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. H¨anggi, Augsburg, Germany G. Hasinger, Garching, Germany W. Hillebrandt, Garching, Germany R. L. Jaffe, Cambridge, MA, USA W. Janke, Leipzig, Germany H. v. L¨ohneysen, Karlsruhe, Germany M. Mangano, Geneva, Switzerland J.-M. Raimond, Paris, France D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany J. Zittartz, K¨oln, Germany

The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg / Germany [email protected]

A. Burnel

Noncovariant Gauges in Canonical Formalism

123

Andr´e Burnel Universite Liege IFPA-AGO Inst. Physique B5 Sart-Tilman 4000 Liege Belgium [email protected]

Burnel, A. Noncovariant Gauges in Canonical Formalism, Lect. Notes Phys. 761 (Springer, Berlin Heidelberg 2009), DOI 10.1007/978-3-540-69921-7

ISBN: 978-3-540-69920-0

e-ISBN: 978-3-540-69921-7

DOI 10.1007/978-3-540-69921-7 Lecture Notes in Physics ISSN: 0075-8450 Library of Congress Control Number: 2008933406 c 2009 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

Gauge theories are, at the present time, the cornerstones of high-energy physics. The whole phenomenology rests on the successes of the standard model of strong, weak and electromagnetic interactions, which is based on the gauge group SU(2) ⊗ U(1) ⊗ SU(3)C . By definition, gauge theories involve more degrees of freedom than required by the physics. The unphysical degrees of freedom (longitudinal and scalar photons and gluons, . . .) are involved in all the steps of the formalism when explicit Lorentz covariance is required. It must be proved that they do not generate unwanted effects or, in other words, that all the contributions of unphysical states finally cancel out. In Quantum Electrodynamics, the solution of this problem rests on the Gupta [8]-Bleuler [5] formalism. In nonabelian gauge theories, the Becchi-Rouet-StoraTyutin [2, 3, 4, 11] (abbreviated as BRST throughout this book) symmetry of the Lagrangian is the key to the solution. This powerful tool and its achievement through the field-antifield formalism1 gives an easy and convincing proof of unitarity and renormalizability of the quantized theory without any use of Feynman graphs. The latter are however unavoidable when applications to particular problems of particle physics are handled. In nonabelian theories, Faddeev-Popov [6] ghost loops complicate the graphical representation and the calculation of amplitudes. In the usual formulation of field theory based on a covariant Lagrangian in the continuous space, they are unavoidable. In order to avoid Faddeev-Popov ghosts, either the Lagrangian (and Hamiltonian) formalism (and the associated energy conservation) or the covariance must be abandoned. Both are physical requirements so that either the lack of energy conservation or the lack of covariance can occur only in an unphysical sector. Here, particular emphasis is given to the last possibility. The most frequently used noncovariant gauges are the Coulomb and axial gauges. 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, . . . occur.

1

See for instance [1]

v

vi

Preface

Axial gauges are thought to be free of Faddeev-Popov ghost loops but they are plagued with the presence of unphysical poles in the propagator which are cumbersome to handle. 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. 1. 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. 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. 2. 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.2 It only deals with the problem of noncovariant gauges. 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. In covariant gauges, the gauge condition involves ∂ μ Aμ = gμν ∂ μ Aν and fixes therefore the time evolution of the A0 -field. Its natural extension consists of Cμν ∂ μ Aν where Cμν is a fixed tensor. Such tensor cannot be completely arbitrary because the Cauchy problem associated with field equations must be well defined. This implies that the differential operator Cμν ∂ μ ∂ ν must be hyperbolic or parabolic. Moreover the gauge condition gives the time evolution of the A0 -field. Therefore, the coordinate frame must be such that C00 = 0. 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. 2

For books on field theory at an elementary level, see, for instance [9, 10]

Preface

vii

Some popular noncovariant gauges like the axial gauge nμ Aμ = 0 do however not fall into this class. 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. 2. 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. The question is more fundamental than to find an accurate prescription for the unphysical pole at n · k = 0 which is appearing in the inversion of the evolution differential operator. 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. Here, it appears that some Feynman integrals show further singularities when det (Cμν ) = 0. 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. Jean-Ren´e Cudell and the referees of “Lecture Notes in Physics” are also acknowledged.

References 1. 2. 3. 4. 5. 6. 7.

Barnich, G., Brandt, F., Henneaux, M.: Phys. Rep. 338, 439 (2000) v Becchi, C., Rouet, A. Stora, R.: Phys. Lett. B52, 344 (1974) v Becchi, C., Rouet, A. Stora, R.: Commun. Math. Phys. 42, 127 (1975) v Becchi, C., Rouet, A., Stora, R.: Ann. Phys. 98, 287 (1976) v Bleuler, K.: Helv. Phys. Acta 23, 567 (1950) v Faddeev, L.S., Popov, V.N.: Phys. Lett. 25B, 29 (1967) v Gribov, V.N.: Nucl. Phys. B139, 1 (1978) v

viii

Preface

8. Gupta, S.N.: Proc. Phys. Soc. (London) A63, 681(1950) v 9. Itzykson, C., Zuber, J.B.: Quantum field theory, McGraw-Hill, New-York (1980) vi 10. Peskin, M.E., Schroeder, D.V.: An introduction to Quantum Field Theory, Addison-Wesley (1995) vi 11. Tyutin, I.V.: “Gauge Invariance In Field Theory And Statistical Physics In Operator Formalism,” LEBEDEV-75-39 v

Contents

1

Canonical Quantization for Constrained Systems . . . . . . . . . . . . . . . . . . 1.1 Canonical Quantization of Mechanical Unconstrained Systems . . . . 1.1.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Lagrangian and Lagrangian Density . . . . . . . . . . . . . . . . . . . . . 1.2.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 The Free Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Solution of the Klein-Gordon Equation . . . . . . . . . . . . . . . . . . 1.3 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Primary and Secondary Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Primary Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Secondary Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 First and Second Class Constraints . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Quantization in the Presence of Second-Class Constraints . . 1.4.5 Quantization in Presence of First-Class Constraints . . . . . . . . 1.4.6 BRST Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Back to the Free Massless Vector Field . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Lagrangian and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Covariant Gauge for the Free Massless Vector Field . . . . . . . . . . . . . . 1.6.1 Commutation Relations for Any Time . . . . . . . . . . . . . . . . . . . 1.6.2 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . 1.6.3 The Gupta-Bleuler Formalism . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 2 2 3 3 3 4 4 5 5 6 8 8 8 9 10 11 13 14 14 14 16 16 18 21 22 23

ix

x

2

3

Contents

Quantization of the Free Electromagnetic Field in General Class III Linear Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lagrangian and Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Derived Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Sketching the Solution of the Cauchy Problem . . . . . . . . . . . . 2.2.5 Particular Cases of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Planar-Type Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constraint Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Canonical Momenta and Primary Constraints . . . . . . . . . . . . . 2.3.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Constraint Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Singular Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Commutation Relations for Any Time . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Canonical Commutation Relations . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Commutation Relations Involving the S Field . . . . . . . . . . . . . 2.5.3 Commutation Relations Involving the S Field . . . . . . . . . . . . 2.5.4 Commutation Relations Involving B = ∂ · A . . . . . . . . . . . . . . 2.5.5 Commutation Relations Between Aμ ’s . . . . . . . . . . . . . . . . . . . 2.5.6 Summary of the Commutation Relations for Any Time . . . . . 2.6 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Momentum Space Expansion of the Fields . . . . . . . . . . . . . . . 2.6.2 Commutation Relations Between Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Summary of the Algebra of Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Gupta-Bleuler Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Covariance Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Translation Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantization of the Free Electromagnetic Field in Class II Axial Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lagrangian and Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Derived Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Summary of the Field Equations . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Constraint Analysis and Effective Hamiltonian . . . . . . . . . . . . . . . . . . 3.4 Solution of Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Solution of n · ∂ S = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 26 26 26 28 28 30 31 32 32 33 33 33 34 34 34 36 38 42 44 45 45 50 55 55 56 56 58 60 61 61 61 61 62 63 63 63 65 65

Contents

4

xi

3.4.2 Solution of (n · ∂ + κ ) S = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Solution of n · ∂ B = (n2 − a)S . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Elementary Solution of n · ∂ A = 0 . . . . . . . . . . . . . . . . . . . . . 3.4.5 Solution of the Cauchy Problem for A(x − z) = Lx Dn (x − z) 3.4.6 Elementary Solution of (n · ∂ )2 A = 0 . . . . . . . . . . . . . . . . . . 3.5 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Equal Time Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Commutation Relations Involving the S-Field . . . . . . . . . . . . 3.5.3 Commutation Relations Involving the S -Field . . . . . . . . . . . . 3.5.4 Commutation Relations Involving ∂ · A . . . . . . . . . . . . . . . . . . 3.5.5 Commutation Relations [Aμ (x), Aν (z) . . . . . . . . . . . . . . . . . . . 3.6 Association of a Feynman Propagator with the Operator n · ∂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Associating Creation and Annihilation Operators with Fields . . . . . . 3.7.1 The S-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Other Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Correct Momentum Space Expansion of the Fields . . . . . . . . . . . . . . . 3.8.1 The S-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 The B-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 The S -Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.4 The Aμ -Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 A Toy Model for the Unphysical S and B Fields . . . . . . . . . . . . . . . . . 3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Interpolating Between Axial and Relativistic Gauges . . . . . . . . . . . . . 3.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 67 67 70 70 71 71 72 72 73 74

Gauge Fields in Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 General Formalism of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 General Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Class III Gauges in Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Building the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Canonical Momenta and Constraints . . . . . . . . . . . . . . . . . . . . 4.3.4 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 The BRST Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Ghost Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Global Internal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.9 Problems of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 88 89 90 91 91 94 94 95 95 97 97 98 98 99

76 77 78 78 80 80 80 82 83 84 85 85 86 86

xii

5

Contents

Perturbation Theory: Renormalization and All That . . . . . . . . . . . . . . . 101 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 The S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3 Perturbative Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.1 Expansion and Consequences of the Unitarity Condition . . . 103 5.3.2 Consequences of the Causality Condition . . . . . . . . . . . . . . . . 104 5.3.3 The First Term T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Nonunicity of the Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3.5 The Fixed Part of Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3.6 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.3.7 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3.8 Feynman Rules in Momentum Space . . . . . . . . . . . . . . . . . . . . 113 5.4 Divergences, Power Counting and Renormalizability . . . . . . . . . . . . . 118 5.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4.2 Power Counting and Superficial Degree of Divergence . . . . . 118 5.4.3 Renormalizability by Power Counting . . . . . . . . . . . . . . . . . . . 120 5.5 Dimensional Regularization of Covariant Divergent Integrals . . . . . . 120 5.5.1 A General One-Loop Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.5.2 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5.3 Use of the Feynman Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5.4 Elimination of the Denominators . . . . . . . . . . . . . . . . . . . . . . . 122 5.5.5 Complex Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.5.6 Tensor Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.6.1 The General One-Loop Integral . . . . . . . . . . . . . . . . . . . . . . . . 124 5.6.2 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.6.3 Elimination of the Denominators . . . . . . . . . . . . . . . . . . . . . . . 126 5.6.4 Calculation of the Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.6.5 Introduction of the Feynman Variables . . . . . . . . . . . . . . . . . . 128 5.6.6 Integration Over λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.6.7 Regularization of Ultraviolet Divergences . . . . . . . . . . . . . . . . 129 5.6.8 Consequences of the Nonsingularity of the B−1 Matrix . . . . . 130 5.7 Computation of the Ghost Loop With Two Legs . . . . . . . . . . . . . . . . . 131 5.8 Renormalization and Counter-Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.8.1 Various Renormalization Schemes . . . . . . . . . . . . . . . . . . . . . . 133 5.8.2 Multiplicative Renormalization . . . . . . . . . . . . . . . . . . . . . . . . 134 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Contents

xiii

6

Slavnov-Taylor Identities for Yang-Mills Theory . . . . . . . . . . . . . . . . . . . 137 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 The Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.3 One-Particle Irreducible Vertex Functions . . . . . . . . . . . . . . . . . . . . . . 138 6.4 Yang-Mills Theory in a General Class III Gauge . . . . . . . . . . . . . . . . . 139 6.4.1 The Lagrangian and Superficially Divergent Processes . . . . . 139 6.4.2 BRST Symmetry, Field Equations and Canonical Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.4.3 Commuting Derivatives and Time-Ordered Products . . . . . . . 143 6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.5.1 Covariant Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.5.2 General Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.5.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.6 Identity for the Three-Gluon Vertex Function . . . . . . . . . . . . . . . . . . . 149 6.6.1 Derivation of the Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.6.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.7 Ghost Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.8 Ghost-Ghost-Gluon Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.8.1 Identity in Coordinate Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.8.2 Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.8.3 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.9 Multiplicative Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7

Field Theory Without Infinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.2 Iterative Construction of the S-Matrix Without Time-Ordering . . . . . 158 7.3 Splitting of Causal Distributions into Advanced and Retarded Parts . 159 7.3.1 Distribution and Fourier Transform . . . . . . . . . . . . . . . . . . . . . 159 7.3.2 Order of Singularity of a Distribution . . . . . . . . . . . . . . . . . . . 160 7.3.3 Splitting of Distribution with Negative Order of Singularity . 161 7.3.4 Nonnegative Singularity Order . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.4 Application to Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.4.1 First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.4.2 Example of a One-Loop Process: The Gluon Self-energy . . . 179 7.4.3 Calculation of I1 (ξ ; κ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.4.4 Calculation of Iˆ2 (k; κ1 , κ2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.4.5 Calculation of the Tensor Distributions . . . . . . . . . . . . . . . . . . 188 7.4.6 The Final Result for the Gluon Self-energy . . . . . . . . . . . . . . . 190 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

xiv

Contents

8

Gauges with a Singular C Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2 The Ghost Loop Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.2.1 The Leibbrandt Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.2.2 Interpolating Between Leibbrandt and Relativistic Gauges . . 197 8.3 Singularities Generated by the Lack of Power Counting in Ultra-Violet Divergent Perturbative Theory . . . . . . . . . . . . . . . . . . . 199 8.3.1 The Loop Integration in a General Gauge . . . . . . . . . . . . . . . . 200 8.3.2 Restriction to Loops with Two External Particles . . . . . . . . . . 201 8.3.3 Further Restriction for m ≤ 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.3.4 Interpolating Between Leibbrandt and Relativistic Gauges . . 205 8.4 Cancellation of Divergences at α = 0 in the Self-Energy . . . . . . . . . . 207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

A

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

B

A Useful Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

C

Generalized Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 C.1 Elementary Solutions of the Klein-Gordon Equation . . . . . . . . . . . . . 215 C.1.1 The Δ Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 C.1.2 The Δ F Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 C.2 The E Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 C.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 C.2.2 Derivation of the Elementary Solution . . . . . . . . . . . . . . . . . . . 216 C.2.3 Zero-Time Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 C.2.4 Integration Over k0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 C.2.5 The Cauchy Problem Associated with 2 . . . . . . . . . . . . . . . . 218 C.3 Generalized Functions Associated with the C Operator . . . . . . . . . . 219 (1) C.3.1 Elementary Solution DC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 C.3.2 Positive Frequency Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 C.3.3 Negative Frequency Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 C.3.4 Elementary Solution with Causal Support . . . . . . . . . . . . . . . . 222 C.3.5 Generalization with a Mass Parameter . . . . . . . . . . . . . . . . . . . 223 C.3.6 Analogous of the Feynman Propagator . . . . . . . . . . . . . . . . . . 224 C.3.7 Solution of the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 225 C.3.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 C.4 Generalized Functions Associated with C2 . . . . . . . . . . . . . . . . . . . . . 226 C.4.1 The Elementary Function with Causal Support . . . . . . . . . . . . 226 C.4.2 Zero Time Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 C.4.3 Integration Over k0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 C.4.4 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 C.5 Generalized Functions Associated with C . . . . . . . . . . . . . . . . . . . 228

Contents

xv

C.5.1 Elementary Solution with Causal Support . . . . . . . . . . . . . . . . 228 C.5.2 Zero-Time Properties of FC and its Time Derivatives . . . . . . . 230 C.5.3 Preferred Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 C.5.4 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 C.6 The G-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Chapter 1

Canonical Quantization for Constrained Systems

1.1 Canonical Quantization of Mechanical Unconstrained Systems 1.1.1 Lagrangian Formalism Because the canonical formalism will be the main tool in the discussion of the topics in this book, let us recall its main features. Let us begin with the canonical quantization of a simple mechanical system characterized by its Lagrangian L. It is a function of the various generalized coordinates qi and of their velocities q˙i L = L(qi , q˙i ).

(1.1)

For the sake of simplicity and because it is the case of most theories of physical interest, explicit time dependence and higher order derivatives are not included in the discussion. Moreover and for the same reasons, the kinetic energy is assumed to be given by 1 T = ∑ q˙i2 . (1.2) 2 i Equations of motion, known as Euler-Lagrange equations, result from the least action principle that the reader is assumed to know. They read d ∂L ∂L − = 0. dt ∂ q˙i ∂ qi

(1.3)

1.1.2 Hamiltonian For quantization purposes, one needs to go to the Hamiltonian formalism. A canonical momentum defined by ∂L (1.4) pi = i ∂ q˙

Burnel, A.: Canonical Quantization for Constrained Systems. Lect. Notes Phys. 761, 1–23 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 1 

2

1 Canonical Quantization for Constrained Systems

is associated to each variable. 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 (1.5) H = pi q˙i − L where summation over repeated indices is understood. 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 (1.5). Let us also recall without proof that the Hamiltonian equations of motion read q˙i =

∂H , ∂ pi

p˙i = −

∂H . ∂ qi

(1.6)

1.1.3 Poisson Brackets For quantization purposes, it is very useful to use the notion of Poisson brackets. If A(pi , qi ) and B(pi , qi ) are differentiable functions without explicit time dependence, these brackets are defined by   ∂A ∂B ∂A ∂B {A, B} = − . (1.7) ∂ qi ∂ pi ∂ pi ∂ qi It is an easy task to derive their following properties {AB,C} = A{B,C} + {A,C}B,

(Leibnitz rule)

{A, B} = −{B, A},

(antisymmetry)

{qi , p

i j} = δ j,

{qi , q j } = {pi , p j } = 0.

(1.8)

(canonical brackets)

Use of these properties allows one to get the brackets between all polynomial functions of pi and qi without explicit use of their definition, a fact that is particularly useful for carrying out the calculations by formal computer algebra.

1.1.4 Quantization It is now easy to proceed to the quantization of the theory through the correspondence principle. It consists of the replacement of classical variables by operators and Poisson brackets by commutators following the rule {A, B} →

−i [A, B]. h¯

(1.9)

1.2 Field Theory

3

Heisenberg evolution equations result from this rule i¯h A˙ = [A, H].

(1.10)

1.2 Field Theory 1.2.1 Lagrangian and Lagrangian Density Let us now extend this formalism to field theory. Formally, a field φ (x) = φ (x,t) can be considered as describing one degree of freedom at each point of the space. The point x is then a continuous extension of the discrete index i and the summation extends to an integration. The correspondence can be summarized in i → x,

∑→

j → y,

(1.11)



d 3 x,

(1.12)

i

δi j → δ (3) (x − y)

(1.13)

where δ (3) (x − y) is the three-dimensional Dirac function. It is assumed that one can write 

L(t) =

d 3 x L (x,t)

(1.14)

where L (x,t) is the Lagrangian density of the theory. 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. Discrete indices can also be included in order to take into account various different fields ui (x). It will again be assumed that the Lagrangian density is a function of the fields and of their first derivatives ∂μ ui (x) where x is a point of space-time x = (x,t). Here, in order to respect invariance under translations, no explicit dependence on x is allowed in the Lagrangian. Because, most practical field theories do not involve them, the case of higher derivatives will also not be considered here. According to the least action principle, it is a straightforward task to get the Euler-Lagrange equations

∂μ

∂L ∂L − = 0. ∂ (∂μ ui ) ∂ ui

(1.15)

1.2.2 Hamiltonian In a similar way as in the case of a discrete number of degrees of freedom, the variable canonically conjugated to ui (x) is defined by

4

1 Canonical Quantization for Constrained Systems

Π i (x) =

∂L . ∂ (∂0 ui (x))

(1.16)

In the same way, the Hamiltonian density is defined by   H (x) = Π i (x)u˙i (x) − L = H ui (x), Π i (x) and the Hamiltonian by



H=

d 3 x H (x).

(1.17)

(1.18)

With the assumption of vanishing fields at infinity, one can add three-divergences to H without modifying H, a property that can be sometimes useful. It is classical to get the Hamiltonian equations of motion u˙i =

∂H , ∂Πi

∂H Π˙ i = − . ∂ ui

(1.19)

1.2.3 Poisson Brackets In the same way as in mechanical systems, it is useful to define Poisson brackets. In order to recover Hamiltonian equation of motions, their definition is   ∂ A(x) ∂ B(y) ∂ A(x) ∂ B(y) (3) − {A(x), B(y)}x0 =y0 = δ (x − y). (1.20) ∂ ui (x) ∂ Π i (y) ∂ Π i (x) ∂ ui (y) It should be noted that they are equal-time brackets in the sense that A and B are taken at the same time. The Poisson brackets between canonical equal-time variables read now

ui (x), Π j (y) x =y = δij δ (3) (x − y), (1.21) 0



ui (x), u j (y) x

0

0 =y0



= Π i (x), Π j (y) x

0 =y0

= 0.

(1.22)

Again, with the same properties (Leibnitz rule and antisymmetry) as those defined for a discrete number of degrees of freedom, it is possible to get Poisson brackets between polynomial functions without the need of expliciting them.

1.2.4 Quantization As for a discrete number of degrees of freedom, quantization is realized through the replacement of Poisson brackets by equal-time commutators multiplied by −i/¯h. In the quantum theory, the Heisenberg equations read

1.2 Field Theory

5

h¯ − A˙ = [A, H]. i

(1.23)

Let us remark that Heisenberg equations involve the Hamiltonian while EulerLagrange equations are derived from the Lagrangian density.

1.2.5 The Free Scalar Field As an elementary example, let us consider the Lagrangian 1 1 L = ∂μ φ ∂ μ φ − m2 φ 2 2 2 describing a free scalar field, the simplest possible case. It is a simple matter of derivation to get

∂L = ∂μφ, ∂ (∂μ φ )

∂L = m2 φ ∂φ

so that the field equation is the Klein-Gordon one ( + m2 ) φ = 0 where  = ∂μ ∂ μ is the d’Alembert operator. By a simple use of the rules of the previous subsection, the conjugate canonical momentum is given by ∂L = ∂0 φ Π= ∂ (∂0 φ ) and the canonical Poisson brackets are obviously {φ (x), φ (y)}x0 =y0 = {Π (x), Π (y)}x0 =y0 = 0, {φ (x), Π (y)}x0 =y0 = δ (3) (x − y). Canonical commutation relations are obtained by the correspondence principle. They read [φ (x), φ (y)]x0 =y0 = [Π (x), Π (y)]x0 =y0 = 0, [φ (x), Π (y)]x0 =y0 = δ (3) (x − y).

1.2.6 Solution of the Klein-Gordon Equation It is important to remark that the canonical commutation relations hold only at equal times. In order to get commutation relations for any time, it is necessary to solve the equations of motion. Such solutions will be carried out in more complicated cases in the following and in the appendices. Here, let us only recall that the solution can be written

6

1 Canonical Quantization for Constrained Systems

φ (x) =

1 (2π )3/2



√

k0 =

|k|2 +m2

d3k  a(k) e−ik·x + a† (k) eik·x . 2k0

(1.24)

This equation shows that the solution of the free Klein-Gordon equation is the superposition of plane waves propagating in both directions. In a quantum theory, plane waves correspond to interaction free particles. This is the reason why the fields with only quadratic terms in the Lagrangian are called free fields. Let us also remark that, in the quantum theory, a(k) and a† (k) are the annihilation and creation operators. Self-adjointness of the φ field implies that a† (k) is the adjoint of a(k). The algebra satisfied by creation and annihilation operators is a key point in building the space of particle states. It can be obtained only by solving the theory and its knowledge is equivalent to the knowledge of commutation relations at any time. The latter ones can be obtained by solving the Cauchy problem associated with the Klein-Gordon equation. This solution is well known and reads f (x) = −



     d 3 y Δ x − y; m2 ∂0y f (y) + d 3 y ∂0y Δ x − y; m2 f (y)

(1.25)

where y0 is arbitrary and Δ (x; m2 ) is the elementary solution with causal support satisfying zero-time properties     Δ x; m2 x =0 = 0, ∂0 Δ x; m2 x =0 = −δ (3) (x). (1.26) 0

0

Using these relations, the commutation relation for any time reads   [φ (x), φ (y)] = iΔ x − y; m2 . Again the explicit solution of this Cauchy problem, the properties of elementary solutions and the derivation of commutation relations for any time are carried out explicitly in more complicated cases in the following.

1.3 The Electromagnetic Field The scalar field is the unique field theory which can be quantized in the simple way described in the previous section. Other systems, including the free Dirac Lagrangian, imply algebraic relations between fields and their conjugate momenta. Such relations are called constraints. The presence of constraints is incompatible with usual commutation relations. Let us consider the example of the electromagnetic field. It is described by the Lagrangian 1 (1.27) L = − F μν Fμν 4

1.3 The Electromagnetic Field

7

where Fμν is an antisymmetric second rank tensor related to electric and magnetic fields by 1 Bi = i jk Fjk . Ei = F0i , 2 These fields can be derived from potentials which are not physical quantities. This is realized by setting Fμν = ∂μ Aν − ∂ν Aμ .

(1.28)

Potentials are obtained from differential equations and are therefore non-unique. The same Fμν tensor results from different potentials related by Aμ = Aμ + ∂μ Λ .

(1.29)

This is called gauge transformation of the potentials and the fact that the physics does not depend on a particular choice of the potentials implies that the Lagrangian must be invariant under the transformation (1.29). This is called gauge invariance. The derivation of field equations from the Lagrangian (1.27) is left as an exercise. They read

∂ μ Fμν = 0

(1.30)

∂L = F μ0. ∂ (∂0 Aμ )

(1.31)

while canonical momenta are

Πμ = Canonical Poisson brackets are

Aμ (x), Aν (y) x

0 =y0

= {Π μ (x), Π ν (y)}x0 =y0 = 0,



Aμ (x), Π ν (y) x

0 =y0

= δμν δ (3) (x − y).

(1.32) (1.33)

They are obviously incompatible with the constraint Π 0 = 0 resulting from (1.31). Another constraint can be derived from the field equations by setting ν = 0 in (1.30). Using (1.31), this field equation becomes

∂ k Π k = 0. This constraint is also incompatible with (1.33). The following section will deal with the quantization in presence of constraints. For the sake of clarity, only a finite number of degrees of freedom will be taken into account in a first time. Extension to field theory is, in most cases, almost obvious. Some exceptions will be explicitly noted when they will be encountered, in particular in noncovariant gauges.

8

1 Canonical Quantization for Constrained Systems

1.4 Primary and Secondary Constraints 1.4.1 Primary Constraints In the quantization process, it is important to obtain all the constraints of the theory. Let us first describe how they can be obtained systematically. In a theory described by a Lagrangian, they occur when, in the course of the Legendre transform between Lagrangian and Hamiltonian formalism, the velocities cannot be solved in terms of momenta. If L(qi , q˙i ) is the Lagrangian, the canonical momenta pi =

∂L ∂ q˙i

(1.34)

are given univoquely but this equation must be solved with respect to the velocities q˙i when performing the Legendre transform. Let k be the rank of the matrix  2  ∂ L . ∂ q˙i ∂ q˙ j If k = n, the solution is unique and there is no problem at all. Constraints occur when k < n. Indeed there are then N = n − k relations between lines of the matrix. Compatibility implies the same combinations between the pi . These relations are called primary constraints. Let us write them as fl (pi , qi ) = 0,

l = 1, . . . , N.

The fact that some velocities cannot be expressed in terms of canonical momenta leads to a non unicity of the Hamiltonian. It can be written N

H = Hcl + ∑ λ j f j j=1

where Hcl is the Hamiltonian when the primary constraints are taken into account (the so-called classical or physical Hamiltonian) and λ j are N unknown Lagrange multipliers corresponding to the undetermined velocities.

1.4.2 Secondary Constraints A constraint must be stable with time. In other words, all its time-derivatives must vanish. Let us therefore first compute f˙l (pi , qi ) = {H, fl (pi , qi )} by taking the constraints into account only once all the Poisson brackets are computed. The result is

1.4 Primary and Secondary Constraints

9



  

  f˙l pi , qi = H, fl pi , qi ≈ Gl pi , qi where the weak equality ≈ means that the constraints are taken into account but only after the computation of the brackets. In general,       f˙l pi , qi = Gl pi , qi + hlk pi , qi fk + · · · (1.35) where the dots are terms at least quadratic in the constraints fl . Stability implies   (1.36) Gl pi , qi ≈ 0. Using (1.35), the second time-derivative can be computed with the help of Leibnitz rule   f¨l pi , qi = {H, Gl } + {H, hlk } fk + Gk hlk + hlk hkm fm + · · · where the dots mean terms at least quadratic into the constraints. When constraints and the stability conditions (1.36) are taken into account,   f¨l pi , qi ≈ {H, Gl }. The stability of fl under the second time-derivative amounts to the one of Gl under first time-derivative. What can Gl be ? Three cases are possible. 1. Gl ≈ 0. Stability holds for the first time-derivative and, of course, for all higher time-derivatives. The process of checking stability stops here. 2. Gl ≈ Gl (pi , qi , λk ) i.e. it is an equation involving the Lagrange multipliers. Then time-derivatives of Gl will involve time-derivatives of the Lagrange multipliers and, if they are known, nothing new can be obtained from higher derivatives. Check of stability again stops here. 3. Gl ≈ Gl (pi , qi ) i.e. a new constraint is appearing. It must also be stable and therefore it must be treated like primary constraints until one of the two above situations is encountered. Once this stability test is realized with all the constraints, compatibility of the equations determining the multipliers must be checked. This check can generate new constraints. They must be treated like the previous ones until full compatibility is obtained. The constraints Gk (pi , qi ), k = 1, . . . Q obtained from stability conditions are called secondary constraints.

1.4.3 First and Second Class Constraints Let us now make a difference between constraints, which will be important for quantization purposes. Let us gather all the constraints, primary and secondary, into a unique set {Fl , l = 1, . . . , N + Q}. A quantity A such that its Poisson brackets with all the constraints are weakly vanishing

10

1 Canonical Quantization for Constrained Systems

{A, Fl } ≈ 0,

l = 1, . . . N + Q,

is called a first-class quantity. In some cases, it is possible to associate to a given quantity a first-class quantity which is weakly equivalent to it according to the following procedure. Let us define A = A + bl Fl ≈ A

(1.37)

where the parameters bl are determined by the requirement that A be first-class. This requirement leads to the linear system bl {Fl , Fk } = −{A, Fk }.

(1.38)

Such a system determines univoquely only a subset of the bl ’s according to the rank of the matrix ({Fl , Fk }). The choice of the determined bl ’s is somewhat arbitrary. One calls second-class constraints1 a particular subset of the constraints such that, for this subset, ({Fl , Fk }) is of maximal rank. The complement of this set can be transformed into a set of constraints which are first-class with respect to the set of second-class constraints. These last constraints are all the first-class constraints. Their presence can be related to a gauge invariance of the theory but this point will not be considered here because one will work directly with models exhibiting already gauge invariance .

1.4.4 Quantization in the Presence of Second-Class Constraints Let us assume that the set of constraints {Fl , l = 1, . . . , P} is second-class. Then the matrix C = (Clk ) = ({Fl , Fk }) can be inverted. With ordinary variables, this matrix is, owing to the antisymmetry of Poisson brackets, antisymmetric. Therefore, its dimension P is even. There is always an even number of second-class constraints.2 To each quantity A, let us associate its first-class partner A given by −1 A = A − {A, Fl }Clk Fk .

By definition, A is such that {A , Fk } ≈ 0, ∀k = 1, . . . , P. The brackets involving only first-class quantities are compatible with the second-class constraints. Using properties of Poisson brackets, the brackets between two first-class quantities can easily be computed. The weak relation 1

This definition is somewhat different from the most frequent one which defines second-class constraints as those whose Poisson brackets with all the constraints are not weakly 0. It is however more precise because in the usual definition, some linear combinations of second-class constraints can be first-class. 2 This is not true for Grassmann odd-parity variables and can also be incorrect in some field theories when derivatives are involved in the brackets.

1.4 Primary and Secondary Constraints

11

−1 {A , B } ≈ {A, B} − {A, Fl }Clk {Fk , B}

(1.39)

results. Defining the Dirac brackets of two quantities A, B by the Poisson brackets between their first-class associates, −1 {A, B}D = {A , B } ≈ {A, B} − {A, Fl }Clk {Fk , B},

(1.40)

these new brackets have the same properties as the Poisson brackets but are compatible with the second-class constraints. Quantization is realized by transforming the Dirac brackets into commutators.

1.4.5 Quantization in Presence of First-Class Constraints All the second-class constraints can be eliminated provided Dirac instead of Poisson brackets are used. Let us therefore assume that only first-class constraints are occurring. In order to have a better understanding of this important problem, let us first work with an academic but very simple and instructive example, the Lagrangian 1 1 L = q˙21 + (q˙2 − q3 )2 . 2 2

(1.41)

This is the simplest case of a theory with a pair of first-class constraints. It reproduces all the features met in usual gauge field theories. In particular, it is invariant under the gauge transformations q2 = q2 + τ ,

q3 = q3 + τ˙ .

Let us proceed with some details and first look at the canonical momenta. We have ∂L ∂L ∂L p1 = = q˙1 , p2 = = q˙2 − q3 , p3 = = 0. ∂ q˙1 ∂ q˙2 ∂ q˙3 A primary constraint p3 ≈ 0 is present in the game and the Hamiltonian becomes H = pi q˙i − L =

1 2 1 2 p + p + p2 q3 + p3 q˙3 . 2 1 2 2

Replacing the undetermined velocity q˙3 by a Lagrange mutiplier λ , the total Hamiltonian is now written as HT =

1 2 1 2 p + p + p2 q3 + λ p3 . 2 1 2 2

(1.42)

A secondary constraint comes from the fact that the time derivative of p3 must also vanish. From the canonical Poisson brackets {qi , p j } = δ ji ,we have

12

1 Canonical Quantization for Constrained Systems

p˙3 = {HT , q3 } = p2 and therefore a constraint p2 ≈ 0. Again p˙2 must vanish but this occurs trivially. It is useful to summarize this search of constraints into a chain as follows .

.

p3 ≈ 0 =⇒ p2 ≈ 0 =⇒ 0 ≈ 0 .

where the symbol =⇒ means that the result after it is obtained by imposing stability of the constraint before it. Here thus p˙3 ≈ 0 implies p2 ≈ 0 and p˙2 ≈ 0 gives rise to a trivial relation. The two constraints are obviously first-class. Let us call, here and in the following, a pair of canonically conjugate variables as a degree of freedom.The two degrees of freedom (q2 , p2 ), (q3 , p3 ) are constrained but not completely because there is no constraint on the coordinates q2 , q3 . Such degrees of freedom are called gauge degrees of freedom. The total Hamiltonian of this system (1.42 ) contains an undetermined parameter λ ≈ q˙3 and all values of this parameter must lead to the same physical theory. A consistent choice of this parameter is called gauge fixing. This choice can be done directly or indirectly. Let us consider the various ways of realizing it. 1. Let us fix q2 by q2 = f (q1 , p1 , p2 ). This defines a new constraint that must be stable. Therefore a new constraint chain appears .

.

q2 − f (q1 , p1 , p2 ) ≈ 0 =⇒ q3 − f˙(q1 , p1 , p2 ) ≈ 0 =⇒ λ − f¨(q1 , p1 , p2 ) ≈ 0. If the original first-class constraints and these new constraints are gathered, a second-class set is obtained and Dirac brackets can be used in order to quantize the theory. The two gauge degrees of freedom can be eliminated and the theory reduces to a theory with only one degree of freedom (q1 , p1 ). The phase space reduces to the physical phase space. This way of doing the choice of the gauge is called here class I gauge fixing. For mathematicians and physicists working in constrained systems, this is simply gauge fixing. For high energy physicists dealing with the Salam-Weinberg model, this is called unitary gauge fixing. Let us note that a class I gauge fixing condition is not completely arbitrary. A general condition like f (q2 , q1 , p1 ) ≈ 0 cannot be used if it does not fix univoquely q2 in terms of the other variables. For instance, a gauge condition like q22 ≈ f (q1 ) is not accurate. None of the quantities q2 , q3 , λ is fixed univoquely. As a general requirement, the resulting Dirac brackets must lead to a consistent quantum theory. They cannot depend on the variables and, in quantum field theory, they must be local. These requirements are generally overlooked and this overlooking can lead to difficulties. This is an example of the Gribov ambiguity[6], first found in Yang-Mills theory. 2. Instead of imposing a condition on q2 , let us introduce a condition fixing q3 like q3 − f (q1 , p1 , p2 ) ≈ 0. Stability implies now the chain .

q3 − f (q1 , p1 , p2 ) ≈ 0 =⇒ λ − f˙(q1 , p1 , p2 ) ≈ 0

1.4 Primary and Secondary Constraints

13

without any condition on q2 but λ is again fixed. As a particular case, let us take q3 ≈ 0. Stability requires . q3 ≈ 0 =⇒ λ ≈ 0. Using Dirac brackets to eliminate the second-class constraints q3 ≈ 0, p3 ≈ 0, the Hamiltonian reads 1 1 H = p21 + p22 . 2 2 This theory can be quantized with two degrees of freedom (q1 , p1 ), (q2 , p2 ) instead of one. Of course, it is a too large theory and the constraint p2 = 0 must be imposed in a weaker way as p2 |ψphys  = 0. The set of states satisfying this condition is a subspace of the whole used space and is called the physical subspace. This method is called Dirac quantization. Please do not confuse it with Dirac brackets. The gauge condition which leaves one unphysical degree of freedom is called class II gauge fixing. The problem with this sometimes useful method lies in the measure of the Hilbert space. Indeed, with two degrees of freedom, the scalar product of two vector states |ψ1 , |ψ2  is given by ψ2 |ψ1  =



dq1 dq2 ψ2 (q1 , q2 )ψ1 (q1 , q2 ).

In the physical subspace, only q1 is a variable and the scalar product is ψ2 |ψ1  =



dq1 ψ2 (q1 )ψ1 (q1 ).

Going from the theory with two degrees of freedom to the physical theory requires the artificial introduction of a δ (q2 )-function in the Hilbert space measure. 3. Finally, the multiplier λ ≈ q˙3 could be fixed directly without any condition on q2 and q3 . This gives an evolution equation for the variables q3 , p3 while the variables q2 , p2 have an evolution given by Hamilton equations. Two unphysical degrees of freedom are involved and it must be assured that the second one is exactly destroying what the first one does wrong. There exists a very elegant and powerful way to realize this goal, the Becchi-Rouet-Stora-Tyutin (BRST) quantization [1, 2, 3, 10]. This way of fixing the gauge is called class III gauge fixing.

1.4.6 BRST Quantization Let us summarize the main points of the method, well described in many textbooks, in all the cases we will consider. Explicit details will be given later on in the particular case of Yang-Mills theory.

14

1 Canonical Quantization for Constrained Systems

1. The gauge invariant Lagrangian is leading to one (or more) chain of two firstclass constraints. 2. The gauge condition is class III leaving thus with two unphysical degrees of freedom. 3. The gauge condition is introduced with a Lagrange multiplier in the Lagrangian. 4. The new Lagrangian is made BRST invariant, the BRST transformations being a subclass of gauge transformations. 5. There are now four unphysical degrees of freedom, the two gauge degrees of freedom and their associated ghosts. 6. An indefinite metric is associated with unphysical degrees of freedom. 7. The physical states are the cohomology classes of the BRST operator, the generator of BRST transformations. It is a conserved nilpotent operator and the physical states are defined by Q|ψphys  = 0,

 |ψphys  ≡ |ψphys  + Q|ψ0 .

8. Under these conditions, it can be proved that only physical states contribute to physical processes.

1.4.7 Summary It is important to understand well how the quantization can be carried out in the presence of constraints. First of all, all the constraints must be obtained and separated into first and second class. The second class constraints are taken into account by using Dirac instead of Poisson brackets. For the first-class constraints, the problem is subtler and the solution can be obtained in three distinct ways according to the number of degrees of freedom which are used. The number of degrees of freedom depends on the way the gauge is fixed. Class I gauges eliminate all of them but will appear to be not useful in Maxwell and Yang-Mills theories. Class II gauges use one unphysical degree of freedom. Again, it is not of pertinent use in relevant theories. Class III gauges work with pairs of unphysical degrees of freedom to which they give an evolution. Because they can allow to conserve covariance, they are very useful. An elegant method, the BRST symmetry[1, 2, 3, 10] allows to assure that no unwanted effects are introduced by the redundant degrees of freedom.

1.5 Back to the Free Massless Vector Field 1.5.1 Lagrangian and Constraints Let us now show how the considerations of the previous section apply to the classical case of the Maxwell field described by the Lagrangian

1.5 Back to the Free Massless Vector Field

15

1 L = − F μν Fμν 4

(1.43)

Fμν = ∂μ Aν − ∂ν Aμ .

(1.44)

where

This Lagrangian is invariant under the gauge transformations Aμ = Aμ + ∂μ Λ .

(1.45)

Canonical momenta are

Πμ =

∂L = F μ0. ∂ (∂0 Aμ )

(1.46)

It is obvious that Π 0 = 0 is a primary constraint. For space-like values of the index μ , Eq. (1.46) leads to ∂0 Ak = Π k − ∂ k A0 . (1.47) This equation relates velocities and canonical momenta. It is left as an exercise to derive the total Hamiltonian 1 1 HT = Π k Π k + Fkl Fkl − Π k ∂ k A0 + Λ Π 0 . 2 4

(1.48)

Canonical Poisson brackets are {Aμ (x), Aν (y)}x0 =y0 = {Π μ (x), Π ν (y)}x0 =y0 = 0,

(1.49)

{Aμ (x), Π ν (y)}x0 =y0 = δμν δ (3) (x − y).

(1.50)

Using the fundamental rules for Poisson brackets (Leibnitz rule and antisymmetry) as well as {∂k A(x), B(y)}x0 =y0 = ∂k {A(x), B(y)}x0 =y0 , (1.51) straightforward but tedious calculations,3 left as an exercise, lead to the constraint chain .

.

Π 0 ≈ 0 =⇒ ∂ k Π k ≈ 0 =⇒ 0 ≈ 0.

(1.52)

Both constraints are first class, a fact related to the presence of the gauge invariance (1.45). From such a first-class constraint chain, it results that the (A0 , Π 0 ) degree of freedom and a scalar combination of the degrees of freedom (Ak , Π k ) are constrained and therefore unphysical degrees of freedom. 3 Most of these calculations can be systematized in a computer program, for instance REDUCE. Such a program has been used to check the calculations.

16

1 Canonical Quantization for Constrained Systems

1.5.2 Gauge Fixing Let us now go to the crucial problem of gauge fixing. It is important to realize that the gauge choice must lead to a consistent solvable theory in the free field case. Let us consider the various usual choices. Two of them are usually considered as working only with physical degrees of freedom, the Coulomb gauge ∂k Ak = 0 and the pure axial gauge A3 = 0. We will not develop the theory here but simply remark that, in both cases, the theory becomes nonlocal, a fact that makes it inaccurate in the quantum case. Actually, this nonlocality finds its roots in the fact that the secondary constraint, which is solved in class I gauges, implies derivatives. Its solution which will express one of the potentials in terms of the other ones is unavoidably nonlocal. Therefore, no consistent class I gauge can be found for massless vector field. Let us now try a class II gauge like the temporal gauge A0 = 0 or its generalization to an axial gauge characterized by a fixed four-vector nμ , n · A = 0. Such gauges played an important role in the search of theories without contributions of FaddeevPopov ghosts[5]. We will devote an entire chapter to the problems they generate. At the present time, let us only simply remark that they are not covariant and also that the contribution of the unphysical degree of freedom must be removed, a fact that is, if possible, not easy to realize. The only consistent gauge choices for the free massless vector field are class III. The most popular of them is the covariant choice ∂ μ Aμ = 0 or its nonhomogeneous generalizations. In the next chapters, we will consider other noncovariant consistent choices. In order to see how they can be developed consistently, it is important to build up with some details the covariant gauge fixing. This is done in the next section.

1.6 Covariant Gauge for the Free Massless Vector Field The class III Lorentz condition

∂ μ Aμ = 0,

(1.53)

as any gauge condition which is not class I, gives an evolution to unphysical degrees of freedom. Such an evolution is not included in the Maxwell equations. Therefore, in order to quantize covariantly electromagnetism, Maxwell equations must be modified. This fact, which is here obvious from elementary principles, is sometimes proved with more or less heavy techniques[9] . Usually, the gauge condition is introduced inside the Lagrangian with the help of a Lagrange multiplier. Allowing for a nonhomogeneous term in it, the Lagrangian reads a 1 (1.54) L = − Fμν F μν + S ∂ μ Aμ + S2 4 2 or, by adding a four-divergence,

1.6 Covariant Gauge for the Free Massless Vector Field

17

1 a L = − Fμν F μν − ∂ μ S Aμ + S2 . 4 2

(1.55)

The parameter a introduces inhomogeneity in the gauge condition. It is called the gauge parameter. The particular value a = 1 simplifies the formalism. On the other hand, invariance with respect to the gauge can be formulated by independence with respect to a so that it is very useful to leave it unfixed. As stated before, a class III gauge involve Faddeev-Popov ghosts introduced through the BRST symmetry. However these ghosts decouple in the free theory so that their contribution will be overlooked now. Field equations are obtained by the Euler-Lagrange principle. They read

∂ μ Fμν − ∂ν S = 0,

(1.56)

∂ μ Aμ + aS = 0.

(1.57)

If the gauge parameter a is different from 0, (1.57) can be solved with respect to S and this field can be eliminated from the Lagrangian. In order to be as general as possible, such an often made substitution will not be considered here. The case a = 0, corresponding to the homogeneous Landau gauge, is here included in the discussion while it is singular if the S-field is eliminated. Canonical momenta are obtained in the standard way. They are

Πμ =

∂L = F μ0, ∂ (∂0 Aμ )

Π=

∂L = −A0 . ∂ (∂0 S)

(1.58)

One immediately sees that two primary constraints

Π 0 = 0,

Π + A0 = 0

(1.59)

are involved. They are obviously second class and, for this reason, they will generate no secondary constraint. In principle, quantization involves the use of Dirac brackets. It is however simpler to consider that A0 is not an independent variable but the opposite of the variable canonically conjugate to S. The nonvanishing canonical commutators are then given by   Aμ (x), S(y) x 

Aμ (x), Π l (y)

0 =y0

x0 =y0

= igμ 0 δ (3) (x − y), = iglμ δ (3) (x − y).

All the other commutators involving Aμ , Π l and S vanish.

(1.60)

18

1 Canonical Quantization for Constrained Systems

1.6.1 Commutation Relations for Any Time In order to get the commutation relations for any time, it is necessary to solve the field equations. Commutation relations at any time are then the solution of a Cauchy problem for which the equal-time commutators are initial values. Let us solve these equations.

1.6.1.1 The Cauchy Problem for the S-Field. First of all, let us get an equation involving only the S-field. Applying ∂ ν to Eq. (1.56) immediately leads to S = 0. (1.61) The S-field obeys the free massless Klein-Gordon equation. Setting D(x − y) = Δ (x − y; 0) for the massless functions, the solution of the Cauchy problem can be written, as explained in the appendices, as S(x) = −



d 3 y D(x − y) ∂0y S(y) +



d 3 y ∂0y D(x − y) S(y)

(1.62)

where y0 is arbitrary. Here, S is a canonical variable while ∂0 S must be related to canonical variables. By taking ν = 0 in (1.56),

∂0 S = ∂μ F μ 0 = ∂k F k0 = ∂k Π k .

(1.63)

After substitution of (1.63) in (1.62) and integration by parts, (1.62) reads S(x) = −∂k



d 3 y D(x − y) Π k (y) − ∂0



d 3 y D(x − y) S(y).

(1.64)

In this equation, the S-field is expressed in terms of canonical variables taken at a given arbitrary time y0 .

1.6.1.2 Commutation Relations Involving the S Field Commutation relations at any time between S(x) and any other field M(z) are then given in terms of equal-time commutators by taking the arbitrary y0 equal to z0 [S(x), M(z)] = − ∂k − ∂0

 

d 3 y D(x − y) [Π k (y), M(z)]y0 =z0 d 3 y D(x − y) [S(y), M(z)]y0 =z0 .

(1.65)

1.6 Covariant Gauge for the Free Massless Vector Field

19

Use of canonical commutation relations immediately leads to [S(x), S(y)] = 0.

(1.66)

Replacing now M by Aμ and using (1.60), [S(x), Aμ (z)] = i(g0μ ∂0 + gkμ ∂k )D(x − z) = i∂μ D(x − z).

(1.67)

1.6.1.3 Solution of the Cauchy Problem for the Aμ -Field Let us now solve the Cauchy problem for the Aμ -field itself. The field equations (1.56) and (1.57) can be combined to get Aμ = −(a − 1)∂μ S,

(1.68)

showing that Aμ obeys an inhomogeneous Klein-Gordon equation. Applying again  to (1.68) leads to the homogeneous fourth order equation 2 Aμ = 0. Solution of the Cauchy problem associated with such an equation is given in appendix C. It reads Aμ (x) = − −

 

d 3 y D(x − y) ∂0y Aμ (y) −



d 3 y E(x − y) ∂0y Aμ (y) −

d 3 y ∂0x D(x − y) Aμ (y)



d 3 y ∂0x E(x − y) Aμ (y).

(1.69)

where y0 is again arbitrary and, from Appendix C,

d E(x) = − Δ (x; κ )

. dκ κ =0 Using (1.68), Aμ (x) = −



d

3

y D(x − y) ∂0y Aμ (y) −

+(a − 1)





d 3 y ∂0x D(x − y) Aμ (y)

d 3 y E(x − y) ∂0 ∂μ S(y) +



 d 3 y ∂0x E(x − y) ∂μ S(y) . (1.70)

It remains to express ∂0 Aμ in terms of canonical variables. For μ = 0, this is done by the gauge condition which reads

∂0 A0 = ∂l Al − aS. For μ = k, the relation (1.47) between velocities and momenta is used.

(1.71)

20

1 Canonical Quantization for Constrained Systems

1.6.1.4 Commutation Relations for the Aμ Fields Commutation relations at any time [Aμ (x), Aν (z)] can be obtained by commuting (1.70) with Aν (z) and taking y0 = z0 . Taking into account that [Aμ (y), Aν (z)]y0 =z0 = 0 as well as Eq. (1.67), [Aμ (x), Aν (z)] = − + i(a − 1)





d 3 y D(x − y) [∂0 Aμ (y), Aν (z)]y0 =z0

  d 3 y E(x − y) ∂0 ∂μ ∂ν D(y − z) + ∂0x E(x − y) ∂μ ∂ν D(y − z) y

0 =z0

.

In order to proceed further in the calculations, particular values of the indices must be taken. Taking first ν = 0 and using [∂0 Aμ (y), A0 (z)]y0 =z0 = − agμ 0 [S(y), A0 (z)]y0 =z0 = iagμ 0 δ (3) (y − z),

∂02 ∂μ D(y − z) y =z = Δ ∂μ D(y − z) y =z = − gμ 0 Δ δ (3) (y − z), 0 0 0 0

k (3)

∂0 ∂μ D(y − z) y =z = − gμ k ∂ δ (y − z), (1.72) 0

0

one gets [Aμ (x), A0 (z)] = − iagμ 0 D(x − z) − i(a − 1)gμ 0 Δ E(x − z) − i(a − 1)gμ k ∂ k ∂ 0 E(x − z).

(1.73)

Taking now

Δ E = ∂02 E − E = ∂02 E − D into account, one obtains [Aμ (x), A0 (z)] = − iagμ 0 D(x − z) + i(a − 1)gμ 0 D(x − z) − i(a − 1)(gμ 0 ∂ 0 + gμ k ∂ k )∂ 0 E(x − z) = − igμ 0 D(x − z) − i(a − 1)∂0 ∂μ E(x − z).

(1.74)

In the same way, [∂0 Ak (y), Al (z)]y0 =z0 = igkl δ (3) (y − z) and [Ak (x), Al (z)] = − igkl D(x − z) − i(a − 1)∂k ∂l E(x − z).

(1.75)

These results can be gathered in a unique equation [Aμ (x), Aν (z)] = − igμν D(x − z) − i(a − 1)∂μ ∂ν E(x − z).

(1.76)

1.6 Covariant Gauge for the Free Massless Vector Field

21

1.6.1.5 The Propagator An important notion in perturbation theory is the Feynman propagator given, in the present case, by

θ (x0 − z0 )[Aμ (x), Aν (z)] − [Aμ (x), Aν (z)](−)

(1.77)

where [Aμ (x), Aν (z)](−) is the negative energy part of the commutator for any time given by (1.76). One must then compute θ (x0 − z0 )D(x − z) − D(x − z)(−) and ∂μ ∂ν [θ (x0 − z0 )E(x − z) − E(x − z)(−) ]. The job is already done in the appendix for the D-function. From the definition of the E-function and its zero-time properties, it results easily that

∂μ ∂ν [θ (x0 − z0 )E(x − z)] = θ (x0 − z0 )∂μ ∂ν E(x − z) and the job is also done for the E-function. The propagator reads then    kμ kν i 1 F 4 Δ μν (x − y) = d k + (a − 1) g e−ik·(x−y) . μν (2π )4 k2 + i k2 + i

(1.78)

(1.79)

The squaring of the distribution 

1 k2 + i

2

is generally ill-defined. Here however, its definition is unambiguously given by 

1 2 k + i

2 =−

1 d 2 2 dk k + i

as it results from the calculation in the appendix.

1.6.2 Creation and Annihilation Operators In order to build the Fock space of states, it is necessary to get the creation and annihilation operators and the algebra they satisfy. Except in the Fermi gauge (a = 1) where the potentials satisfy the massless Klein-Gordon equation, this is a cumbersome task first realized by Rideau[8]. Here, because we concentrate our attention mainly on noncovariant gauges, the derivation is not made explicitly. Only the results are recalled. The momentum space expansion of the fields are S(x) =

i (2π )3/2



d3k  s(k) e−ik·x + s† (k) eik·x , 2|k|

(1.80)

22

1 Canonical Quantization for Constrained Systems

Aμ (x) =



d3k 2|k|



2

s(k) n·k i=1  (ik · n x · n + 1)s(k) −ik·x + kμ g(k) − (a − 1)kμ e 2(n · k)2  2 s† (k) (i) − kμ g† (k) + ∑ μ (k)a†i (k) − nμ n · k i=1   (ik · n x · n − 1)s† (k) ik·x + (a − 1)kμ . e 2(n · k)2 1 (2π )3/2

∑ μ

(i)

(k)ai (k) + nμ

(1.81)

The nonvanishing commutation relations between creation and annihilation operators are    ai (k), a†j (k  ) = δi j 2|k|δ (3) k − k  ,     s(k), g† (k  ) = 2|k|δ (3) k − k  ,    (a − 3)|k| (3)  g(k), g† (k, ) = k−k . δ 2 (n · k) All the other commutators vanish. The fact that [s(k), s† (k)] vanish imply the presence of an indefinite metric. This is a blessing for unitarity of the theory.

1.6.3 The Gupta-Bleuler Formalism The Gupta[7]-Bleuler[4] formalism is a method allowing to construct the physical Fock space from the more general space including unphysical particles needed by the introduction of a class III gauge condition. It can easily be summarized in the following way. Physical states are those generated by a†i (k), (i = 1, 2) while unphysical states are generated by s† (k) and g† (k) 1. Let us first consider a state containing a particle created by g† . It is an unphysical state that must be eliminated by a subsidiary condition. Because s and g† do not commute, the condition s(k)g† (k  )|0 = 0 is not trivial. Therefore or, in coordinate space,

s(k)|Ψphys  = 0

(1.82)

S(−) (x)|Ψphys  = 0

(1.83)

is a condition allowing to eliminate, from the whole set of states, some unphysical ones namely those containing particles created by g† (k).

References

23

2. States containing an arbitrary number of particles created by s† (k) have zero norm. By introducing an equivalence relation  |Ψphys  ≡ |Ψphys  + s† (k)|Ψ0 ,

(1.84)

those states are equivalent to states with no unphysical particle at all. It is remarkable that these features still hold when the field Aμ is coupled to a conserved current. Indeed the S field still remains a free field. It still commutes with itself at any time. The whole structure of the unphysical sector is unchanged by the interaction with a conserved current and the Gupta-Bleuler formalism can therefore be reproduced as in the free field case. By restricting the physically acceptable states through the conditions (1.83) and (1.84), the longitudinal and scalar photons introduced by the use of the four components of the potential do not contribute to physical processes.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Becchi, C., Rouet, A. Stora, R.: Phys. Lett. B52, 344 (1974) 13, 14 Becchi, C., Rouet, A. Stora, R.: Commun. Math. Phys. 42, 127 (1975) 13, 14 Becchi, C., Rouet, A., Stora, R.: Ann. Phys. 98, 287 (1976) 13, 14 Bleuler, K.: Helv. Phys. Acta 23,567 (1950) 22 Faddeev, L.S., Popov, V.N.: Phys. Lett. 25B, 29 (1967) 16 Gribov, V.N.: Nucl. Phys. B139, 1 (1978) 12 Gupta, S.N.: Proc. Phys. Soc. (London) A63, 681(1950) 22 Rideau, G.: Lett. Math. Phys. 1, 17 (1975) 21 Strocchi, F.: Phys. Rev. 162, 1429 (1967) 16 Tyutin, I.V.: “Gauge Invariance In Field Theory And Statistical Physics In Operator Formalism,”LEBEDEV-75-39 13, 14

Chapter 2

Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

2.1 Introduction In the previous chapter, the standard quantization of the Maxwell theory in a covariant gauge was carried out. Let us now generalize this method to general class III gauges, the only consistent ones in the noncovariant case. The method was first used by the author [3] in order to get a consistent canonical formalism for the axial gauge regularized with the Leibbrandt [7]-Mandelstam [8] prescription. Later on, it was generalized to an arbitrary class III gauge [4]. Here, the method is further generalized to nonpreferred coordinate frames, a notion that will appear useful and important in the following. The philosophy underlying this chapter consists in generalizing the method used in covariant gauges recalled in the previous chapter to a more general Lagrangian containing fixed four-vectors. Lorentz invariance can then be broken. It is found that the theory can be consistently defined under two conditions. 1. The second-order differential operator governing the evolution of unphysical degrees of freedom must be hyperbolic or parabolic in order to have a well-defined Cauchy problem. 2. The quantization must be carried out in a frame which is not singular. This notion of singularity of frames is found neither in gauge theories with a finite number of degrees of freedom nor in covariant gauges. In the last case, covariance indeed implies that all frames are equivalent. In noncovariant gauges, they are not so that one must take care of the consistency of the whole theory. Such care is generally overlooked and this overlooking generates troubles like, for instance, those found with the Coulomb gauge in Yang-Mills theory. Cases of historical interest are briefly considered. They include the Coulomb gauge as well as the improperly called axial gauge with Leibbrandt [7]-Mandelstam [8] prescription. After the solution of the system in coordinate space,the construction of creation and annihilation operators and the derivation of their algebra, the problem of covariance of the theory is discussed. Its solution is simple. As Maxwell equations, covariance holds only in the physical sector which is a subspace of the general Fock space. This subspace is obtained by the Gupta-Bleuler formalism.

Burnel, A.: Quantization of the Free Electromagnetic Field in General Class III Linear Gauges. Lect. Notes Phys. 761, 25–60 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 2 

26

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

Many of the sections of this chapter are very technical. Details are necessary for a good understanding but the reader who is not interested in these details can find a summary of the results at the end of the main technical sections.

2.2 Lagrangian and Field Equations 2.2.1 The Lagrangian The generalization of the Lagrangian to an arbitrary linear class III gauge can be written 1 a 1 L = − Fμν F μν −Cμν ∂ μ SAν + aS2 + a SC μ ∂μ S + ∂ μ S ∂μ S 4 2 2

(2.1)

where Cμν and Cμ are arbitrary fixed, therefore not depending on coordinates or derivatives, tensors of rank 2 and 1 respectively. The gauge parameters a and a introduce inhomogeneity in the gauge condition Cμν ∂ μ Aν = 0. With respect to the relativistic case where Cμν = gμν , a second rank tensor is introduced. It is usually built with the help of fixed four-vectors. This expression of the Lagrangian was obtained from various trials induced by cases of interest discussed previously in the literature. These particulars cases will be described later on. Let us also note that a further generalization can be done by replacing the last term in  ∂ μ S ∂ ν S /2. Only one gauge of this type with a second the Lagrangian by Cμν fixed second-rank tensor has been used. It mimics the Coulomb gauge as a class II gauge but in a singular frame [2]. Such a generalization will not be discussed here. In order to simplify the writing of equations, it is useful to set vˆμ = Cν μ vν ,

v˜μ = Cμν vν

(2.2)

for any four-vector v. In a covariant gauge, of course, vˆμ = v˜μ = vμ . In a general noncovariant gauge, the vectors vˆμ and v˜μ can be considered as independent of the vector vμ . With such a notation, the second term of the Lagrangian reads ˜ −Cμν ∂ μ SAν = −∂ˆ S · A = −∂ S · A.

2.2.2 Euler-Lagrange Equations Let us first derive the equations of motion for the various fields involved in the story.

2.2 Lagrangian and Field Equations

27

2.2.2.1 Variation with Respect to Aν By a careful derivation,

∂L = −F μν , ∂ (∂μ Aν )

∂L = −∂ˆ ν S ∂ Aν

(2.3)

so that the corresponding field equations read

∂ μ Fμν − ∂ˆν S = 0.

(2.4)

By expliciting Fμν , they can be rewritten as Aν − ∂ν ∂ · A − ∂ˆν S = 0.

(2.5)

These equations generalize the Maxwell equations by giving an evolution to all the components of the potential.

2.2.2.2 Variation with Respect to S In the same way,

∂L = −A˜ μ , ∂ (∂ μ S)

∂L = aS + aC · ∂ S ∂S

(2.6)

so that the corresponding field equation is

∂ˆ · A + aS + aC · ∂ S = 0.

(2.7)

It represents the gauge condition with two gauge parameters a and a . The presence of the second parameter a is related to the presence of the fixed four-vector Cμ which is zero in a truly covariant gauge. It is important to remark here that one restricts oneself to class III gauges i.e. gauges giving a time evolution to the A0 degree of freedom. This implies C00 = 0. 2.2.2.3 Variation with Respect to S Again

∂L = a SCμ + a ∂μ S , ∂ (∂ μ S )

∂L =0 ∂S

(2.8)

so that the corresponding field equation is aC · ∂ S + a S = 0. This equation relates the two Lagrange multipliers S and S .

(2.9)

28

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

2.2.3 Derived Field Equations In order to simplify a little more the writing of the equations, let us set C = ∂ · ∂ˆ .

(2.10)

This defines a second-order operator generalizing the d’Alembert operator. Let us now handle the equations in order to be able to solve them successively as explained in the following subsection. First of all, by applying ∂ ν to Eq. (2.5), one gets an equation involving only the S-field C S = 0.

(2.11)

This is a first step. Now, when one looks at (2.5), one sees that it will be useful to get the auxiliary field B = ∂ · A before the complete Aμ -field. This can be obtained in the following way. Applying ∂ˆ ν to Eq. (2.5), ∂ˆ · A − C ∂ · A − ∂ˆ · ∂ˆ S = 0.

(2.12)

∂ˆ · A + aS + aC · ∂ S = 0.

(2.13)

a (C · ∂ )2 S + aC · ∂ S = 0.

(2.14)

Applying  to (2.7), Applying C · ∂ to (2.9),

By adding (2.12), (2.14) and subtracting (2.13), C B = C ∂ · A = KS

(2.15)

where the operator K is given by K = a (C · ∂ )2 − a − ∂ˆ · ∂ˆ .

(2.16)

2.2.4 Sketching the Solution of the Cauchy Problem The Cauchy problem is defined by field equations and initial conditions for fields and their first time derivatives. This problem will be solved explicitly at the level of commutation relations in order to get them for any time. Let us briefly explain the path which will be followed for getting the solution. The unknown fields are Aμ , S and S and the auxiliary field B ≡ ∂ · A. The solution is calculated first by using only the coordinate space. The solution in momentum space is given later when discussing creation and annihilation operators.

2.2 Lagrangian and Field Equations

29

1. First the equation C S = 0 is solved. It is a homogeneous second-order equation governed by the operator C . In order that the Cauchy problem be well defined, it must be imposed that this operator be hyperbolic or parabolic i.e. that the eigenvalues of the matrix (Cμν ) are such that one of them (corresponding to time) is of opposite sign with respect to the other three or vanish. Solution of (2.11) gives S in terms of the initial values of S and ∂0 S. 2. Once S is known, (2.9) S = −C · ∂ S must be solved in order to get S . It is an inhomogeneous second-order equation whose differential operator is the usual hyperbolic d’Alembert operator. Explicit solution of such an equation is given in Appendix C. In addition to the value of S for any time, initial values of S and ∂0 S are required. 3. Once S and S are known, (2.15) C B = KS must be solved. The solution requires the knowledge of S for any time and of the initial values of B and ∂0 B where the auxiliary field B is given by B = ∂ · A. Note  =g . that S plays no role in the assumed case Cμν μν The initial values of B and ∂0 B are obtained in the following way. a. Because B = ∂0 A0 − ∂k Ak , its initial value is obtained from those of ∂0 A0 and Ak which are assumed to be given. b. Using ∂0 B = ∂02 A0 − ∂k ∂0 Ak , its initial value is determined from ∂0 Ak which is given and from ∂02 A0 . The latter is determined by time-differentiating the gauge condition (2.7) C00 ∂02 A0 = C0k ∂02 Ak +Ck0 ∂0 ∂k A0 −Ckl ∂0 ∂k Al − a∂0 S − a ∂0C · ∂ S . (2.17) In this equation, S and S are known at any time while initial values of ∂0 ∂k Aμ are given by initial conditions. To get the initial value of ∂02 Ak , Eq. (2.5) is used with ν = k. This gives

∂02 Ak = Δ Ak + ∂k B + ∂ˆk S. Initial values of ∂02 Ak are then related to given initial values. It is important to recall here that C00 = 0 is assumed. 4. Finally, (2.5) Aν = ∂ν B + ∂ˆν S is solved in order to get Aν . This is again an inhomogeneous second-order equation the solution of which is given in the appendix.

30

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

2.2.5 Particular Cases of Interest Let us now discuss particular choices of the tensors Cμν and Cμ which can be found in the literature. Actually, it is the reverse way i.e. starting from these cases that led to the general formulation. 1. As already noted, the covariant gauges are described by the choice Cμν = gμν . If a fixed four-vector is introduced, the system is invariant only under transformations leaving this vector unchanged and the general Lorentz covariance is broken. Therefore Cμ = 0 in a covariant gauge. 2. If a single four-vector nμ is introduced, one takes Cμ = nμ and Cμν = nμ nν . If the fixed four-vector nμ is time-like, the particular coordinate frame in which nμ = (1, 0, 0, 0) is useful. This particular gauge is called static temporal gauge. It leads to some troubles with respect to the formulation given here. Indeed, in momentum space, the free field equation will read k02 f˜(k) = 0. Only zero frequencies are involved and one cannot separate positive from negative frequency contributions. In order to be able to use the formalism developed here, such a gauge must be regularized in order to recover positive and negative frequencies. 3. This regularization is done by interpolating between static temporal and relativistic gauges. The Cμν tensor is Cμν = nμ nν − α gμν

(2.18)

where α is a new parameter. It can be noted that the operator Cμν ∂ μ ∂ ν is hyperbolic for α < 0 and α > 1. The limiting case α = 0 is the static temporal gauge while relativistic gauges are obtained in the limit α → ±∞. The case α = 1 merits some special attention. Let us take a = a = 0 and consider nμ as a time-like four-vector. The gauge condition (2.7) becomes n · ∂ n · A − ∂ · A = 0.

(2.19)

In the frames where n = (1, 0, 0, 0), it reduces to the Coulomb gauge

∂k Ak = 0.

(2.20)

Note that these frames are singular frames in the sense that C00 = 0. The gauge condition which is class III in a general frame becomes class I in these particular frames. In such frames, at least for the Coulomb gauge, locality is lost. This is the unique defect in the abelian case. In Yang-Mills theory, many other troubles occur such as operator ordering and Gribov ambiguity [6]. Though the evolution operator C is parabolic in the Coulomb gauge, quantization can be realized but only in a nonsingular frame. If one persists in use of this particular frame, the various difficulties must be solved in a somewhat cumbersome way. See for instance the work of Christ and Lee [5].

2.2 Lagrangian and Field Equations

31

4. It is sometimes interesting to introduce two given four-vectors nμ and n∗μ . Setting Cμν = n∗μ nν ,

Cμ = n∗μ ,

(2.21)

such a gauge corresponds, for a = 0, to the axial gauge and, for a = n2 , to the planar gauge both regularized with what is known as the Leibbrandt [7]Mandelstam [8] prescription. This terminology is misleading because, for a = 0 for instance, the result is no longer the class II axial gauge n · A = 0 but a class III one n∗ · ∂ n · A = 0. These gauges will be called Leibbrandt gauges in the following because the Leibbrandt form of the prescription corresponds to the poles associated, in momentum space, with the operator C . Some particular frames merit special attention. a. If n∗2 = n2 , a frame in which n = (n0 , n), can be found. Then

n∗ = (n0 , −n)

C = n20 ∂02 − (n.∂ )2

is in diagonal form. This is called a preferred frame. b. If n∗2 = −n2 = 0, a frame in which n∗ = (1, 0, 0, 0),

n = (0, 0, 0, 1)

can be found. The role of n and n∗ can be inverted. In such a frame, C00 = 0. This frame is singular and cannot be used for quantizing the theory. 5. It will also be very useful to interpolate between the Leibbrandt and relativistic gauges by choosing Cμν = n∗μ nν + α gμν . Utility of these interpolating gauges where the operator C is hyperbolic for α ≥ 0 will be discussed later in the framework of renormalization of Yang-Mills theory. 6. Some other particular cases are sometimes met. For instance, Cμν = α (nμ nν − gμν ) + nμ n∗ν − n∗μ nν has been considered in order to avoid infra-red problems in quantum electrodynamics [1].

2.2.6 Planar-Type Gauges In the case of relativistic gauges, the particular Fermi gauge, characterized by the gauge parameter a = 1 is very often used because the propagator involves only a

32

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

single pole while a double pole is present in any other gauge. In the case of Leibbrandt gauges, the corresponding situation is also met. Indeed, only a simple unphysical pole is involved with the choice a = 0,

a = n2 .

It corresponds to light-cone gauge for n2 = 0 and to planar gauges for n2 = 0. Gauges in which the operator K is equal to zero will be called planar-type gauges. Let us give the realization of the planar-type gauges when interpolating between Leibbrandt and relativistic gauges. It can be checked that, when n2 = n∗2 , the choice Cμν = n∗μ nν

α − nμ n∗ν + α gμν , α + n2 + n · n∗

Cμ = n∗μ

 2 α n2 + nμ (2.22) α + n2 + n · n∗

leads to K = 0 provided that, at the same time, a = α 2 , a = n2 . When n2 = n∗2 , no planar-type gauge can be found. In the case of a single fixed four-vector n, a planar-type gauge is given by Cμν = nμ nν − α gμν ,

Cμ = nμ ,

a = n2 − 2α ,

a = −α 2 .

(2.23)

For n2 = 1, this is an interesting interpolation between static temporal, Coulomb and Fermi gauge.

2.3 Constraint Analysis 2.3.1 Canonical Momenta and Primary Constraints Let us go back to the Lagrangian (2.1) and proceed to the constraint search. The definition of canonical momenta leads to   (2.24) Π μ = F μ 0 , ΠS = −Aˆ 0 , ΠS = a SC0 + ∂0 S . The case a = 0 can be treated separately and is simpler. It is included in the following discussion by setting in the same time ΠS = S = a = 0. There are two primary constraints in the game

Π 0 = 0,

ΠS + Aˆ 0 = 0.

(2.25)

The time derivatives of Ak and S , the “velocities”, can be obtained from the definition of canonical momenta

∂0 Ak = Π k − ∂ k A0 ,

a ∂0 S = ΠS − a SC0 .

(2.26)

On the other hand, the time derivative of A0 and S are undetermined because the corresponding momenta are constrained.

2.4 Singular Frames

33

2.3.2 The Hamiltonian It is a matter of elementary but tedious calculations to obtain the Hamiltonian density H =

 1 k k 1 1 Π Π + Fkl Fkl − Π k ∂ k A0 − ∂k SA˜ k + aC02 − a S2 2 4 2 1 2 a    C0 S + a SCk ∂k S + Π − Π ∂k S  ∂k S   S 2a S 2   + Λ0 Π 0 + ΛS ΠS + Aˆ 0 ,

+

(2.27)

where the undetermined velocities ∂0 A0 and ∂0 S are replaced respectively by the Lagrange multipliers Λ0 and ΛS .

2.3.3 Constraint Chains Again, the constraints must be stable in time. By taking their Poisson brackets with the Hamiltonian, one gets the constraint chains .

Π 0 ≈ 0 =⇒ C00ΛS + ∂ k Π k −Ck0 ∂k S ≈ 0,

(2.28)

.

ΠS + A˜ 0 ≈ 0 =⇒ C00Λ0 + aS +Ckl ∂k Al − (Ck0 +C0k )∂k A0 − C0k Π k +C0 ΠS − aCk ∂k S .

(2.29)

These chains are stopped if C00 = 0 because the multipliers are then determined. The number of independent pairs of field and conjugate momenta reduces then to five, corresponding to the three space-components of the potential and the auxiliary fields S and S . The time-component of the potential and its momentum are constrained. If one chooses a frame in which C00 = 0, there are additional constraints ∂ k Π k − Ck0 ∂k S ≈ 0 and aS + Ckl ∂k Al − (Ck0 + C0k )∂k A0 which must themselves be stable in time. The number of degrees of freedom is still reduced but this occurs only in particular frames.

2.4 Singular Frames From the previous section, it is clear that, for noncovariant gauges, not all the coordinate frames are equivalent. In coordinate frames in which C00 = 0, the number of degrees of freedom is less than in general frames. These frames are singular and they do not allow a consistent quantization. Without entering into cumbersome details in the general case, let us recall that the Coulomb gauge becomes class I in

34

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

such a frame with the consequence of a nonlocal theory. In the Yang-Mills case the situation is even worse. Here, in order to keep consistency, never will quantization be carried out in such a frame. Singularity of some coordinate frames is known in the formal theory of partial differential equations. In such frames, the system of PDE cannot be brought into an involutive form. Both notions of singular frames are probably related although no proof of it has been given at the present time.

2.5 Commutation Relations for Any Time 2.5.1 Canonical Commutation Relations In a nonsingular frame, there are only two constraints

Π 0 = 0,

ΠS + A˜ 0 = 0.

(2.30)

They are second class and fix the pair of conjugate variables A0 and Π 0 . One may use Dirac brackets in order to quantize the theory but it is simpler to consider the constrained pair as given by the constraints and take only as canonical pairs (Al , Π l ), (S, ΠS ) and (S , ΠS ). Canonical quantization implies 

Ak (x), Π l (y)

x0 =y0

= iδkl δ (3) (x − y),

[S(x), ΠS (y)]x0 =y0 = iδ (3) (x − y),    S (x), ΠS (y) x =y = iδ (3) (x − y), 0

(2.31)

0

and the vanishing of all other commutators between canonical variables at equal time.

2.5.2 Commutation Relations Involving the S Field Let us now proceed to the search of commutation relations at any time and begin with the S-field satisfying the homogeneous equation C S = 0. It is a second order partial differential equation whose elementary solutions are obtained in the appendix. In particular, the solution of the Cauchy problem can be written as S(x) = −



↔

d 3 y DC (x − y) ∂0y S(y) +



d 3 y ∂Z DC (x − y)S(y)

(2.32)

2.5 Commutation Relations for Any Time

35

where the DC is given in the appendix and Zl =

C0l +Cl0 , 2C00

∂Z = 2Z.∂ ,

↔

a ∂0y b = a∂0 b − ∂0 ab.

As for the solution of the Klein-Gordon equation, y0 is arbitrary. Of course, C00 = 0 since we work in a nonsingular frame. Let us also note the presence of an additional term, the last one, with respect to the covariant case. Another way of writing this solution is S(x) = −(∂0 − ∂Z )



d 3 y DC (x − y)S(y) −



d 3 y DC (x − y)∂0 S(y)

(2.33)

where now, the derivatives in the first term are taken with respect to x. In order to compute commutation relations, ∂0 S must be expressed in terms of canonical variables. Using (2.4) with ν = 0,

∂0 S =

 1  Cl0 ∂l S + ∂k Π k C00

(2.34)

in a nonsingular frame. From (2.34) and canonical commutation relations, both S and ∂0 S commute with S and S at equal time. Commuting S(x) given by (2.33) with S(z) and taking the arbitrary y0 equal to z0 , one easily gets [S(x), S(z)] = [S(x), S (z)] = 0

(2.35)

for any time. In the same way, canonical commutation relations and use of (2.34) imply [S(x), Al (y)]x0 =y0 = 0,  1 ∂k Π k (x), Al (y) C00 x0 =y0 i =− ∂l δ (3) (x − y). C00

(2.36)

[∂0 S(x), Al (y)]x0 =y0 =

(2.37)

Therefore, at any time, [S(x), Al (z)] =

i ∂l DC (x − z). C00

(2.38)

To get the commutation relations involving A0 , the constraint A˜ 0 + ΠS = 0 must be solved. In a nonsingular frame, this solution reads A0 =

1 (C0l Al − ΠS ). C00

Canonical commutation relations imply then

(2.39)

36

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

1 [S(x), ΠS (y)]x0 =y0 C00 i (3) =− δ (x − y), C00 1 C0l [∂0 S(x), Al (y)]x0 =y0 = C00

− Cl0 ∂l [S(x), ΠS (y)]x0 =y0 i ∂Z δ (3) (x − y). =− C00

[S(x), A0 (y)]x0 =y0 = −

[∂0 S(x), A0 (y)]x0 =y0

(2.40)

(2.41)

Therefore, in the same way as for other commutators, i i (∂0 − ∂Z )DC (x − z) + ∂Z DC (x − z) C00 C00 i = ∂0 DC (x − z). C00

[S(x), A0 (z)] =

(2.42)

Gathering the results (2.38) and (2.42),   i S(x), Aμ (z) = ∂μ DC (x − z). C00

(2.43)

All the commutators at any time involving the S-field are obtained.

2.5.3 Commutation Relations Involving the S Field Let us now do the same job for the S -field satisfying the inhomogeneous equation S = −C · ∂ S. If M(z) is an arbitrary field operator,    S (x), M(z) = −C · ∂ [S(x), M(z)]. This is a Cauchy problem associated with the d’Alembert operator and for which the initial values are the corresponding equal-time commutators. Remaining cases of interest to which all other ones are related are M = S and M = Aμ . In the case M = S , [S (x), S (z)] satisfies, according to (2.35), a homogeneous Klein-Gordon equation. The solution associated with the Cauchy problem is therefore  ↔      S x), S (z) = − d 3 y D(x − y) ∂0y S (y), S (z) (2.44) where y0 is still arbitrary and will be taken equal to z0 .

2.5 Commutation Relations for Any Time

37

The equal time commutator [S (y), S (z)] vanishes according to canonical commutation relations. It remains to compute the equal-time commutator [∂0 S (y), S (z)]y0 =z0 . In order to express ∂0 S in terms of canonical variables, the definition of the canonical momentum associated with S

ΠS = aC0 S + a ∂0 S is used. If the parameter a is equal to 0, the field S does not occur in the story. Let us therefore assume a = 0 and write

∂0 S  =

1 Π  −C0 S. a S

(2.45)

Canonical commutation relations imply then    S (x), S (y) x

0 =y0

   ∂0 S (x), S (y) x

= 0,

0 =y0

i = −  δ (3) (x − y). a

(2.46)

Therefore, from (2.44),    i S (x), S (z) =  D(x − z). a

(2.47)

In the case M is replaced by Aμ , use of (2.43) leads to   i  S (x), Aμ (z) = − ∂μ C · ∂ DC (x − z). C00 Such an equation is solved in Appendix C where the various functions and their properties are obtained. Its solution is    i S (x), Aμ (z) = − ∂μ C · ∂ FC (x − z) C00

i ∂μ C · ∂ FC (y − z)

− d C00 y0 =z0

 ↔

− d 3 y D(x − y) ∂0y [S (y), Aμ (z)]

. 

3

↔ y D(x − y) ∂0y

(2.48)

y0 =z0

Zero-time properties of FC are

∂0p FC (x, 0) = 0 for p ≤ 2,

∂03 FC (x, 0) = −δ (3) (x)

while, from canonical commutation relations,    S (x), Aμ (y) x =y = 0, 0 0   ∂0 S (x), Ak (y) x =y = 0, 0 0    iC0 (3) ∂0 S (x), A0 (y) x =y = δ (x − y). 0 0 C00

(2.49) (2.50) (2.51)

38

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

It is then a straightforward task to get [S (x), Aμ (z)] = −

i ∂μ C · ∂ FC (x − z). C00

(2.52)

2.5.4 Commutation Relations Involving B = ∂ · A Let us now derive the commutation relations at any time for the auxiliary B-field. As usual, they will be obtained by successive operations : 1. The solution of the Cauchy problem. 2. The expression of B and its time-derivative in terms of canonical variables. 3. The expression of equal-time commutators implying B and its time-derivative and the Aμ - field itself. 4. Finally, the calculation of the commutators. All these steps are now carried. Since they are very technical and involves long details, the reader who is not interested in these may directly go to the final result.

2.5.4.1 The Cauchy Problem The B field satisfies C B = KS where K is a second-order differential operator given by (2.16). Using the field equation satisfied by S, C2 B = 0. The Cauchy problem associated with this equation is again solved explicitly in Appendix C. The solution reads B(x) = −



+∂Z

↔

d 3 y DC (x − y) ∂0y B(y) −





↔

d 3 y EC (x − y) ∂0y KS(y)

d 3 y[DC (x − y)B(y) + EC (x − y)KS(y)].

(2.53)

By using the fact that ∂Zx DC (x − y) = −∂Zy DC (x − y) and the same for the EC - function and the time derivatives, by integrating by parts and assuming that S and B vanish at infinity, it is easy to get the following equivalent expression B(x) = −∂0 −∂0

 

d 3 y DC (x − y)B(y) −



d 3 y EC (x − y)KS(y) −



d 3 y DC (x − y)(∂0 − ∂Z )B(y) d 3 y EC (x − y)(∂0 − ∂Z )KS(y).

(2.54)

Such an expression is more useful than the previous one for computing commutators.

2.5 Commutation Relations for Any Time

39

2.5.4.2 Expression of B and ∂ 0 B in Terms of Canonical Variables In a class III gauge, the gauge condition gives the evolution equation for A0 . By expliciting (2.7), it can be written C00 ∂0 A0 = C0l ∂0 Al +Cl0 ∂l A0 −Ckl ∂k Al − aS − a (C0 ∂0 − C.∂ )S .

(2.55)

By using the relations between momenta and velocities

∂0 Al = Π l − ∂ l A0 , a ∂0 S = ΠS − aC0 S, so that, in terms of canonical variables,one gets B = ∂Z A0 − ∂k Ak 1  C0l Π l −Ckl ∂k Al + (aC02 − a)S −C0 ΠS + a C.∂ S . + C00

(2.56)

It remains to do the same for ∂0 B. Differentiating (2.55) with respect to time, C00 ∂02 A0 = C0l ∂02 Al +Cl0 ∂l ∂0 A0 −Ckl ∂k ∂0 Al   − a∂0 S − a C0 ∂02 − C.∂ ∂0 S .

(2.57)

Some first and second-order time derivatives can be obtained from field equations, for instance,

∂02 Al = Δ Al + ∂l B +C0l ∂0 S −Ckl ∂k S, ∂02 S = Δ S −C0 ∂0 S + C.∂ S,  1  ∂0 S = ∂k Π k +Cl0 ∂l S . C00 Other time-derivatives are given by the relations between velocities and momenta. It is then a matter of straightforward calculations to get the expression of (∂0 − ∂Z )B in terms of canonical variables (∂0 − ∂Z )B =

1 [C0l Δ Al +Cl0 ∂l ∂k Ak −Clk ∂l ∂k A0 −C00 Δ A0 C00  Ckl − aC0 Δ S + C.∂ ΠS + (Z − 1)∂k Π k − ∂k Π l + ZCl0 ∂l S C00  1  C0l Ckl ∂k S + 2aC0 C.∂ S − (2.58) C00

where Z=

aC02 − a +C0l C0l . 2 C00

(2.59)

40

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

2.5.4.3 Equal-Time Commutators From (2.56) and (2.58), the expression of A0 in terms of canonical variables and canonical commutation, it is straightforward to get [B(x), Ak (y)]x0 =y0 = −i

C0k (3) δ (x − y), C00

[B(x), A0 (y)]x0 =y0 = −iZ δ (3) (x − y) and [(∂0 − ∂Z )B(x), Ak (y)]x0 =y0 [(∂0 − ∂Z )B(x), A0 (y)]x0 =y0

(2.60) (2.61)

  Clk = − i (Z − 1)∂k − ∂l δ (3) (x − y), (2.62) C00  C0k =i ∂k − Z ∂Z C00   (3) 2   + 2 C0l Ckl ∂k + a C0 C.∂ δ (x − y). C00 (2.63)

In order to simplify the writing, let us introduce the quantities Ul ≡ aC0Cl −C00Cl0 +ClkC0k , Then [(∂0 − ∂Z )B(x), A0 (y)]x0 =y0

∂U ≡

U.∂ . 2 C00

  Ck0 = i 2∂U − (Z − 1)∂Z + ∂k δ (3) (x − y). C00

(2.64)

(2.65)

One is now able to compute a part of the commutator of B given by (2.54) with Aμ . Setting this part (1)

Iμ = − −

 

d 3 y ∂0 DC (x − y)[B(y), Aμ (z)] d 3 y DC (x − y)[(∂0 − ∂Z )B(y), Aμ (z)],

(2.66)

one gets 

(1) Ik

(1)

I0

 C0k Clk =i ∂0 + (Z − 1)∂k − ∂l DC (x − z) C00 C00   ∂ˆk + (Z − 1)∂k DC (x − z), =i C00   ∂ˆ0 =i + (Z − 1)(∂0 + ∂Z ) − 2∂U DC (x − z). C00

(2.67) (2.68)

2.5 Commutation Relations for Any Time

41

2.5.4.4 Equal-Time Values of [∂ n0 KS(x), Aμ (y)] There is another part of B in (2.54). It involves KS and its time-derivatives. Commutation relations for any time between S and Aμ are known and given by (2.43). So, one gets i [∂0n KS(x), Aμ (y)] = ∂μ ∂0n KDC (x − y). C00 Such commutators are involved at equal-time in the solution of the Cauchy problem. Let us thus compute these equal-time values. First, it is useful to expand K in the form 2 K = C00 [(Z − 1)∂02 − 2∂0 U.∂ +U] (2.69) where U=

a (C.∂ )2 + aΔ +Ckl Cml ∂k ∂m −Cl0Ck0 ∂l ∂k 2 C00

(2.70)

while ∂U is defined by Eq. (2.64). Then the use of zero-time properties of the DC function and of its time derivatives given in Appendix C leads to 2 KDC (x, 0) = C00 [2U.∂ − (Z − 1)∂Z ] δ (3) (x),   2 (Z − 1)(W − ∂Z2 ) + 2U.∂ ∂Z −U δ (3) (x), ∂0 KDC (x, 0) = C00

(2.71) (2.72)

2 ∂02 KDC (x, 0) = C00 [(Z − 1)(2W − ∂Z2 )∂Z

−2U.∂ (W − ∂Z2 ) −U ∂Z ]δ (3) (x)

(2.73)

from which it is easy to derive 2 (∂0 − ∂Z )KDC (x, 0) = C00 [(Z − 1)W −U] δ (3) (x), 2 (∂0 − ∂Z )∂0 KDC (x, 0) = C00 [(Z − 1)∂Z − 2U.∂ ]W δ (3) (x).

(2.74) (2.75)

Let us now introduce these results in the second part of the commutator of B given by (2.54) with Aμ : (2)

Iμ = − −

 

d 3 y ∂0 EC (x − y)[KS(y), Aμ (z)] d 3 y EC (x − y)[(∂0 − ∂Z )KS(y), Aμ (z)]

(2.76)



i d 3 y ∂0 EC (x − y)K ∂μ DC (y − z) C00  i − d 3 y EC (x − y)K(∂0 − ∂Z )∂μ DC (y − z) C00

=−

where y0 = z0 . Straightforward calculations give Ik = −iC00 {∂0 [2U.∂ − (Z − 1)∂Z ] + (Z − 1)W −U} ∂k EC (x − z). (2)

(2.77)

42

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

Replacing U in terms of K, (2)

Ik =

  i K ∂k EC (x − z) − i(Z − 1)∂kC00 ∂02 − ∂0 ∂Z +W EC (x − z). C00

(2.78)

Recalling that C = C00 (∂02 − ∂0 ∂Z +W ) and EC = DC , (2)

Ik =

i K ∂k EC (x − z) − i(Z − 1)∂k DC (x − z). C00

(2.79)

In the same way, (2)

I0 =

i K ∂0 EC (x − z) − i(Z − 1)(∂0 + ∂Z )DC (x − z) + 2iU.∂ DC (x − z). (2.80) C00

The reader who tries to reproduce these calculations should be aware that care must be taken with the transfer by integration by parts of the operator K from the variable z to the variable x. This transfer generates nontrivial contributions.

2.5.4.5 The Commutator [B(x), Aμ (z)] (1)

(2)

It is now sufficient to sum Iμ and Iμ to get, after straightforward but tedious calculations left to the reader, a simple expression for the commutator [B(x), Aμ (z)] =

i i ˆ K ∂μ EC (x − z) + ∂μ DC (x − z). C00 C00

(2.81)

2.5.5 Commutation Relations Between Aμ ’s 2.5.5.1 The Cauchy Problem Let us now tackle the last step in obtaining commutation relations between potentials. Using (2.5), the equation which must be solved for the potentials reads

where

Aμ = Vμ

(2.82)

Vμ = ∂μ B + ∂ˆ μ S.

(2.83)

Commuting Vν with Aμ , one gets, from Eqs. (2.81) and (2.43), [Aμ (x),Vν (y)] =

  i  K ∂μ ∂ν EC (x − z) + ∂ˆ μ ∂ν + ∂μ ∂ˆν DC (x − z) . C00

It is useful to write this commutation relation as

(2.84)

2.5 Commutation Relations for Any Time

43

[Aμ (x),Vν (y)] = Lμν ΔC (x − y; κ ) κ =0 where the operator Lμν given by   i d ˆ ˆ Lμν = ∂ μ ∂ν + ∂ μ ∂ν − K ∂ μ ∂ν C00 dκ

(2.85)

(2.86)

is symmetric.The solution of the Cauchy problem related to this nonhomogeneous Klein-Gordon equation is given in Appendix C

 ↔

[Aμ (x), Aν (z)] = Lμν FC (x − y; κ ) + d 3 y D(x − y) ∂0y Lμν FC (y − z; κ )

−



d

3

↔ y D(x − y) ∂0y

κ =0

[Aμ (y), Aν (z)]

(2.87)

where y0 = z0 .

2.5.5.2 Equal-Time Values and Commutation Relations for Any Time It remains to compute the equal-time values and make the integrations. Using zerotime properties of FC and its time derivatives given in Appendix C, it is an easy task to derive Lkl FC (x, 0) = 0,

∂0 Lkl FC (x, 0) = 0,

Lk0 FC (x, 0) = 0,

∂0 Lk0 FC (x, 0) = −i

L00 FC (x, 0) = 0,

∂0 L00 FC (x, 0) = −i(Z + 1)δ (3) (x).

(2.88) C0k (3) δ (x), C00

(2.89) (2.90)

On the other hand, equal-time commutation relations between Aμ and Aν can be obtained from canonical commutation relations and the gauge condition. They read [Ak (x), Al (y)]x0 =y0 = 0,

[∂0 Ak (x), Al (y)]x0 =y0 = igkl δ (3) (x − y),

[Ak (x), A0 (y)]x0 =y0 = 0,

[∂0 Ak (x), A0 (y)]x0 =y0 = −i

[A0 (x), A0 (y)]x0 =y0 = 0,

[∂0 A0 (x), A0 (y)]x0 =y0 = −iZ δ (3) (x − y).

(2.91)

C0k (3) δ (x − y), (2.92) C00 (2.93)

One then computes the various commutators [Ak (x), Al (y)], [Ak (x), A0 (y)], [A0 (x), A0 (y)] separately and combines the results in

[Aμ (x), Aν (z)] = Lμν FC (x − z; κ ) κ =0 − igμν D(x − z). (2.94)

44

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

Expliciting Lμν , [Aμ (x), Aν (z)] = − igμν D(x − z) + +

 i ˆ ∂μ ∂ν + ∂μ ∂ˆν FC (x − z) C00

i K ∂μ ∂ν GC (x − z). C00

(2.95)

The case K = 0 (planar type gauges) gives a simpler expression where the GC function disappears.

2.5.5.3 The Propagator The Feynman propagator is easily derived from the commutation relations. As in covariant gauges, owing to the equal-time properties of the FC function, the differential operators commute with the step function. The Feynman functions associated with FC and GC are derived in Appendix C. It is then easy to get the propagator in a general class III gauge as    ˜ kˆ μ kν + kμ kˆ ν K(k)k i 1 μ kν F 4 ik·x e Δ μν (x) = − d k 2 gμν − . (2.96) + 2 (2π )4 k + i kC2 + i (kC + i)2 As in covariant gauges, the squaring of the distribution 1/(kC2 +i) does not generate any trouble. It is again defined by differentiation.

2.5.6 Summary of the Commutation Relations for Any Time One summarizes here the results of the section. [S(x), S(z)] = [S(x), S (z)] = 0,    i S (x), S (z) =  D(x − z), a   i S(x), Aμ (z) = ∂μ DC (x − z), C00    i S (x), Aμ (z) = − ∂μ C · ∂ FC (x − z), C00    i ˆ Aμ (x), Aν (z) = − igμν D(x − z) + ∂μ ∂ν + ∂μ ∂ˆν FC (x − z) C00 +

i K ∂μ ∂ν GC (x − z). C00

2.6 Creation and Annihilation Operators

45

2.6 Creation and Annihilation Operators 2.6.1 Momentum Space Expansion of the Fields 2.6.1.1 The S-Field Let us now tackle the problem of the definition of creation and annihilation operators through momentum space expansion of the fields. The solution is again obtained step by step. One begins with the unphysical sector and, in particular, the operators associated with the S-field satisfying the homogeneous equation C S = 0. This equation is not invariant in noncovariant gauges so that the corresponding particles will be unphysical ones. Owing to the fact that, in momentum space, the field ˜ = 0, one can write equation becomes kC2 S(k) S(x) =

i (2π )3/2



d 4 k s(k)δ (kC2 )e−ik·x

(2.97)

where, as in covariant gauges, the i factor is introduced for convenience. In order to perform the integration over k0 , one proceeds as in Appendix C and sets  Ckl kk kl V (k) = , Y (k) = −V (k) + (Z.k)2 (2.98) C00 before making the change of integration variables k = k ,

k0 = k0 + Z.k.

The result is S(x) = where



i (2π )3/2 |C00 |



d 4 k s(k )δ (k02 −Y 2 (k  ))e−ik+ ·x

(2.99)

   k+ = k0 + Z.k  , k  .

As in the relativistic case, a factor 1 = θ (k0 ) + θ (−k0 ) can be inserted and the change of integration variables k = k in the first integral and k = −k in the second one can be made. This leads to   i 4 2 2 −ik+ ·x ik+ ·x . S(x) = d k θ (k ) δ (k −Y (k) ) s(k)e + s(−k)e 0 0 (2π )3/2 |C00 | (2.100) Setting again s† (k) = s(−k) and integrating over k0 ,

46

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

S(x) =



i (2π )3/2 |C00 |

d3k  . s(k)e−ik+ ·x + s† (k)eik+ ·x 2Y (k) k0 =Y (k)

(2.101)

One obtains so an annihilation operator s(k), the adjoint of which is a creation operator s† (k).

2.6.1.2 The B-Field Let us do the same job for the auxiliary B-field satisfying the inhomogeneous equation C B = KS. The solution is written as B(x) = B1 (x) + B2 (x) where B1 is the general solution of the homogeneous equation and B2 a particular solution of the inhomogeneous one. Their momentum space expansions are respectively 

i d 4 k b(k)δ (kC2 )e−ik·x , (2π )3/2  i ˜ s(k)δ  (kC2 )e−ik·x B2 (x) = d 4 k K(k) (2π )3/2

B1 (x) =

(2.102) (2.103)

ˆ The homogeneous part is obtained in the same ˜ where K(k) = ak2 − a (C · k)2 + kˆ · k. way as in the case of the S-field while it is easy to check that the other part B2 (x) is indeed a particular solution. In order to carry out the integration over k0 , it is useful to extract the derivatives by setting (2.104) B2 (x) = KS (x) where 

i S (x) = d 4 k s(k)δ  (kC2 )e−ik·x (2π )3/2



d i 4 2 −ik·x =− d k s(k) δ (k − κ )e . C

(2π )3/2 d κ κ =0

(2.105)

Now, the creation and annihilation operators can be introduced as here above and the integration over k0 can be carried out as in Appendix C. This leads to 

d3k  −ik+ ·x † ik+ ·x b(k)e + b (k)e , (2.106) k0 =Y (k) (2π )3/2 |C00 | 2Y (k)  d3k  i S (x) = (ix0 k0 + 1)s(k)e−ik+ ·x (2π )3/2 |C00 | 4Y 3 (k) −(ix0 k0 − 1)s† (k)eik+ ·x . (2.107) B1 (x) =

i

k0 =Y (k)

2.6 Creation and Annihilation Operators

47

One obtains so a second pair of annihilation b(k) and creation operator b† (k) associated with the B-field. 2.6.1.3 The S -Field The S -field satisfies the inhomogeneous equation S = −C · ∂ S. As above, the solution is written as S (x) = S1 (x) + S2 (x) where S1 is the general solution of the homogeneous equation and S2 a particular solution of the inhomogeneous one. As usual for free fields satisfying the Klein-Gordon equation, the momentum space expansion of S1 is S1 (x) =

1 (2π )3/2



  d 4 k s (k)δ k2 e−ik·x .

(2.108)

In the inhomogeneous part, it is again useful to extract the derivative by setting S2 (x) = −C · ∂ S  (x)

(2.109)

with   S  (x) = DH ∗ S (x) = −

i (2π )3/2



  d4k s(k)δ kC2 e−ik·x 2 k

(2.110)

where DH (x) = δ (4) (x). The writing of S  (x) as a product of convolution will be useful in the following. It is again easy to check that one obtains so a particular solution of the inhomogeneous equation. Here and in similar situations in the following, no prescription for the pole at k2 = 0 is needed if it is assumed that kC2 and k2 have no common zero. Covariant gauges are therefore excluded from the present discussion. The now familiar method of introducing creation and annihilation operators and the integration over k0 give S1 (x) =



1 (2π )3/2 |C00 |

S  (x) = −

i (2π )3/2 |C00 |

d3k   s (k)e−ik·x + s† (k)eik·x , 2|k| k0 =|k| 

d3k  −ik+ ·x † ik+ ·x s(k)e + s (k)e . 2 k0 =Y (k) 2Y (k)k+

Again, a pair of creation and annihilation operators is obtained.

(2.111)

(2.112)

48

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

2.6.1.4 The Aμ -Field Let us now tackle the difficult problem of the momentum expansion of the Aμ -field itself. It satisfies the inhomogeneous equation Aμ = Vμ where

Vμ = ∂μ B + ∂ˆ μ S.

The general solution of this equation can again be written as the superposition of a general solution of the homogeneous equation and of a particular solution of the inhomogeneous one (1) (2) Aμ = Aμ + Aμ . (2.113) The homogeneous part can be developed in momentum space as (1)

Aμ (x) =

1 (2π )3/2



  d 4 k A˜ μ (k)δ k2 e−ik·x .

(2.114)

(i) As in covariant gauges, A˜ μ (k) is expanded on the basis formed by the vectors μ (k), k = 1, 2, kμ and kˆ μ . 2

(i) A˜ μ (k) = ∑ ai (k)μ (k) + a3 (k)kˆ μ + a4 (k)kμ .

(2.115)

i=1

The vector kˆ μ now plays the role of the fixed four-vector nμ , so that all the properties of the basis and of the projection operator Pμν are similar. They are     kˆ ·  (i) δ k2 = k ·  (i) δ k2 = 0, and

 (i) ·  ( j) = −δ i j ,

(i)

(i)

μ (−k) = μ (k) (2.116)

  2 (i) (i)   δ k2 ∑ μ ν = −Pμν δ k2

(2.117)

i=1

where Pμν = gμν −

kˆ μ kν + kˆ ν kμ kμ kν kˆ · kˆ + . (kC )2 kC2

(2.118)

It is easy to check that the operator Pμν is the projection operator on the space orthogonal to kˆ μ and kμ when k2 = 0. In other words,     kˆ μ Pμν δ k2 = k μ Pμν δ k2 = 0,

(2.119)

    Pμν Pλν δ k2 = Pμλ δ k2 .

(2.120)

The inhomogeneous part can be written

2.6 Creation and Annihilation Operators

49

      (2) Aμ (x) = DH ∗Vμ (x) = ∂μ DH ∗ B (x) + ∂ˆ μ DH ∗ S (x).

(2.121)

By using properties of the convolution product, it can easily be shown to satisfy

∂ μ Aμ (x) = B(x), (2.122)  H   H  (2) μ ∂ˆ Aμ (x) = D ∗ KS (x) + ∂ˆ · ∂ˆ D ∗ S (x)    = K + ∂ˆ · ∂ˆ DH ∗ S (x) = −aC · ∂ S2 (x) − aS(x). (2.123) (2)

The condition ∂ · A = B implies then ∂ · A(1) = 0 and, from the properties of polarization vectors,   kC2 a3 (k)δ k2 = 0. In other words, a3 (k) = 0. No new operator is needed. It is now important to relate the a4 operator to previous ones. The gauge condition ∂ˆ · A = −aS − aC · ∂ S and (2.123) imply

∂ˆ · A(1) = −aC · ∂ S1 .

(2.124)

    kC2 a4 (k)δ k2 = −aC · k s (k)δ k2

(2.125)

In momentum space,

so that a4 is related to s . According to (2.124), it is useful to write Aμ (x) = Aμ (x) + a ∂μ C · ∂ A (0) (x) (1)

(0)

(2.126)

with ∂ˆ · A(1) = 0. Therefore, by using the properties of the projection operator Pμν (0)

Aμ (x) =

1 (2π )3/2



d4k

2

∑ ai (k)μ

(i)

  (k)δ k2 e−ik·x .

(2.127)

i=1

In the same way as here above, one can write the second part as  2   1 (0) H  4 s (k)δ k A (x) = −(DC ∗ S1 )(x) = d k e−ik·x . kC2 (2π )3/2

(2.128)

(2)

Playing the same game with Aμ (x), one can write (2) Aμ (x) = −i∂μ A (a) (x) − i∂ˆ μ A (b) (x) − iK ∂μ A (c) (x)

where A

(a)

  (x) = i DH ∗ B1 (x) =

1 (2π )3/2



  b(k)δ k2 −ik·x d k e , kC2 4

(2.129)

(2.130)

50

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

A

(b)

1 (x) = i(D ∗ S)(x) = (2π )3/2 H

A (c) (x) =

1 (2π )3/2



d4k



  s(k)δ k2 −ik·x d k e , kC2 4

s(k)δ  (kC2 ) −ik·x e . k2

(2.131)

(2.132)

Now, familiar methods of introducing creation and annihilation operators and integration over k0 give, after some calculations left to the reader,

 3  d k s (k)e−ik·x + s† (k)eik·x

1 A (0) (x) = , (2.133)

2k0 kC2 (2π )3/2 k =|k| 0

A A

(a)

(b)

(x) =

(x) =

A (c) (x) =

1



(2π )3/2 |C00 | 1



(2π )3/2 |C00 |

d 3 k b(k)e−ik+ ·x + b† (k)eik+ ·x

2 2k0 k+ k

d 3 k s(k)e−ik+ ·x + s† (k)eik+ ·x

2 2k0 k+ k

,

(2.134)

,

(2.135)

0 =Y (k)

0 =Y (k)



d3k  2 [k+ (ix0 k0 + 1) + 2k0 k0+ ]s(k)e−ik+ ·x 2 )2 (2π )3/2 |C00 | 4k03 (k+  2 −[k+ (ix0 k0 − 1) − 2k0 k0+ ]s† (k)eik+ ·x |k0 =Y (k) . (2.136) 1

It remains to gather all these results in the expression of Aμ (x) but the result is too heavy to be written explicitly.

2.6.2 Commutation Relations Between Creation and Annihilation Operators Let us now get the algebra satisifed by all the creation and annihilation operators which have been introduced. 2.6.2.1 Commutation Relation Between s and s† Let us begin with the operators s and s† associated with the S-field. Because [S(x), S(y)] = 0 at any time, it is obvious that [s(k), s(k  )] = [s† (k), s† (k  )] = [s(k), s† (k  )] = 0.

(2.137)

2.6 Creation and Annihilation Operators

51

As in covariant gauges, this implies that the metric is not positive defined since states created by s† (k) have zero norm. 2.6.2.2 Commutation Relation Between s and s † Because

  S(x), S (y) = 0

at any time, it is again obvious that       s(k), s (k  ) = s† (k), s† (k  ) = s(k), s† (k  ) = 0.

(2.138)

2.6.2.3 Commutation Relation Between s and s † From the fact that S and S commute at any time and the decomposition of S , the commutation relation at any time    i S (x), S (y) =  D(x − y) a reduces to

   i S1 (x), S1 (y) =  D(x − y). a The problem is the same as for the free scalar field and the solution is obviously     s (k), s (k  ) = s† (k), s† (k  ) = 0,

    2|k| s (k), s† (k  ) =  δ (3) k − k  . a (2.139)

2.6.2.4 Commutation Relation Between s and b† From the decomposition of B into homegeneous B1 and inhomogeneous B2 parts, it is clear that [S(x), B2 (y)] = 0. Assuming, as usual, that annihilation operators commute with all other annihilation operators and the same for creation ones     s(k), b(k  ) = s† (k), b† (k  ) = 0 (2.140) and setting     s(k), b† (k  ) = − s† (k), b(k  ) = α (k)δ (3) (k − k  ), it is easy to get

(2.141)

52

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

[S(x), B(y)] = [S(x), B1 (y)] =

1 2 (2π )3C00



 d3k α (k) eik+ ·(x−y) − e−ik+ ·(x−y) . 2 4Y (k)

(2.142)

From (2.43) and the definition of DC , i DC (x − y) C00  d 3 k 2  ik+ ·(x−y) 1 −ik+ ·(x−y) k =− − e e . (2.143) (2π )3C00 2Y (k) +

[S(x), B(y)] = −

The expression of α (k) is obtained by comparing these equations so that 2 (3) δ (k − k  ) [s(k), b† (k  )] = −2C00Y (k)k+

(2.144)

with 2 k+ = (Y (k) + Z.k)2 − |k|2 .

2.6.2.5 Commutation Relation Between b and b† From the commutation relation for any time (2.81) and the definition of B, it comes [B(x), B(y)] = −

i KEC (ξ ) C00

where ξ = x − y. Again, one assumes that   b(k), b(k  ) = 0,   b(k), b† (k  ) = β (k)δ (3) (k − k  ).

(2.145)

(2.146) (2.147)

From the decomposition of B into homogeneous and inhomegeneous parts and the commutation of S with itself, [B(x), B(y)] = [B1 (x), B1 (y)] + K {[B1 (x), S (y)] + [S (x), B1 (y)]} .

(2.148)

According to (2.147) and the momentum space expansion of B1 , [B1 (x), B1 (y)] =

1 2 (2π )3C00



 d3k ik+ ·(x−y) −ik+ ·(x−y) . β (k) e − e 4Y 2 (k)

(2.149)

On the other hand, [B1 (x), S (y)] =



d 3 k α (k) 1 2 8Y 4 (k) (2π )3C00  (iy0 k0 − 1)e−ik+ ·ξ + (iy0 k0 + 1)eik+ ·ξ .

(2.150)

2.6 Creation and Annihilation Operators

53

Therefore by a straightforward calculation, 

 d 3 k α (k)  ik+ ·ξ −ik+ ·ξ − e e 8Y 4 (k)  3 d k α (k)  1 ik+ ·ξ −ik+ ·ξ − ξ k − 1)e + (i ξ k + 1)e . (2.151) (i 0 0 0 0 2 8Y 4 (k) (2π )3C00

[B1 (x), S (y)] + [S (x), B1 (y)] =

1 2 (2π )3C00

Let us now compute iEC (ξ ) by using  2  (iξ0 k0 ± 1) e∓ik+ ·ξ = −k+ (iξ0 k0 ± 1)e∓ik+ ·ξ ± 2k0 k0+ e∓ik+ ·ξ . The result is 

  d3k + ik+ ·ξ −ik+ ·ξ 2k k − e e 0 0 4Y 3 (k)   d3k 2  1 ik+ ·ξ −ik+ ·ξ (i . (2.152) k ξ k − 1)e + (i ξ k + 1)e − 0 0 0 0 + (2π )3 4Y 3 (k)

iEC (ξ ) = −

1 (2π )3

By replacing α (k) in (2.151) by its computed value in the previous subsection and using (2.152), i EC (ξ ) C00    d3k  2 1 k+ + 2k0 k0+ eik+ ·ξ − e−ik+ ·ξ . − 3 3 (2π ) C00 4Y (k)

[B1 (x), S (y)] + [S (x), B1 (y)] = −

(2.153)

Gathering the results, [B(x), B(y)] = −

  d 3 k  ik+ ·ξ 1 i −ik+ ·ξ KEC (ξ ) + − e e C00 (2π )3C00 4Y 3 (k)    β (k)Y (k)  2 ˜ +) . − k+ + 2k0 k0+ K(k C00

Comparison with (2.145) gives the value of β (k)

β (k) =

 C00  2 ˜ + ) k+ + 2k0 k0+ K(k . k0 =Y (k) Y (k)

(2.154)

2.6.2.6 Commutation Relations Involving ai The polarization vectors are never involved in the results of the commutators between Aμ and S or S or B. This implies that the operators ai and a†i commute with b, s, s and their adjoints. Setting

54

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

  ai (k), a j (k  ) = 0,

 ai (k), a†j (k  ) = 2|k|δi j δ (3) (k − k  ),

(2.155)

it is easy to compute the commutator 



(0) (0) Aμ (x), Aν (y)

1 = (2π )3



 d3k Pμν eik·(x−y) − e−ik·(x−y) . 2|k|

It reproduces the value −igμν D(x − y) but it also contains the terms    3  kˆ μ kν + kμ kˆ ν kμ kν kˆ · kˆ  ik·ξ 1 d k −ik·ξ e . − + − e (2π )3 2|k| kC2 (kC2 )2 Setting

1 1

d = , d κ kC2 − κ κ =0 (kC2 )2

these terms read

 

  d ∂μ ∂ν ∂ˆ · ∂ˆ ΔCH ∗ D (ξ )

i ∂ˆ μ ∂ν + ∂μ ∂ˆν + dκ κ =0

(2.156)

where (C + κ )ΔCH (ξ ) = δ (4) (ξ ). This last term is a part of the terms occuring in the commutation relation for any time (3.52). For obtaining all the other terms, let us now compute the commutator  A (0) (x), A (0) (y) . From its definition and (2.139), 

A (0) (x), A (0) (y) =

1 a (2π )3



d 3 k eik·(x−y) − e−ik·(x−y) . 2|k| (kC2 )2

It can be rewritten as



i d (0) (0) H A (x), A (y) =  (Δ ∗ D)(ξ )

. a dκ C κ =0 Using (C + κ )ΔCH (x; κ ) = δ (4) (x) and the definition of D, 



i d H ΔC (x; κ ) ∗ D(x)

. A (0) (x), A (0) (y) =  a dκ κ =0

Combining this last result with (2.156) and using D = 0, one obtains the contribution from homogeneous terms

2.7 The Gupta-Bleuler Formalism



d − igμν D(ξ ) + i ∂ˆ μ ∂ν + ∂μ ∂ˆν − K ∂μ ∂ν dκ

 H 

= − igμν D(ξ ) + C00 Lμν ΔC ∗ D (ξ ) κ =0 .

55



 H  ΔC ∗ D (ξ )

κ =0

(2.157)

The contribution of the inhomogeneous part of the solution is easy to compute. Commutation relations for any time indeed give   Vμ (x),Vν (y) = Lμν ΔC (ξ ; κ )|κ =0 . This contributes to [Aμ (x), Aν (y)] with

  Lμν ΔC ∗ DH (ξ ) κ =0 . Combined with (2.157), this gives the desired contribution Lμν FC (ξ ; κ )|κ =0 =

 i  ˆ ∂μ ∂ν + ∂μ ∂ˆν FC (ξ ) − K ∂μ ∂ν GC (ξ ) . C00

2.6.3 Summary of the Algebra of Creation and Annihilation Operators Let us recall here which are the nonvanishing commutators between creation and annihilation operators without giving their expression which is not useful for the following purposes.   s(k), b† (k  ) ,   s (k), s† (k  ) ,   b(k), b† (k  ) ,  ai (k), a†j (k  ) i = 1, 2. Let us also recall that the s(k) and s (k) are respectively associated with the unphysical field S and S while b(k) is related to the auxiliary but also unphysical B = ∂ · A. The only physical particles are associated with the operators ai (k), i = 1, 2. For completeness,   (2.158) s(k), s† (k  ) = 0.

2.7 The Gupta-Bleuler Formalism Here, as in covariant gauges and owing to (2.158), the metric is not positive-definite and the Gupta-Bleuler formalism is used to restrict the description to physical particles. It works almost exactly as in the covariant case.

56

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

1. Because s and b† do not commute, the states generated by b† are eliminated by a subsidiary condition s|Ψphys  = 0. 2. Because of the presence of zero norm states generated by s† , these ones are eliminated by the equivalence relation  |Ψphys  ≡ |Ψphys  + s† |Ψ0 .

3. The only change with respect to covariant case is that states generated by s† must, when present, be eliminated. The condition is an additional subsidiary condition s |Ψphys  = 0. Let us note that, in coordinate space, the subsidiary condition can be written S(−) |Ψphys  = 0. Such a condition eliminates both the homogeneous and the inhomogeneous parts. When all these conditions are imposed, the corresponding space contains only physical states even if the theory contains more states from the beginning.

2.8 Covariance Problems In noncovariant gauges, it must be assured that covariance holds in the physical sector, the one obtained from the Gupta-Bleuler formalism. The way to prove it consists in showing that the Poincar´e algebra holds in this sector. This is the aim of this section.

2.8.1 Translation Invariance The Lagrangian (2.1) does not depend explicitly on coordinates. Under the infinitesimal transformations

δ xμ = μ ,

δ A μ = λ ∂ λ A μ ,

δ S = λ ∂ λ S,

the variation of the Lagrangian is

δ L = λ ∂ λ L .

δ S  = λ ∂ λ S  ,

2.8 Covariance Problems

57

Application of the usual Noether theorem leads to the conservation of the energymomentum tensor1 T νμ =

∂L ∂L ∂L ∂ ν Aτ + ∂νS + ∂ ν S − gμν L , ∂ (∂μ Aτ ) ∂ (∂μ S) ∂ (∂μ S )

(2.159)

∂ ν Tν μ = 0.

(2.160)

From this conservation, the charges 

d 3 x T0μ (x)

Pμ =

(2.161)

are conserved. It can be checked that, with F standing for any canonical variable, [Pμ , F(x)] = −i∂μ F(x).

(2.162)

This is an almost obvious task but, for μ = 0 one must use, in addition to field equations, relations between velocities and momenta as well as constraints. The Leibnitz rule, which holds for both commutation and derivation, implies that (2.162) holds for any polynomial expression of fields and canonically conjugate momenta. The operator Pμ is the generator of translations. Note however that, when Pμ is built in this way, it commutes with xν . As a particular value, let us replace F by T0ν [Pμ , T0ν (x)] = −i∂μ T0ν (x).

(2.163)

Integration over x leads, for μ = k, to [Pk , Pν ] = −i



d 3 x∂k T0ν (x) = 0

because the fields vanish at infinity. In the same way, using energy-momentum conservation, [P0 , Pν ] = −i



d 3 x∂0 T0ν (x) = −i



d 3 x∂l Tl ν (x) = 0.

The generators of translation commute [Pμ , Pν ] = 0.

(2.164)

1 With this definition, the energy-momentum tensor is not symmetric. In covariant theories, it could be symmetrized by adding a four-divergence. The tensor so obtained is called the improved energymomentum tensor. Here the lack of covariance prevents the symmetrization. For this reason, only the ordinary energy-momentum tensor is used.

58

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

2.8.2 Lorentz Transformations 2.8.2.1 The Transformations Infinitesimal Lorentz transformations are given by xμ → xμ + μν xν . The fields S and S are assumed scalar fields while Aμ is assumed to transform as a vector 1 1 μν (xμ ∂ ν − xν ∂ μ ) S, δ S = μν (xμ ∂ ν − xν ∂ μ ) S , (2.165) 2 2   1 μ (2.166) δ Aλ = μν (xμ ∂ ν − xν ∂ μ ) Aλ + gλ Aν − gνλ Aμ . 2

δS =

Invariance of the Lagrangian can hold only if Cμ and Cμν transform respectively like a vector and a second-rank tensor. Unlike the metric tensor, their components change from frame to frame. These changes are however not generated by a change of coordinates and must be introduced by hand. Actually, the presence of such fixed tensors breaks the Lorentz symmetry and the breaking can be obtained by considering them as fixed parameters. The variation of the Lagrangian can then be computed. It reads 1 δ L = μν [(xμ ∂ ν − xν ∂ μ ) L + K μν − K ν μ ] (2.167) 2 where K μν = A˜ μ ∂ ν S − Aμ ∂ˆ ν S − a SC μ ∂ ν S . (2.168) This tensor K μν vanishes obviously in the covariant case Cμν = gμν , Cμ = 0. In this case, the Lagrangian is a scalar.

2.8.2.2 Nonconservation of the Kinetic Momentum Tensor Let us now apply the Noether theorem to the general case. It results that the kinetic momentum tensor M μνρ = T μν xρ − T μρ xν +

∂L ∂L Aν − Aρ ∂ (∂μ Aρ ) ∂ (∂μ Aν )

= T μν xρ − T μρ xν − F μρ Aν + F μν Aρ satisfies

∂ μ Mμνρ = Kνρ − Kρν .

It is not conserved in noncovariant gauges.

(2.169) (2.170) (2.171)

2.8 Covariance Problems

59

In spite of this nonconservation, let us define the charges 

Mμν =

d 3 x M0μν .

(2.172)

Because conservation does not hold, they are time-dependent

∂0 Mμν =



d 3 x ∂0 M0μν =



d 3 x ∂ ρ Mρ μν =



  d 3 x Kμν − Kν μ = 0.

(2.173)

2.8.2.3 The Algebra of Charges Let us consider the algebra generated by the charges Pμ and Mμν . First of all, let us consider the commutator [Pμ , M0νρ (x)] = [Pμ , T0ν (x)]xρ − [Pμ , T0ρ (x)]xν + [Pμ , Πρ (x)Aν (x) − Πν (x)Aρ (x)]. From the fact that Pμ is the generator of translations, [Pμ , M0νρ (x)] = −i[∂μ T0ν (x)]xρ + i[∂μ T0ρ (x)]xν   − i∂μ Πρ (x)Aν (x) − Πν (x)Aρ (x) . It is easy to rewrite it as [Pμ , M0νρ (x)] = −i∂μ M0νρ (x) + i[gμν T0ρ (x) − gμρ T0ν (x)].

(2.174)

For μ = k, the rapid vanishing of the fields at infinity implies the vanishing of

 3 d x ∂k Ml νρ (x). Using also

∂0 M0νρ (x) = ∂ τ Mτνρ (x) + ∂k Mkνρ (x), integration over x implies [Pμ , Mνρ ] = i(gμν Pρ − gμρ Pν ) − igμ 0



d 3 x ∂ τ Mτνρ (x).

(2.175)

In the same way but after rather cumbersome calculations, left as an exercise, or a symmetry argument, the commutation relations between two M charges can be written as   [M λ ρ , M μν ] = i gλ μ M ρν + gρν M λ μ − gρ μ M λ ν − gλ ν M ρ μ    + i d 3 x xν gμ 0 − xμ gν 0 ∂ τ Mτλ ρ (x)    (2.176) − i d 3 x xρ gλ 0 − xλ gρ 0 ∂ τ Mτ μν (x).

60

2 Quantization of the Free Electromagnetic Field in General Class III Linear Gauges

The algebra of the charges Pμ and Mμν does not close when the kinetic momentum tensor is not conserved. This is the manifestation, at the level of commutation relations between charges, of the violation of Lorentz invariance.   Let us note that the new “charges” d 3 x ∂ τ Mτ μν (x), d 3 x xρ ∂ τ Mτ μν (x), when commuted with the generators Pμ , Mμν and themselves, also generate other charges ad infinitum. We are not interested here in obtaining the full algebra.

2.8.2.4 Poincar´e Invariance in the Physical Sector The Poincar´e algebra is violated by the nonvanishing of K μν = A˜ μ ∂ ν S − Aμ ∂ˆ ν S − a SC μ ∂ ν S . Let us note that this expression involves the fields S and S which vanish in the physical subsector of the Fock space. Therefore, if the normal product is taken,    Ψphys | : Kμν (x) : |Ψphys = 0. (2.177) This shows that, as Maxwell equation, Poincar´e invariance holds in the physical sector. The proof of this last relation is easier when BRST quantization is used. Indeed, Kμν can then be written as a commutator with the BRST charge. Explicitly construction is realized later in the case of Yang-Mills theory, of which Maxwell theory is a particular case.

References 1. 2. 3. 4. 5. 6. 7. 8.

Bagan, E., Fiol, B., Lavelle, M., McMullan, D.: Mod. Phys. Lett. A12, 1815 (1997) 31 Burnel, A.: Phys. Rev. D32,450 (1985) 26 Burnel, A.: Phys. Rev. D40, 1221 (1989) 25 Burnel, A. Kobes, R., Kunstatter, G., Mak, K.: Ann. Phys. 204, 247 (1990) 25 Christ, N.H., Lee, T.D.: Phys. Rev. D22, 939 (1980) 30 Gribov, V.N.: Nucl. Phys. B139, 1 (1978) 30 Leibbrandt, G.: Phys. Rev. D29, 1699 (1984) 25, 31 Mandelstam, S.: Nucl. phys. B213, 149 (1983) 25, 31

Chapter 3

Quantization of the Free Electromagnetic Field in Class II Axial Gauges

3.1 Introduction In this chapter, noncovariant axial gauges of the type n·A = 0 or their planar partners are considered. The fixed four-vector n can be of any nature, space-like, time-like or light-like. Such gauges are class II in a general coordinate frame. This fact implies new features with respect to class III gauges. In particular, neither the BRST nor the Gupta-Bleuler formalism apply because there is only one unphysical degree of freedom and no indefinite metric, at least if one works consistently. The evolution equation for the unphysical degree of freedom is of first order. This fact prevents the usual association between a field and a pair of creation and annihilation operators. If the Cauchy problem can be univoquely solved in coordinate space, the Feynman propagator cannot be defined in the usual way. Actually, the Hamiltonian involving the unphysical degree of freedom is not bounded from below. Therefore, no vacuum can be defined in the entire Fock space including unphysical states. All these features make axial gauges unsuitable for perturbative calculations, at least in the usual way. In order to prove all of these facts, the canonical formalism is developed in the same way as in the previous chapter. We first derive field equations and solve them in coordinate space. A second step consists in obtaining commutation relations for any time. A quantum theory is thus obtained. Its interpretation in terms of particles is done by going to momentum space where the defects of the model governed by first-order equations are exhibited.

3.2 Lagrangian and Field Equations 3.2.1 The Lagrangian The Lagrangian describing the Maxwell field in a class II axial gauge is [2] 1 a L = − Fμν F μν + Sn · A + aSS + ∂ μ S ∂μ S 4 2

(3.1)

Burnel, A.: Quantization of the Free Electromagnetic Field in Class II Axial Gauges. Lect. Notes Phys. 761, 61–86 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 3 

62

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

where n is the fixed four-vector and a a parameter introducing inhomogeneity in the gauge condition. The values a = 0 and a = n2 correspond respectively to axial and planar gauges. No particular value for n2 is chosen so that light-cone (n2 = 0) and temporal (n = (1, 0, 0, 0)) gauges can be considered as particular cases of the general discussion. The case n2 < 0, usually called axial gauge, deserves some special attention. It is included in the following discussion but we shall see that the frames in which n0 = 0 are singular.

3.2.2 Euler-Lagrange Equations 3.2.2.1 Variation With Respect to Aν From

∂L = −F μν , ∂ (∂μ Aν )

∂L = nν S ∂ Aν

(3.2)

the corresponding Euler-Lagrange field equations are

∂ μ Fμν + nν S = 0.

(3.3)

By expliciting Fμν , they can be rewritten as Aν − ∂ν ∂ · A + nν S = 0.

(3.4)

As in class III gauges, they generalize the Maxwell equations.

3.2.2.2 Variation With Respect to S In the same way,

∂L = 0, ∂ (∂ μ S)

∂L = n · A + aS ∂S

(3.5)

so that the corresponding field equation reads n · A + aS = 0.

(3.6)

It represents the gauge condition. The case a = 0 is obviously the axial gauge while the parameter a introduces some inhomogeneity which will appear useful in the following. 3.2.2.3 Variation With Respect to S Again

∂L = a∂μ S , ∂ (∂ μ S )

∂L = aS ∂ S

(3.7)

3.3 Constraint Analysis and Effective Hamiltonian

leads to

63

a(S − S) = 0.

(3.8)

As in class III gauges, this equation relates the two Lagrange multipliers S and S .

3.2.3 Derived Field Equations It is also interesting to get field equations in an order that can help their solution. Let us thus again make some handling of them. By applying ∂ ν to Eq. (3.3), n · ∂ S = 0.

(3.9)

n · A − n · ∂ ∂ · A + n2 S = 0.

(3.10)

Multiplying Eq. (3.4) by nν ,

Application of  to (3.6) leads to n · A + aS = 0.

(3.11)

The combination (3.10)–(3.11)+(3.8) leads to n · ∂ ∂ · A = (n2 − a)S.

(3.12)

The choice a = n2 (planar gauge) obviously simplifies the last equation which expresses the auxiliary field ∂ · A in terms of S.

3.2.4 Summary of the Field Equations Let us now gather the field equations in the order in which they should be solved. Setting B = ∂ · A, they are n · ∂ S = 0, n · ∂ B = (n2 − a)S, S = S, Aμ = ∂μ B − nμ S.

3.3 Constraint Analysis and Effective Hamiltonian With self-explanatory notations, the definition of canonical momenta gives

Π μ = F μ0,

ΠS = 0,

ΠS = a∂0 S .

(3.13)

64

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

As in the class III case, the case a = 0 can be treated separately and is simpler. It is included in the following discussion if, in addition to a = 0, the field S and its canonically conjugate variable ΠS are dropped out of the equations. There are obviously two primary constraints

Π 0 = 0,

ΠS = 0

(3.14)

while time derivatives of Ak and S can be related to canonical variables by a∂0 S = ΠS .

∂0 Ak = Π k − ∂ k A0 ,

(3.15)

Through a Legendre transform, it is then a straightforward task to write the total Hamiltonian 1 k k 1 1 a Π Π + Fkl Fkl − Π k ∂ k A0 + ΠS2 + ∂k S ∂k S 2 4 2a 2 −S (n · A + aS ) + Λ1 Π 0 + Λ2 ΠS .

(3.16)

The canonical nonvanishing brackets are obtained as usual. They are   Aμ (x), Π ν (y) x =y = δμν δ (3) (x − y),

(3.17)

HT =

0

0

{S(x), ΠS (y)}x0 =y0 = δ (3) (x − y),    S (x), ΠS (y) x =y = δ (3) (x − y). 0

0

(3.18) (3.19)

Now, because there are constraints, one must impose their stability and take their Poisson brackets with the Hamiltonian. After elementary calculations, one gets thus the chains .

.

Π 0 ≈ 0 =⇒ ∂k Π k + n0 S ≈ 0 =⇒ n0Λ2 − n.∂ S ≈ 0,

(3.20)

ΠS ≈ 0 =⇒ aS + n · A ≈ 0 =⇒ n0Λ1 + ΠS − n.Π − n.∂ A0 ≈ 0. .

.

If n0 = 0, the last two relations fix the Lagrange multipliers and one gets a system of four second-class constraints. On another hand, when n0 = 0, the multipliers are not determined. Additional constraints appear and the number of degrees of freedom is then lesser in the frames where n0 = 0. These frames are singular and they cannot occur in the quantization process. A consequence of this fact is that, in order to quantize the axial gauge A3 = 0 and its inhomogeneous partners, a boost must first be performed in such a way that the gauge condition reads n · A = 0 with n0 = 0. If this is not realized, the solution of the differential equation ∂k Π k = 0 will imply nonlocality, a fact that makes the quantum theory inconsistent. Assuming now and in the following that n0 = 0, the set of second-class constraints can be implemented into the Hamiltonian by eliminating the fields A0 and S whose canonically conjugate momenta vanish. After an integration by parts, the effective Hamiltonian reads

3.4 Solution of Field Equations

65



Heff =

d 3 x Heff (x)

(3.21)

where 1 1 1 a 1 Heff (x) = Π k Π k + Fkl Fkl + ∂ k Π k (n.A − aS ) + ΠS2 + ∂k S ∂k S . (3.22) 2 4 n0 2a 2

3.4 Solution of Field Equations As in the previous chapter, the solution of the Cauchy problem associated with the field equations is necessary to obtain commutation relations between fields at any time and the algebra of creation and annihilation operators. Here, the various evolution operators are n · ∂ , (n · ∂ )2 ,  and their products. For each of the involved evolution operator, elementary solutions will be derived and the Cauchy problem will be solved. All these developments are made in complete analogy with the class III case.

3.4.1 Solution of n · ∂ S = 0 Let us solve this equation by going to momentum space and setting 

S(x) = Then implies

˜ eik·x . d 4 k S(k)

(3.23)

n·∂ S = 0

(3.24)

˜ = h(k) δ (n · k) S(k)

(3.25)

where h(k) is arbitrary. Assuming that the coordinate frame is not singular, an elementary solution associated to the operator n · ∂ can be written, after normalization, as  |n0 | Dn (x) = d 4 k δ (n · k)eik·x . (3.26) (2π )3 There is an important difference with respect to a class III gauge. For a given k, the sign of energy is uniquely defined while, in class III gauges, it was left free. Obvious manipulations using properties of the Dirac-function are leading successively to     1 n.k ik·x 1 4 d k δ − = d 3 k e−ik.(x−nx0 /n0 ) . (3.27) Dn (x) = k e 0 (2π )3 n0 (2π )3

66

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges



Therefore Dn (x) = δ

(3)

nx0 x− n0

 .

(3.28)

Owing to (3.25), the general solution of the Eq. (3.24) is given by the three dimensional convolution of the elementary solution Dn (x) with an arbitrary function H(x), the Fourier transform of the arbitrary h(k)   3 (3.29) S(x) = Dn ∗ H (x).   n(x0 − y0 ) S(x, x0 ) = H x − , y0 n0

Therefore

(3.30)

where y0 is arbitrary. At x0 = y0 , it results that S(x, y0 ) = H(x, y0 ) so that the Cauchy problem is easily solved. Given the initial value S(x, 0), the value at the time x0 is given by   nx0 ,0 . (3.31) S(x, x0 ) = S x − n0

3.4.2 Solution of (n · ∂ + κ ) S = 0 As in class III gauges, in order to solve equations with the operator (n · ∂ )2 , it is useful to introduce a parameter κ which will have here the dimension of a mass and consider this operator as the limit of κ going to 0 of (n · ∂ )(n · ∂ + κ ). One must then solve the equation (3.32) (n · ∂ + κ ) S = 0. As in the previous subsection, the elementary solution can be obtained successively as 

|n0 | d 4 k δ (n · k − iκ )eik·x (2π )3  1 = d 3 k e−ik.(x−nx0 /n0 ) e−κ x0 /n0 (2π )3   nx0 −κ x0 /n0 (3) =e δ x− = e−κ x0 /n0 Dn (x). n0

Dn (x; κ ) =

The solution of the Cauchy problem is again obvious   n(x0 − y0 ) −κ (x0 −y0 )/n0 S(x, x0 ) = e S x− , y0 n0 once S(x, y0 ) is given.

(3.33)

(3.34)

3.4 Solution of Field Equations

67

It is left as an exercise to show that the elementary solution associated with the operator (n · ∂ )2 which vanishes at zero time is   d Dn (x; κ ) = x0 Dn (x). (3.35) En (x) = −n0 dκ κ =0 This solution is of restricted interest here.

3.4.3 Solution of n · ∂ B = (n2 − a)S The application of n · ∂ to leads to

 n · ∂ B = n2 − a S

(3.36)

(n · ∂ )2 B = 0.

(3.37)

Again one considers this equation as the limit for κ → 0 of n · ∂ (n · ∂ + κ ) B = 0. Then, by solving the Cauchy problem for the operator n · ∂ ,   nx0 (n · ∂ + κ ) B(x, x0 ) = [(n · ∂ + κ ) B] x − ,0 , n0

(3.38)

(3.39)

and, for the operator (n · ∂ + κ ), −κ x0 /n0

n · ∂ B(x, x0 ) = e



 nx0 (n · ∂ B) x − ,0 . n0

(3.40)

After subtracting these equations and dividing by κ , the limit for κ → 0 leads to     nx0 x0 nx0 B(x, x0 ) = B x − , 0 + (n · ∂ B) x − ,0 (3.41) n0 n0 n0 where the last term can be replaced by using (3.36). The solution of the Cauchy problem for (3.36) is then   nx0 x0 B(x, x0 ) = B x − , 0 + (n2 − a) S(x, x0 ). (3.42) n0 n0

3.4.4 Elementary Solution of n · ∂ A = 0 As in class III gauges, an elementary solution of the equation n · ∂ A = 0

(3.43)

68

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

satisfying definite zero-time properties, i.e. the vanishing of the solution and of its first time derivative at x0 = 0 is sought. Let us start from  (a) Fn (x) = DH ∗ Dn (x)

(3.44)

which is obviously a solution of (3.43). Here, DH is a particular solution of DH (x) = δ (4) (x), for instance 1 D (x) = − (2π )4



H

d4k

eik·x . k2 + i

It is clear that the chosen prescription should play no role at all in the story. Therefore, it will be omitted in the following but, of course, at the end of the calculation, it must be checked that a definite prescription is indeed not needed. The elementary solution will then be defined in a unique way. From (3.26) and properties of Fourier transforms, (a)

Fn (x) = −

|n0 | (2π )3



δ (n · k) ik·x e . k2

(3.45)

e−ik.(x−nx0 /n0 ) . (n.k)2 − |k|2 n20

(3.46)

d4k

Integration over k0 leads to (a)

Fn (x) = −

n20 (2π )3



d3k

Choosing the reference frame in such a way that n = (0, 0, 1), the denominator can be written  −n2 k32 − n20 k12 + k22 . (a)

Therefore no prescription is needed to define Fn (x) in the light-like and time-like cases (n2 ≥ 0) since then, the denominator never vanishes when k = 0. In these cases, all the integrations can be carried out but this is not a necessary task for the following. (a) If Fn (x) is an elementary solution, it does not satisfy accurate zero-time properties. In order to fulfil this requirement, one adds, as in noncovariant class III gauges, another elementary solution proportional to D ∗ DH n where (4) (x). n · ∂ DH n (x) = δ

(3.47)

Such a contribution can be written, with α as a parameter which will be fixed later on,  2  α (b) 4 (k0 )δ k eik·x . d k (3.48) Fn (x) = (2π )3 n·k Integration over k0 leads to

3.4 Solution of Field Equations (b)

Fn (x) = =

69

α 2(2π )3 α (2π )3







d 3 k −ik.x e−i|k|x0 ei|k|x0 e + |k| n0 |k| − n.k n0 |k| + n.k



d 3 k n0 |k| cos |k|x0 + in.k sin |k|x0 −ik.x e . |k| |k|2 n20 − (n.k)2 (a)

(3.49) (3.50)

(b)

The sum of the two contributions Fn (x) + Fn (x) is Fn (x; α ) =



d3k 1 e−ik.x (2π )3 |k|[|k|2 n20 − (n.k)2 ]  n.kx0 + α n0 |k| cos |k|x0 n20 |k| cos n0

 n.kx0 + i n20 |k| sin + α n.k sin |k|x0 . n0

(3.51)

By imposing the vanishing at x0 = 0, the value α = −n0 is obtained. Therefore a convenient elementary solution is 

Fn (x) =

d3k n0 e−ik.x 3 2 (2π ) |k|[|k| n20 − (n.k)2 ]

  n.kx0 n.kx0 − cos |k|x0 + i n0 |k| sin − n.k sin |k|x0 . n0 |k| cos n0 n0 (3.52)

It is a trivial task to check that Fn (x, 0) = ∂0 Fn (x, 0) = 0,

∂02 Fn (x, 0) = δ (3) (x).

(3.53)

It is also obvious that the numerator vanishes when the denominator does so that no prescription is needed to define the Fn -function. The elementary solution of the equation (n · ∂ + κ )A = 0

(3.54)

is obtained by the replacement n.k −→ n.k + iκ in (3.52). Such a function will be denoted by Fn (x; κ ). All its properties are not necessary. So, only the useful ones are derived hereunder. From Fn (x; κ ) = DH (x) ∗ Dn (x; κ ) − n0 DH n (x; κ ) ∗ D(x), it is obvious that Fn (x; κ ) = Dn (x; κ ),

(n · ∂ + κ )Fn (x; κ ) = −n0 D(x).

(3.55)

70

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

The zero-time properties n.∂ − κ (3) δ (x), n

0   n.∂ − κ 2 (3) 4 ∂0 Fn (x, 0; κ ) = Δ + δ (x) n0

∂03 Fn (x, 0; κ ) =

(3.56)

are easily deduced from (3.53) and (3.55) .

3.4.5 Solution of the Cauchy Problem for A(x − z) = Lx Dn (x − z) The calculation made in Appendix C about the solution of the Cauchy problem associated with the equation A(x − z) = Lx DC (x − z), where Lx is a derivation operator with respect to x, can be reproduced when DC , FC are replaced by Dn , Fn . Indeed the calculation does not imply the use of properties of these functions but only the fact that they are elementary functions associated with the evolution operator accompanying  in the equations. Therefore, the solution of the Cauchy problem for the equation A(x) = Lx Dn (x; κ )

(3.57)

is A(x) = Lx Fn (x; κ ) +



↔

d 3 y D(x − y) ∂0y Ly Fn (y; κ )   ↔  − d 3 y D(x − y) ∂0y A(y) .

(3.58)

y0 =x0

3.4.6 Elementary Solution of (n · ∂ )2 A = 0 As in class III gauges, the equation n · ∂ (n · ∂ + κ )A = 0

(3.59)

is considered and the limit for κ → 0 is taken. This leads to consider the elementary solution Gn of (3.59) as the solution of n · ∂ Gn (x) = n0 Fn (x; κ ),

(n · ∂ + κ )Gn (x) = n0 Fn (x).

(3.60)

3.5 Commutation Relations

71

By subtracting these equations, dividing by κ and taking the limit κ → 0 by L’Hospital theorem,   d Gn (x) = −n0 Fn (x; κ ) . (3.61) dκ κ =0 Zero-time properties of Gn are easily obtained from this definition

∂0m Gn (x, 0) = 0,

m = 0, 1, 2,

∂03 Gn (x, 0) = δ (3) (x),

n.∂ (3) δ (x), n0

   n.∂ 2 (3) 5 ∂0 Gn (x, 0) = Δ + 3 δ (x). n0

∂04 Gn (x, 0) = 2

(3.62) (3.63)

(3.64)

3.5 Commutation Relations Because one will consider a quantum field theory, it is necessary to tackle the problem of commutation relations between field operators. Since the theory is a free one, commutation relations for any time can be obtained by solving the Cauchy problem associated with the required field equation with initial values given by equal-time commutators.

3.5.1 Equal Time Commutators Let us thus begin with equal time commutators. From the constraint analysis based on (3.1), it is clear that the nonconstrained canonical pairs are (Al , Π l ), (S , ΠS ). Nonvanishing canonical commutators are then [S (x), ΠS (y)]x0 =y0 = iδ (3) (x − y). (3.65) The constrained variables A0 and S are, in a nonsingular frame, given respectively by ∂k Π k n.A − aS , S=− . (3.66) A0 = n0 n0 [Ak (x), Π l (y)]x0 =y0 = iδkl δ (3) (x − y),

The relations between velocities and momenta are

∂0 Al = Π l + ∂l A0 ,

∂0 S  =

ΠS . a

All these relations allow us to obtain the desired equal-time commutators.

(3.67)

72

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

3.5.2 Commutation Relations Involving the S-Field Because S satisfies the homogeneous equation n · ∂ S = 0, commutators for any time are immediately given by the three-dimensional convolution of the Dn function with the corresponding equal-time commutators 3

[S(x), K(y)] = Dn (x) ∗ [S(x), K(y)]x0 =y0

(3.68)

for any operator K. From the canonical commutation relations and the constraints, it is easy to get   [S(x), S(y)]x0 =y0 = 0, S(x), S (y) x =y = 0, (3.69) 0

0

[S(x), Ak (y)]x0 =y0 =

i ∂k δ (3) (x − y), n0

(3.70)

[S(x), A0 (y)]x0 =y0 =

i n.∂ δ (3) (x − y). n20

(3.71)

Using (3.68), the commutation relations for any time are then given by   [S(x), S(y)] = S(x), S (y) = 0,

  i S(x), Aμ (y) = ∂μ Dn (x − y). n0

(3.72)

3.5.3 Commutation Relations Involving the S -Field The field S satisfies the equation S (x) = S(x).

(3.73)

When commuted with S (z), it gives the homogeneous equation [S (x), S (z)] = 0. The solution of the Cauchy problem is now well known and given by       S (x), S (z) = − d 3 y ∂0 S (y), S (z) y

−



  d 3 y S (y), S (z) y

0 =z0

0 =z0

D(x − y)

∂0 D(x − y).

(3.74)

After the use of equal-time commutators and integration over y, i [S (x), S (z)] = D(x − z). a

(3.75)

3.5 Commutation Relations

73

Let us now take the commutator of (3.73) with Aμ (z). By taking (3.72) into account, it satisfies the inhomogeneous equation   i  S (x), Aμ (z) = [S(x, Aμ (z)] = ∂μ Dn (x − z). n0

(3.76)

Using (3.58) where the operator Lz is given by Lz = −

i z ∂ n0 μ

as well as zero-time properties of Fn and the equal-time commutators [S (x), Aμ (y)]y0 =x0 = 0,

[∂0 S (x), Aμ (y)]y0 =x0 =

i gμ 0 δ (3) (x − y), n0

(3.77)

straightforward calculations lead to [S (x), Aμ (z)] =

i ∂μ Fn (x − z). n0

(3.78)

3.5.4 Commutation Relations Involving ∂ · A Setting B ≡ ∂ · A, this field satisfies the equation n · ∂ B = (n2 − a)S. The solution of the Cauchy problem is given by Eq. (3.42) which, when commutators are involved, can be usefully written as    d 3 3 B(x) = B(x, 0) ∗ Dn (x) − n2 − a S(x, 0) ∗ Dn (x; κ ) dκ κ =0  x0 3 = B(x, 0) ∗ Dn (x) + n2 − a S(x). n0

(3.79)

(3.80)

In order to get equal-time commutation between B(x) and Aμ (y), B(x) must be expressed in terms of canonical variables. By differentiating the gauge condition (3.6) with respect to time and using relations between velocities and momenta, B=

n0 nk Π k + n.∂ (n.A − aS ) − n0 ΠS − ∂l Al . n20

Use of canonical commutation relations gives, after elementary calculations,

(3.81)

74

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

nk (3) δ (x − y), n0 n2 + a − n2 (3) = −i 0 2 δ (x − y). n0

[B(x), Ak (y)]x0 =y0 = −i

(3.82)

[B(x), A0 (y)]x0 =y0

(3.83)

From (3.72) and (3.80), one easily gets [B(x), Ak (y)] = −i

nk n2 − a Dn (x − y) + i 2 ∂k [(x0 − y0 )Dn (x − y)], n0 n0

[B(x), A0 (y)] = −i

n20 + a − n2 n2 − a D (x − y) + i (x0 − y0 )∂0 Dn (x − y) n n20 n20

= −iDn (x − y) + i

n2 − a ∂0 [(x0 − y0 )Dn (x − y)]. n20

(3.84)

(3.85)

These results are summarized in   nμ n2 − a B(x), Aμ (y) = −i Dn (x − y) + i 2 ∂μ [(x0 − y0 )Dn (x − y)]. n0 n0

(3.86)

3.5.5 Commutation Relations [Aμ (x), Aν (z)] Let us now tackle the problem of the commutation relations between the potentials themselves. Field equations allow to write Aμ = Vμ with

Vμ = ∂μ B − nμ S.

Commutation relations for any time between Vμ (x) and Aν (y) can be obtained from (3.72) and (3.86). They read   nμ ∂ν + nν ∂μ Vμ (x), Aν (y) = −i Dn (x − y) n0 n2 − a +i 2 ∂μ ∂ν [(x0 − y0 )Dn (x − y)] n0  

 nμ ∂ν + nν ∂μ n2 − a d ∂μ ∂ν Dn (x − y; κ ) + = −i n0 n0 d κ κ =0  y = Lμν Dn (x − y; κ )κ =0 (3.87)

where Lyμν

 nμ ∂ν + nν ∂μ n2 − a d ∂ μ ∂ν , =i − n0 n0 d κ

(3.88)

3.5 Commutation Relations

75

all the derivatives being taken with respect to y. It results that the commutator [Aμ (x), Aν (y)] satisfies the inhomogeneous equation  [Aμ (x), Aν (z)] = Lzμν Dn (x − z; κ )κ =0 . The Cauchy problem is solved in Eq. (3.58) so that  [Aμ (x), Aν (z)] = Lzμν Fn (x − z; κ )κ =0  ↔  + d 3 y D(x − y) ∂0y Lzμν Fn (y − z; κ )κ =0   ↔ − d 3 y D(x − y) ∂0y [Aμ (y), Aν (z)] .

(3.89)

y0 =z0

It remains now to get the necessary equal-time commutators from canonical commutation relations. After elementary calculations, they are   Aμ (x), Aν (y) x =y = 0, (3.90) 0

0

[∂0 Ak (x), Al (y)]x0 =y0 = igkl δ (3) (x − y),

(3.91)

[∂0 Ak (x), A0 (y)]x0 =y0 = −i

nk (3) δ (x − y), n0

(3.92)

[∂0 A0 (x), A0 (y)]x0 =y0 = −i

a + |n|2 (3) δ (x − y). n20

(3.93)

On the other hand, zero-time properties of Fn imply    Lyμν Fn (x − y)x =y  = 0, 0

0

 y ∂0y Lkl Fn (x − y)x

   0 =y0

 y ∂0y Lk0 Fn (x − y)x

   0 =y0

 y ∂0y L00 Fn (x − y)x

   0 =y0

(3.94)

κ =0

κ =0

κ =0

κ =0

= 0,

(3.95) nk (3) δ (x − y), n0

(3.96)

= −iδ (3) (x − y) − i

a + |n|2 (3) δ (x − y). (3.97) n20

= −i

It is then an obvious task to get the final result [Aμ (x), Aν (z)] = − igμν D(x − z) − i +i

nμ ∂ν + nν ∂μ Fn (x − z) n0

n2 − a ∂μ ∂ν Gn (x − z). n20

(3.98)

76

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

3.6 Association of a Feynman Propagator with the Operator n · ∂ In usual theories, the Feynman propagator associated with any function K(x) whose support is causal is defined by K F (x) = θ (x0 )K(x) − K (−) (x)

(3.99)

where K (−) (x) is the negative frequency part of K(x). Let us do the work for Dn (x) by assuming that the theory is behaving as a usual second order theory. Assuming that the coordinate frame is chosen in such a way that n0 > 0, the negative frequency part is  n0 (−) d 4 k θ (k0 )δ (n · k)eik·x . (3.100) Dn (x) = (2π )3 Its Fourier transform is (−) D˜ n (k) =



d 4 x D(−) (x)e−ik·x = 2π n0 δ (n · k)θ (n.k).

(3.101)

The computation of the Fourier transform of θ (x0 )Dn (x) gives successively 

d 4 x θ (x0 )Dn (x)e−ik·x =

 ∞ −∞

dx0 θ (x0 )e−ix0 (k0 −k.n/n0 )

−in0 −i = k0 − k.n/n0 − i n · k − i   1 = −in0 P + π n0 δ (n · k) n·k =

(3.102) (3.103) (3.104)

where P is the principal value. Therefore, using properties of distributions recalled in Appendix A, 

D˜ Fn (k) =

d 4 x DFn (x)e−ik·x

(3.105)



 1 = −in0 P + π n0 δ (n · k) [1 − 2θ (n.k)] n·k 

  1 = −in0 P − iπδ (n · k)(n.k) n·k =

−in0 . n · k + i sign(n.k)

(3.106) (3.107) (3.108)

From this point of view, the pole at n · k = 0 is regularized in a well-defined way by1 1

This prescription was, for the first time, obtained in ref [1] for the case n2 = 0 and generalized to any n2 by Lazzizzera [7]. They have shown that, in the sense of distribution theory, it is equiv-

3.7 Associating Creation and Annihilation Operators with Fields

77

1 1 −→ . n·k n · k + i sign(n.k)

(3.109)

This is obviously compatible with the Wick rotation and, moreover, it is the only way to get a solution of n · ∂ DFn (x) = δ (4) (x) compatible with Wick rotation. Some remarks are however in order. 1. Such a prescription can only hold if n = 0. The temporal gauge in the frame in which n = (1, 0, 0, 0) is therefore excluded from the discussion. There is however no fundamental reason, when solving the Cauchy problem, to reject this case.2 ˜ =0 In fact, in the temporal gauge with n = 0, the S-field satisfies ∂0 S = 0, k0 S(k) and has only zero frequencies. 2. A calculation with this prescription in Yang-Mills theory [4] leads to results which are not consistent with a renormalizable gauge. The self-energy is nonlocal and does not satisfy Ward identities. This means that additional contributions to the self-energy are needed. Such contributions look like Faddeev-Popov ghosts but these ghosts are not required in axial gauges. 3. If the same decomposition into positive and negative frequency parts is made at the level of field operators and if creation and annihilation operators are associated with this decomposition, the number of degrees of freedom in momentum space is different from the number of degrees of freedom in coordinate space. An indefinite metric is also involved while it has nothing to do in the story. This will be developed in the next section. For all these reasons, it is clear that something is wrong with the application of standard methods to the present case. There is a fundamental difference with the usual approach. The unphysical fields S and B are governed by first-order evolution equations and not second-order ones. In the light of these facts, the definition of operators in momentum space will be reconsidered.

3.7 Associating Creation and Annihilation Operators with Fields As in ref. [1], let us try to associate creation and annihilation operators to the various fields involved in the problem.

alent to the Leibbrandt-Mandelstam prescription. It must however be stressed that not all the uses of the prescription are done with well-behaved test-functions so that equivalence in the sense of distribution theory has no meaning at all. 2 A solution of the Cauchy problem is given in [3] for the similar case ∂ 2 S = 0 but the problem of 0 the definition of the vacuum was not considered in this paper.

78

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

3.7.1 The S-Field Let us first do the job with the field S(x) satisfying the homogeneous equation n · ∂ S = 0. If n = 0, its Fourier expansion S(x) =

1 (2π )3/2



d 3 k s(k)eik.(x−nx0 /n0 )

can be decomposed into positive- and negative-frequency parts 1 S(x) = (2π )3/2



    n.k n.k d k θ +θ − s(k)eik.(x−nx0 /n0 ) . n0 n0 3

(3.110)

The change k into −k in the second integral leads to   n.k  d kθ s(k)eik.(x−nx0 /n0 ) + s(−k)e−ik.(x−nx0 /n0 ) . n0 (3.111) † As usual, hermiticity of the S field leads to s(−k) = s (k). Like in standard gauges, it is tempting to interpret the operators s and s† respectively as annihilation and creation operators. If such an interpretation is made, the vanishing of the commutator between two S-fields leads to the commutation relations 1 S(x) = (2π )3/2





3

    s(k), s(k  ) = s(k), s† (k  ) = 0.

(3.112)

As in standard class III gauges, an indefinite metric would then be associated with the vanishing of commutation relations between s and s† . Indeed     ||s† (k)|0 || = 0|s(k)s† (k)|0 = 0|[s(k), s† (k)]|0 = 0 because the vacuum is annihilated by s.

3.7.2 Other Fields Let us only sketch the problem by following the same reasoning as in class III gauges for which the equations are very similar. A second pair of creation and annihilation operators (g, g† ) is associated with the B-field, a third one with the S -field. Finally, the equation Aμ = ∂μ B − nμ S

3.7 Associating Creation and Annihilation Operators with Fields

79

admits the solution Aμ (x) =

  1 (1) d 4 k Aμ (k)δ k2 eik·x 3 (2π )    |n0 | (2) (3) 4  + d k δ (n · k)A (k) + δ (n · k)A (k) . μ μ (2π )3 (2)

(3.113)

(3)

The last two contributions Aμ (k) and Aμ (k) are related, by the inhomogeneous equation, to the degrees of freedom associated with B and S. As usual, the first part (1) Aμ (k) can be written as 2

Aμ (k) = ∑ ai (k)μ (k) + a3 (k)kμ + a4 (k)nμ (1)

(i)

i=1

(i)

where the polarization vectors μ (k) satisfy   δ k2 k ·  (i) (k) = δ k2 n ·  (i) (k) = 0. Use of all field equations imposes 2

Aμ (k) = ∑ ai (k)μ (k) − a (1)

i=1

(i)

s (k)kμ n·k

where s (k) is associated with S . Therefore, as in the class III case, five pairs of creation and annihilation operators are needed      ai (k), a†i (k) , i = 1, 2, s (k), s† (k) , s(k), s† (k) , g(k), g† (k) while the number of pairs of canonical variables is 4 :     S , ΠS . Al , Π l , (l = 1, 2, 3) and The other pairs of variables (A0 , Π 0 ) and (S, ΠS ) are constrained in contrast with the class III case where only the first pair was constrained. This problem of different numbers of degrees of freedom in momentum and coordinate spaces clearly shows that something is wrong in the above development. Actually, the field S(x) is not an ordinary field. It indeed obeys a first order equation and not a second order one. Therefore, in momentum space, a frequency with well-defined sign (k0 = k.n/n0 ) corresponds to a given momentum k. With such a feature, the change of the sign of k changes at the same time as the sign of energy. It is therefore hard to interpret an antiparticle as a particle of opposite momentum and energy. The operators s(k) and s† (k) are not creation and annihilation operators in the usual sense.

80

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

3.8 Correct Momentum Space Expansion of the Fields 3.8.1 The S-Field Let us now make the Fourier expansion of the S-field without necessarily associating creation and annihilation operators to it. The expansion reads S(x) =

i (2π )3/2



d 3 k s(k)e−ik.(x−nx0 /n0 )

(3.114)

where the factor i is again introduced for convenience. At zero time, S(x, 0) =

i (2π )3/2



d 3 k s(k)e−ik.x .

(3.115)

Inversion of the three-dimensional Fourier transform leads to s(k) = −

i (2π )3/2



d 3 x S(x, 0)eik.x .

(3.116)

In contrast to usual fields, the time derivative is not involved in such a relation so that only one operator in momentum space is associated with S. From equal-time commutators, [s(k), s(k  )] = 0. (3.117)

3.8.2 The B-Field In the same way, the general solution of the homogeneous equation n · ∂ B(0) = 0 is written as B(0) (x) =

1 (2π )3/2



d 3 k g(k)e−ik.(x−nx0 /n0 ) .

(3.118)

The inversion of the Fourier transform gives g(k) = Because

1 (2π )3/2



d 3 x B(0) (x, 0)eik.x .

 x0 B(x) = B(0) (x) + n2 − a S(x), n0

(3.119)

3.8 Correct Momentum Space Expansion of the Fields

81

it can also be written as g(k) =

1 (2π )3/2



d 3 x B(x, 0)eik.x

(3.120)

with B solution of the inhomogeneous equation. The commutator between s and g is easily computed from the commutation relations between S and B = ∂ · A. Using (3.116) and (3.120), i [s(k), g(k )] = − (2π )3 





d 3 x d 3 y [S(x, 0), B(y, 0)]ei(k.x+k .y) .

(3.121)

By using (3.72), [s(k), g(k  )] = −

1 n0 (2π )3





d 3 x d 3 y Dn (x − y)|x0 =y0 ei(k.x+k .y) .

(3.122)

From the equal-time properties of Dn ,   s(k), g(k  ) = −

1 (2n0 π )3



   d 3 x d 3 y (n.∂ )2 − n20 Δ δ (3) (x−y)ei(k.x+k .y) (3.123)

and, by integrating over y,   s(k), g(k  ) = −

1 (2n0 π )3



   d 3 x eik.x (n.∂ )2 − n20 Δ eik .x .

(3.124)

The derivations can be carried out explicitly so that [s(k), g(k  )] =

  1 (n.k  )2 − n20 |k  |2 3 (2n0 π )





d 3 x ei(k+k ).x

(3.125)

and, by the definition of the Dirac-function, [s(k), g(k  )] =

 1  (n.k)2 − n20 |k|2 δ (3) (k + k  ). n30

(3.126)

The same kind of calculation leads to [g(k), g(k  )] = −

2(n2 − a) n.k δ (3) (k + k  ). n30

(3.127)

Note the sign in the argument of the δ -function, which is different from the sign occurring in commutation relations between creation and annihilation operators. Here, we have not yet associated the words “creation” or “annihilation” to the operators s and g. As we have seen it above, the associations s† (k) = s(−k), g† (k) = g(−k) which give back the correct sign are also not allowed because the field equation is first-order.

82

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

3.8.3 The S -Field The S -field satisfies the inhomogeneous equation (when a = 0) S = S. Its general solution can be written as  S (x) = S(0) (x) + DH ∗ S (x)

(3.128)

where S(0) (x) is the solution of the homogeneous equation and DH (x) = δ (4) (x). The product of convolution used here is the four-dimensional product 

 DH ∗ S (x) = d 4 y DH (x − y)S(y).

(3.129)

The momentum space expansion of the S-field can be written S(x) =



i|n0 | (2π )3/2

d 4 k δ (n · k)s(k)eik·x

and DH (x) = −

1 (2π )4



d 4 k ik·x e k2

(3.130)

(3.131)

up to a prescription for the poles at k0 = ±|k|. As previously, the result must be independent of it so that it is omitted. Properties of the Fourier transform of a convolution product followed by integration over k0 immediately imply that  4  H d k −i|n0 | D ∗ S (x) = δ (n · k)s(k)eik·x 3/2 k2 (2π )

=

−in20 (2π )3/2



d3k (n.k)2 − n20 |k|2

s(k)e−ik.(x−nx0 /n0 ) .

It can be checked that no prescription is actually needed for the denominator. On the other hand, the momentum expansion of the homogeneous part is straightforward. It reads  3   d k  1 (0) −ik·x † ik·x  s (k)e + s (k)e (3.132) S (x) =  (2π )3/2 2k0 k0 =|k| where s (k) and s† (k) can respectively be interpreted as annihilation and creation operators. It is left as an exercise to get the commutation relations

3.8 Correct Momentum Space Expansion of the Fields

83

    s(k), s (k) = s (k), s (k) = 0,   2|k| (3) s (k), s† (k) = δ (k − k  ). a

(3.133) (3.134)

3.8.4 The Aμ -Field The Aμ -field satisfies the equation Aμ = ∂μ B − nμ S. Again, the general solution is written as the sum of the general solution of the homogeneous equation and of a particular solution of the inhomogeneous one (0)

Aμ (x) = Aμ (x) + ∂μ (DH ∗ B)(x) − nμ (DH ∗ S)(x).

(3.135)

As in class III gauges, the homogeneous part can be written as (0)

Aμ (x) = with

1 (2π )3/2

2



 (0) d 4 k Aμ (k)δ k2 eik·x

Aμ (k) = ∑ ai (k)μ (k) + a3 (k)kμ + a4 (k)nμ (0)

(i)

i=1

(i)

where the polarization vectors μ (k) satisfy   δ k2 k ·  (i) (k) = δ k2 n ·  (i) (k) = 0. Use of all field equations imposes 2

Aμ (k) = ∑ ai (k)μ (k) − a (0)

(i)

i=1

s (k)kμ . n·k

As in class III gauges, creation and annihilation operators are introduced with the same commutation relations [ai (k), a†j (k  )] = 2|k|δi j δ (3) (k − k  ). They commute with all the other operators. It remains to compute the momentum space expansion of (DH ∗ x0 S)(x). From properties of Fourier transforms, x0 S(x) = −

|n0 | (2π )3/2



d4k

∂ δ (n · k)s(k)eik·x ∂ k0

(3.136)

84

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

and (DH ∗ x0 S)(x) =

1|n0 | (2π )3/2

=−



n20 (2π )3/2

d4k ∂ δ (n · k)s(k)eik·x k2 ∂ k0



d 3 k s(k)e−ik.(x−nx0 /n0 )

ix0 [(n.k)2 − n20 |k|2 ] − 2n0 n.k . [(n.k)2 − n20 |k|2 ]2 As in class III gauges, it is now a matter of straightforward calculations to show that commutation relations in coordinate space are correctly reproduced. Such a calculation follows the same lines as in class III gauges. It is left as an exercise.

3.9 A Toy Model for the Unphysical S and B Fields In order to better understand the difficulties generated by the presence of fields obeying first order scalar equations, let us consider a toy model where only such fields are present. It is given by the Lagrangian u L (x) = B(x) n · ∂ S(x) − S2 (x). 2

(3.137)

It is an easy task to derive the field equations n · ∂ S = 0,

n · ∂ B = uS.

(3.138)

They are identical to those obtained previously for the S and B fields. Canonical commutation relations are also similar. There are two primary constraints ΠB = 0, ΠS = n0 B. (3.139) When n0 = 0, these constraints are second-class and the B field and its canonicallyconjugate variable ΠB are not true degrees of freedom. Canonical commutation relations are [S(x), S(y)]x0 =y0 = [B(x), B(y)]x0 =y0 = 0, [S(x), B(y)]x0 =y0 =

i (3) δ (x − y). n0 (3.140)

Let us now consider the Hamiltonian 1 Hc (x) = B(x)n.∂ S(x) + uS2 (x). 2

(3.141)

Even with u ≥ 0, such a Hamiltonian is not bounded from below so that no ground state, no vacuum can be defined.

3.11 Interpolating Between Axial and Relativistic Gauges

85

A similar part is present in the axial gauge Hamiltonian so that the conclusion on the lack of ground stat also holds if the unphysical contribution is included.

3.10 Conclusions In momentum space, class II axial gauges and their inhomogeneous partners involve three pairs of creation and annihilation operators (ai , a†i ), i = 1, 2, (s , s† ) and a pair (s, g) which is not associated with creation and annihilation of a unphysical particle and for which no vacuum can be defined. No normal product for the (s, g) pair is defined. The absence of a vacuum, in the usual sense, in the whole space implies that all the features of usual field theory are lost in class II axial gauges : the perturbative expansion, the Gupta-Bleuler formalism and the usual definition of the propagator as the vacuum expectation value of time-ordered product of free fields cannot be defined. The whole theory must be reconsidered with new tools where the interpretation of unphysical contributions as an unphysical particle is lost. This is a heavy price to pay for the absence of Faddeev-Popov ghosts in nonabelian theories. Lack of a theory behaving as a usual one does not mean that models of propagators for class II axial gauges cannot be built and meet with some success. Propagators with a definite prescription for handling the unphysical pole at n · k = 0 can perfectly work in particular problems. Many examples can be shown. The difficulty is that such phenomenological propagators are not the universal propagator in axial gauges. Therefore, counter-examples of their validity can also be exhibited in other processes than those for which they have been built. The simplest example is the principal-value prescription which was used as the first prescription for the handling of the unphysical poles. Though it gave some consistent results, it lacked the factorization in Wilson loop calculations.

3.11 Interpolating Between Axial and Relativistic Gauges In order to get a propagator in axial gauges, it is sometimes suggested to interpolate between the axial gauge n · A = 0 and relativistic gauge ∂ · A = 0 through a gauge condition like3 n · A + α∂ · A = 0. The α = 0 limit is then taken. The gauge condition so obtained is not included in the previous discussion of various gauges. It is however class III because it involves time-derivative of the timecomponent of the potential. The limiting case α = 0 is a class II gauge. Because the 3 Use of this gauge can be found in [6]. Precursors but in a singular frame were Chan and Halpern [5].

86

3 Quantization of the Free Electromagnetic Field in Class II Axial Gauges

nature of the gauge is changing in the limit, there is no warranty that the limit α → 0 will be smooth. This remark shows that the only way to study the class II axial gauge consists in finding a consistent way to define the physical subspace and the related way of obtaining a propagator. With our knowledge, this remains an open question.

3.12 Summary The axial gauge Lagrangian a 1 L = − Fμν F μν + Sn · A + aSS + ∂ μ S ∂μ S 4 2 can be quantized leading to commutation relations for any time [Aμ (x), Aν (y)] = − igμν D(x − y) − i +i

nμ ∂ν + nν ∂μ Fn (x − y) n0

n2 − a ∂μ ∂ν Gn (x − y) n20

where F and G are generalized functions defined in the text. The field equations are n · ∂ S = 0, n · ∂ B = (n2 − a)S, S = S, Aμ = ∂μ B − nμ S. Owing to the first-order nature of the field equations in the unphysical sector, the usual particle-antiparticle interpretation of field theory is lost. The unbounded Hamiltonian does not allow a definition of a vacuum and the whole usual machinery of field theory is in trouble.

References 1. 2. 3. 4. 5. 6. 7.

Bassetto, A., Dalbosco, M., Lazzizzera, I. Soldati, R.: Phys. Rev. D31,2012 (1985) 76, 77 Bassetto, A., Lazzizzera, I., Nardelli, G., Soldati, R.: Phys. Lett. 227B, 427 (1989) 61 Burnel, A.: Nucl. Phys. B198, 531 (1982) 77 Burnel, A. Caprasse, H.: Int. J. Mod. Phys. A7, 6509 (1992) 77 Chan, H.S., Halpern, M.B.: Phys. Rev. D33, 540 (1985) 85 Joglekar, S.D., Misra, A.: Int. J. Mod. Phys. A15, 1453 (2000) 85 Lazzizzera, I.: Phys. Lett. 210B,188 (1988) 76

Chapter 4

Gauge Fields in Interaction

4.1 Introduction Free field theory is essential in understanding consistency of the gauge choice. However, particles are not free but interact. In this chapter, we recall how interactions involving gauge fields are built up according to the principle of gauge invariance. This principle finds its roots in the fact that electromagnetic interactions can be recovered in the following way : 1. The free Lagrangian describing, for instance, an electron   LD = Ψ¯ iγ μ ∂μ − m Ψ

(4.1)

is invariant under the U(1) transformations

Ψ  = eieθ Ψ .

(4.2)

This invariance leads to the conservation of the electromagnetic current Jμ = eΨ¯ γμ Ψ ,

∂ μ Jμ = 0.

(4.3)

2. If the invariance is required for local transformations θ = θ (x), the derivative ∂μ must be replaced with ∂μ − ieAμ where Aμ is the electromagnetic potential submitted to its usual gauge transformation Aμ → Aμ + ∂μ θ .

(4.4)

3. The free, gauge invariant, part of the electromagnetic field 1 − F μν Fμν 4 is added to the Lagrangian which reads now   1 LD = Ψ¯ iγ μ (∂μ − ieAμ ) − m Ψ − F μν Fμν . 4

(4.5)

It is invariant under the local gauge transformations (4.2), (4.4). Burnel, A.: Gauge Fields in Interaction. Lect. Notes Phys. 761, 87–99 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 4 

88

4 Gauge Fields in Interaction

The dynamics of electromagnetic interactions is obtained, in this way, from the conservation laws. This formalism is applied to other symmetries and leads to YangMills theories.1 Applications of this formalism to the Goldstone model, lead to the Higgs mechanism.2 Application to the symmetry group of weak and electromagnetic interactions leads to the Salam [9]-Weinberg [11] model and, when the principle is applied to the color group SU(3)C the result is Quantum Chromodynamics [5] (QCD).

4.2 General Formalism of Gauge Invariance 4.2.1 General Transformations The general formalism of gauge invariance can be built up in a very simple way. Let the Lagrangian   (4.6) L = L Ψ , ∂μ Ψ be a function of the fields and their first derivatives only. It is assumed to be invariant under the transformations Ψ  = UΨ (4.7) belonging to a Lie group G . In other words, ∀U ∈ G ,     L  Ψ  , ∂μ Ψ  = L Ψ , ∂μ Ψ .

(4.8)

If the transformations U are allowed to become local ones U = U(x),

(4.9)

the derivative of the Ψ -field transforms as   ∂μ Ψ  = ∂μ (UΨ ) = ∂μ U Ψ +U ∂μ Ψ .

(4.10)

Invariance is broken by the last term. To restore the invariance, the derivative is replaced by a new one, the covariant derivative (4.11) Dμ = ∂μ − igAμ where Aμ is a matrix acting in the representation in which Ψ lies and transforming in such a way that   (4.12) (Dμ Ψ ) = U Dμ Ψ . 1

This formalism was for the first time used by O. Klein in 1939 in the framework of KaluzaKlein theories. The actual denomination follows the paper of Yang and Mills [12] who applied the formalism to the isospin SU(2) group. 2 Different authors made this step at almost the same time [7, 4, 6]

4.2 General Formalism of Gauge Invariance

Setting

89

Aμ = UAμ U −1 + Xμ

(4.13)

where the first term is the usual transformation of a matrix under a group, the Xμ will be fixed by the condition (4.12). By computing the action of the covariant derivative on the field Ψ , (Dμ Ψ ) = ∂μ Ψ  − igAμ Ψ  = (∂μ U)Ψ +U ∂μ Ψ − igUAμ U −1UΨ − igXμ UΨ   = UDμ Ψ + ∂μ U Ψ − igXμ UΨ . (4.14) Comparison with (4.12) leads to i Xμ = − (∂μ U)U −1 g

(4.15)

Therefore the transformation law of the matrix Aμ is Aμ = UAμ U −1 −

 i ∂μ U U −1 . g

(4.16)

In order to introduce a gauge invariant kinetic term for the new vector fields, a generalization of Fμν is needed. The simplest way to obtain it consists in computing the commutator of two covariant derivatives      Dμ , Dν Ψ = −ig ∂μ Aν − ∂ν Aμ − ig Aμ , Aν Ψ . (4.17) The tensor

  Fμν = ∂μ Aν − ∂ν Aμ − ig Aμ , Aν

(4.18)

transforms under the local group as  Fμν = UFμν U −1 .

(4.19)

The minimal gauge invariant Lagrangian therefore can be written 1 L = L (Ψ , Dμ Ψ ) − tr Fμν F μν 2

(4.20)

when the generators of the associated Lie algebra are normalized by 1 tr Ti T j = δi j . 2

4.2.2 Infinitesimal Transformations Let us now go to the Lie algebra by setting U(x) = exp[igTi θ i (x)]

(4.21)

90

4 Gauge Fields in Interaction

where the Ti are the representations of the generators of the Lie algebra. They are assumed to satisfy the commutation relations [Ti , T j ] = ici jk Tk

(4.22)

where the ci jk are the structure constants. The matrix Aμ can be expanded in the basis given by the generators Aμ = Aiμ Ti ,

(4.23)

introducing thus the gauge potentials Aiμ . If only infinitesimal transformations are considered, the transformation law (4.16) becomes, after elementary calculations,   Ti Aiμ = Ti Aiμ + ∂μ θ i + gci jk Aμj θ k . (4.24) Therefore the gauge potentials Aiμ transform as Aiμ = Aiμ + ∂μ θ i + gci jk Aμj θ k .

(4.25)

Under global transformations ∂μ θ i = 0, the gauge potentials transform according to the adjoint representation. In the same way, the matrix Fμν can be expanded in the basis of the Lie algebra generators by setting i . (4.26) Fμν = Ti Fμν It is easy to see that the field strengths transform as the adjoint representation j i i Fμν = Fμν + gci jk Fμν θ k.

(4.27)

The kinetic part of the gauge vector fields can be written 1 i μν 1 i μν 1 F tr Ti T j = − Fμν Fi . − tr Fμν F μν = − Fμν 2 2 4

(4.28)

4.2.3 Remarks 4.2.3.1 Universality In the case of nonabelian groups, the pure gauge field term in the Lagrangian contains self-interactions between gauge fields. The coupling constant is gci jk . It is the same coupling constant that occurs in the transformation law (4.25) of the vector field and in covariant derivatives. If the group is semi-simple, two different matter fields Ψ1 and Ψ2 will couple to the potentials with the same coupling constant g. The coupling constant is said to be universal.

4.3 Class III Gauges in Yang-Mills Theory

91

If the group is not semi-simple, a different coupling constant is associated with each of its factors. If the group is abelian, the coupling constant can vary with the different matter fields.

4.2.3.2 Geometric Formulation This way of building the gauge field Lagrangian finds a very elegant and simple formulation when differential geometry is used. The potential is then a one-form connection on a principal vector bundle and the field strength is the curvature form. Both forms are Lie-algebra-valued forms. If this formulation is very powerful for the study of classical gauge theories, it cannot be generalized to quantum theories. Indeed quantization requires the choice of a particular coordinate frame in which time is treated on a different footing than the other coordinates. The notion of pform in Minkowski space is then losing its interest and the heavier formulation of the theory built up here cannot be avoided.

4.3 Class III Gauges in Yang-Mills Theory 4.3.1 Building the Lagrangian 4.3.1.1 The Yang-Mills Part Let us consider the Yang-Mills Lagrangian

where

1 a μν LYM = − Fμν Fa 4

(4.29)

a Fμν = ∂μ Aaν − ∂ν Aaμ + g f abc Abμ Acν .

(4.30)

It is the common part to various gauge invariant field theories (QCD, SalamWeinberg, Higgs and massive gauge invariant models) and everything that can be said about it can be transferred without any difficulty to these theories. It is sufficient to add the corresponding part and to particularize the global group. In order to have no problem with the covariant or contravariant positions of the indices, the global group is assumed compact SU(N). The Lagrangian (4.29) is invariant under the gauge transformations

where

δ Aaμ = (Dμ θ )a

(4.31)

(Dμ θ )a = ∂μ θ a + g f abc Abμ θ c .

(4.32)

92

4 Gauge Fields in Interaction

This expression of the covariant derivative applies to any field used in this section because pure Yang-Mills theory lives in the adjoint representation. According to this gauge invariance and in the same way as in the free field case, a chain of first class constraints .

Πa0 = 0 =⇒ (Dk Π k )a = 0

(4.33)

results. Except that they are as many as the dimension of the adjoint representation and that the covariant derivative replaces the ordinary one, these constraints have the same expressions as in the free field case.

4.3.1.2 Gauge Fixing In order to quantize this theory, a gauge must be chosen. Because the free field theory is a copy with N 2 − 1 fields of the free massless vector boson field, a consistent gauge can only be class III. Taking the most general case into account, the gauge fixing term reads 1 a Lgf = −C μν ∂μ Sa Aaν + aSa Sa + a SaC μ ∂μ Sa + ∂ μ Sa ∂μ Sa . 2 2

(4.34)

Note that the derivatives in the gauge fixing term are ordinary and not covariant ones.

4.3.1.3 BRST Transformations In order to reduce the theory to the physical degrees of freedom, Becchi-RouetStora-Tyutin [1, 2, 3, 10] invariance must be imposed. For the gauge potentials, BRST transformations are particular gauge transformations for which θ a (x) = ca (x)λ where the fields ca (x) and the parameter of the transformation λ are odd Grassmann variables. They anticommute instead of commuting. The BRST transformations of the potentials read

δ Aaμ = (Dμ c)a λ .

(4.35)

The Lagrange multipliers Sa and Sa are, by assumption, invariant under BRST transformations δ Sa = δ Sa = 0. (4.36) In order to get the transformation of the ghost field c, one imposes nilpotency of the BRST transformations δ 2 Aaμ = 0 which, of course, implies

δ (Dμ c)a = 0.

(4.37)

4.3 Class III Gauges in Yang-Mills Theory

93

After a careful calculation, one obtains this transformation law as

δ ca =

g fabc cb cc λ . 2

(4.38)

It can be checked that, under these transformations,

δ LYM = 0,

δ Lgf = −∂μ SaC μν (Dν c)a λ .

(4.39)

4.3.1.4 The Ghost Term In order to restore invariance, a new term is required Lgh = −i∂μ c¯aC μν (Dν c)a .

(4.40)

It makes the whole Lagrangian invariant if the antighost field c¯a transforms as

δ c¯a = −iSa λ .

(4.41)

The factor i is introduced in order that the ghost fields be hermitean c†a = ca ,

c¯†a = c¯a .

(4.42)

4.3.1.5 The Complete Lagrangian Using the previously introduced conventions aˆμ = Cν μ aν ,

a˜μ = Cμν aν ,

(4.43)

the complete BRST-invariant Lagrangian under consideration is 1 a μν 1 L = − Fμν Fa − ∂ μ Sa A˜ aμ + aSa Sa + a SaC μ ∂μ Sa 4 2 a + ∂ μ Sa ∂μ Sa − i∂μ c¯a (D˜ μ c)a 2 1 1 a μν ˆ μ Fa − ∂ Sa Aaμ + aSa Sa + a SaC μ ∂μ Sa = − Fμν 4 2 a μ  a + ∂ Sa ∂μ S − i∂ˆ μ c¯a (Dμ c)a . 2

(4.44)

(4.45)

Both expressions reduce to the same one when A˜ aμ and ∂ˆ μ Sa are explicitly written.

94

4 Gauge Fields in Interaction

4.3.2 Field Equations When Grassmann variables are involved, it is important to take a convention for the derivation with respect to these variables. Here, one will use the convention of left-derivation     ∂ AB ∂A ∂B = B + (−1)PA A (4.46) ∂ξ ∂ξ ∂ξ where PA is the Grassmann parity of A PAμ = PS = PS = 0,

Pc = Pc¯ = 1

(4.47)

and Pξ = 1. It is then an easy task left as an exercise to derive the following field equations a  μ D Fμν − ∂ˆν Sa + ig fabc ∂ˆν c¯b cc = 0, ∂ · A˜ a + aSa + aC · ∂ S = 0, aC · ∂ Sa + a Sa = 0, ∂μ (D˜ μ c)a = 0, ¯ a = 0. (Dμ ∂ˆ μ c) Because they contain interaction terms, they cannot be solved explicitly.

4.3.3 Canonical Momenta and Constraints Again the derivation of canonical momenta is left as an exercise. The result is

∂L ∂L = Faμ 0 , ΠSa = = −A˜ a0 , a ∂ (∂0 Aμ ) ∂ (∂0 Sa )   ∂L = a SaC0 + ∂0 Sa , ΠSa =  ∂ (∂0 Sa )  a ∂L ∂L = i∂ˆ0 c¯a , π¯a == = −i D˜ 0 c . πa = a a ∂ (∂0 c ) ∂ (∂0 c¯ )

Πaμ =

(4.48) (4.49) (4.50)

There are obviously two sets of constraints

Πa0 = 0,

ΠSa + A˜ a0 = 0

(4.51)

which, for each value of the index a, are identical to those encountered in the free field case. As in this case, they are second class and therefore the variables Aa0 and Πa0 will not be considered as canonical variables but given by the constraints. The nonvanishing canonical commutation relations are therefore

4.3 Class III Gauges in Yang-Mills Theory

 = iδba δkl δ (3) (x − y), Aak (x), Πbl (y) x0 =y0  = [Sa (x), ΠSb (y)]x0 =y0 = iδba δ (3) (x − y), Sa (x), ΠSb (y) x0 =y0

{ca (x), πb (y)}x0 =y0 = {c¯a (x), π¯b (y)}x0 =y0 = iδba δ (3) (x − y).

95

(4.52) (4.53) (4.54)

Except for the ghosts which were not introduced, these commutation relations are identical to those of the free field case.

4.3.4 The Hamiltonian After straightforward calculations following the same lines as in the free field case and therefore left as an exercise, the Hamiltonian density reads H =

1 k k 1 a a 1 Πa Πa + Fkl Fkl − Πak (Dk A0 )a − ∂k Sa A˜ ak + (−a + aC02 )(Sa )2 2 4 2  1 a +  (ΠSa )2 − ΠSa C0 Sa + a SaCk ∂k Sa + ∂k Sa ∂k Sa 2a 2 i Ck0 k C0k k + π¯a πa − g fabc Ab0 cc πa − ∂ c¯a π¯a − (D c)a πa C00 C00 C00 C0k k +i (D c)aCl0 ∂ l c¯a . (4.55) C00

4.3.5 The BRST Charge Let us now derive the BRST charge and its properties. According to Noether’s theorem, invariance under BRST transformations implies the existence of a conserved current

∂L ∂L ∂L + δ ca + δ c¯a Jμ λ = δ Aνa ∂ (∂ μ Aνa ) ∂ (∂ μ ca ) ∂ (∂ μ c¯a )    a ig a (Dν c)a − fabc ∂ˆ μ c¯a cb cc + Sa D˜ μ c λ. = −Fμν 2

(4.56)

It is easily obtained from the definition of this current provided care is taken when dealing with odd-parity Grassmann variables. By integration over space, a time-independent charge is associated with a conserved current. Here, it is called the BRST charge which, in terms of canonical variables reads

96

4 Gauge Fields in Interaction



QB =

 g d 3 x Πak (Dk c)a − fabc π a cb cc + iSa π¯ a . 2

(4.57)

By using canonical commutation relations, it is easy to check the following commutation or anticommutation relations   ig QB , Aaμ (x) = −i(Dμ c)a (x), {QB , ca (x)} = − fabc cb cc , 2 {QB , c¯a (x)} = −Sa (x).

(4.58)

It is another way to express the transformation laws of the various fields under BRST. Of course, for μ = 0, the constraint Aa0 =

1 (−ΠSa +C0k Aak ) C00

is used. Various other commutation or anticommutation relations can also be deduced after some elementary calculations  QB , Πak = −ig fabc Πbk cc , {QB , π¯a } = 0,  {QB , πa } = −i(Dk Π k )a − ig fabc cb π c , {QB , (Dk c)a } = QB , fabc cb cc = 0. (4.59) With the use of the Jacobi identity, they allow to prove the nilpotency of the BRST operator Q2B = {QB , QB } = 0. (4.60) This proof is left as an exercise. They also allow to check that the BRST charge commutes with the Hamiltonian [QB , H] = 0.

(4.61)

Of course, this was expected from the conservation of QB . It is interesting to note that, with the help of the BRST operator, the Lagrangian can be written   a a 1 a μν Fa + QB , Sa c¯a + a c¯aC · ∂ Sa + ∂μ c¯a ∂ μ Sa − ∂ˆ μ c¯a Aaμ . (4.62) L = − Fμν 4 2 2 Here c¯a is a new ghost field such that {QB , c¯a } = Sa . Again the check is left to the reader. With such a notation, the BRST transformation can be extended to the gauge parameters a and a which were, up to now, considered as BRST invariant [8].

4.3 Class III Gauges in Yang-Mills Theory

97

4.3.6 Ghost Number For completeness, let us remark that there is an additional symmetry of the Lagrangian. It is indeed invariant under ca → e−θ ca ,

c¯a → eθ c¯a ,

(4.63)

the other fields being unchanged under these transformations. By the Noether theorem, it leads to the conserved charge 

Q=

d 3 x [−ca πa + c¯a π¯ a ] .

(4.64)

With this charge, one gets the commutation relations [Q, cb ] = −icb , [Q, Q] = 0,

[Q, c¯b ] = ic¯b ,

[Q, Abμ ] = [Q, Sb ] = [Q, Sb ] = 0,

[Q, QB ] = −iQB .

(4.65) (4.66)

One calls ghost number carried by an operator φ the value α such that [Q, φ ] = −iαφ .

(4.67)

With this convention, the fields Aaμ , Sa , Sa and their canonically conjugate momenta carry ghost number 0. The ghost ca and the momentum canonically conjugate to the antighost π¯ a carry ghost number 1 while the antighost c¯a and π a carry ghost number −1. The BRST operator carries ghost number 1.

4.3.7 Global Internal Symmetry By construction, the Lagrangian is invariant under the global transformations

φ a (x) → φ a (x) + g fabc φ b (x)θ c

(4.68)

where θ c are coordinate independent. Here φ a is any of the fields Aaμ , Sa , Sa , ca or c¯a . Using the expression (4.62) of the Lagrangian, the associated conserved current is    a a a b ν     b . (4.69) Jμ = g fabc −Fμν Ac + QB , ∂μ Sb c¯c + ∂μ c¯b Sc − Aμ c¯c 2 2 This internal symmetry plays no special role in the story of quantization.

98

4 Gauge Fields in Interaction

4.3.8 Physical States It is important to see how the theory with its numerous ghosts can be reduced to the physical theory. Because, in the case of a nonabelian gauge invariance, the Lagrange multiplier Sa is no longer a free field, the Gupta-Bleuler formalism cannot be built. Only the BRST formalism can apply. Physical states are therefore defined by the cohomology classes of the BRST operator i.e. they satisfy QB |Ψphys  = 0,  |Ψphys  ≡ |Ψphys  + QB |Ψ0 .

(4.70)

This formalism rests only on BRST invariance and therefore holds for any class III, covariant or noncovariant, gauge. Further details will not be given here.

4.3.9 Problems of Covariance In the noncovariant case, it must be proved that the physical sector is covariant. These problems of covariance are treated exactly in the same way as in the free field case but with the addition of the ghost term. They are even simpler. Therefore, the whole machinery will not be repeated and only the results are given. Because the Lagrangian does not depend explicitly on coordinates, the energy momentum tensor T νμ =

∂L ∂L ∂L ∂L ∂ ν Aτa + ∂ ν Sa + ∂ ν Sa + ∂ ν ca ∂ (∂μ Aaτ ) ∂ (∂μ Sa ) ∂ (∂μ Sa ) ∂ (∂μ ca ) ∂L − gμν L (4.71) + ∂ ν c¯a ∂ (∂μ c¯a )

is conserved and the generators of translations 

Pμ =

d 3 x T0μ (x)

(4.72)

commute. Invariance under Lorentz transformations does not however hold in the general case. As in the Maxwell theory, the kinetic momentum tensor M μνρ = T μν xρ − T μρ xν +

∂L ∂L Aν − Aρ ∂ (∂μ Aaρ ) a ∂ (∂μ Aaν ) a

= T μν xρ − T μρ xν − Faμρ Aνa + Faμν Aρa satisfies

∂ μ Mμνρ = Kνρ − Kρν .

(4.73) (4.74) (4.75)

References

99

With respect to the free field case, Kμν contains now a ghost contribution and reads K μν = A˜ aμ ∂ ν Sa − Aaμ ∂ˆ ν Sa − a SaC μ ∂ ν Sa − i∂ ν c¯a (D˜ μ )a +i∂ˆ ν c¯a (Dμ c)a . It can easily be checked that one can write   K μν = QB , ∂ ν c¯a A˜ aμ − ∂ˆ ν c¯a Aaμ − a c¯aC μ ∂ ν Sa .

(4.76)

(4.77)

The fact that it is given by an anticommutator with the BRST charge immediately implies    Ψphys |K μν |Ψphys =0 (4.78) and the covariance in the physical sector immediately results. This covariance, of course, holds in the entire space for relativistic gauges Cμν = gμν , Cμ = 0, in which case ∂ τ Mτ μν = 0. The algebra of the operators Pμ and Mμν is completely identical to the algebra found in the free field case of Chap. 2. Moreover, one can prove in a straightforward way that (4.79) [QB , Pμ ] = [QB , Mμν ] = 0. This means that the operators Pμ and Mμν map the physical subspace onto itself. This is an important point in the proof of covariance of the physical theory in noncovariant gauges.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Becchi, C., Rouet, A. Stora, R.: Phys. Lett. B52, 344 (1974) 92 Becchi, C., Rouet, A. Stora, R.: Commun. Math. Phys. 42, 127 (1975) 92 Becchi, C., Rouet, A., Stora, R.: Ann. Phys. 98, 287 (1976) 92 Englert, F., Brout, R.: Phys. Rev. Lett. 13, 321 (1964) 88 Fritzsch, H., Gell-Mann, M., Leutwyler, H.: Phys. Lett. 47B, 365 (1973) 88 Guralnik, G., Hagen, C., Kibble, T.: Phys. Rev. Lett. 13, 585 (1964) 88 Higgs, P.W.: Phys. Lett. 12, 132 (1964) 88 Piguet, O., Sibold, K.: Nucl. Phys. B253, 517 (1985) 96 Salam, A.: Elementary Particle Theory. In: Svartholm, N. (ed.) (Stockholm: Almquist) p. 367 (1968) 88 10. Tyutin, I.V.: “Gauge Invariance In Field Theory And Statistical Physics In Operator Formalism,”LEBEDEV-75-39 92 11. Weinberg, S.: Phys. Rev. Lett. 19, 1264 (1967) 88 12. Yang, C.N., Mills, R.: Phys. Rev. 96, 191 (1954) 88

Chapter 5

Perturbation Theory: Renormalization and All That

5.1 Introduction In this chapter, the S-matrix is built perturbatively from the causality condition in contrast with most standard approaches to perturbative theory. This approach, due to Bogoliubov and collaborators [1], makes clearer the question of renormalization and allows a comparison with another construction, without divergences, which will be considered in a forthcoming chapter. Moreover, when the vacuum expectation values of the fields vanish, no tadpole graphs are needed. The problem of ultraviolet divergences associated with closed loops in Feynman graphs is discussed. These divergences are dealt with using the dimensional regularization method, which is extended to noncovariant integrals occurring in general class III gauges (allowing preferred frames). As an example, the ghost loop contributing to the gluon self-energy is computed in a general gauge allowing preferred frames. The problem of renormalization is also briefly discussed.

5.2 The S-Matrix 5.2.1 Definition and Properties Let us consider a scattering process from an initial state to a final state. |in → |out. Particles in the |in and |out states are free physical particles and one assumes that the time separation between in and out states is big enough to be considered infinite: the |in and |out states are taken respectively at t = −∞ and at +∞. Let g(x) be a function with compact support in space-time and such that 0 < g(x) < 1. In other words, g(x) vanishes outside a compact region of space-time. The operator S(g) is defined as the operator acting on |in to give |out when the interaction is switched on with an intensity g

Burnel, A.: Perturbation Theory: Renormalization and All That. Lect. Notes Phys. 761, 101–136 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 5 

102

5 Perturbation Theory: Renormalization and All That

|out = S(g)|in whereas the physical S-matrix is its limit when the g-function becomes 1 in the full space-time S = lim S(g). g→1

It is natural to assume the following properties of the physical S-matrix.

5.2.1.1 Covariance The definition must be independent of the reference frame. If L is a Lorentz transformation represented by UL when acting on the space of states, the covariance condition reads, as usual, S(Lg) = UL S(g)UL† . 5.2.1.2 Unitarity The norm of the |in and |out states is equal so that S† (g)S(g) = 1|. 5.2.1.3 Causality An event occurring at a time t cannot be affected by events occurring at times greater than t. If the supports of g1 and g2 are disconnected in time in such a way that the time in g2 is always greater than r and the time in g1 smaller than r, what is written as supp (g1 ) < supp (g2 ), the following relations can be written |Ψr  = S(g1 )|in,

|out = S(g2 )|Ψr 

where |Ψr  denotes the wave-function at the time r. It easily results that S(g1 + g2 ) = S(g2 )S(g1 ).

(5.1)

This is the way causality will be taken into account.

5.2.1.4 Important Remarks The properties of the S-matrix hold for physical states. Class III gauges introduce unphysical states, and physics does not impose any condition on them. Therefore

5.3 Perturbative Expansion

103

the properties of covariance and unitarity are not required in the full space of states but only in the physical subspace. The causality property is however assumed still to hold. Let us briefly show how the covariance and unitarity conditions must be handled in presence of unphysical states. 1. Class III gauges of Maxwell or Yang-Mills theory are plagued with an indefinite metric. Unitarity does not hold in the whole space but only in the physical subspace. Because indefinite metric is required to make the Hamiltonian (the generator of time-translation) hermitean, pseudo-unitarity is required in the whole spaces. This means that the unitarity condition holds but with the adjoint defined with respect to the indefinite inner product. Let us recall that, in mathematics, the adjoint operator of a given operator A is, when < a|b > is the inner product of any two vectors |a > and |b >, the operator A† defined by < a|A† b >=< b|A|a >∗ where the star means complex conjugation. 2. In the case of noncovariant gauges, the covariance condition is of course not assumed in the whole space but only in the physical subspace. As seen in the previous chapter, BRST quantization guarantees that this property indeed holds when only physical states are involved.

5.3 Perturbative Expansion 5.3.1 Expansion and Consequences of the Unitarity Condition Let us write the following perturbative expansion of the S(g) operator ∞

1 n! n=1

S(g) = 1| + T = 1| + ∑



d 4 x1 . . . d 4 xn Tn (x1 , . . . , xn ) g(x1 ) . . . g(xn ).

(5.2)

Here Tn (x1 , . . . , xn ) is symmetric with respect to its arguments. In the same way, the perturbative development of S−1 (g) is written as ∞

1 n! n=1

S−1 (g) = 1| + ∑



d 4 x1 . . . d 4 xn T˜n (x1 , . . . , xn ) g(x1 ) . . . g(xn )

(5.3)

where T˜n (x1 , . . . , xn ) is also a symmetric operator. If the unitarity condition S−1 (g) = S† (g) is used, one obviously gets T˜n = Tn† . By multiplying S(g) by S−1 (g),

(5.4)

104

5 Perturbation Theory: Renormalization and All That

S(g)S−1 (g) =

∞

∞

1 1

∑ ∑ m! k!



d 4 y1 . . . d 4 ym d 4 x1 . . . d 4 xk Tk (x1 , . . . , xk )

m=0 k=0

T˜m (y1 , . . . , ym )g(x1 ) . . . g(xk ) g(y1 ) . . . g(ym ).

(5.5)

Setting m + k = n and yi = xk+i , S(g)S−1 (g) =

∞

n

1 1 m! (n − m)! n=0 m=0

∑∑



d 4 x1 . . . d 4 xn T˜m (x1 , . . . , xm )

Tn−m (xm+1 , . . . , xn )g(x1 ) . . . g(xn ).

(5.6)

This is the content of the unitarity condition. A further restriction is often made: perturbative unitarity or the assumption that the condition S(g)S−1 (g) = 1| holds at any order of perturbation. It implies, ∀n ≥ 1, n



1 1 d 4 x1 . . . d 4 xn m=0 k! (n − m)! T˜m (x1 , . . . , xk ) Tn−m (xk+1 , . . . , xn )g(x1 ) . . . g(xn ) = 0.

∑

(5.7)

Symmetrizing this relation by summing over the n! permutations of the variables xi and using the fact that the g-functions are arbitrary, the integration and the gfunctions can be dropped out as in distribution theory, thus yielding

∑ Tm (X)T˜n−m (Y ) = 0.

(5.8)

P0

The sum runs over all partitions of the set {x1 , . . . , xn } into two disjoint sets X and Y including empty sets. For n = 1, the relation is particularly interesting. It reads T1 (x) + T˜1 (x) = 0.

(5.9)

It is important to remark that unitarity order, by order, in perturbative expansion is only a sufficient but not a necessary condition to guarantee that (5.6) holds. For instance, in the electroweak theory, the requirement of perturbative unitarity implies the need of a scalar Higgs boson exchange. If this boson is not discovered by experiment, this does not mean that the whole Salam-Weinberg model will fail. The contribution of higher orders can restore unitarity.

5.3.2 Consequences of the Causality Condition Let us first apply the development to S(g1 + g2 ) where supp (g1 ) < supp (g2 ) ∞

S(g1 + g2 ) =

1 ∑ n! n=0



d 4 x1 . . . d 4 xn Tn (x1 , . . . , xn )

[g1 (x1 ) + g2 (x1 )] . . . [g1 (xn ) + g2 (xn )] .

(5.10)

5.3 Perturbative Expansion

105

Let us consider the term of order n in the g’s. It contains 2n terms which, using the symmetry of the T -function, can be ordered by permuting integration variables in the form g2 (x1 ) . . . g2 (xm ) g1 (xm+1 ) . . . g1 (xn ). The number of possible permutations is n!/m!(n − m)!. Therefore [g1 (x1 ) + g2 (x1 )] . . . [g1 (xn ) + g2 (xn )] can be replaced by n

n! g2 (x1 ) . . . g2 (xm ) g1 (xm+1 ) . . . g1 (xn ) m!(n − m)! m=0

∑

and ∞

S(g1 + g2 ) =

n

1 ∑ ∑ m!(n − m)! n=0 m=0



d 4 x1 . . . d 4 xn Tn (x1 , . . . , xn )

g2 (x1 ) . . . g2 (xm ) g1 (xm+1 ) . . . g1 (xn ).

(5.11)

On the other hand, the product of the expansions of S(g2 ) and S(g1 ) gives ∞

S(g2 ) S(g1 ) =

∞

1 1 ∑ ∑ m! k! m=0 k=0



d 4 y1 . . . d 4 ym d 4 x1 . . . d 4 xk

Tk (x1 , . . . , xk ) Tm (y1 , . . . , ym )g2 (x1 ) . . . g2 (xk ) g1 (y1 ) . . . g1 (ym ). (5.12) Setting m + k = n and yi = xk+i , ∞

S(g2 ) S(g1 ) =

n

1 1 m! (n − m)! n=0 m=0

∑∑



d 4 x1 . . . d 4 xn Tm (x1 , . . . , xm )

Tn−m (xm+1 , . . . , xn )g2 (x1 ) . . . g2 (xm ) g1 (xm+1 ) . . . g1 (xn ). (5.13) Using the causality condition with {x1 , . . . , xk } > {xk+1 , . . . xn }, Tn (x1 , . . . , xn ) = Tk (x1 , . . . , xk ) Tn−k (xk+1 , . . . , xn ).

(5.14)

If times are ordered in such a way that x10 > x20 > · · · > xn0 , iterating this equation gives (5.15) Tn (x1 , . . . , xn ) = T1 (x1 ) T1 (x2 ) . . . T1 (xn ). In the case times are not ordered, the right-hand-side is replaced by a time-ordered product Tn (x1 , . . . , xn ) = T (T1 (x1 ) T1 (x2 ) . . . T1 (xn )) . (5.16) The S-matrix expansion can then be written  n  ∞ 1 4 S(g) = 1| + ∑ T d x T1 (x)g(x) . n=1 n!

(5.17)

106

5 Perturbation Theory: Renormalization and All That

Such results rely only on the causality condition. Unitarity order by order in perturbation theory is not used. An alternative way, avoiding the problem of time ordering, will be discussed later on.

5.3.3 The First Term T1 In usual quantum mechanics, the evolution problem in the Schr¨odinger picture is given by the Schr¨odinger equation i

∂Ψ = HΨ . ∂t

If the interaction picture

Ψ (t) = e−iH0 t φ (t),

H = H0 + HI ,

is used, the Schr¨odinger equation becomes i where Its solution is

∂φ = HI (t)φ (t) ∂t

HI (t) = eiH0 t HI e−iH0 t .

φ (t) = e−i

t

−∞ HI (t

 ) dt 

φ (−∞).

In other words, the S-matrix is given by S = e−i

∞

−∞ HI (t) dt

.

In agreement with this result, the first-order term of the S-matrix expansion will be taken as T1 (x) = iLint (x) (5.18) where Lint (x) is the interaction part of the Lagrangian. The conditions of causality, covariance and unitarity imposed on the S-matrix imply that the Lagrangian be a local, covariant and self-adjoint operator, at least in the physical sector. It is a function of the fields and of their derivatives. Although the correspondence with the interaction Hamiltonian holds only for interactions which are free of derivatives, derivatives of fields can be included inside interactions by taking (5.18) as the first term of the expansion. In the interaction picture, all the fields are free fields which can be expanded in momentum space in terms of creation and annihilation operators. Because 0|S|0 = 1 implies 0|Lint |0 = 0, the product of fields occurring in the interaction Lagrangian is a normal product in which the annihilation operators are always at the

5.3 Perturbative Expansion

107

right of creation ones, assuring the vanishing of the vacuum expectation values. Let us write this interaction Lagrangian in the following generic form: Lint (x) = −i ∑



α1 ···α pk

d 4 x1 . . . d 4 x pk Vk

(x − x1 , . . . , x − x pk )

k

: φα1 (x1 ) · · · φα pk (x pk ) :

(5.19)

α1 ···α p

k where the Vk (x − x1 , . . . , x − x pk ) functions, called vertex functions, are products of a constant, the coupling constant, and of pk δ -functions or, if derivative couplings are present, derivatives of the δ -function. Various vertices can be considered and αi can be multi-indices.

5.3.3.1 Example: Vertex Functions in Yang-Mills Theory In order to be as pedagogical as possible, let us take an example of this. In YangMills theory including ghosts, the interaction Lagrangian is g g2 μ Lint (x) = − fabc (∂ μ Aνa − ∂ ν Aaμ ) Abμ Acν − fabc fade Ab Aνc Adμ Aeν 2 4 − ig fabc ∂ˆ μ c¯a Ab cc . μ

(5.20)

It contains three terms leading to three different vertex functions. The three-gluon term1 in which all fields depend on the variable x can be written − g fabc ∂ μ Aνa Abμ Acν = −g fa1 a2 a3



d 4 x1 d 4 x2 d 4 x3 ∂μx21 Aaμ11 (x1 )Aaμ22 (x2 )Aaμ33 (x3 )

gμ1 μ3 δ (4) (x − x1 )δ (4) (x − x2 )δ (4) (x − x3 ). An integration by parts leads to 

g f a1 a2 a3

d 4 x1 d 4 x2 d 4 x3 Aaμ11 (x1 )Aaμ22 (x2 )Aaμ33 (x3 )gμ1 μ3

∂μx21 δ (4) (x − x1 )δ (4) (x − x2 )δ (4) (x − x3 ). The three-gluon vertex function is therefore Vaμ11aμ22aμ3 3 (x − x1 , x − x2 , x − x3 ) = ig fa1 a2 a3 gμ1 μ3

∂xμ12 δ (4) (x − x1 )δ (4) (x − x2 )δ (4) (x − x3 ). (5.21) In the same way, the four-gluon vertex function is

1

This is, of course, a useful abuse of language. Gluons occur only in Quantum Chromodynamics while our considerations apply to any Yang-Mills theory.

108

5 Perturbation Theory: Renormalization and All That

g2 faa1 a2 faa3 a4 gμ1 μ3 gμ2 μ4 4

Vaμ11aμ22aμ3 a34μ4 (x − x1 , x − x2 , x − x3 , x − x4 ) = −i 4

∏ δ (4) (x − xi )

(5.22)

i=1

while the antighost-gluon-ghost vertex function is Vaμ12a2 a3 (x − x1 , x − x2 , x − x3 ) = −g fa1 a2 a3 ∂ˆxμ12 δ (4) (x − x1 )δ (4) (x − x2 )δ (4) (x − x3 ). (5.23) It should be kept in mind that the derivative operator is associated with the antighost field.

5.3.4 Nonunicity of the Tn Besides covariance (in the physical sector) and symmetry, the nth order S-matrix element satisfies the causality condition Tn (x1 , . . . , xn ) = Tk (x1 , . . . , xk )Tn−k (xk+1 , . . . , xn ).

(5.24)

This last condition determines completely Tn from T1 in the region {x1 , . . . , xk } > {xk+1 , . . . , xn }. Nothing is said about the value of Tn when the various points are identical. Therefore a term Λn (x1 , . . . , xn ) which is a combination of products of derivatives of the Dirac functions δ (xi − x j ) can be added to the solution imposed by the causality condition. If the additional sufficient but not necessary condition of perturbative unitarity is used, in other words, if one imposes Tn (x1 , . . . , xn ) + T˜n (x1 , . . . , xn ) = − ∑ Tk (X)T˜n−k (Y )

(5.25)

P

where the summation runs over all the partitions of the set {x1 , . . . , xn } into two disjoint nonempty sets, the Λn (x1 , . . . , xn ) function is antihermitean. Details on this arbitrariness will be given later on in a more mathematical way but it is important to note that this function is local because it contains only Dirac functions and their derivatives.

5.3.5 The Fixed Part of Tn The part of Tn which is fixed by the causality condition reads Tn (y1 , . . . , yn ) = in T (L (y1 ) . . . L (yn ))

(5.26)

5.3 Perturbative Expansion

109

where T is the time-ordered product. For the sake of simplicity, let us first consider the case where only one vertex function is present so that Lint (x) = −i



d 4 x1 . . . d 4 x p V α1 ···α p (x − x1 , . . . , x − x p ) : φα1 (x1 ) · · · φα p (x p ) : .

(5.27)

Then, using the fact that the vertex function is a c-number, Tn can be written as ! !  n p n 1 α1 j ···α p j j 4 j j (y j − x1 , . . . , y j − x p ) Tn (y1 , . . . , yn ) = ∏ ∏ d xi ∏ V n! j=1 i=1 j=1 ! p   n j (5.28) T ∏ : ∏ φαi j xi : . j=1

i=1

The last factor is an ordinary product of normal products but not itself a normal product. Let us now transform it into a normal product.

5.3.6 Wick’s Theorem 5.3.6.1 Relation Between Normal and Ordinary Products of Two Fields Let us write the product of two fields in terms of their decomposition into positive and negative frequency parts (+)

(+)

(+)

(−)

φα1 (x1 )φα2 (x2 ) = φα1 (x1 )φα2 (x2 ) + φα1 (x1 )φα2 (x2 ) (−)

(+)

(−)

(−)

+ φα1 (x1 )φα2 (x2 ) + φα1 (x1 )φα2 (x2 ).

(5.29)

If η12 is defined as the product of the Grassmann parities of the fields φαi , i = 1, 2, the normal product can be written as (+)

(+)

(+)

(−)

: φα1 (x1 )φα2 (x2 ) : = φα1 (x1 )φα2 (x2 ) + φα1 (x1 )φα2 (x2 ) (+)

(−)

(−)

(−)

+ η12 φα2 (x2 )φα1 (x1 ) + φα1 (x1 )φα2 (x2 ). (5.30) Defining

[A, B]η = AB − ηAB BA

which, according to the Grassmann parity, is nothing other than the commutator or the anticommutator and subtracting (5.30) from (5.29), one obtains  (−) (+) φα1 (x1 )φα2 (x2 ) =: φα1 (x1 )φα2 (x2 ) : + φα1 (x1 ), φα2 (x2 ) (5.31) η

where the last term of the right-hand-side is a c-number and not an operator.

110

5 Perturbation Theory: Renormalization and All That

5.3.6.2 Relation Between Normal and Time-Ordered Products of Two Fields Using

: φα1 (x1 )φα2 (x2 ) :

= η12 : φα2 (x2 )φα1 (x1 ) :,

inverting the role of the indices and multiplying (5.31) by η12 , one gets  (−) (+) η12 φα2 (x2 )φα1 (x1 ) =: φα1 (x1 )φα2 (x2 ) : + + η12 φα2 (x2 ), φα1 (x1 ) . η

(5.32)

Therefore,    (−) (+) T (φα1 (x1 )φα2 (x2 )) = : φα1 (x1 )φα2 (x2 ) : +θ x10 − x20 φα1 (x1 ), φα2 (x2 ) η   0  (−) (+) 0 + θ x2 − x1 η12 φα2 (x2 ), φα1 (x1 ) . (5.33) η

Taking into account the fact that commutators are c-numbers, "   # (−) (+) (−) (+) φα1 (x1 ), φα2 (x2 ) = 0| φα1 (x1 ), φα2 (x2 ) |0 . η

η

(5.34)

(−)

The negative frequency part annihilates the vacuum φαi |0 = 0 and one also has (+)

0|φαi = 0. It easily comes    (−) (+) (−) φα1 (x1 ), φα2 (x2 ) = 0|φα1 (x1 )φα2 (x2 )|0 = 0|φα1 (x1 )φα2 (x2 )|0 . (5.35) η

The relation between time-ordered and normal products reads then T (φα1 (x1 )φα2 (x2 )) =: φα1 (x1 )φα2 (x2 ) : + 0|T (φα1 (x1 )φα2 (x2 )) |0.

(5.36)

where the second term of the right-hand-side is in fact the Feynman propagator.

5.3.6.3 Wick’s Theorem Relations (5.31) and (5.36) are very similar and can be written in a unique way as P (φα1 (x1 )φα2 (x2 )) =: φα1 (x1 )φα2 (x2 ) : +C (φα1 (x1 )φα2 (x2 ))

(5.37)

where P is either the ordinary or the time-ordered product and the contraction or pairing C is, respectively, the commutator between negative and positive frequency parts or the vacuum expectation value of the time-ordered product. In both cases, the pairing is a c-number. Extension of this relation to products of any number of factors is known as Wick’s theorem. It reads

5.3 Perturbative Expansion

111

P (φα1 (x1 ) . . . φαn (xn )) = : φα1 (x1 ) . . . φαn (xn ) : +C (φα1 (x1 )φα2 (x2 )) : φα3 (x3 ) . . . φαn (xn ) : + terms with all possible pairings + two and more pairings. (5.38) Grassmann parity of the factors must be taken into account when pairing between two nonconsecutive operators is realized. The proof by induction can be found in many books and is left to the reader.

5.3.7 Feynman Rules Let us introduce Wick’s theorem inside Tn . It is used twice. First of all, normal products are expressed in terms of P-products. This introduces pairings between fields with variables in the same vertex function. Secondly, all the P-products are expressed in terms of normal products. Pairings between fields with variables in the same vertex function are again present but with the opposite sign and they will cancel out. The global effect of this double use of Wick’s theorem amounts to applying directly the theorem to Tn but without pairings on fields in the same vertex function. Moreover, the fact that the pairing between fields of different kinds vanishes whatever the pairing is can also be taken into account. Finally a sum of terms like ! !  n p n 1 α1 j ···α p j j l 4 j j (y j − x1 , . . . , y j − x p ) Tn (y1 , . . . , yn ) = ∏ ∏ d xi ∏ V n! j=1 i=1 j=1 l

: ∏ φαi (xi ) : j=1

k

∏C i=1



 φβ j (x j )φβ j (xj )

is obtained. The set of indices {αi , i = 1, . . . l, β j , β j , j = 1, . . . k} constitute a partition of the set {1 j , . . . , p j , j = 1, . . . n}. If only one kind of vertex is taken into account, 2k + l = np. There are as many terms as there are possible partitions. Let us associate a point with each fixed variable y j and lines with integration variables xij . Two kinds of lines are considered according to the way the variable xij occurs. If it occurs in a normal product, the corresponding line starts from the point y j and does not end. It is called an external line. If the variable xij1 is associated with a pairing, the corresponding line is called internal. It is identical with the line corresponding to the other variable xij2 involved in the pairing and joins the points y j1 and y j2 . Because a line is associated with a field variable, different kinds of lines can be associated with different kinds of fields. Building all the possible graphs with n points and the number of lines allowed by the vertex functions and summing them is equivalent to the building of Tnl . The following rules give the equivalence. 1. A vertex function is linked to each point. 2. A pairing function is associated with each internal line.

112

5 Perturbation Theory: Renormalization and All That

3. A field operator is associated with each external line. All these field operators are gathered inside a normal product. 4. Integration over each variable associated with a line is carried out. As an example, let us consider the following particular diagram. x3

y3 x

x1

y1

y y2

x2

where wavy lines correspond to Yang-Mills particles. Its contribution to T2 (x, y) reads 

3

∏ d 4 xi d 4 yi Vaμ11aμ22aμ3 3 (x − x1 , x − x2 , x − x3 )C

i=1 ν1 ν2 ν3 Vb1 b2 b3 (y − y1 , y − y2 , y − y3 )



 Aaμ11 (x1 )Abν11 (y1 )

a

a

: Aaμ22 (x2 )Aμ33 (x3 )Aaν22 (y2 )Aν33 (y3 ) : .

Note that this contribution is symmetric in x and y. Similar graphs with different positions of the variables xi and yi occur, according to the way the pairing is realized. Indeed, the pairings (x1 , y2 ), (x1 , y3 ), (x2 , y1 ), . . . are also present in the expansion. They can be taken into account by making a cyclic permutation of the variables x1 , x2 , x3 and y1 , y2 , y3 . The whole set of these contributions can be represented by a unique graph with symmetrized vertices. Symmetrization of the vertices consists of a complete permutation of lines of the same nature. In the Feynman rules, a symmetry factor, here 1/2 for each vertex, is then appearing because, in the pairings, only cyclic permutations are considered. For each set of lines either external or internal that can be permuted, the inverse of the number of the possible permutations appears as a symmetry factor. In Yang-Mills theory, the symmetrized vertex functions are    μ μ μ VSa11 a22 a33 (x − x1 , x − x2 , x − x3 ) = ig fa1 a2 a3 gμ1 μ3 ∂xμ12 − ∂xμ32     + gμ2 μ3 ∂xμ31 − ∂xμ21 + gμ1 μ2 ∂xμ23 − ∂xμ13 3

∏ δ (4) (x − xi ),

(5.39)

i=1

μ μ μ μ

VSa11 a22 a33a44 (x − x1 , x − x2 , x − x3 , x − x4 ) = −ig2  faa1 a2 faa3 a4 (gμ1 μ3 gμ2 μ4 − gμ2 μ3 gμ1 μ4 ) + faa1 a3 faa2 a4 (gμ1 μ2 gμ3 μ4 − gμ2 μ3 gμ1 μ4 ) + faa1 a4 faa2 a3 (gμ1 μ2 gμ3 μ4 − gμ1 μ3 gμ2 μ4 )



4

∏ δ (4) (x − xi ). i=1

(5.40)

5.3 Perturbative Expansion

113

Because all the fields are different, the antighost-Yang-Mills particle-ghost vertex needs no symmetrization. With symmetric vertex functions, the Feynman rules involve a symmetry factor, equal to the inverse of the number of possible permutations, which is associated with each set of identical lines. Let us recall that these rules allow the construction of the symmetric Tn functions from all the possible graphs with n points and their corresponding lines.

5.3.8 Feynman Rules in Momentum Space 5.3.8.1 Fourier Transform of Vertex Functions Let us consider a vertex function V α1 ···α p (x − x1 , . . . , x − x p ). Setting ξi = x − xi , i = 1, . . . p, its Fourier transform is defined by V α1 ···α p (ξ1 , . . . , ξ p ) = ˜ α1 ···α p

V

1 (2π )4p 

(k1 , . . . , k p ) =



d 4 k1 · · · d 4 k p V˜ α1 ···α p (k1 , . . . , k p ) ei ∑i ki ·ξi , (5.41)

d ξ1 · · · d ξ p V 4

4

α1 ···α p

−i ∑i ki ·ξi

(ξ1 , . . . , ξ p ) e

.

(5.42)

For instance, the symmetrized vertex functions of the Yang-Mills theory in momentum space are

− g f a1 a2 a3

μ μ μ (5.43) V˜Sa11 a22 a33 (p1 , p2 , p3 ) =  μμ μ2 μ1 μ3  μ2 μ3 μ1 μ2 1 3 g (p1 − p3 ) + g (p3 − p2 ) + g (p2 − p1 ) ,

μ μ μ μ V˜Sa11 a22 a33a44 (p1 , p2 , p3 , p4 ) = −ig2



faa1 a2 faa3 a4 (gμ1 μ3 gμ2 μ4 − gμ2 μ3 gμ1 μ4 )

+ faa1 a3 faa2 a4 (gμ1 μ2 gμ3 μ4 − gμ2 μ3 gμ1 μ4 )

 + faa1 a4 faa2 a3 (gμ1 μ2 gμ3 μ4 − gμ1 μ3 gμ2 μ4 ) , (5.44) μ V˜aμ12a2 a3 (p1 , p2 , p3 ) = −ig fa1 a2 a3 pˆ1 2 .

(5.45)

The three- and four-Yang-Mills particle vertices are symmetrized over all lines. Momenta are associated with particles and one should remember that antiparticles have momenta opposite to those of the corresponding particles. In the ghost-antighostYang-Mills vertex, the involved momentum is the one of the antighost. It is not opposite to the one of the ghost because the antighost is a (fictitious) particle and not the antiparticle of the ghost.

114

5 Perturbation Theory: Renormalization and All That

Because the vertices are symmetric, the momenta are all flying in or flying out the vertex. The convention which is taken here is that they are all flying in. Indeed, the corresponding S- matrix element at first order reads S1 = −i



d 4 xd 4 x1 d 4 x2 d 4 x3 Vaμ11aμ22aμ3 3 (x − x1 , x − x2 , x − x3 ) a

: Aaμ11 (x1 )Aaμ22 (x2 )Aμ33 (x3 ) : .

(5.46)

Each of the field Aaμii (xi ) can be expanded in momentum space. By considering only the physical part and dropping the internal symmetry indices, this expansion reads Aμi (xi ) =

1 (2π )3/2



d 3 qi 2q0i

 ( j) † −iqi ·xi iqi ·xi .  (q ) a (q )e + a (q )e ∑ μi i j i j i 2

(5.47)

j=1

If the Yang-Mills particles are flying in the vertex, the corresponding S-matrix element is 0|S1 a†j1 (p1 )a†j2 (p2 )a†j3 (p3 )|0. Commutation with a j (qi ) gives a factor δ (3) (qi − pi ). All the permutation between the qi or pi must be taken into account. This amounts to symmetrizing the vertex function. A factor eipi ·xi appears from this δ -function and the momentum space expansion of the fields. Another factor e−iki ·xi comes from the Fourier expansion of the vertex function. Integration over xi then gives a factor δ (4) (ki − pi ) which implies pi = ki . The ki ’s, which are the momenta involved in the vertex function, fly in the same direction as the pi ’s, which are, by assumption, the momenta associated with incoming particles. In the vertex function, the momenta are therefore associated with incoming particles.

5.3.8.2 Momentum Conservation at Each Vertex In the S-matrix, integration of Tnl (y1 , . . . , yn ) over the various yi is carried out. This amounts to integrating each vertex function V α1 ···α p (y − x1 , . . . , y − x p ) over y. No other factor shows a dependence on y. From the Fourier transform of the vertex function, 

d 4 yV α1 ···α p (y − x1 , . . . , y − x p ) =

1 (2π )4p



d 4 y d 4 k1 · · · d 4 k p

V˜ α1 ···α p (k1 , . . . , k p ) ei ∑i ki ·(y−xi ) . The integration over y can be performed and leads to a factor

(5.48)

5.3 Perturbative Expansion

115 p

!

∑ ki

(2π )4 δ (4)

i=1

which manifests the energy-momentum conservation at each vertex. This is the first rule in momentum space.

5.3.8.3 Integration Over the Auxiliary Space Variables An auxiliary space variable noted xi in the following is associated with each line of a vertex. Fourier transform of the vertex function associates a momentum to each line and a factor e−iki ·xi (2π )4 in the contribution to the S-matrix. Integration over the variables xi must also be performed but these variables also occur either in pairing functions or in fields. In order to carry out the integrations over the xi -variables, fields and pairings are expanded in momentum space.

φα (xi ) =

1 (2π )3/2



d 3 pi 2p0i

∑ α

( j)

 (pi ) a j (pi )e−ipi ·xi + a†j (pi )eipi ·xi .

(5.49)

j

1 C(φαi (xi )φα j (x j )) = (2π )4



d 4 ri Δαi α j (ri )eiri ·(xi −x j ) .

(5.50)

Integration over the various variables xi leads to a factor 1.

(2π )4 δ (4) (pi − ki ) if the momentum pi is associated with a creation operator,

2.

(2π )4 δ (4) (pi + ki ) if the momentum pi is associated with an annihilation operator,

3.

(2π )4 δ (4) (ri ± ki ) if the variable xi corresponds to an internal line.

There are as many factors as there are xi -variables so that the (2π )4 factors cancel out with those occurring from the Fourier transform of the vertex functions.

5.3.8.4 Integration Over the Momenta Associated With Vertices Integration over the various ki will now be carried out. These variables occur in vertex functions and in δ factors.

116

5 Perturbation Theory: Renormalization and All That

1. Internal lines. Let us first consider the δ -functions associated with ri , a momentum associated with the Fourier transform of a pairing. Two such factors occur. Therefore if ri is associated with an internal line joining two vertices, it is equal to the momentum flying into the first vertex and opposite to the momentum flying into the second vertex. Integration over the corresponding momenta flying in the vertices amounts to associating a unique momentum with each internal line. From (5.50), integration over it and a factor 1/(2π )4 are involved. This is the second rule. 2. External lines. Owing to the δ (4) (pi ± ki ) factor, integration over a momentum associated with an external line of a vertex amounts to replacing this momentum either by the one associated with a creation operator flying into the vertex (δ (4) (pi − ki )) or by the one associated with an annihilation operator flying out of the vertex (δ (4) (pi + ki )). Three-dimensional integration over this momentum is understood and a factor 1 α (p), 2p0 (2π )3/2 where α (p) is the polarization factor of the particle, also occurs. Integration over p is not yet realized.

5.3.8.5 Matrix Element A physical process involves an S-matrix element with given initial and final states. If a particle of momentum P is present in the initial state, it is associated with a creation operator a†j (P) while, if it occurs in the final state, it is associated with the corresponding annihilation operator and one must compute matrix elements of the form % $ 0| ∏ ai (Pi ) : i

∏ Aαk (pk ) : ∏ a†j (P j )|0 k

(5.51)

j

where A is either a creation or an annihilation operator. Of course, in the normal product, there are as many creation operators as there are particles in the final state and as many annihilation operators as there are particles in the initial state. Let us give an example. If there are two particles of a given type both in the initial and in the final states, (5.51) reads   0|a(P3 )a(P4 )a† (k1 )a† (k2 )a(k3 )a(k4 )a† (P1 )a† (P2 )|0 . It can be computed from the commutation relations [a(p), a† (p  )] = 2p0 δ (3) (p − p  ). The result is 2P10 2P20 2P30 2P40 δ (3) (k3 − P1 )δ (3) (k4 − P2 )δ (3) (k1 − P3 )δ (3) (k2 − P4 ) +(P4 ↔ P3 , P1 ↔ P2 ).

5.3 Perturbative Expansion

117

For particles of different nature, there are as many such factors as there are different kinds of particles. Integration over, e.g., k1 amounts to replacing this momentum by the corresponding momentum of one of the final particles with symmetrization over particles in both the final and initial states. Because the vertices are symmetric, a factor, here equal to 2, occurs. This factor cancels the similar factor required by the symmetrization of the vertex on external lines. Let us also note that the factors 2Pi0 cancel out with those coming from the momentum space expansion of the fields. The factors (2π )3/2 disappears if the one-particle state is defined as |q = (2π )3/2 a† (q)|0. The normalization of the state is then q|q  = 2q0 (2π )3 δ (3) (q − q  ). This normalization will be used in the following.

5.3.8.6 Summary of the Rules in Momentum Space For a given physical process at a given order n in the S-matrix expansion, all the possible topologically different graphs with n vertices are drawn. The corresponding matrix element is built up with the following rules: α ···α

p 1. A symmetric vertex factor VS 1 p (k1 , . . . , k p )(2π )4 δ (4) (∑i=1 ki ) is associated with each vertex. 2. A pairing Δα1 α2 (q) is associated with each internal line. Integration over q and a factor 1/(2π )4 are also present. In the usual perturbation theory, the pairing function is the Feynman propagator. 3. To each external line is associated a factor α (pα ). 4. When fermions are considered, a sign −1 is associated with each fermion loop. It results from the Wick theorem. 5. Finally, a symmetry factor must be taken into account. For each vertex, one must divide by the number of possible permutations of internal lines.

Of course, some integrations over internal lines are very easy. Taking into account the various (2π )4 factors and δ -functions, there remains an overall (2π )4 δ (∑i pi ) factor and an integration  1 d4l (2π )4 over each momentum which is not fixed by momentum conservation. Such momenta occur in closed loops.

118

5 Perturbation Theory: Renormalization and All That

5.4 Divergences, Power Counting and Renormalizability 5.4.1 Example Let us consider the Yang-Mills particle contribution to its self-energy at the second order. It is given by the Feynman diagram p+q → p

p

← q

where wiggles represent Yang-Mills particles. Dropping out the overall momentum conservation δ -factor, the Feynman rules give 1 I(p) = 2(2π )4



d4q V

μ1 μ2 μ3 a3 b3 a2 b2 Sa1 a2 a3 (p, q, −p − q)Δ F μ2 ν2 (p + q)Δ F μ3 ν3 (q) ν1 ν2 ν3 (−p, −q, p + q)μ1 (p)ν1 (p). VSb 1 b2 b3

If only the high-momentum behaviour of the integrant is taken into account, after dropping out the indices, V (p, q, −p − q) ≈ q, and I(p) ≈



ΔF ≈

1 q2

d4q . q2

The integral is obviously divergent. It is clear that, for getting this result, the propagator must be assumed to have the same asymptotic behaviour in all space-time directions. Such a situation is not always realized in noncovariant gauges. In the following, it is assumed that the asymptotic behaviour of the propagator is isotropic. From this example, it is clear that ultra-violet divergences are associated with some Feynman graphs in field theory.

5.4.2 Power Counting and Superficial Degree of Divergence In order to handle these divergences, let us first classify them and introduce the superficial degree of divergence ω of a graph. Let us consider a graph with

5.4 Divergences, Power Counting and Renormalizability

• • • • •

119

bI internal boson lines, bE external boson lines, fI internal fermion lines, fE external fermion lines, ni vertices of type (i) with ∂i derivatives.

The asymptotic behaviour is obtained from a counting of the momenta. Such a counting leads to the definition

ω = 4( fI + bI ) − 4(∑ ni − 1) − ( fI + 2bI ) + ∑ ni ∂i . i

(5.52)

i

In this equation, • the first term takes into account integrations over momenta associated with internal lines; • the second term takes into account the Dirac functions expressing energy-momentum conservation at the vertices; of course, the overall energy-momentum conservation is taken out; • the third term takes the propagators into account; there is a difference in the behaviour between a fermion and a boson propagator; in noncovariant gauges, the same behaviour in all directions is assumed; • the last term counts the derivatives in the vertices. Denoting by • bi the number of bosonic fields in the vertex of type (i), • fi the one of fermionic fields, it is obvious that 2bI + bE = ∑ ni bi , i

2 fI + fE = ∑ ni fi . i

On the other hand, dimensional analysis allows one to write [g(i) ] = 4 − bi −

3 f i − ∂i 2

where g(i) is the coupling constant occurring in the interaction term (i) and [g(i) ] its mass dimension. Elementary algebra allows then to write the superficial degree of divergence of the graph as 1 ω = 4 − bE − fE − ∑ ni [g(i) ]. (5.53) 2 i A graph is said to be superficially convergent if ω < 0 and superficially divergent if ω ≥ 0. This notion is important because it can be proven that a graph is convergent if itself and all its subgraphs are superficially convergent.2 2

This is the power-counting theorem. It was first mentionned by Dyson [2] and mathematically proven in various papers [4, 5, 6, 7, 8].

120

5 Perturbation Theory: Renormalization and All That

5.4.3 Renormalizability by Power Counting Let us first assume that [g(i) ] = 0 as is the case in Yang-Mills theory even when coupled to fermion fields. Then, from (5.53), the superficial degree of divergence is the same for all the graphs with the same external lines. The number of processes leading to divergences is finite and limited to graphs with a small number of external lines. Such a theory is said to be renormalizable by power counting. Let us now assume, for the sake of clarity, that the coupling constant is unique. If its dimension d is positive, the superficial degree of divergence diminishes with the number of vertices present in the process so that at sufficiently high order, all the contributions to a given process become convergent. The theory is said to be, superrenormalizable by power counting. In contrast, if d < 0, the superficial degree of divergence increases with the order and, at a sufficiently high order, all the processes are divergent. Such a theory is said to be nonrenormalizable by power counting. For the above reasoning, the boson propagators are assumed to behave like 1/k2 for large k. This is not the case for the vector boson propagator in the Proca formulation   kμ kν −i Dμν (k) = 2 gμν − 2 . k − m2 + i m It behaves asymptotically as a constant so that a term 2bVI must be added to Eq. (5.53). Then, for a given process, the superficial degree of divergence increases with the number of vector boson propagators and at a sufficiently high order, all the processes become divergent. Again, the theory is nonrenormalizable by power counting.

5.5 Dimensional Regularization of Covariant Divergent Integrals 5.5.1 A General One-Loop Integral The presence of loops in Feynman graphs implies in general the occurrence of divergent integrals. In order to handle these, a method of regularization must be used. At present, the most popular one is dimensional regularization.3 Let us first explain this method in the case of covariant integrals. Let us take the example of the vector contribution to the vector-particle selfenergy in Yang-Mills theory. By making the vertex functions and the vector-boson propagators explicit, one obtains the following integrals 

p μ1 . . . p μm d4 p . (2π )4 [p2 + i]a1 [((p + q)2 + i]a2

Such expressions can be generalized for a general one-loop integral into 3

Note however that it can run into trouble in the presence of a γ5 coupling.

5.5 Dimensional Regularization of Covariant Divergent Integrals ···an Iμa11 ... μm =



121

p μ1 . . . p μm d4 p (2π )4 ∏ni=1 [(p + qi )2 + i]ai

(5.54)

where the integers ai are positive.

5.5.2 Euclidean Space The first step in the calculation of such integrals consists in the transition to Euclidean space by a Wick rotation and in the change of variable p0 = ip0 . When only covariant gauges are considered, poles in the p0 plane are symmetric with respect to the origin and there is no trouble with the Wick rotation. In order to carry out the transition to Euclidean space, it is useful to consider all the four-vectors in this Euclidean space. Then (a · b)M = −(a · b)E . If a vector occurs at the numerator of an integral, e.g. aμ , it is considered as the projection of the vector a on a unit vector eμ aμ = eμ · a so that E aM μ = −aμ .

Then, with these rules, ···an m+∑i=1 ai Iμa11 ... μm = i(−1) n



p μ1 . . . p μm d4 p (2π )4 ∏ni=1 [(p + qi )2 ]ai

n = i(−1)m+∑i=1 ai Eμa11 ···a ...μm . n

(5.55)

5.5.3 Use of the Feynman Formula The Feynman formula 1 ∏ni=1 bi

= (n − 1)!

 1 0

!

n

∏ d ξi i=1

is first used to reduce the integral to the form !   n Eμa11 ···a ...μm

=

1

4

d p 0

n

∏ d ξi i=1

δ

n

1 − ∑ ξi i=1

δ (1 − ∑ni=1 ξi ) (∑ni=1 bi ξi )n

!

p μ1 . . . p μm

(p2 + 2p · q + M 2 )λ

(5.56)

(5.57)

122

5 Perturbation Theory: Renormalization and All That

where λ = ∑ni=1 ai is a positive integer, n

n

q = ∑ ξi qi ,

M 2 = ∑ ξi q2i .

i=1

i=1

Momenta can be replaced by derivatives over q by using 

d4 p

pμ

=−

(p2 + 2p · q + M 2 )λ

∂ 1 2(λ − 1) ∂ qμ



d4 p

1 (p2 + 2p · q + M 2 )λ −1

.

(5.58)

Therefore the only remaining momentum integral is 

d4 p

I=

1

(5.59)

(p2 + 2p · q + M 2 )λ

where λ is still an integer but can take positive or negative values. It is first assumed that λ is still positive. The result is extended to λ ≤ 0 by analytic continuation.

5.5.4 Elimination of the Denominators When A > 0, the definition of the Γ -function allows to write the formula 1 1 = AN Γ (N)

 ∞ 0

d α α N−1 e−α A .

(5.60)

for any integer positive value of N. By using this formula and analytically continuing it to an arbitrary dimension 2ω , one gets I=

1 Γ (λ )

 ∞ 0

d α α λ −1



d 2ω p e−α ( p

2 +2p·q+M 2

).

(5.61)

For integer values of the space dimension, the momentum integral is a gaussian integral that can be performed. Then I=

πω Γ (λ )

 ∞ 0

d α α λ −ω −1 e−α (M

2 −q2

).

(5.62)

5.5.5 Complex Dimension The analytic continuation to complex ω can now be realized on the basis of the last formula where the integration over α can be carried out using the definition (5.60) of the Γ -function. The result is

5.5 Dimensional Regularization of Covariant Divergent Integrals



d 2ω p

1 (p2 + 2p · q + M 2 )λ

=

πω Γ (λ )

123

ω −λ  Γ (λ − ω ) M 2 − q2 .

(5.63)

If λ = 2 for instance, the Γ -function has a pole at ω = 2. The divergence in the momentum integral is regularized by a pole in a parameter obtained by the analytic continuation of the dimension to complex values. Let us remark that the analytic continuation for non-positive integer values of λ gives, owing to the Γ (λ ) function in the denominator, a vanishing result.

5.5.6 Tensor Integrals Integrals involving momenta in the numerator can be obtained by derivation with respect to qμ according to Eq. (5.58). In the simplest cases, various other tricks can be used. Let us first compute 

d 2ω p

pμ 2 (p + 2p · q + M 2 )λ

.

This regularized integral is convergent so that the change of variable pμ = kμ − qμ , which is forbidden in divergent integrals, can be performed. It gives 

d 2ω p

pμ = (p2 + 2p · q + M 2 )λ



d 2ω k

kμ − qμ (k2 + M 2 )λ

where M 2 = M 2 − q2 . The first integral, which involves kμ , vanishes because the integrant is an odd function of k. The second integral is already computed and one finally gets 

d 2ω p

ω −λ  pμ qμ π ω = − Γ (λ − ω ) M 2 − q2 . Γ (λ ) (p2 + 2p · q + M 2 )λ

(5.64)

In the same way, the change of variable pμ = kμ − qμ gives 

d 2ω p

pμ pν = 2 (p + 2p · q + M 2 )λ



d 2ω k

kμ kν + qμ qν 2 (k + M 2 )λ



d 2ω k . (k2 + M 2 )λ

Again the integrals with an odd number of k in the numerator vanish. Tensor analysis allows to compute the first integral. Setting 

d 2ω k

kμ kν = Agμν (k2 + M 2 )λ

because there is no momentum dependence of the result and contracting with the metric tensor, one gets

124

5 Perturbation Theory: Renormalization and All That



Obviously, 2ω A =

d 2ω k



k2 (k2 + M 2 )λ

μ

= Agμ = 2ω A.

d 2ω k − M 2 2 (k + M 2 )λ −1



d 2ω k (k2 + M 2 )λ

and, after some elementary algebra, 

d 2ω p

 pμ pν πω  2 2 ω −λ M = − q Γ (λ ) (p2 + 2p · q + M 2 )λ   2 M − q2 gμν Γ (λ − ω − 1) + qμ qν Γ (λ − ω ) . (5.65) 2

5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame 5.6.1 The General One-Loop Integral Let us now generalize these standard results to noncovariant gauges. In the case of a general class III gauge, the general one-loop integral reads ···a2n Iμa11 ... μm =



p μ1 . . . p μm d4 p (2π )4 ∏ni=1 [(p + qi )2 + i]ai [(p + qi )C2 + i]ai+n

(5.66)

where the integers ai are either positive or zero and the notations are those of Chap. 2 and Appendix C i.e pC2 = Cμν pμ pν . Let us proceed to the calculations of such integrals in the same way as for covariant ones. First of all, it is important to go to the Euclidean space through a Wick rotation. The possibility of performing it restricts considerably the choice of coordinate frames and not necessarily in the same way as in the quantization procedure. In order to develop a method which is as general as possible, one restricts the development to gauges where a preferred frame can be found. This includes Leibbrandt gauges and their interpolations to relativistic gauges when n2 = n∗2 but neither those with n2 = n∗2 nor the Coulomb gauge. For these last gauges, where the calculation method can depend on the gauge, the reader can consult the original papers.4

4

For a large collection of references, see [3].

5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame

125

5.6.2 Euclidean Space When only covariant gauges are considered, poles in the p0 plane are symmetric with respect to the origin. When noncovariant gauges are considered, there are, in nonsingular frames, for each noncovariant denominator (p + qi )C2 + i, two poles symmetric with respect to the point Z.p where the notations are those defined in Chap. 2 and in Appendix C pC2 = Cμν pμ pν ,

Zl =

C0l +Cl0 . 2C00

Then, of course, a Wick rotation is guaranteed not to cross the pole only if Z.p = 0. In other words, in order to be able to perform the Wick rotation in nonsingular frames, it must be assumed that a preferred frame can be found in which calculations are performed. The transition from Minkowski to Euclidean space must be made at the level of three quantities p2 , pμ and pC2 for any four-vector p. The first two expressions are extended as in the covariant case. Under the assumption of a preferred frame, one gets 2 = C00 p20 +Ckl pk pl . pC,M E Going to Euclidean space through pM O = iP0 , one gets 2 = −C00 p20,E +Ckl pEk pEl . pC,M

By imposing 2 2 pC,M = −pC,E ,

it becomes E M C00 = C00 ,

E M Ckl = −Ckl .

In the same way, the condition pμ Cμν qν = −(p,CE q)E implies μν CE

=

C00

−iC0k

−iCk0

−(Ckl )

! .

This is the expression of the Cμν fixed tensor in Euclidean space. Provided the (Ckl ) matrix is itself hermitian, in a preferred frame, the CE matrix in Euclidean space is a hermitian matrix and (p,CE q)E = (CE p, q)E .

126

5 Perturbation Theory: Renormalization and All That

Then, under the assumption of a preferred frame, ···a2n m+∑i=1 ai Iμa11 ... μm = i(−1) 2n



p μ1 . . . p μm d4 p n 4 (2π ) ∏i=1 [(p + qi )2 ]ai [(p + qi )C2 ]ai+n

2n = i(−1)m+∑i=1 ai Eμa11 ···a ...μm . 2n

(5.67)

5.6.3 Elimination of the Denominators Instead of introducing Feynman variables before eliminating the denominators, let us proceed in the reverse order and first use the formula (5.60) 2n Eμa11 ···a ...μm =



d 4 p p μ1 . . . p μm (2π )4 ∏2n i=1 Γ (ai )

 ∞ 2n 

∏

0 i=1

n

αiai −1 d αi



∏ e−[αi (p+qi ) +αn+i (p+qi )C ] . 2

2

(5.68)

i=1

Permuting the integrations and arranging the exponential factors, !    ∞ 2n n αiai −1 d αi a1 ···a2n Eμ1 ...μm = ∏ Γ (ai ) exp − ∑ (qi , (αi 1| + αn+iC)qi ) 0 i=1 i=1    n d4 p pμ . . . pμm exp −(p, ∑ (αi 1| + αn+iC)p) (2π )4 1 i=1   n

exp −2(p, ∑ (αi 1| + αn+iC)qi )

(5.69)

i=1

where (a, b) denotes the scalar product in Euclidean space and C is the matrix (Cμν ) in the same space. Setting B−1 =

n

∑ (αi 1| + αn+iC),

(5.70)

i=1 n

Q=

∑ (αi 1| + αn+iC)qi ,

(5.71)

i=1

and generalizing to an arbitrary dimension d, the integral over momenta reads 

Iμ1 ...μm =

−1 dd p pμ . . . pμm e−(p,B p) e−2(p,Q) . (2π )d 1

5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame

Using pμ e−2(p,Q) = −

127

1 ∂ −2(p,Q) e , 2 ∂ Qμ

the computed integral becomes    −1 m ∂m d d p −(p,B−1 p) −2(p,Q) Iμ1 ...μm = e e . μ μ m 1 2 ∂Q ···∂Q (2π )d This integration is a gaussian integration in a d-dimensional space. It can be performed if the matrix B−1 can be diagonalized. Because C is hermitean, it can be assumed that the reference frame is such that C is in diagonal form. Then B−1 is also diagonal and   −1 m ∂m π d/2 & (5.72) Iμ1 ...μm = e(Q,BQ) . μ μ m 1 2 ∂Q ···∂Q det(B−1 ) Of course, det(B−1 ) = 0 is required. This is a capital assumption in order to extend the method of calculation to noncovariant gauges.

5.6.4 Calculation of the Derivatives Let us now compute the various derivatives with respect to Qμi . One gets successively

∂ (Q,BQ) e = 2(BQ)μ1 e(Q,BQ) , ∂ Q μ1

(5.73)

  ∂2 e(Q,BQ) = 2Bμ1 μ2 + 4(BQ)μ1 (BQ)μ2 e(Q,BQ) , ∂ Q μ1 ∂ Q μ2

(5.74)

∂3 ∂ Q μ1 ∂ Q μ2 ∂ Q μ3

  e(Q,BQ) = 4 (BQ)μ1 Bμ2 μ3 + (BQ)μ2 Bμ1 μ3 + (BQ)μ3 Bμ2 μ1

(5.75) + 8(BQ)μ1 (BQ)μ2 (BQ)μ3 e(Q,BQ) , 4   ∂ e(Q,BQ) = 4 Bμ1 μ2 Bμ3 μ4 + Bμ1 μ3 Bμ2 μ4 + Bμ1 μ4 Bμ2 μ3 ∂ Q μ1 ∂ Q μ2 ∂ Q μ3 ∂ Q μ4  +8 Bμ1 μ2 (BQ)μ3 (BQ)μ4 + Bμ1 μ3 (BQ)μ2 (BQ)μ4 + Bμ1 μ4 (BQ)μ2 (BQ)μ3 + Bμ2 μ3 (BQ)μ1 (BQ)μ4  + Bμ2 μ4 (BQ)μ1 (BQ)μ3 + Bμ3 μ4 (BQ)μ1 (BQ)μ2

+ 16(BQ)μ1 (BQ)μ2 (BQ)μ3 (BQ)μ4 e(Q,BQ) . (5.76)

128

5 Perturbation Theory: Renormalization and All That

These results can be gathered in the equation m/2 or (m−1)/2 ∂m (Q,BQ) (Q,BQ) e = e 2m−k ∑ ∂ Q μ1 · · · ∂ Q μm k=0 

Bμm μm−1 · · · Bμm−2k+2 μm−2k+1



m−2k

∏ (BQ)μi + sym.

(5.77)

i=0

where the summation runs over all integer values from 0 to m/2 if m is even and from 0 to (m − 1)/2 if m is odd. Setting ω = d/2, the integral can be rewritten as m 2n Eμa11 ···a ...μm = (−1)



πω ∏2n i=1 Γ (ai )

 ∞ 2n ai −1 d αi −A m/2 or (m−1)/2 −k ∏i=1 αi & 2 e 0

∑

det(B−1 )

Bμm μm−1 · · · Bμm−2k+2 μm−2k+1

k=0



m−2k

∏ (BQ)μi + sym.

(5.78)

i=0

where

n

A = ∑ (qi , (αi 1| + αn+iC)qi ) − (Q, BQ).

(5.79)

i=1

5.6.5 Introduction of the Feynman Variables Let us set

αi = λ ξi ,

2n

i = 1, . . . , 2n,

∑ ξi = 1.

(5.80)

i=1

The Jacobian of this change of variables is equal to λ 2n−1 . The variables ξi vary from 0 to 1 and the variable λ from 0 to ∞. Setting B−1 0 = Q0 =

n

n

i=1 n

i=1

∑ (ξi 1| + ξn+iC) = 1| − (1| −C) ∑ ξn+i ,

(5.81)

∑ (ξi qi + ξn+iCqi ),

(5.82)

∑ (qi , (ξi 1| + ξn+iC)qi ) − (Q0 , B0 Q0 ),

(5.83)

Q = λ Q0 ,

(5.84)

i=1 n

A0 =

i=1

it is easy to see that B = λ −1 B0 ,

BQ = B0 Q0 ,

A = λ A0 .

5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame

129

The one-loop integral becomes 2n Eμa11 ···a ...μm



πω

 1 2n ai −1 d ξi ∏i=1 ξi

2n

!

 δ 1 − ∑ ξi  i=1 det B−1 0   ∞ m/2 or (m−1)/2  2 −k ∑2n ai −k−1−ω −λ A0 dλ e λ i=1 ∑ μ2 0 k=0 

= (−1)m

(2π )2ω ∏2n i=1 Γ (ai )

B0μm μm−1 · · · B0μm−2k+2 μm−2k+1



0

m−2k

∏ (B0 Q0 )μi + sym.

(5.85)

i=0

Such an expression can be analytically continued to the complex ω -plane.

5.6.6 Integration Over λ Integration over λ is done by using (5.60). It gives

2n Eμa11 ···a ...μm

m

= (−1) m/2



πω (2π )2ω ∏2n i=1 Γ (ai )

or (m−1)/2

∑

2−k Γ

 1 2n ai −1 d ξi ∏i=1 ξi



0

2n

B0μm μm−1 · · · B0μm−2k+2 μm−2k+1

i=1

ω +k−∑2n i=1 ai

A0

i=1

k=0

!

1 − ∑ ξi

δ

det(B−1 0 ) !

∑ ai − k − ω

2n



m−2k

∏ (B0 Q0 )μi + sym.

.

(5.86)

i=0

In this result, in principle, all the ai are strictly positive integers. This equation can however be applied to zero values of some ai by simply dropping out the vanishing ai and setting the corresponding ξi equal to zero. Covariant results can then be recovered by taking either C = 1| or ξn+i = 0. The B0 matrix is then the identity matrix and ! n

A0 = ∑ ξi q2i − i=1

n

∑ ξi qi

2

.

(5.87)

i=1

5.6.7 Regularization of Ultraviolet Divergences Under the assumption that the matrix B−1 is never singular, in both covariant and noncovariant Feynman integrals, Eq. (5.85) shows that all the problems of divergences are summarized in poles of the Γ -functions. These poles occur for zero or

130

5 Perturbation Theory: Renormalization and All That

negative integer values of their arguments thus for integer values of ω such that ω ≥ ∑2n i=1 ai . The ultraviolet divergences in four-dimension space are replaced with poles of the Γ -function.

5.6.8 Consequences of the Nonsingularity of the B−1 Matrix Nonsingularity of the B−1 matrix amounts to requiring that det(γ 1| + β C) = 0 for any positive or zero values of γ and β ; γ = β = 0 is, of course, excluded. Here γ is occurring for any covariant denominator while β occurs for any noncovariant one. When there are only covariant denominators in the Feynman integral, β = 0, γ = 0. The matrix B−1 is never singular so that it can be claimed that covariant integrals have only poles in the dimensional parameter ω . Problems with singular B−1 can occur only in noncovariant integrals. In particular, they can occur for ghost loops where only noncovariant factors are present in the denominator and γ = 0. Then B−1 = C. The general method of computing Feynman integrals through dimensional regularization implies that the C matrix be nonsingular. Typical examples of singular C matrices are Leibbrandt and Coulomb gauges. For Leibbrandt gauges, Cμν = n∗μ nν . In the case n2 = n∗2 = 0, for instance, the nonsingular Minkowskian frame in which n = (1, 0, 0, 1), n∗ = (1, 0, 0, −1) exhibits the singularity of the C matrix in an obvious way. For the Coulomb gauge where, however, the above calculations cannot be applied because no preferred nonsingular M = n n −g frame can be found, Cμν μ ν μν and the choice of the frame n = (1, 0, 0, 0) clearly shows the singularity of the C-matrix. In both cases, the situation can be cured by modifying the C matrix through an interpolation between the gauge taken into account and a relativistic gauge. For instance, for the Leibbrandt gauges, M = n∗μ nν − α gμν Cμν

or another such interpolation can be used. With such a modified Cμν , B−1 is no longer singular. Singularity of the C-matrix is easily related to anisotropy in power counting. Indeed, in the frame in which C is diagonal, some of its components are vanishing. Therefore, pˆ μ and p˜ μ vanish in the corresponding direction. Power counting in integrals involving pˆ μ or p˜ μ thus depends on the chosen direction. Regularization of the C-matrix through interpolation with relativistic gauges restores isotropic power counting. For this reason, such an interpolation will be called power counting regularization. It should not be confused with ultra-violet regularization which is done through poles in the parameter ω . Chapter 8 is devoted to the discussion of power counting singularities.

5.7 Computation of the Ghost Loop With Two Legs

131

5.7 Computation of the Ghost Loop With Two Legs As an illustrative example of the general way of computing dimensionally regularized loop integrals in any class III gauge, let us consider the ghost loop with two Yang-Mills legs. p q

q

p+q

It leads to the integral 

Iμν =

d 4 p pˆ μ ( pˆ + q) ˆν = Cλ μ Cρν 2 4 (2π ) pC (p + q)C2



d 4 p pλ (p + q)ρ . (2π )4 pC2 (p + q)C2

Two integrals 

Eλ ρ = 

Eλ =

pλ pρ d4 p , (2π )4 pC2 (p + q)C2

(5.88)

d4 p pλ 2 4 (2π ) pC (p + q)C2

(5.89)

must be computed. Using dimensional regularization and assuming a nonsingular C, both integrals are particular cases of the general integral with n = 2,

a1 = a2 = 0,

a3 = a4 = 1,

ξ1 = ξ2 = 0,

q1 = 0,

q2 = q.

Then, using the fact that C is diagonal and real, one easily gets B−1 0 = C,

Q0 = ξ4Cq,

B0 Q0 = ξ4 q,

A0 = (q, ξ4Cq) − ξ42 (Cq, q) = ξ4 (1 − ξ4 )(q,Cq) = ξ4 (1 − ξ4 )qC2 . Simple transposition of the formula gives   ω −2  1 ξ (1 − ξ )qC2 π ω μ 2(ω −2) √ Eλ ρ = d ξ Γ (2 − ω ) ξ 2 qλ qρ μ2 (2π )2ω detC 0   ω −1 ξ (1 − ξ )qC2 μ2 −1 + Γ (1 − ω ) Cλ ρ . (5.90) 2 μ2 The integration over ξ is easily done in terms of the Euler B-function

132

5 Perturbation Theory: Renormalization and All That

Cλ μ Cρν Eλ ρ =

2(ω −2)  π ω qC √ Γ (2 − ω )qˆμ qˆν B(ω + 1, ω − 1) (2π )2ω detC  1 + qC2 Γ (1 − ω )Cμν B(ω , ω ) . 2

(5.91)

In the same way,

π ω μ 2(ω −2) √ Eλ = − (2π )2ω detC

 1 0



ξ (1 − ξ )qC2 d ξ Γ (2 − ω ) μ2

ω −2

ξ qλ

2(ω −2)

=−

π ω qC √ Γ (2 − ω )qλ B(ω , ω − 1). (2π )2ω detC

(5.92)

Adding both contribution and using known properties of Γ and B-functions, Iμν

  2(ω −2) π ω qC qC2 Cμν Γ 2 (ω ) √ = Γ (2 − ω ) qˆμ qˆν + . Γ (2ω ) 2(1 − ω ) (2π )2ω detC

(5.93)

This is a general result holding only under the assumption that C is nonsingular. When C is singular, the ghost loop is not defined by this way of computing it.

5.8 Renormalization and Counter-Terms For the sake of clarity, let us first work with the second order only. Once all the contributing Feynman graphs have been computed, the second-order contribution T2 (x1 , x2 ) is given by an expression where ultraviolet divergences are regularized by poles of the Γ -functions Let us write this expression as T2 (x1 , x2 ) = Γ (2 − ω )A2 (x1 , x2 ) + B2 (x1 , x2 ) where both A2 (x1 , x2 ) and B2 (x1 , x2 ) are finite for ω = 2 but Γ (2 − ω ) is infinite for ω = 2. It is however known that the second order of the S-matrix is not completely determined by the iterative construction given by the Feynman rules. Functions Λ2 (x1 , x2 ) proportional to a linear combination of derivatives of δ (x1 −x2 ) can be present. They are completely undetermined. This freedom can be used to eliminate the ultraviolet divergences by taking, for instance,

Λ2 (x1 , x2 ) = −Γ (2 − ω )A2 (x1 , x2 ). Then the second-order S-matrix is S2 (x1 , x2 ) = B2 (x1 , x2 ).

5.8 Renormalization and Counter-Terms

133

Because of the constraint on Λ2 (x1 , x2 ), this of course implies that A2 (x1 , x2 ) be a linear combination of derivatives of δ (x1 − x2 ). Such an expression of Λ2 (x1 , x2 ) is called a counter-term. Going back to the S-matrix expansion, 

S=

d 4 x Lint (x) +

1 2!



d 4 x1 d 4 x2 [T2 (x1 , x2 ) + Λ2 (x1 , x2 )].

Owing to the presence of the Dirac function in Λ2 (x1 , x2 ), integration over one of the variables, say x2 , can be performed. Setting

Λ2 (x1 ) =

1 2



d 4 x1 Λ2 (x1 , x2 ),

the S-matrix can be written 

S=

d 4 x [Lint (x) + Λ2 (x)] +

1 2!



d 4 x1 d 4 x2 T2 (x1 , x2 )

where Λ2 (x) is a local but divergent expression. The counter-term can be incorporated in the Lagrangian. The new Lagrangian LR = L + Λ2 (x) is called the renormalized Lagrangian. Of course, this reasoning is easily generalized to any order. A renormalizable theory has a number of counter-terms which are not increasing with the order of perturbation. In a nonrenormalizable theory, the number of counter-terms increases with the order of perturbation while in a superrenormalizable theory, counter-terms are polynomials in the coupling constant.

5.8.1 Various Renormalization Schemes The above way of eliminating ultraviolet divergences is not unique. Indeed finite terms can still be added to Λ2 (x1 , x2 ) provided they are still proportional to a linear combination of derivatives of the Dirac function δ (x1 − x2 ). This freedom gives rise to various ways of defining a finite S-matrix. These various ways are called renormalization schemes. In order to illustrate the question, let us take a particular example. The photon self-energy in a covariant gauge gives rise to the result  ω −2  2  Γ (2 − ω )π ω −q2 Π q = . (2π )2ω

(5.94)

134

5 Perturbation Theory: Renormalization and All That

Setting 2 − ω = ,

Π (q2 ) =

2 2 1 Γ ()e− ln(−q /4π μ ) . (4π )2

(5.95)

In order to respect dimensional analysis, a regularization mass μ is introduced. It is required by the fact that the action is dimensionless in a 2ω = 4 − 2 dimensional space. In order to respect this constraint, the coupling constant g must be replaced by gμ  . The most used renormalization schemes are the following: 1. The minimal substraction scheme MS; the result is developed into a Laurent series      2 2 1 1 − + O() (5.96) Π q2 = γ + ln(4 π ) − ln −q / μ (4π )2  where γ is the Euler constant; the pole term in 1/ is removed. 2. The modified minimal subtraction scheme MS; in addition to the pole, the constants −γ + ln(4π ) which are present in any integral are also removed. 3. The momentum subtraction scheme MOM in which   Π q2 q2 =−μ 2 = 0 (5.97) is imposed; μ is called the renormalization mass. 4. The on-shell renormalization in which   Π q2 q2 =0 = 0

(5.98)

is imposed. Counter-terms and Green functions depend on the renormalization scheme. Although very useful in practical calculations, this is outside the scope of these notes.

5.8.2 Multiplicative Renormalization In renormalizable theories, the structure of counter-terms is the same as the structure of some terms of the original Lagrangian. For instance, in covariant gauges of Quantum Electrodynamics, the original Lagrangian is   1 1 ¯ μ Aμ ψ . L = − F μν Fμν − ∂ μ SAμ + aS2 + ψ¯ iγ μ ∂μ − m ψ + eψγ 4 2 The following counter-terms are found   ¯ μ Aμ ψ . Λ (x) = C3 F μν Fμν +C2 ψ¯ iγ μ ∂μ − (m + δ m) ψ +C1 eψγ

(5.99)

(5.100)

5.9 Summary

135

Adding both contributions and setting Z3−1 = 1 − 4C3 ,

Z2−1 = 1 +C2 ,

Z1−1 = 1 +C1 ,

mr = m + δ m,

the renormalized Lagrangian becomes   1 1 ¯ μ Aμ ψ . LR = − Z3 F μν Fμν − ∂ μ SAμ + aS2 + Z2 ψ¯ iγ μ ∂μ − m ψ + eZ1 ψγ 4 2 (5.101) It can be written 1 μν (0) 1 (0) 2 + ψ¯ (0) (iγ μ ∂μ − m(0) )ψ(0) LR = − F(0) Fμν − ∂ μ S(0) Aμ + a(0) S(0) 4 2 (0) + e(0) ψ¯ (0) γ μ Aμ ψ(0) . (5.102) if the normalization of the fields is changed by setting (0)

1/2

Aμ = Z3 Aμ ,

1/2

ψ(0) = Z2 ψ ,

−1/2

S(0) = Z3

S.

(5.103)

In the same time, new parameters are defined by a(0) = Z3 a,

m(0) = m + δ m,

−1/2

e(0) = Z1 Z2−1 Z3

e.

(5.104)

This change of normalization is the origin of the name “renormalization”. Original parameters and fields are called bare parameters or fields while the new ones are called renormalized parameters or fields.

5.9 Summary The causality condition allows to write the S-matrix perturbative expansion as  n  ∞ 1 d 4 x T1 (x)g(x) S(g) = 1| + ∑ T n=1 n! where T1 is equal to i times the interacting Lagrangian. The nth-order term is however not univoquely determined because no condition is given when all the points xi are identical. Using Wick’s theorem, the perturbative expansion can be represented by Feynman graphs. Ultraviolet divergences are associated with some of them. The superficial degree of divergence of a graph is given by

ω = 4 − bE −

1 fE − ∑ ni [g(i) ]. 2 i

136

5 Perturbation Theory: Renormalization and All That

Dimensional regularization is used here to deal with the ultraviolet divergences. It holds in both covariant and noncovariant gauges. For the latter however, the calculation carried out here only holds when the C-matrix is nonsingular and when a preferred frame can be found. The ultraviolet divergences can be dropped with the help of the nonunicity of the Tn . Counterterms can be added to the Lagrangian and absorbed in a renormalization of the fields and parameters of the theory. It is important to note that these counterterms are always local.

References 1. Bogoliubov, N.N., Shirkov, D.V.: Introduction to the theory of quantized fields, Interscience, New-York (1959) 101 2. Dyson, F.J.: Phys. Rev. 75,736 (1949) 119 3. Physical and Nonstandard Gauges. In: Gaigg, P. Kummer, W., Schweda, M. (eds.) Lecture Notes in Physics 361, Springer, Berlin (1990) 124 4. Hahn, Y., Zimmermann, W.: Comm. Math. Phys. 10,330 (1968) 119 5. Nakanishi, N.: Prog. Theor. Phys. 17,401 (1957) 119 6. Nakanishi, N.: J. Math. Phys. 4, 1385 (1963) 119 7. Weinberg, S.: Phys. Rev. 118, 838 (1960) 119 8. Zimmermann, W.: Comm. Math. Phys. 11, 1 (1968) 119

Chapter 6

Slavnov-Taylor Identities for Yang-Mills Theory

6.1 Introduction Slavnov-Taylor identities [1, 2] generalize to nonabelian theories the WardTakahashi identities obtained in Quantum Electrodynamics. They are derived in many textbooks for covariant and even general linear gauges [3]. In contrast with quantization, the derivation of these identities is much easier with path-integrals. Within the canonical formalism, the job becomes soon very tedious.1 Here, only a few identities will be discussed in order to stress the difference between covariant and noncovariant gauges. The reader who is not interested in the technical details can directly read the conclusion in the last section of this chapter. There is no fundamental difference between covariant and noncovariant class III gauges, a fact that is almost obvious from the path-integral formalism [3].

6.2 The Reduction Formula Let us call connected a S-matrix element in which all the particles occurring in initial or final states are really involved in the interaction. In field theory, connected S-matrix elements are related to Green functions 0|T (φα1 (x1 ), . . . φαn (xn ))|0 by the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula [4] and !  n  n i d 4 xi out|inC = ∏ √Zi (2π )3/2 i=1 ! !

∏ αii (pi )e−ipi ·xi (j )

i∈in n

∏ K αi β i

!

∏

i∈out

¯αii (pi )eipi ·xi (j )

0|T (φβ1 (x1 ), . . . φβn (xn ))|0.

(6.1)

i=1

1

A complete derivation of the identities in covariant gauges can be found in [5].

Burnel, A.: Slavnov-Taylor Identities for Yang-Mills Theory. Lect. Notes Phys. 761, 137–156 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 6 

138

6 Slavnov-Taylor Identities for Yang-Mills Theory

In this equation, • the index ji describes the polarization of the particle i, • Zi is a renormalization constant, • K αβ is the second-order derivation operator involved in the field equation satisfied by the field φβ , • the fields inside the time-ordered product are interacting fields, • the free fields are expanded in momentum space according to 

d 3 pi 1 φα (xi ) = (2π )3/2 2p0i  ( j) ( j) † −ipi ·xi ipi ·xi (p )a (p )e +  ¯ (p )a (p )e  . i j i α α ∑ i j i

(6.2)

j

Setting (j )

out|inC =

(j )

αii (pi ) ¯αii (pi ) α1 ...αn 1 √ √ T (p1 , . . . , pn ) ∏ ∏ 3n/2 Zi i∈out Zi (2π ) i∈in (2π )4 δ (4) (∑ pi ),

(6.3)

i

using the definition of the propagator K αβ Dβ γ (x) = −iδγα δ (4) (x)

(6.4)

and assuming all the particles are entering the process, Dβ1 α1 (p1 ) · · · Dβn αn (pn )T α1 ...αn (p1 , . . . , pn )(2π )4 δ (4) (∑ pi ) = 

(−1)n

i

n

∏ d 4 xi e−ipi ·xi 0|T (φβ1 (x1 ), . . . φβn (xn ))|0. i=1

The quantity T α1 ...αn (p1 , . . . , pn ) is the central object of the theory. It is called a proper vertex-function and, in perturbative theory, is connected to Feynman rules.

6.3 One-Particle Irreducible Vertex Functions Let us consider a Feynman graph. It will be said to be one-particle irreducible if, by cutting anyone of its internal lines, it cannot be separated into two disjoint graphs. The corresponding Green function is also called irreducible. One-particle irreducible vertex functions are the central objects in the discussion of renormalization. As a simple example, let us consider the two-particle vertex function, the propagator. Graphically,

6.4 Yang-Mills Theory in a General Class III Gauge

=

+

+

139

+ ...

where hatched blobs denote the one-particle irreducible part. In equations, P = PF + PF Σ P

(6.5)

where P is the propagator, PF the free one and Σ the one-particle irreducible twopoint function, the self-energy. By multiplying on the right by P−1 , on the left by PF−1 , PF−1 = P−1 + Σ .

(6.6)

6.4 Yang-Mills Theory in a General Class III Gauge 6.4.1 The Lagrangian and Superficially Divergent Processes Let us consider only pure Yang-Mills theory in a class III general gauge. Inclusion of fermions or other matter fields does not lead to any new feature. The free Lagrangian is 1 1 L0 = − (∂ μ Aνa − ∂ ν Aaμ )(∂μ Aaν − ∂ν Aaμ ) − ∂ˆ μ Sa Aaμ + aSa2 + a SaC · ∂ Sa 4 2 a + ∂μ Sa ∂ μ Sa − i∂ˆ μ c¯a ∂μ ca (6.7) 2 while the interaction part contains three terms   1 μ Lint = −g f abc ∂ μ Aνa Abμ Acν − g2 fabc f ab c Abμ Acν Ab Aνc − ig fabc ∂ˆ μ c¯a Abμ cc . (6.8) 4

The coupling constant g is dimensionless. Therefore, as an application of the previous chapter, if isotropic power counting holds, the superficial degree of divergence of a Feynman graph is ω = 4 − bE where bE is the number of external lines. It is easy to see that the superficial degree of divergence is greater or equal to zero for seven processes: 1. The gluon self-energy process for which ω = 2,

140

6 Slavnov-Taylor Identities for Yang-Mills Theory

2. The ghost self-energy process for which ω = 2,

3. The three-gluon process for which ω = 1,

4. The gluon-ghost-ghost process for which ω = 1,

5. The four-gluon process for which ω = 0,

6. The gluon-gluon-ghost-ghost process for which ω = 0,

6.4 Yang-Mills Theory in a General Class III Gauge

141

7. The four-ghost process for which ω = 0.

Most of these processes can however be less divergent than expected by the superficial analysis. For instance, in the first four processes, dimensional analysis accounts for the presence of external momenta through factors. Then, of course, the degree of divergence is diminished by one unit for each external factor of mass dimension 1. The result after extraction of all external momenta contribution is called the effective degree of divergence. In the present case, the first five processes have ωeff = 0 while the last two ones have strictly negative effective degrees of divergence. In order to illustrate this situation, let us consider the gluon self-energy at second order. Two graphs contribute, the exchange of a gluon pair and that of a ghost pair. Both are second-rank tensors for which there is an integration over one momentum. Each of the two vertices introduces a momentum in the numerator and both propagators introduce a squared momentum in the denominators. Integrals are thus of the type  pμ pν Iμν (q) = d 4 p 2 p (p + q)2 leading to ω = 4 + 2 − 4 = 2. Since Iμν (q) is a second-rank tensor depending only on q, the tensor method implies Iμν (q) = A(q2 )qμ qν + B(q2 )gμν . Dimensional analysis imposes that A(q2 ) is a dimensionless quantity while the dimension of B(q2 ) is M 2 . In order to have a function with zero dimension, one sets B(q2 ) = q2 B (q2 ). Then both A(q2 ) and B (q2 ) have a superficial degree of divergence equal to 0. The same reasoning can be repeated for the first four processes. For the last three ones, the superficial degree of divergence is vanishing and no mass dimensioned factor can appear. The fact that the effective degree of divergence of the last two processes is less than the superficial one is a consequence of Slavnov-Taylor identities.

6.4.2 BRST Symmetry, Field Equations and Canonical Commutation Relations The whole machinery of deriving Slavnov-Taylor identities in Yang-Mills theory relies on the BRST symmetry. In the canonical formalism, field equations and

142

6 Slavnov-Taylor Identities for Yang-Mills Theory

equal-time commutation relations are relevant in many steps of the derivation. In order to be as clear as possible, let us briefly recall the results previously derived. In addition, we discuss how derivatives and time-ordered products can be commuted, this commutation being involved in most of the calculations.

6.4.2.1 BRST Transformations By construction, the total Lagrangian is invariant under the BRST transformations

δ Aaμ = λ (Dμ c)a ,

λ δ ca = − g f abc cb cc , 2

δ c¯a = iλ Sa ,

δ Sa = 0,

δ Sa = 0. (6.9)

6.4.2.2 Field Equations Field equations are easily obtained as the Euler-Lagrange equations of the variational principle. They read (Dν F μν )a + ∂ˆ μ Sa − ig f abc ∂ˆ μ c¯b cc = 0,

(6.10)

∂ˆ μ Aaμ

(6.11)



a

+ aS + a C · ∂ S = 0, a

aC · ∂ Sa + a Sa = 0, (Dμ ∂ˆ μ c) ¯ a = 0,

(6.13)

∂ˆ μ (Dμ c)a = 0.

(6.14)

(6.12)

6.4.2.3 Canonical Momenta From the usual rules, canonical momenta are given by

Πaμ = Faμ 0 , ΠSa = −C0ν Aνa , Πca = i∂ˆ0 c¯a , Πc¯a = −iC0ν (Dν c)a , ΠSa = aC0 Sa + a ∂0 Sa .

(6.15)

As usual for the vector fields, only the space components are considered as canonical variables. The time components are constrained and given by

Πa0 = 0,

Aa0 =

C0k Aak − ΠSa C00

in a nonsingular frame. The nonvanishing canonical commutation relations are therefore [Aak (x), Πbl (y)]x0 =y0 = iδab δkl δ (3) (x − y),

(6.16)

6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy

Sa (x), ΠSb (y) x =y  0 0 b ca (x), Πc (y) x =y 0 0  b c¯a (x), Πc¯ (y) x =y  0 0  b Sa (x), ΠS (y) 

x0 =y0

143

= iδab δ (3) (x − y),

(6.17)

= iδab δ (3) (x − y),

(6.18)

= iδab δ (3) (x − y),

(6.19)

= iδab δ (3) (x − y).

(6.20)

6.4.3 Commuting Derivatives and Time-Ordered Products In the way identities for Green functions will be derived, the main problem consists in permuting derivatives and time-ordered products. Let Ai (xi ) be fermionic or bosonic operators. By definition, the time-ordered product is T (A1 (x1 ) · · · An (xn )) =

∑

i1 ···in

sign(P) θ (xi01 − xi02 ) · · · θ (xi0n−1 − xi0n ) Ai1 (xi1 ) · · · Ain (xin )

(6.21)

where the sum runs over all the permutation of the indices and sign(P) is the signature of the permutation of the involved fermionic fields. It is then a matter of systematic calculations to get

∂xμ1 T (A1 (x1 ) · · · An (xn )) = T (∂ μ A1 (x1 ) · · · An (xn )) +g

μ0

n

∑

δ (x10 − xi0 )[A1 (x1 ), Ai (xi )]± T

i=2

n

∏

! A j (x j )

(6.22)

j=2, j=i

where [ , ]± is either the anticommutator if both fields are fermionic or the commutator in the opposite case. Such an equation shows that derivatives and time-ordering do not commute in general. Additional terms involving equal-time commutators are introduced.

6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy 6.5.1 Covariant Gauges 6.5.1.1 Derivation of the Identity for the Two-Gluon Green Function Let us first derive, in covariant gauges, the Slavnov-Taylor identity for the two-point gluon function or propagator 0|T (Aμ (x)a Aν (y)b )|0.

144

6 Slavnov-Taylor Identities for Yang-Mills Theory

The method consists in starting with a trivial relation 0|T (∂ · Aa (x)c¯b (y))|0 = 0

(6.23)

where |0 is the physical vacuum which is invariant under BRST transformations. From the BRST transform of each field one gets 0|T (∂ · (D c)a (x)c¯b (y))|0 + i 0|T (∂ · Aa (x)Sb (y))|0 = 0.

(6.24)

Using field equations and assuming, as in the whole section, a = 0, 0|T (∂ · Aa (x)∂ · Ab (y))|0 = 0.

(6.25)

The following step consists in extracting the derivatives from the time-ordered product. By applying (6.22),

∂xρ 0|T (Aaλ (x)Abτ (y))|0 = 0|T (∂ ρ Aaλ (x)Abτ (y))|0

+gρ 0 δ (x0 − y0 ) 0|[Aaλ (x), Abτ (y)]|0

∂xρ ∂yσ 0|T (Aaλ (x)Abτ (y))|0

= 0|T (∂ ρ Aaλ (x)Abτ (y))|0, =

0|T (∂ ρ Aaλ (x)∂ σ Abτ (y))|0 −gσ 0 δ (x0 − y0 ) 0|[∂ ρ Aaλ (x), Abτ (y)]|0.

(6.26) (6.27)

The above relevant commutator is an equal-time commutator which is given by canonical commutation relations and field equations. One then obtains easily

∂xλ ∂yτ 0|T (Aaλ (x)Abτ (y))|0 = 0|T (∂ · Aa (x)∂ · Ab (y))|0 − iaδ (4) (x − y). (6.28) Using the identity (6.25),

∂xλ ∂yτ 0|T (Aaλ (x)Abτ (y))|0 = −iaδ (4) (x − y).

(6.29)

This is the Slavnov-Taylor identity for the two-gluon Green function. It is obviously satisfied by the free propagator 

δ ab d4k (0)a (0)b 0|T (Aμ (x)Aν (y))|0 = −i 4 2 (2π ) k + i   kμ kν gμν + (a − 1) 2 e−ik·(x−y) . k + i

(6.30)

6.5.1.2 Momentum Space Let us set 0|T (Aaμ (x)Abν (y))|0 = −

i (2π )4



d 4 k e−ik·(x−y) Dab μν (k).

(6.31)

6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy

145

The identity then gives rise to ab k μ kν Dab μν (k) = aδ .

(6.32)

Because Dab μν (k) is, in Minkowski space a second-rank tensor depending only on k and, in internal symmetry space, also a second-rank tensor, tensor analysis allows to write ab 2 2 Dab μν (k) = δ [A(k )gμν + B(k )kμ kν ] with, from the identity, B(k2 ) = −

A(k2 ) a + 2 2. k2 (k )

This result is usually written as Dab μν (k) =

δ ab k2



kμ kν gμν − 2 k



kμ kν 1 +a 2 2 1 + Π (k ) k

 (6.33)

where the denominators are regularized according to the causal prescription k2 → k2 + i. This expression can be inverted as   kμ kν ab ab 2 2 (k) = δ Π (k )](k g − k k ) + D−1 [1 + . μν μ ν μν a

(6.34)

6.5.1.3 Self-Energy λρ

Let us now introduce the gluon self-energy Πcd (k) through (0)ab

λρ

(0)ac

db Dab μν (k) = Dμν (k) + Dμλ (k)Πcd (k)Dρν (k).

(6.35)

By contraction with k μ and use of (6.33), ρν

ac (k)Dcb (k). 0 = k μ Πμρ

(6.36)

After multiplication by D−1bd ντ , ad (k) = 0. k μ Πμτ

(6.37)

The gluon self-energy, which is the quantity computed by the Feynman rules in momentum space, is transverse. It is a matter of simple calculations to get (0)−1 ab

ab Πμν (k) = Dμν

ab 2 ab 2 (k) − D−1 μν (k) = (−k gμν + kμ kν )δ Π (k ).

(6.38)

146

6 Slavnov-Taylor Identities for Yang-Mills Theory

6.5.2 General Gauges 6.5.2.1 The Propagator Let us generalize the above derivation of Slavnov-Taylor identities to a general class III gauge. No assumption on isotropic power counting is done. Instead of the c¯a which is used in the trivial relation (6.23), a slightly modified field given by c¯a = [−a + a (C · ∂ )2 ]c¯a

(6.39)

is introduced. The modification is done in order to get, under BRST transformations,

δ c¯a = iλ ∂ˆ · A,

(6.40)

a fact that is easily checked from field equations. Following the same method as in covariant gauges, the starting point is the trivial relation (6.41) 0|T (∂ˆ · Aa (x)c¯b (y))|0 = 0 where |0 is again the physical vacuum invariant under BRST transformations. The BRST transform of this relation gives, after use of field equations, 0|T (∂ˆ · Aa (x)∂ˆ · Ab (y))|0 = 0.

(6.42)

Permuting the derivatives and the time-ordered product, using again field equations as well as commutation relations, one gets

∂ˆxλ ∂ˆyτ 0|T (Aaλ (x)Abτ (y))|0 = 0|T (∂ˆ · Aa (x)∂ˆ · Ab (y))|0 − i(a − aC02 )δ (4) (x − y). (6.43) This implies after some calculations and use of (6.42) ∂ˆxλ ∂ˆyτ 0|T (Aaλ (x)Abτ (y))|0 = −i[a − a (C · ∂ )2 ]δ (4) (x − y).

(6.44)

This is the Slavnov-Taylor identity for the gluon propagator in a general class III gauge.

6.5.2.2 Self-energy Defining the Fourier transform by (6.31), it is easy to get 2  2 ab k2 kˆ μ kˆ ν Dab μν (k) = [ak − a (C · k) ]δ .

(6.45)

The free propagator (0)ab Dμν (k) =

  kˆ μ kν + kμ kˆ ν ak2 − a (C · k)2 + kˆ · kˆ δ ab gμν − kμ kν + k2 + i kC2 + i (kC2 + i)2

(6.46)

6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy

147

satisfies this identity. That both the interacting and free time-ordered products satisfy the same identity can be obtained without any calculation by noting that, except for the presence of a ghost term which plays no role in commutation relations, the field equations in terms of canonical variables have almost the same structure. Expanding again the gluon propagator in terms of the free propagator and the vacuum polarization, using the property of the free propagator k2 kˆ μ Dμν (k) = (0)ab

ak2 − a (C · k)2 kν δ ab , kC2

(6.47)

it is easy to derive ak2 − a (C · k)2  bρ ab λ ac kˆ μ Dab δ + k Π (k)D (k) k ν c ν μν (k) = λρ k2 kC2

(6.48)

ac k μ Πμλ (k)Dλcbν (k)kˆ ν = 0.

(6.49)

and also Inserting (6.48) into (6.49), one gets ρν

ab ac cd (k) + k μ Πμν (k)kτ Πτρ (k)Dbd (k) = 0. k μ kν Πμν

(6.50)

The unique solution of this equation is the transversality of the vacuum polarization ab (k) = 0. k μ Πμν

(6.51)

In Feynman rules, at most, two types of vectors kμ and kˆ μ are involved. Writab (k) as a second-rank tensor depending on these two vectors and using the ing Πμν conservation law (6.51), it becomes       2 k2 k2 ˆ ˆ Π1 (k) −k gμν + kμ kν + Π2 (k) kμ − kμ 2 kν − kν 2 kC kC (6.52) ˆ where Πi , i = 1, 2 are dimensionless functions of the invariants k2 , kC2 and kˆ · k. A consequence of the transversality of the self-energy can easily be obtained from (6.48). It reads ab Πμν (k) = δ ab

ak2 − a (C · k)2 ˆ μ (0)ab kν δ ab . kˆ μ Dab μν (k) = k Dμν (k) = k2 kC2

(6.53)

Of course, in all these equations, denominators are regularized by the causal i. One uses again tensor methods to write Dab μν (k) in terms of the two independent fourˆ vectors kμ and kμ . This involves, after the use of (6.53), two independent functions ˆ The result can be written as of the invariant which can be built from k and k.

148

6 Slavnov-Taylor Identities for Yang-Mills Theory

Dab μν (k)



 kˆ μ kν + kμ kˆ ν 1 kˆ · kˆ gμν − kμ kν + 2 2 2 1 + Π (k) kC + i (kC + i) ! kˆ μ kˆ ν ak2 − a (C · k)2 + . (6.54) kμ kν + D(k) gμν − 2 2 (kC + i) kˆ · kˆ

δ ab = 2 k + i

The relations between Π1 , Π2 , Π and D can be obtained after some calculations. They are

Π1 =

D(1 + Π ) − Π , D(1 + Π ) − 1

Π2 = −

(1 + Π )2 D [(1 + Π )D + 1][(1 + Π )D + 1]

where D =

(6.55)

D(kC2 )2 . k2 kˆ · kˆ

6.5.3 Renormalization From power counting, it results that the vacuum polarization is divergent with an effective degree of divergence equal to zero. This means that, in dimensional regularization, there is a simple pole at ω = 2. In covariant gauges, counter-terms of the form C3 Aaμ (gμν − ∂ μ ∂ ν )Aaν which, up to a four-divergence, are equal to −C3 (∂ μ Aνa − ∂ ν Aaμ )(∂μ Aaν − ∂ν Aaμ ) are introduced. As in QED, no counter-terms affect the gauge fixing. In noncovariant gauges, the vacuum polarization can contain a second conserved tensor    k2 k2 Π2 (k) kμ − kˆ μ 2 kν − kˆ ν 2 . kC kC If this term is divergent, a counter-term    CAaμ ∂ μ − ∂ˆ μ C−1 ∂ ν − ∂ˆ ν C−1 Aaν must be added to the Lagrangian. However, the presence of C−1 makes such a term nonlocal and the renormalized Lagrangian would lose its meaning for describing a consistent canonical field theory. Therefore, the divergent part of Π2 vanishes and, as in covariant gauges, ab 2 Πμν div (k) = (kμ kν − k gμν )C3

(6.56)

6.6 Identity for the Three-Gluon Vertex Function

149

where C3 contains the simple pole at ω = 2. The counter-terms needed for the renormalization of the gluon self-energy have the same nature as in covariant gauges. Gauge-fixing terms are not renormalized. The fact that counter-terms are local can easily be checked without calculations at the one-loop level. Let us indeed take the formula (5.85). Nonlocal infinities can occur only when ω + k − ∑2n i=1 ai is an integer greater than 0. Here ω = 2, k is an integer whose maximal value is half the number of momenta in the numerators of the integrand and ∑2n i=1 ai is the number of denominators. In noncovariant gauges, the propagator contains three kinds of terms with 0, 2 or 4 momenta in the numerator. Combination of these terms in the gluon self-energy gives the values summarized in Fig. 6.1 for ∑2n i=1 ai and the effective number of momenta in the numerators m − 2. Note that what matters is the effective number and not the superficial one when considering divergences. 0 2 4 0 2 3 4 2 3 4 5 4 4 5 6

0 2 4 0 0 2 4 2 2 4 6 4 4 6 8

Fig. 6.1 Values of ∑2n i=1 ai on the left and m − 2 on the right for the different terms characterized by the number of involved momenta in the propagator for the gluon contribution to gluon self-energy. Ghost contribution is given by the 00 case. At the one-loop level, there are two propagators

In all cases, the maximal value of ω + k − ∑2n i=1 ai is obtained for k = (m − 2)/2 and vanishes. Let us also go back to (6.54) and remark that, if D(k) = 0, there is a pole in the propagator at kˆ · kˆ = 0 which is not required by field equations. The presence of such a pole is not natural and it should be avoided by requiring D = 0 and Π2 = 0. All these arguments show that covariant and noncovariant gauges do not behave differently at the level of the self-energy.

6.6 Identity for the Three-Gluon Vertex Function The simplest identity after the transversality of the gluon self-energy concerns the three-gluon Green function 0|T (Aμ (x)a Aν (y)b Aρ (z)c )|0. Here and in the following, the identity will be derived in the general case and its implications will be discussed for both covariant and noncovariant gauges.

150

6 Slavnov-Taylor Identities for Yang-Mills Theory

6.6.1 Derivation of the Identity Again a trivial identity 0|T (∂ˆ · Aa (x)∂ˆ · Ab (y)c¯c (z))|0 = 0

(6.57)

is the starting point. Its BRST transform is taken and, after the use of field equations, 0|T (∂ˆ · Aa (x)∂ˆ · Ab (y)∂ˆ · Ac (z))|0 = 0.

(6.58)

Again, the derivatives will be commuted with the time-ordered product but, here, this commutation does not generate any other term because, in addition to equaltime commutators, the vacuum expectation value of a single field is taken. Such an expectation value always vanishes. Therefore, (6.58) leads to ∂ˆxμ ∂ˆyν ∂ˆzρ 0|T (Aμ (x)a Aν (y)b Aρ (z)c )|0 = 0.

(6.59)

This is the Slavnov-Taylor identity satisfied by the three-gluon Green function. Defining the proper three-gluon vertex function in momentum space Taμ11aμ2 a2 μ3 3 (p1 , p2 , p3 ) by 

d 4 x d 4 y d 4 z e−ip1 ·x e−ip2 ·y e−ip3 ·z 0|T (Aμ (x)a Aν (y)b Aρ (z)c )|0 = μ μ μ

a a

a a

a a

−Ta 1a a2  3 (p1 , p2 , p3 )Dμ1 μ1 (p1 ) Dμ2 μ2 (p2 ) Dμ3 μ3 (p3 ) 1 1

1 2 3

2 2

(2π ) δ

4 (4)

3 3

(p1 + p2 + p3 ),

(6.60)

Eq. (6.59) gives μ

μ

μ

a a

a a

a a

μ μ μ

p23 pˆ1 1 pˆ2 2 pˆ3 3 Dμ1 μ1 (p1 )Dμ2 μ2 (p2 )Dμ3 μ3 (p3 )Ta 1a a2  3 (p1 , p2 , p3 ) = 0. 1 1

From (6.53),

2 2

μ

μ

μ

3 3

(6.61)

1 2 3

p1 1 p2 2 p3 3 Tμa11μa22μa33 (p1 , p2 , p3 ) = 0

(6.62)

in both covariant and noncovariant gauges. This is the identity imposed by BRST invariance on the proper vertex function.

6.6.2 Renormalization The vertex function is symmetric. It is an antisymmetric tensor in internal space where the tensor structure is f a1 a2 a3 . In Minkowski space, it is also an antisymmetric tensor depending on three four-vectors p1 , p2 , p3 and on fixed tensors like the metric

6.7 Ghost Propagator

151

tensor. In covariant gauges where the metric tensor is the only fixed tensor, tensor methods allow to build up the structure   f a1 a2 a3 gμ1 μ2 (p1 − p2 )μ3 + gμ1 μ3 (p3 − p1 )μ2 + gμ3 μ2 (p2 − p3 )μ1 . It is unique and satisfies automatically the identity. When noncovariant gauges are taken into account, more terms are allowed. The fixed tensor Cμν involved in the gauge fixing must be taken into account. Introducing the four-vectors pˆ1 , pˆ2 , pˆ3 , the symmetric structure   f a1 a2 a3 gμ1 μ2 ( pˆ1 − pˆ2 )μ3 + gμ1 μ3 ( pˆ3 − pˆ1 )μ2 + gμ3 μ2 ( pˆ2 − pˆ3 )μ1 can occur at the same time as terms involving the products of two or three momenta. Terms involving two or three momenta are finite by power counting. The additional term which is linear in the momenta does not satisfy the identity (6.62). Therefore, it cannot occur in the divergent part. In conclusion, the structure of the counter-term implied by the ultra-violet divergence of the three-gluon vertex is the same as the three-gluon term in the original Lagrangian for both covariant and noncovariant gauges. It is written as μ

C1 fabc ∂μ Aaν Ab Aνc .

6.7 Ghost Propagator The BRST transform of the trivial identity 0|T (Aaμ (x)c¯b (y))|0 = 0

(6.63)

gives 0|T (∂μ ca [−a + a (C · ∂ )2 ]c¯b (y))|0 + g fabc 0|T (Acμ (x)cd (x)c¯b (y))|0 + i 0|T (Aa (x)∂ˆ · Ab (y))|0 = 0. (6.64) μ

The commutation of derivatives and time-ordered product is again performed. In this case, such a calculation is far from obvious. Various contributions occur but there is a cancellation so that the result is simple   −a + a (C · ∂ )2 ∂μ 0|T (ca (x)c¯b (y))|0 + g fabc 0|T (Acμ (x)cd (x)c¯b (y))|0 + i∂ˆyν 0|T (Aaμ (x)Abν (y))|0 = 0. (6.65) Defining the ghost propagator by 

 d 4 xd 4 y 0|T (ca (x)c¯b (y))|0e−ik·x e−ik ·y = iD˜ ab (k)(2π )4 δ (4) (k + k )

(6.66)

152

6 Slavnov-Taylor Identities for Yang-Mills Theory

and setting −g





n b d 4 xd 4 y e−ik·x e−ik ·y famn 0|T (Am μ (x)c (x)c¯ (y))|0 =

(2π )4 δ (4) (k + k )Σ μac (k)D˜ cb (k),

(6.67)

[−ak2 + a (C · k)2 ][kμ δ ac + Σ μac (k)]D˜ cb (k) + ik2 kˆ ν Dab μν (k) = 0.

(6.68)

one easily gets

Using (6.53),  kμ D˜ ab (k) − δ ab

 i + Σ μac (k)D˜ cb (k) = 0. kC2 + i

(6.69)

In the free field theory or at the lowest order of the perturbation theory,

δ ab (0) D˜ ab (k) = i 2 . kC + i Eq. (6.69) can then be written  kμ D˜ ab (k) − D˜ (0)ab (k) + Σ μac (k)D˜ cb (k) = 0.

(6.70)

(6.71)

As for the gluon propagator, the ghost propagator is decomposed into a sum over one-particle reducible and irreducible parts

Then

(0) (0) D˜ ab (k) = D˜ ab (k) − D˜ ac (k)Π˜ cd (k)D˜ db (k).

(6.72)

(0) Σ μac (k)D˜ cb (k) = kμ D˜ ac (k)Π˜ cd (k)D˜ db (k)

(6.73)

and, after multiplying by D−1 bd (k),

Σ μac (k) = i

kμ ˜ ac Π (k)). 2 kC + i

(6.74)

It also results from this identity that kˆ μ Σ μac (k) = iΠ˜ ac (k)).

(6.75)

Note that Σ μac (k) is related to the gluon-ghost-ghost vertex. This identity therefore introduces a relation between the coefficients of the counter-terms for the ghost propagator and the ghost-ghost-gluon vertex.

6.8 Ghost-Ghost-Gluon Vertex

153

6.8 Ghost-Ghost-Gluon Vertex 6.8.1 Identity in Coordinate Space In order to avoid cumbersome calculations in commuting derivatives and timeordered product, the starting point is now the trivial identity 0|T (Aaμ (x)∂ˆ · Ab (y)c¯c (z))|0 = 0. Its BRST transform gives, after use of field equations, 0|T ((Dμ c)a (x)∂ˆ · Ab (y)c¯c (z))|0 + +i 0|T (Aaμ (x)∂ˆ · Ab (y)Sc (z))|0 = 0. (6.76) Here, derivatives obviously commute with the time-ordering so that the above identity reads n b c ∂μx ∂ˆyν 0|T (ca (x)Abν (y)c¯c (z))|0 + ∂ˆyν g f amn 0|T (Am μ (x)c (x)Aν (y)c¯ (z))|0 + i∂ˆ ν 0|T (Aa (x)Ab (y)Sc (z))|0 = 0. (6.77) y

μ

ν

The differential operator [−a + a (C · ∂ )2 ]z is now applied. It commutes with the time-ordering in the last term. Using field equations, [−a + a (C · ∂ )2 ]Sc (z) = ∂ˆ · Ac (z). The differential operator commutes with the time-ordering and the identity reads [−a + a (C · ∂ )2 ]z [∂μx ∂ˆyν 0|T (ca (x)Abν (y)c¯c (z))|0 + ∂ˆ ν g f amn 0|T (Am (x)cn (x)Ab (y)c¯c (z))|0] y

μ

ν

+ iz ∂ˆzλ ∂ˆyν 0|T (Aaμ (x)Abν (y)Acλ (z))|0 = 0.

(6.78)

Applying ∂ˆ μx to (6.78) and using (6.69) leads to [−a + a (C · ∂ )2 ]z

 Cy 0|T (Aaμ (x)cb (y)c¯c (z))|0

n c + g f bmn ∂ˆyν 0|T (Aaμ (x)Am ν (y)c (y)c¯ (z))|0 = 0.

(6.79)

6.8.2 Momentum Space Let us now introduce the proper ghost-ghost-gluon vertex function through 

d 4 xd 4 yd 4 z e−ip·x e−iq·y e−ik·z 0|T (Aaμ (x)cb (y)c¯c (z))|0 

   μ = −gGa c b (p, k, q)Daμ aμ (p)D˜ b b (q)D˜ c c (k)(2π )4 δ (4) (k + p + q).

(6.80)

154

6 Slavnov-Taylor Identities for Yang-Mills Theory

Let us also define 

n b c d 4 xd 4 yd 4 z e−ip·x e−iq·y e−ik·z famn 0|T (Am μ (x)c (x)Aν (y)c¯ (z))|0    = iGμν b c a (k, q, p)D˜ c c (q)Dν  ν (p)(2π )4 δ (4) (k + p + q).

(6.81)

The identity (6.79) leads to  c b 2 a  c b qˆμ Ga (p, k, q)D˜ b b (q). μν (p, k, q) = qC Gν

(6.82)

   

The function Gμν b c a (q, k, p) is again decomposed into its one-particle reducible and irreducible parts 

bca aa bca” ˜ Gbca μν (p, k, q) = Gμν (p, k, q) + Σ μ (q)Da a” (q)Gν (p, k, q).

(6.83)

Use of (6.69) leads to the result acb qˆμ Gacb μν (p, k, q) = iGν (p, k, q).

(6.84)

6.8.3 Remark The identity (6.78) contains more information than (6.79). It relates Gacb μν (p, k, q), acb ab Gμ (p, k, q), Σ μ (q), the tree-gluon proper vertex function and the propagators. Using the fact that Σ μab (q) is proportional to qμ in both covariant and noncovariant gauges, it is possible to get a relation between Gacb μν (p, k, q), the three-gluon proper vertex function and the propagators. Again, this relation has the same structure in covariant and noncovariant gauges.

6.9 Multiplicative Renormalization Derivation of additional Slavnov-Taylor identities will not be performed here. For covariant gauges, it can be found, for instance, in the book of Pascual and Tarrach [5]. The examples which have been handled clearly show that identities for linear covariant and noncovariant gauges are very similar. When going to noncovariant gauges, there is only a replacement of ∂μ by ∂ˆ μ or the corresponding momentum when the gauge condition and the antighost c¯ are involved. If path integrals instead of canonical methods are used, such a replacement can easily be obtained from the identity on the generating functional of proper irreducible vertex function. See for instance [3] for the case a = 0, a = 0.

6.9 Multiplicative Renormalization

155

This similarity between covariant and noncovariant gauges means that the renormalization can be carried out in a similar way in both cases. Counter-terms only occur for the gauge-invariant part of the gluon propagator, the ghost propagator, the three- and four-gluon vertices and the ghost-ghost-gluon vertex. These counterterms have the same structure as the corresponding terms in the original Lagrangian so that 1 1 L + Lct = − Z3 (∂ μ Aνa − ∂ ν Aaμ )(∂μ Aaν − ∂ν Aaμ ) − ∂ˆ μ Sa Aaμ + aSa2 4 2 a + a SaC · ∂ Sa + ∂μ Sa ∂ μ Sa − iZ˜ 3 ∂ˆ μ c¯a ∂μ ca − gZ1 f abc ∂ μ Aνa Abμ Acν 2   1 μ − Z4 g2 fabc f ab c Abμ Acν Ab Aνc − igZ˜ 1 fabc ∂ˆ μ c¯a Abμ cc . (6.85) 4 Introducing new fields and parameters through (0)a

Aμ

1/2

= Z3 Aaμ ,

(0) 1/2 ca = Z˜ 3 ca ,

(0)

Sa = ZS Sa , a(0) = Za a,

(0)

Sa

= ZS Sa ,

a(0) = Za a ,

(0)

1/2

c¯a = Z˜ 3 c¯a ,

g(0) = Zg g,

(6.86) (6.87)

the renormalized Lagrangian can be written in the same way as the original Lagrangian but in terms of the bare fields and parameters 1 (0)ν (0)μ (0)a (0)a L + Lct = − (∂ μ Aa − ∂ ν Aa )(∂μ Aν − ∂ν Aμ ) 4 1 (0) (0)a (0)2 (0) − ∂ˆ μ Sa Aμ + a(0) Sa + a(0) S(0)aC · ∂ Sa 2 a(0) (0) (0) ∂μ Sa ∂ μ S(0)a − i∂ˆ μ c¯(0)a ∂μ ca 2 (0)b (0)ν (0)b (0)c − ig(0) fabc ∂ˆ μ c¯(0)a Aμ c(0)c − g(0) f abc ∂ μ Aa Aμ Aν +

  (0)b (0)c (0) μ (0)ν 1 − g(0)2 fabc f ab c Aμ Aν Ab Ac 4

(6.88)

provided −1/2

ZS = ZS = Z3 Zg =

−3/2 Z1 Z3

,

Za = Za = Z3 , −1/2 −1 = Z˜ 1 Z˜ 3 Z3 , Zg2 = Z4 Z3−2 .

(6.89)

Such an identification implies a relation between the renormalization constants Z˜ 3 Z1 Z3 = = . Z4 Z1 Z˜ 1

(6.90)

It can be shown that such a relation is implied by the Slavnov-Taylor identities.

156

6 Slavnov-Taylor Identities for Yang-Mills Theory

6.10 Summary Only a few Slavnov-Taylor identities are derived here from the canonical formalism. They take the same form in noncovariant and in covariant gauges provided pˆ μ is used instead of pμ when pμ is the momentum of an antighost. In particular, transversality of the gluon self-energy ab k μ Πμν (k) = 0

and of the three-gluon vertex function μ

μ

μ

a a a

p1 1 p2 2 p3 3 Tμ11μ22μ33 (p1 , p2 , p3 ) = 0 hold in both covariant and noncovariant class III gauges. As a consequence of these identities, the renormalization program can be performed in the same way in both cases.

References 1. 2. 3. 4. 5.

Slavnov, A.: Theor. Math. Phys. 10, 99 (1972) 137 Taylor, J.C.: Nucl. Phys. B33, 436 (1971) 137 Itzykson, C., Zuber, J.B.: Quantum field theory, McGraw-Hill, New-York (1980) 137, 154 Lehmann, H., Symanzik, K., Zimmermann, W.: Nuovo Cim. 6, 309 (1957). 137 Pascual, P., Tarrach, R.: Renormalization for the practitioner, Lecture Notes in, Physics, 194 Springer, Berlin (1984) 137, 154

Chapter 7

Field Theory Without Infinities

7.1 Introduction In the usual formulation of perturbative field theory, the S-matrix is constructed from the causality condition which implies, iteratively, Tn (x1 , . . . , xn ) = T (T1 (x1 ) . . . Tn (xn )). The time-ordered product involves the multiplication of a product of fields by a stepfunction. Because products of fields are, in general, distributions and not functions, the time-ordered product is not always defined. As an example, let us assume that the product of two fields gives rise to the δ (4) (x) function. Time-ordered product involves the multiplication of the Dirac function by a step function θ (x0 ). This is not defined in usual distribution theory. Divergences in loop integrals can be traced back to this lack of consistent definition.1 An alternative less popular way of calculating perturbatively S-matrix elements has been initiated by Epstein and Glaser [1] and used intensively by Scharf and his collaborators [2]. It is free of divergences. Therefore no regularization is needed but, as in the standard method, S-matrix elements are not univoquely defined by the construction. Arbitrary local terms, which were fixed by the renormalisation, are still present. Here, this method is briefly summarized and used in Yang-Mills theory. Modifications due to the use of noncovariant gauges are taken into account. Moreover, in covariant gauges, the absence of mass is used to carry out, in an easier way, the distribution splitting for any value of the gauge parameter in the case of the gluon self-energy. This new result has been carried out in order to test the package CANTENS [3] for handling tensors in REDUCE. The remainder of the topic in Fermi covariant gauges is covered in [2] and not recopied here. In the next chapter, the method will also be used in order to discuss ghost loops in noncovariant gauges. It will make clear the fact that, in Leibbrandt gauges, ghost loops are not decoupling.

1 Justification of the method can however be given by defining correctly the controversial products. See [4]

Burnel, A.: Field Theory Without Infinities. Lect. Notes Phys. 761, 157–191 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 7 

158

7 Field Theory Without Infinities x0 x2 > 0, x0 > 0

x2 < 0

x2 < 0

x1

x2 > 0, x0 < 0

Fig. 7.1 The light-cone in a two-dimensional Minkowski space. Here x2 = x02 − x12 . The light-cone is given by the two straight lines x2 = 0. A causal distribution vanishes for x2 < 0. Its support is the region x2 ≥ 0. It can be split into the sum of an advanced part whose support is the region x2 ≥ 0, x0 > 0 and a retarded part whose support is x2 ≥ 0, x0 < 0

7.2 Iterative Construction of the S-Matrix Without Time-Ordering Let us define An (x1 , . . . , xn ) = ∑ T˜m (X) Tn−m (Y, xn ), P

Rn (x1 , . . . , xn )

= ∑ Tn−m (Y, xn ) T˜m (X) P

where Tn and T˜n are respectively the nth order terms of the S and S−1 matrices and P is a partition of the set {x1 , . . . , xn−1 } into two subsets X et Y such that X is not empty. The number of points in X is m. It is clear that Rn and An at a given order n can be built from the assumed known Tm , m < n. Let us also define in a similar way An (x1 , . . . , xn ) = ∑ T˜m (X) Tn−m (Y, xn ), P0

Rn (x1 , . . . , xn ) = ∑ Tn−m (Y, xn ) T˜m (X) P0

where P0 is a similar partition but allowing empty X. In an obvious way, An (x1 , . . . , xn ) = An (x1 , . . . , xn ) + Tn (x1 , . . . , xn ),

(7.1)

7.3 Splitting of Causal Distributions into Advanced and Retarded Parts

Rn (x1 , . . . , xn ) = Rn (x1 , . . . , xn ) + Tn (x1 , . . . , xn ).

159

(7.2)

From these relations, Tn = An − An = Rn − Rn

(7.3)

and, by subtracting them, one defines a new distribution Dn = Rn − An = Rn − An .

(7.4)

The iterative construction of Tn results from a theorem that will not be proved here. This theorem states that the support of the distribution Dn is causal and Rn , An are respectively its retarded and advanced parts. Definitions of a causal distribution and its advanced and retarded parts is given by Fig. 7.1. Knowing the n − 1 first orders, An and Rn and their difference Dn can be built. It remains to get its advanced and retarded parts in order to build iteratively the S-matrix from this theorem. The splitting of a distribution into advanced and retarded parts must be defined in a consistent way. This is the aim of the next section.

7.3 Splitting of Causal Distributions into Advanced and Retarded Parts 7.3.1 Distribution and Fourier Transform Let us first recall known facts and consider a distribution d(x) in an m dimensional space. Strictly speaking, it is a functional which associates to any test-function ϕ (x) a number d(x), ϕ (x). Though it is not, in general, a true integral, it is usual and useful to note  d(x), ϕ (x) =

d m x d(x)ϕ (x).

In the same way, its Fourier and inverse Fourier transforms are noted ˆ d(p) =

1 (2π )m/2

˘ d(p) =

1 (2π )m/2

 

d m x d(x) eip·x ,

(7.5)

d m x d(x) e−ip·x

(7.6)

where again integrals are true integrals only for integrable functions. The Parseval theorem can be checked with the help of the integral representation and gives   ˆ d(x), ϕ (x) = d(p), ϕ˘ (p) .

(7.7)

160

7 Field Theory Without Infinities

7.3.2 Order of Singularity of a Distribution 7.3.2.1 Definition ˆ Let d(x) be a distribution with Fourier transform d(p). Let us assume that the ultraˆ violet behaviour of d(p) is isotropic. Then, it is obvious that, for any positive scale factor a, one has ˆ ˆ = aω lim d(p). lim d(ap) p→∞

p→∞

The parameter ω so introduced is called the order of singularity of the distribution d. It is useful to associate to any distribution d(x) with order of singularity ω another distribution d0 (x) defined in momentum space by ˆ lim a−ω d(ap) = dˆ0 (p).

a→∞

(7.8)

It is obvious that d(x) and d0 (x) have the same order of singularity and that dˆ0 (p) is the high momentum behaviour of d(p). From the property of Fourier transform in m dimensional space, ˆ ˆ d(ap) = a−m d(x/a). Therefore

d0 (x) = lim a−(ω +m) d(x/a). a→∞

(7.9)

The order of singularity of a distribution with isotropic ultra-violet behaviour characterizes thus both the ultra-violet behaviour in momentum space and the behaviour near the origin in coordinate space.

7.3.2.2 Examples The order of singularity of a distribution can be easily computed. Let us consider some examples. 1. d(x) = 1. It is clear that d(x/a) = d(x) = d0 (x). Therefore Eq. (7.9) reads lim a−(ω +m) d(x/a) = d(x)

a→∞

and obviously implies ω = −m. One can also determine the order of singularity from a simple calculation in momentum space. The Fourier transform of the function 1 reads ˆ = (2π )m/2 δ (m) (p) d(p) and an elementary property of the Dirac-function gives

7.3 Splitting of Causal Distributions into Advanced and Retarded Parts

161

δ (m) (ap) = a−m δ (m) (p), confirming that the order of singularity is −m. 2. d(x) = δ (m) (x). Then δ (m) (x/a) = am δ (m) (x) and, from (7.9), ω= 0. 3. d(x) = (x0 )δ x2 in a four dimensional Minkowski space. Then again with the help of properties of the Dirac-function, d(x/a) = a2 d(x) and, again from (7.9) with m = 4,

ω + 4 = 2,  4. d(p) = θ (p0 )δ p2 − m2 . Here

ω = −2.



  1 ˆ d(ap) = 2 θ (p0 )δ p2 − m2 /a2 . a Therefore

  ˆ lim a2 d(ap) = θ (p0 )δ p2 = dˆ0 (p).

a→∞

Here, in contrast with the previous examples and due to the presence of a mass, the function d0 is different from the function d. Both have ω = −2 as results from the definition of the order of singularity. 5. If a derivative of a distribution whose order of singularity is ω is taken, the order of singularity of the derivative is increased by one unit. This is obvious because, in momentum space, the derivative ∂μ is replaced by ipμ . In the same way, multiplication by x decreases the order of singularity by one unit.

7.3.3 Splitting of Distribution with Negative Order of Singularity Distributions with negative order of singularity are regular at the origin. Therefore, the product of the distribution with the step-function is well-defined. The splitting is then realized with d(x) = r(x) − a(x) where the retarded and advanced parts, respectively r(x) and a(x) are given by r(x) = θ (x · v)d(x),

a(x) = −θ (−x · v)d(x).

Here v is a time-like vector in the m dimensional space. This is the usual timeordering. Such a splitting is unique. Indeed, if another splitting d(x) = r (x) − a (x) was allowed, the relations r (x) = r(x) for x · v > 0 and a (x) = a(x) for x · v < 0 would

162

7 Field Theory Without Infinities

hold. This means that r (x) can differ from r(x) only at x = 0. Therefore, one can write r (x) = r(x) + ∑ Ca Da δ (x) a

where a = (a1 , . . . , an ) can be a multi-index and Da represents a multiple derivative. The singularity order is conserved in the splitting so that r(x) and r (x) have the same order of singularity ω . This implies that the order of singularity of the difference is at most ω . Then |a| = ∑i ai ≤ ω and, for negative ω , no value is allowed.

7.3.4 Nonnegative Singularity Order 7.3.4.1 Example Let us first consider the question in a one-dimensional space with, as particular case, the Dirac function δ (x) whose singularity order is 0. The multiplication of the Dirac function with the step function has no meaning. Indeed, by the definition of the Dirac distribution, θ (x)δ (x), ϕ (x) = δ (x), θ (x)ϕ (x) = θ (0)ϕ (0) and θ (0) is not defined. Such a relation can however take a meaning if the test-function ϕ (x) is restricted in such a way that ϕ (0) = 0. On these test-functions, the retarded part can be defined as r(x) = θ (x)d(x) for any distribution d(x) whose singularity order is 0. In distribution language, r(x), ϕ (x) = θ (x)d(x), ϕ (x) = d(x), θ (x)ϕ (x),

ϕ (0) = 0.

Let us set ϕ (x) = xϕ˜ (x) where ϕ˜ (x) is not submitted to any restriction other than those usually imposed on test-functions. Then xr(x), ϕ˜ (x) = θ (x)xd(x), ϕ˜ (x). When the singularity order of d(x) is 0, the singularity order or the distribution xd(x) is −1. Therefore r1 (x) = θ (x)xd(x) is defined in a unique way and the equation satisfied by r(x) is xr(x) = r1 (x). With the Dirac function, r1 (x) = 0 and the general solution of the equation is r(x) = Cδ (x) with arbitrary C.

7.3 Splitting of Causal Distributions into Advanced and Retarded Parts

163

This elementary example shows that the decomposition of a distribution whose support is causal into an advanced and a retarded part is not unique when the singularity order is nonnegative. This can be generalized to other distributions. For a general nonnegative singularity of order ω , the test-functions ϕ (x) must be restricted by ϕ (x) = xω +1 ϕ˜ (x). In other words, it is assumed that it vanishes at the origin with all its derivatives of order less than or equal to ω

∂ a ϕ (x)

= 0, a ≤ ω. ∂ xa x=0 7.3.4.2 Generalization In an m dimensional space, let us set am , xa = x1a1 . . . xm

Da =

|a| = a1 + · · · + am ,

∂ |a|

am . ∂ x1a1 . . . ∂ xm

If the singularity order of the distribution d(x) is ω , the usual time-ordering by multiplication with the step function θ (x · v) is well-defined on test-functions ϕ (x) restricted by |a| ≤ ω . Da ϕ (x)|x=0 = 0, It is now necessary to define a relation between a general test-function and a reduced test-function on which the distribution can be multiplied by a step function. The division by x has no meaning in more than one dimension. The construction is done in the following way. Let us introduce an auxiliary test-function w(x) such that w(0) = 1,

Da w(0) = 0,

Let us set Wϕ (x) = ϕ (x) − w(x)

1 ≤ a ≤ ω.

ω

xa a (D ϕ ) (0) a! |a|=0

∑

where a! = a1 ! · · · an !. These test-functions Wϕ satisfy

DaWϕ (x) x=0 = 0, |a| ≤ ω and, on this subspace of test-functions, r(x),Wϕ (x) = d(x), θ (x · v)Wϕ (x). The retarded part of the distribution d(x) is therefore defined by r(x), ϕ (x) = r(x),Wϕ (x) = d(x), θ (x · v)Wϕ (x).

(7.10)

164

7 Field Theory Without Infinities

As in the above example, the decomposition is not unique. Indeed the distributions r (x) = r(x) + ∑ Ca Da δ (x) a

and r(x) are identical for x · v > 0. They differ only at the origin. Both r(x) and d(x) will have the same order of singularity if a is restricted by |a| ≤ ω . All the Ca are arbitrary constants.

7.3.4.3 Practical Use In most cases, the distributions are known in momentum space and the Fourier transform is very difficult to carry out. A general procedure to handle the causal splitting of the distributions in momentum space is described in [2]. Normalization i.e. fixing the arbitrary local terms, is also discussed there. Fortunately, for the applications which are handled here, the Fourier transform can be performed and, in coordinate space, the distribution, at the second order of perturbation, takes the form   d(x) = (x0 )δ  x2 . Its splitting into advanced and retarded parts is trivial.     r(x) = θ (x0 )δ  x2 , a(x) = −θ (−x0 )δ  x2 . The splitting of derivatives of this distribution is also trivial. Arbitrary terms δ (4) (x) and derivatives up to the order of singularity must however be taken into account.

7.4 Application to Yang-Mills Theory 7.4.1 First Order As in the usual approaches with divergent loop integrals and renormalization, the first order T1 is given by the interaction Lagrangian but the latter is here limited to the first order in the coupling constant. A second difference lies in the fact that fields occurring in T1 are now free fields. In pure Yang-Mills theory,   T1 (x) = ig fabc : Aaμ (x) Abν (x) ∂ ν Acμ (x) − iAaμ (x)∂ˆ μ c¯b (x)cc (x) : (7.11) where normal ordering is again used. In Yang-Mills theory, one imposes invariance under BRST transformations in order to assure that the theory describes only two physical degrees of freedom. Because fields are free fields and T1 is restricted to first

7.4 Application to Yang-Mills Theory

165

order in the coupling constant, BRST-like requirement must be imposed but only on T1 . This implies that the BRST charge is slightly different from the usual case. It is given by 

QB =

d 3 x [Sa (x) ∂˜0 ca (x) − ∂ˆ0 Sa (x) ca (x)]

(7.12)

where only free fields are involved. Using free field equations, it is easy to check that the BRST charge QB defined by (7.12) is conserved. It is also easy to show that     QB , Aaμ (x) = −i∂μ ca (x), [QB , Sa (x)] = QB , Sa (x) = 0, {QB , c¯a (x)} = −Sa (x),

{QB , ca (x)} = 0.

(7.13)

These commutation relations define the BRST transformations of the fields. From these commutation relations and free field equations, it is possible after some elementary calculations to write   i [QB , T1 ] = −g fabc ∂μ : Faμν Abν cc + ∂ˆ μ c¯a cb cc : . (7.14) 2 Though T1 itself is not BRST-invariant but its transformation is a four-divergence, 

d 3 x [QB , T1 (x)] = 0.

(7.15)

The first order contribution to the S-matrix is invariant under BRST transformations.

7.4.1.1 Second Order According to the general theory, the second order is given by T2 (x1 , x2 ) = A2 (x1 , x2 ) − A2 (x1 , x2 ) = R2 (x1 , x2 ) − R2 (x1 , x2 )

(7.16)

where, taking also into account the unitarity condition T˜1 = −T1 , R2 (x1 , x2 ) = T1 (x2 ) T˜1 (x1 ) = −T1 (x2 ) T1 (x1 ),

(7.17)

A2 (x1 , x2 ) = T˜1 (x1 ) T1 (x2 ) = −T1 (x1 ) T1 (x2 )

(7.18)

and R2 and A2 are respectively the retarded and advanced parts of the causal distribution (7.19) D2 (x1 , x2 ) = R2 (x1 , x2 ) − A2 (x1 , x2 ) = [T1 (x1 ), T1 (x2 )]. There are various terms corresponding to the Feynman graphs with two points that can be built from the two vertices containing respectively three gluons and two ghosts and one gluon. The various graphs are

166

1. external ghost, no external gluon

2. No external ghost, two external gluons

3. No external ghost, four external gluons

and those obtained by permutation of lines 4. No external ghost, six external gluons

including permutations of lines. 5. Two external ghosts, no external gluon

7 Field Theory Without Infinities

7.4 Application to Yang-Mills Theory

167

6. Two external ghosts, two external gluons

7. Two external ghosts, four external gluons

8. Four external ghosts, no external gluon

9. Four external ghosts, two external gluons

In order to compute these various terms, the Feynman rules in coordinate space can be applied. In contrast to the standard theory, the pairing function is now given by  (−) (+) (7.20) C(A1 (x1 )A2 (x2 )) = A1 (x1 ), A2 (x2 ) ±

168

7 Field Theory Without Infinities (±)

where Ai are respectively the positive and negative frequency parts of the field Ai and either the commutator or the anticommutator is taken according to bosonic or fermionic character of the fields. Note that these Feynman rules give A2 and R2 but not directly the S-matrix at the second order. To get it, the causal decomposition of D2 = R2 − A2 must be performed. In contrast with the usual perturbation theory, this decomposition is not always given by a direct time-ordering. For this reason, use of Feynman rules is of little help. The various terms can be obtained more easily directly from Wick’s theorem. Because the calculations are systematic, the reader can also use his preferred computer algebra program and get the result in a form that can be directly translated in the usual writing. Here is the result for A2 obtained by using REDUCE. Each term must be multiplied by the factor g2 fa1 b1 c1 fa2 b2 c2 . 1. No external ghost, no external gluon a. Exchange of three gluons     b1 b2 ν 1 μ1 c 1 A1a (x1 ), Aaμ22 (x2 ) 2 = C Aν1 (x1 ), Aν2 (x2 ) C ∂ A   C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 )   + C Abν11 (x1 ), Abν22 (x2 ) C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 ))   C Aaμ11 (x1 ), Aaμ22 (x2 )     + C Abν11 (x1 ), Abν22 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 )   C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 )     + C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 )   C Aaμ11 (x1 ), Aaμ22 (x2 )     + C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 )   C Aaμ11 (x1 ), Abν22 (x2 )     + C Abν11 (x1 ), Aaμ22 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 )   C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 )   + C Abν11 (x1 ), Aaμ22 (x2 ) C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 ))   C Aaμ11 (x1 ), Abν22 (x2 ) (7.21) b. Exchange of two ghosts and one gluon     c1 ˆ μ2 b 2 A1b C i∂ˆ μ1 c¯b1 (x1 ), cc2 (x2 ) 2 = C c (x1 ), i∂ c¯ (x2 )   C Aaμ11 (x1 ), Aaμ22 (x2 )

(7.22)

7.4 Application to Yang-Mills Theory

169

2. No external ghost, two external gluons a. Gluon exchange b1 b2 A2a 2 = : Aν1 (x1 )Aν2 (x2 ) :

  C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 ))C Aaμ11 (x1 ), Aaμ22 (x2 )

+ : Abν11 (x1 )Abν22 (x2 ) :     : C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 ) C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) + : Abν11 (x1 )∂ ν2 Aμ2 c2 (x2 ) :     C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 ) C Aaμ11 (x1 ), Aaμ22 (x2 ) + : Abν11 (x1 )∂ ν2 Aμ2 c2 (x2 ) :     C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 ) C Aaμ11 (x1 ), Abν22 (x2 ) + : Abν11 (x1 )Aaμ22 (x2 ) :     C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 ) C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) + : Abν11 (x1 )Aaμ22 (x2 ) :

  C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 ))C Aaμ11 (x1 ), Abν22 (x2 )

+ : ∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 ) :     C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) C Aaμ11 (x1 ), Aaμ22 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 ) :     C Abν11 (x1 ), Aaμ22 (x2 ) C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )∂ ν2 Aμ2 c2 (x2 ) :     C Abν11 (x1 ), Abν22 (x2 ) C Aaμ11 (x1 ), Aaμ22 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )∂ ν2 Aμ2 c2 (x2 )) :     C Abν11 (x1 ), Aaμ22 (x2 ) C Aaμ11 (x1 ), Abν22 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )Aaμ22 (x2 ) :     C Abν11 (x1 ), Abν22 (x2 ) C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )Aaμ22 (x2 ) :     C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) C Aaμ11 (x1 ), Abν22 (x2 ) + : Aaμ11 (x1 )Abν22 (x2 ) :     C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 )

170

7 Field Theory Without Infinities

+:

Aaμ11 (x1 )Abν22 (x2 ) 

C

:



Abν11 (x1 ), Aaμ22 (x2 )

C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 ))

+ : Aaμ11 (x1 )∂ ν2 Aμ2 c2 (x2 ) :     C Abν11 (x1 ), Abν22 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 ) + : Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) :     C Abν11 (x1 ), Aaμ22 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 ) + : Aaμ11 (x1 )Aaμ22 (x2 ) :   C Abν11 (x1 ), Abν22 (x2 ) C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 )) + : Aaμ11 (x1 )Aaμ22 (x2 ) :     C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 ) (7.23) b. Ghost exchange   a1 a2 c1 ˆ μ2 b 2 A2b 2 =: Aμ1 (x1 )Aμ2 (x2 ) : C c (x1 ), i∂ c¯ (x2 )   C i∂ˆ μ1 c¯b1 (x1 ), cc2 (x2 ) 3. No external ghost, four external gluons b1 ν 1 μ1 c 1 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 ) : A3 2 = : Aν1 (x1 )∂ A   C Aaμ11 (x1 ), Aaμ22 (x2 )

+ : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 )Aaμ22 (x2 ) :   C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) + : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) :   C Aaμ11 (x1 ), Abν22 (x2 ) + : Abν11 (x1 )Aaμ11 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 ) :   C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 ) + : Abν11 (x1 )Aaμ11 (x1 )Abν22 (x2 )Aaμ22 (x2 ) : C (∂ ν1 Aμ1 c1 (x1 ), ∂ ν2 Aμ2 c2 (x2 )) + : Abν11 (x1 )Aaμ11 (x1 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) :   C ∂ ν1 Aμ1 c1 (x1 ), Abν22 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )Aaμ11 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 ) :   Abν11 (x1 ), Aaμ22 (x2 )

(7.24)

7.4 Application to Yang-Mills Theory

171

+ : ∂ ν1 Aμ1 c1 (x1 )Aaμ11 (x1 )Abν22 (x2 )Aaμ22 (x2 ) :   C Abν11 (x1 ), ∂ ν2 Aμ2 c2 (x2 ) + : ∂ ν1 Aμ1 c1 (x1 )Aaμ11 (x1 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) :   C Abν11 (x1 ), Abν22 (x2 )

(7.25)

4. No external ghost, six external gluons b1 ν 1 μ1 c 1 (x1 )Aaμ11 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) : A4 2 =: Aν1 (x1 )∂ A

(7.26)

5. Two external ghosts, no external gluon   c2 c1 ˆ μ1 b 1 ˆ μ2 b 2 A5 2 = : i∂ c¯ (x1 )c (x2 ) : C c (x1 ), i∂ c¯ (x2 ) C(Aaμ11 (x1 ), Aaμ22 (x2 ))  + : cc1 (x1 )i∂ˆ μ2 c¯b2 (x2 ) : C i∂ˆ μ1 c¯b1 (x1 ), cc2 (x2 )   C Aaμ11 (x1 ), Aaμ22 (x2 ) 

(7.27)

6. Two external ghosts, two external gluons c2 ˆ μ2 b 2 A6 2 = : c (x2 )i∂ c¯ (x2 ) :    : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 ) : C Aaμ11 (x1 ), Aaμ22 (x2 )   + : Abν11 (x1 )Aaμ11 (x1 ) : C ∂ ν1 Aμ1 c1 (x1 ), Aaμ22 (x2 )   + : ∂ ν1 Aμ1 c1 (x1 )Aaμ11 (x1 ) : C Abν11 (x1 ), Aaμ22 (x2 )

+ : cc1 (x1 ), i∂ˆ μ1 c¯b1 (x1 ) :    : Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 ) : C Aaμ11 (x1 ), Aaμ22 (x2 )   + : Abν22 (x2 )Aaμ22 (x2 ) : C Aaμ11 (x1 ), ∂ ν2 Aμ2 c2 (x2 )   + : ∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) : C Aaμ11 (x1 ), Abν22 (x2 ) + : i∂ˆ μ1 c¯b1 (x1 )cc2 (x2 ) : : Aaμ11 (x1 )Aaμ22 (x2 ) :   C cc1 (x1 ), i∂ˆ μ2 c¯b2 (x2 ) + : cc1 (x1 )i∂ˆ μ2 c¯b2 (x2 ) : : Aaμ11 (x1 )Aaμ22 (x2 ) :   C i∂ˆ μ1 c¯b1 (x1 ), cc2 (x2 )

(7.28)

7. Two external ghosts, four external gluons (7)

A2

= : cc2 (x2 )i∂ˆ μ2 c¯b2 (x2 ) : : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 )Aaμ11 (x1 )Aaμ22 (x2 ) : + : cc1 (x1 )i∂ˆ μ1 c¯b1 (x1 ) : : Aaμ11 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) : (7.29)

172

7 Field Theory Without Infinities

8. Four external ghosts, no external gluon (8)

A2

=: cc1 (x1 )i∂ˆ μ1 c¯b1 (x1 )cc2 (x2 )i∂ˆ μ2 c¯b2 (x2 ) : C(Aaμ11 (x1 ), Aaμ22 (x2 ))

(7.30)

9. Four external ghosts, two external gluons (9)

A2

=: cc1 (x1 )i∂ˆ μ1 c¯b1 (x1 )cc2 (x2 )i∂ˆ μ2 c¯b2 (x2 ) : : Aaμ11 (x1 )Aaμ22 (x2 ) :

(7.31)

The corresponding expressions for R2 are obtained by the change 1 ↔ 2 in all the indices and subindices.

7.4.1.2 Splitting of the Tree Graphs The connected tree-graph contribution to D2 contains four external fields and the exchange of either a gluon or a ghost. Three different contributions are obtained. They are, up to the factor g2 fa1 b1 c1 fa2 b2 c2 , 1. Four external gluons, exchange of a gluon   (3) D2 = : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 )) : Aaμ22 (x2 ), Aaμ11 (x1 )   + : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 )Aaμ22 (x2 ) : ∂ ν2 Aμ2 c2 (x2 ), Aaμ11 (x1 )  + : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) : Abν22 (x2 ), Aaμ11 (x1 )   + : Abν11 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 )Aaμ11 (x1 ) : Aaμ22 (x2 ), ∂ ν1 Aμ1 c1 (x1 ) + : Abν11 (x1 )Abν22 (x2 )Aaμ22 (x2 )Aaμ11 (x1 ) : [∂ ν2 Aμ2 c2 (x2 ), ∂ ν1 Aμ1 c1 (x1 )]  + : Abν11 (x1 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 )Aaμ11 (x1 ) : Abν22 (x2 ), ∂ ν1 Aμ1 c1 (x1 )  + : ∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 )Aaμ11 (x1 ) : Aaμ22 (x2 ), Abν11 (x1 )  + : ∂ ν1 Aμ1 c1 (x1 )Abν22 (x2 )Aaμ22 (x2 )Aaμ11 (x1 ) : ∂ ν2 Aμ2 c2 (x2 ), Abν11 (x1 )  + : ∂ ν1 Aμ1 c1 (x1 )∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 )Aaμ11 (x1 ) : Abν22 (x2 ), Abν11 (x1 ) (7.32) 2. Two external gluons, two external ghosts a. Exchange of a gluon (6a)

D2

= : cc2 (x2 )i∂ˆ μ2 c¯b2 (x2 ) :    : Abν11 (x1 )∂ ν1 Aμ1 c1 (x1 ) : Aaμ22 (x2 ), Aaμ11 (x1 )   + : Abν11 (x1 )Aaμ11 (x1 ) : Aaμ22 (x2 ), ∂ ν1 Aμ1 c1 (x1 )   + : ∂ ν1 Aμ1 c1 (x1 )Aaμ11 (x1 ) : Aaμ22 (x2 ), Abν11 (x1 )

7.4 Application to Yang-Mills Theory

173

+ : cc1 (x1 )i∂ˆ μ1 c¯b1 (x1 ) :    : Abν22 (x2 )∂ ν2 Aμ2 c2 (x2 ) : Aaμ22 (x2 )Aaμ11 (x1 )   + : Abν22 (x2 )Aaμ22 (x2 ) : ∂ ν2 Aμ2 c2 (x2 ), Aaμ11 (x1 )   + : ∂ ν2 Aμ2 c2 (x2 )Aaμ22 (x2 ) : Abν22 (x2 ), Aaμ11 (x1 )

(7.33)

b. Exchange of a ghost (6b)

D2

= − : Aaμ22 (x2 )Aaμ11 (x1 ) :   cc2 (x2 ), i∂ˆ μ1 c¯b1 (x1 ) : cc1 (x1 )i∂ˆ μ2 c¯b2 (x2 ) :   + i∂ˆ μ2 c¯b2 (x2 ), cc1 (x1 ) : i∂ˆ μ1 c¯b1 (x1 )cc2 (x2 ) : (7.34)

3. Four external ghosts, exchange of a gluon   (8) D2 = Aaμ22 (x2 ), Aaμ11 (x1 ) : cc1 (x1 )i∂ˆ μ1 c¯b1 (x1 )cc2 (x2 )i∂ˆ μ2 c¯b2 (x2 ) :

(7.35)

All of these contributions involve a normal product of four fields and either a commutator between two gluon fields or an anticommutator between a ghost and an anti-ghost field. The commutators or anticommutator involve free fields and are known from free field theory. Their singularity order is −2 for all linear and class III gauges. Introduction of one derivative leaves the singularity order negative. Therefore, for almost all terms contributing to tree graphs, the splitting can be realized with usual time-ordering. The corresponding contribution to T2 can therefore be obtained with the commutators replaced by the Feynman propagators. For all these terms, usual Feynman rules hold. There is an exception to this rule, the occurence of two derivatives in g2 fa1 b1 c1 fa2 b2 c2 : Abν11 (x1 )Abν22 (x2 )Aaμ22 (x2 )Aaμ11 (x1 ) : [∂ ν2 Aμ2 c2 (x2 ), ∂ ν1 Aμ1 c1 (x1 )]. The singularity order of the commutator is now 0. Therefore, the splitting is not unique though trivial [∂ ν2 Aμ2 c2 (x2 ), ∂ ν1 Aμ1 c1 (x1 )]ret = ∂xν11 ∂xν22 [Aμ1 c1 (x1 ), Aμ2 c2 (x2 )]ret +δ (4) (x1 − x2 )δ c1 c2 Cν1 ν2 μ1 μ2

(7.36)

The first term is given by usual time-ordering and, like the other contributions to tree graphs, it is given by the Feynman rules. The second term is a local interaction at order g2 . Remembering the structure of the Yang-Mills Lagrangian, such term must reproduce the four-gluon contact interaction. This condition fixes the arbitrary constants Cν1 ν2 μ1 μ2 in such a way that the contribution to T2 of this term reads

174

7 Field Theory Without Infinities

i

g2 fa b c fa b c δ (4) (x1 − x2 ) : Aaμ1 (x1 )Abν1 (x1 )Aaμ2 (x2 )Aνb2 (x2 ) : . 2 1 1 2 2

(7.37)

In the usual interacting theory, the four-gluon interaction is a consequence of nonabelian gauge invariance. Here, the perturbation expansion is made order by order in the coupling constant g. The gauge is fixed but BRST invariance must still hold. It is satisfied at zeroth and first orders in g. Requiring that it also holds at order g2 , [QB , T2 ] = ∂xμ1 Aμ (x1 , x2 ) + ∂xμ2 Bμ (x1 , x2 ). In the next section, the explicit check of the BRST invariance of the tree-graph contribution to T2 will be realized. Readers who are not interested in those technical and cumbersome details may skip this section. With respect to the work of Scharf and collaborators [2], this calculation is made in a general class III linear gauge and not only in the covariant Fermi gauge.

7.4.1.3 BRST Invariance of the Tree-Graph Contribution to T 2 BRST invariance of the first-order term implies [QB , T1 (x)] = ∂ μ φμ (x) where φμ (x) stands for the heavier expression given by (7.14) but does not need to be explicited for the reasoning. By using the definition of D2 as the commutator between T1 at two different times and the Jacobi identity, one obtains [QB , D2 (x1 , x2 )] = [[QB , T1 (x2 )], T1 (x1 )] − [[QB , T1 (x1 )], T1 (x2 )] = ∂xμ2 [φμ (x2 ), T1 (x1 )] − ∂xμ1 [φμ (x1 ), T1 (x2 )].

(7.38) (7.39)

It is a combination of four-divergences. This does not mean that the second-order contribution to the S-matrix T2 is a four-divergence. Indeed, causal splitting does not necessarily respect the four-divergences. At the level of tree-graphs, D2 involves commutators of free fields and their derivatives while T2 involves the corresponding Feynman propagators with, if the order of singularity is not negative, possible local terms. With the generalized functions defined in appendix C, the involved commutators are,  i ˆ (∂μ ∂ν + ∂μ ∂ˆν )FC (x − z) [Aaμ (x), Aν (z)b ] = δ ab −igμν D(x − z) + C00  i K ∂μ ∂ν GC (x − z) , (7.40) + C00   iδ ab ca (x), ic¯b (z) = DC (x − z). C00

(7.41)

7.4 Application to Yang-Mills Theory

175

The corresponding propagators are Fab Δ μν (x1 − x2 ) = −

DCFab (x1 − x2 ) =

i δ ab (2π )4

i δ ab (2π )4



d4k

eik·(x1 −x2 ) k2 + i 

 ˜ kˆ μ kν + kμ kˆ ν K(k)k μ kν , gμν − + 2 kC2 + i (kC + i)2 (7.42)



d4k

eik·(x1 −x2 ) kC2 + i

.

(7.43)

The main difference between commutators and propagators rests in the fact that D = C DC = 0

(7.44)

leading to    i ˆ  Aaμ (x), Aν (z)b = δ ab ∂μ ∂ν + ∂μ ∂ˆν DC (x − z) C00  i K ∂μ ∂ν EC (x − z) , + C00   a b C c (x), c¯ (z) = 0

(7.45) (7.46)

while Fab Δ μν (x1 − x2 ) = igμν δ ab δ (4) (x1 − x2 )    ˆ ˆ ˜ K(k)k i μ kν ab 4 ik·(x1 −x2 ) kμ kν + kμ kν δ d ke , + + 2 (2π )4 kC2 + i (kC + i)2

C DCFab (x1 − x2 ) = −iδ ab δ (4) (x1 − x2 ).

(7.47) (7.48)

Local terms which are absent in commutators appear in propagators when the d’Alembert operator or its noncovariant extension are applied. These local terms can violate the four-divergence character of the BRST transform of the tree-graph contribution to T2 . They must cancel out. Let us check this cancellation. Let us set μ (7.49) Cbc (x) = Abν (x)Fcν μ (x) + icc (x)∂ˆ μ c¯b (x), and consider the sum of all contributions to tree graphs at the second-order. Up to a factor g2 fab1 c1 fab2 c2 , it reads μ

μ

T2tree = −ΔF μ1 μ2 (x1 − x2 ) : Cb11c1 (x1 )Cb22c2 (x2 ) : μ

+ ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : Cb11c1 (x1 )Abν22 (x2 )Acμ22 (x2 ) : μ

+ ∂ ν1 ΔF μ1 μ2 (x1 − x2 ) : Cb22c2 (x2 )Abν11 (x1 )Acμ11 (x1 ) : − ∂ ν1 ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : Abν22 (x2 )Acμ22 (x2 )Abν11 (x1 )Acμ11 (x1 )

176

7 Field Theory Without Infinities

− Cν1 ν2 μ1 μ2 δ (4) (x1 − x2 ) : Abν22 (x2 )Acμ22 (x2 )Abν11 (x1 )Acμ11 (x1 ) + ∂ˆ ν1 ΔF (x1 − x2 ) : Abν11 (x1 )cc1 (x1 )Abν22 (x2 )i∂ˆ ν2 c¯c2 (x2 ) : − ∂ˆ ν2 ΔF (x1 − x2 ) : Abν22 (x2 )cc2 (x2 )Abν11 (x1 )i∂ˆ ν1 c¯c1 (x1 ) : where the derivatives with index i are taken with respect to xi . Setting νμ Abc (x) = cb (x)Fcν μ (x)

(7.50)

(7.51)

and using free field equations, the BRST transforms of the various products of fields are   μ νμ fabc QB , Cbc (x) = −i∂ν Abc (x), (7.52)   fabc QB , Abν (x)Acμ (x) = −i fabc ∂ν cb (x)Acμ (x) − ∂μ cb (x)Acν (x) (7.53)     = i fabc −∂ν cb (x)Acμ (x) + ∂μ cb (x)Acν (x) + Aνbcμ (x) ,    i fabc QB , Abν (x)cc (x) = − fabc ∂ν cb (x)cc (x) , 2    μ fabc QB , Abν (x)i∂ˆ ν c¯c (x) = i fabc ∂μ Cbc (x) − ∂μ Abν ∂ ν Acμ .

(7.54) (7.55) (7.56)

Using these relations, the BRST transform of the tree-graph contribution can be written, up to a factor ig2 fab1 c1 fab2 c2 , as  ν μ μ [QB , T2tree ] = ΔF μ1 μ2 (x1 − x2 ) : ∂ν1 Ab11c1 1 (x1 )Cb22c2 (x2 ) + 1 ↔ 2  ν μ − ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : ∂ν1 Ab11c1 1 (x1 )Abν22 (x2 )Acμ22 (x2 ) : +1 ↔ 2    μ + ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : Cb11c1 (x1 ) −∂ν2 cb2 (x2 )Acμ22 (x2 )    μ +∂ μ2 cb2 (x2 )Acν22 (x2 ) + Aν22b2 c2 (x2 ) : +1 ↔ 2    −∂ ν1 ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : −∂ν2 cb2 (x2 )Acμ22 (x2 )   μ + ∂ μ2 cb2 (x2 )Acν22 (x2 ) + Aν22b2 c2 (x2 ) Abν11 (x1 )Acμ11 (x1 ) :  +1 ↔ 2 +Cν1 ν2 μ1 μ2 δ (4) (x1 − x2 )   : ∂ν1 cb1 (x1 )Acμ11 (x1 ) − ∂μ1 cb1 (x1 )Acν11 (x1 ) Abν22 (x2 )Acμ22 (x2 ) :  +1 ↔ 2

7.4 Application to Yang-Mills Theory

177

  1 − ∂ˆ ν1 DCF (x1 − x2 ) : ∂ν1 cb1 (x1 )cc1 (x1 ) Abν22 (x2 )i∂ˆ ν2 c¯c2 (x2 ) : 2  +1 ↔ 2  + ∂ˆ ν1 DCF (x1 − x2 ) : Abν11 (x1 )cc1 (x1 )   ∂ν2 Cbν22c2 (x2 ) − ∂μ2 Abν22 (x2 )∂ ν2 Acμ22 (x2 ) : + 1 ↔ 2 . (7.57) νμ

The terms containing Abi ci i i (xi ) obviously occur as four-divergences. Dropping them away, it remains to consider the local terms coming from    μ ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : Cb11c1 (x1 ) −∂ν2 cb2 (x2 )Acμ22 (x2 )    +∂ μ2 cb2 (x2 )Acν22 (x2 ) : +1 ↔ 2      + ∂ ν1 ∂ ν2 ΔF μ1 μ2 (x1 − x2 ) : ∂ν2 cb2 (x2 )Acμ22 (x2 ) − ∂ μ2 cb2 (x2 )Acν22 (x2 )  Abν11 (x1 )Acμ11 (x1 ) : +1 ↔ 2   + Cν1 ν2 μ1 μ2 δ (4) (x1 − x2 ) : ∂ν1 cb1 (x1 )Acμ11 (x1 ) − ∂μ1 cb1 (x1 )Acν11 (x1 )  Abν22 (x2 )Acμ22 (x2 ) : + 1 ↔ 2    1 b2 ν1 b1 c1 ν2 c 2 ˆ ˆ − ∂ ΔF (x1 − x2 ) : ∂ν1 c (x1 )c (x1 ) Aν2 (x2 )i∂ c¯ (x2 ) : + 1 ↔ 2 2  + ∂ˆ ν1 DCF (x1 − x2 ) : Abν11 (x1 )cc1 (x1 )   ∂ν2 Cbν22c2 (x2 ) − ∂μ2 Abν22 (x2 )∂ ν2 Acμ22 (x2 ) : +1 ↔ 2 . (7.58) They are obtained by letting the derivatives ∂νi act on the ΔF functions when a derivative ∂ νi or ∂ˆ νi already acts on it and extracting the resulting local terms. All the other terms will lead to four-divergences, sometimes in a cumbersome way. Let us remark that all these other terms are not involved in the Fermi gauge. After this operation, the local terms are, up to a factor −g2 fab1 c1 fab2 c2 ,  μ δ (4) (x1 − x2 ) : Cb1 c1 (x1 )cb2 (x2 )Acμ2 (x2 ) : +1 ↔ 2  − ∂ ν2 δ (4) (x1 − x2 ) : cb1 (x1 )Acμ1 (x1 )Abν22 (x2 )Acμ2 (x2 ) : + 1 ↔ 2  1 − δ (4) (x1 − x2 ) : cb1 (x1 )cc1 (x1 )Abμ2 (x2 )i∂ˆ μ c¯c2 (x2 ) : +1 ↔ 2 2   − iCν1 ν2 μ1 μ2 δ (4) (x1 − x2 ) : ∂ν1 cb1 (x1 )Acμ11 (x1 ) − ∂μ1 cb1 (x1 )Acν11 (x1 )  Abν22 (x2 )Acμ22 (x2 ) : +1 ↔ 2 . (7.59)

178

7 Field Theory Without Infinities

The derivative ∂xν22 δ (4) (x1 − x2 ) can be changed into −∂xν12 δ (4) (x1 − x2 ). Then a fourdivergence can be added and this term can be written up to the four-divergence as  − δ (4) (x1 − x2 ) : ∂ ν cb1 (x1 )Acμ1 (x1 )Abν2 (x2 )Acμ2 (x2 ) : + : cb1 (x1 )∂ ν Acμ1 (x1 )Abν2 (x2 )Acμ2 (x2 ) : . The same term with 1 ↔ 2 must of course also to be taken into account. Owing to the presence of the δ (4) (x1 − x2 ), all the fields can be taken at the same point. Changing the indices b1 ↔ b2 , c1 ↔ c2 and using the fact that Cν1 ν2 μ1 μ2 = Cgν1 ν2 gμ1 μ2 is the unique nontrivial contribution, the change 1 ↔ 2 only introduces a factor 2. The coefficient of the −2g2 δ (4) (x1 − x2 ) term is  fab1 c1 fab2 c2 Abν1 Fcν1μ cb2 Acμ2 + icc1 ∂ˆ μ c¯b1 cb2 Acμ2 1 − cb1 ∂ ν Acμ1 Abν2 Acμ2 + cb1 cc1 Abμ2 i∂ˆ μ cc2 2 − ∂ ν cb1 Acμ1 Abν2 Acμ2   − iC ∂ ν cb1 Acμ1 − ∂ μ cb1 Acν1 Abν2 Acμ2 .

(7.60)

There are three kinds of terms. 1. Three gluons and one ghost, the derivative acting on a gluon field  fab1 c1 fab2 c2 Abν1 Fcν1μ cb2 − cb1 ∂ ν Acμ1 Abν2 Acμ2 . Renaming the dummy indices,   Abν1 ∂ ν Acμ1 cb2 Acμ2 fab1 c1 fab2 c2 − fac2 c1 fab2 b1 − fab2 c1 fab1 c2 . This term vanishes owing to the Jacobi identity satisfied by the structure constants. 2. Three ghost fields and one gluon field   1 b 1 c1 b 2 ˆ μ c2 c1 ˆ μ b 1 b 2 c2 i fab1 c1 fab2 c2 c ∂ c¯ c Aμ + c c Aμ ∂ c¯ . 2 Renaming again the dummy indices,   1 icc1 ∂ˆ μ c¯b1 cb2 Acμ2 fab1 c1 fab2 c2 + fab2 c1 fac2 b1 . 2 The Jacobi identity allows us to write it as

7.4 Application to Yang-Mills Theory

179



icc1 ∂ˆ μ c¯b1 cb2 Acμ2 fab1 c1 fab2 c2 −



1 1 fac2 c1 fab1 b2 − fab1 c1 fab2 c2 . 2 2

Changing the dummy indices c1 ↔ b2 in the second term, the vanishing is obvious. 3. Three gluons and one ghost, the derivative acting on the ghost field    fab1 c1 fab2 c2 ∂ ν cb1 Acμ1 Abν2 Acμ2 − iC ∂ ν cb1 Acμ1 − ∂ μ cb1 Acν1 Abν2 Acμ2 . The vanishing of this term leads to i C= . 2

(7.61)

and fixes univoquely the four-gluon contact interaction.

7.4.2 Example of a One-Loop Process: The Gluon Self-energy 7.4.2.1 The Ghost Loop Contribution In pure Yang-Mills theory, two kinds of loops contribute to the gluon self-energy, the ghost loop and the gluon loop. In Quantum Chromodynamics, a fermion loop must also be included. Because of its technical simplicity, let us first consider the ghost loop contribution. At the level of A2 , it is given by (2b)

A2

= g2 fa1 b1 c1 fa2 b2 c2 : Aμ1 a1 (x1 )Aμ2 a2 (x2 ) : ∂ˆ μ2 C(cb1 (x1 ), ic¯c2 (x2 ))∂ˆ μ1 C(ic¯c1 (x1 ), cb2 (x2 ))

(7.62)

where, as previously, derivatives with the index μi are taken with respect to xi . Using the definition of the pairings as anticommutators and their expressions from free field theory,   i (−) (+) (−) δb c D (x1 − x2 ), C(cb1 (x1 ), ic¯c2 (x2 )) = cb1 (x1 ), ic¯c2 (x2 ) = C00 1 2 C (7.63)   i (−) (+) (+) C(ic¯c1 (x1 ), cb2 (x2 )) = ic¯c1 (x1 ), cb2 (x2 ) = δb c D (x2 − x1 ) C00 2 1 C i (−) =− δb c D (x1 − x2 ), (7.64) C00 2 1 C (7.62) becomes (2b)

A2

= − : Aμ1 a1 (x1 )Aμ2 a2 (x2 ) :

1 2 g f a 1 b 1 c1 f a 2 b 1 c1 2 C00

(−) (−) ∂ˆxμ22 DC (x1 − x2 ) ∂ˆxμ11 DC (x1 − x2 ).

(7.65)

180

7 Field Theory Without Infinities

Setting Pμ1 μ2 (x) =

1 ˆ μ2 (−) (−) ∂ DC (x) ∂ˆ μ1 DC (x), 2 C00

(7.66)

it can be written as (2b)

A2

= g2 fa1 b1 c1 fa2 b1 c1 : Aμ1 a1 (x1 )Aμ2 a2 (x2 ) : Pμ1 μ2 (x1 − x2 ).

(7.67)

The quantity Pμ1 μ2 (x) is symmetric on its indices. Setting Pˆ μ1 μ2 (k) =



d 4 x e−ik·x Pμ1 μ2 (x)

(7.68)

(−)

and using the explicit expression of DC (x) given in appendix C, Pˆ μ1 μ2 (k) =

1 (2π )2



      d 4 q θ q0 − Z.q δ qC2 θ k0 − q0 − Z.(k − q)   μ  δ kC2 − k˜ · q − q · kˆ qˆμ2 kˆ − qˆ 1 .

(7.69)

This result holds in any linear class III gauge where Pˆ μ1 μ2 (k) is a second-rank ˜ If the gauge is covaritensor depending on three independent momenta k, kˆ and k. ant, k = kˆ = k˜ and the number of independent momenta is one. The vector Z also vanishes. The use of tensor methods will, of course, be much simpler in a covariant gauge to which one will now be restricted.

7.4.2.2 Covariant Gauges Let us now compute Pˆ μ1 μ2 (k) in a covariant gauge. It reads Pˆ μ1 μ2 (k) =

1 (2π )2



        d 4 q θ q0 δ q2 θ k0 − q0 δ k2 − 2k · q qμ2 (k − q)μ1 . (7.70)

Because it is a second-rank tensor depending only on the vector k, one can write     Pˆ μ1 μ2 (k) = A k2 gμ1 μ2 + B k2 k μ1 k μ2 .

(7.71)

Multiplying this expression by kμ1 kμ2 , one gets  2 Ak2 + B k2 =



        d 4 q θ q0 δ q2 θ k0 − q0 δ k2 − 2k · q   k · q k2 − k · q  2 2      k 1 = d 4 q θ (q0 ) δ (q2 ) θ k0 − q0 δ k2 − 2k · q . (2π )2 2 (7.72) 1 (2π )2

7.4 Application to Yang-Mills Theory

181

Taking the trace of (7.71), it comes 1 (2π )2

4A + Bk2 =



        d 4 q θ q0 δ q2 θ k0 − q0 δ k2 − 2k · q

(k · q − q2 )          1 k2 d 4 q θ q0 δ q2 θ k0 − q0 δ k2 − 2k · q . (7.73) = (2π )2 2

7.4.2.3 Computation of I(k) These handlings show that only one scalar integral 

        d 4 p θ p0 δ p2 θ k0 − p0 δ k2 − 2k · p



 d3 p  2 δ k − 2|p|k0 + 2p.k θ (k0 − |p|) 2|p|

I(k) = =

(7.74)

must be computed. The work can be done by using spherical coordinates. The integral over the azimutal angle is trivial and gives a factor 2π . The one on the polar angle is handled by setting, as usual, u = cos θ while integration over the radius is limited by the step-function. It becomes thus I(k) = πθ (k0 )

 k0 0

 1

pd p

−1

  du δ k2 − 2pk0 + 2p|k|u

(7.75)

where the θ (k0 ) factor is introduced to take into account the fact that the integral vanishes if k0 < 0. Inserting step functions, the integral over u can be extended from −∞ to +∞ I(k) = πθ (k0 )

 k0

 ∞

pd p

0

−∞

  du δ k2 − 2pk0 + 2p|k|u θ (1 − u)θ (u + 1)

(7.76)

so that the Dirac-function can be used to carry it out. It becomes I(k) =

πθ (k0 ) 2|k|

 k0 0

    d p θ k2 − 2p(k0 − |k|) θ −k2 + 2p(k0 + |k|) .

(7.77)

The second step function can be written as θ [(k0 + |k|)(2p − k0 + |k|)]. Because k0 + |k| is always positive, it reduces to θ (2p − k0 + |k|). In the same way, the first step function reads θ [(k0 − |k|)(k0 + |k| − 2p)]. It reduces to θ (k0 + |k| − 2p) if k0 > |k| and to θ (2p − k0 − |k|) if k0 < |k|. In the case k0 < |k|, this step function does not vanish only for p > (k0 + |k|)/2, a value which is greater than the upper limit of integration k0 . Therefore, the integral vanishes for k0 < |k|. One can take this vanishing into account by the introduction of a factor θ (k2 ). Then, using both step functions in the case k0 > |k|,

182

7 Field Theory Without Infinities

I(k) =

 πθ (k0 )  2  (k0 +|k|)/2

2|k|

θ k

(k0 −|k|)/2

dp =

  π θ (k0 ) θ k2 . 2

(7.78)

7.4.2.4 Fourier Transform and Splitting Let us remark that I(k) is the convolution product in four dimensions of θ (p0 ) δ (p2 ) with itself. It is thus the Fourier transform of the products of the inverse Fourier transforms of these functions. Taking careful account of all the factors, one can also write  (7.79) I(k) = −(2π )2 d 4 x e−ik·x D(−) (x)D(−) (x). The Fourier transform of θ (k0 ) θ (k2 ) is computed in Appendix A. Inserting the result into (7.79), one easily gets the result corresponding to the A part, noted HA ,       1 1 (−) (−)   2 HA (x) = −D (x)D (x) = P − iπ (x0 )δ x . (7.80) (2π )4 x2 Because HR (x) = HA (−x),

(7.81)

the corresponding causal distribution is HD (x) = HR (x) − HA (x) =

  1 2iπ (x0 )δ  x2 . 4 (2π )

(7.82)

Its splitting into advanced and retarded part is trivial HA (x) =

  1 2iπθ (−x0 )δ  x2 +Cδ (4) (x) 4 (2π )

(7.83)

where the arbitrary local term Cδ (4) (x) occurs because the singularity order of the distribution HD (x) is zero. At the level of the S-matrix, the resulting distribution is, up to the local term,       1 1   2 HT (x) = HA (x) − HA (x) = −P + iπδ x (2π )4 x2   1 d 1 =− . (2π )4 dx2 x2 + i

(7.84)

7.4.2.5 Back to the Ghost Contribution Replacing now A and B in (7.71) by their computed values,   1 (k2 gμ1 μ2 + 2k μ1 k μ2 )θ (k0 ) θ k2 . Pˆ μ1 μ2 (k) = − 96π

(7.85)

7.4 Application to Yang-Mills Theory

183

and, by Fourier transform, Pμ1 μ2 (x) =

1 (gμ1 μ2 + 2∂ μ1 ∂ μ2 ) HA (x) 12

(7.86)

The causal distribution is 1 (gμ1 μ2 + 2∂ μ1 ∂ μ2 )HD (x). 12 Its singularity order is 2 which means that the splitting into advanced and retarded parts will contain an arbitrary linear combination of derivatives of the Dirac function δ (4) (x), up to the second order. Owing to this fact, the splitting can be realized at the level of the HD function. Therefore, up to local terms containing second-order derivatives of the Dirac function, (2b)

T2

(x1 , x2 ) = −

g2 fa b c fa b c : Aμ1 a1 (x1 )Aμ2 a2 (x2 ) : 12(2π )4 1 1 1 2 1 1  2 1 (gμ1 μ2 + 2∂ μ1 ∂ μ2 ) (7.87) ξ 2 + i

where ξ = x1 − x2 .

7.4.2.6 The Gluon Contribution to the Gluon Self-energy Setting again ξ = x1 − x2 and   C Aaμ11 (x1 ), Aaμ22 (x2 ) = δ a1 a2 Δ μ1 μ2 (ξ )

(7.88)

where Δ μ1 μ2 (ξ ) is symmetric on μ1 and μ2 , renaming dummy indices, the gluon contribution can be written up to a factor −g2 fab1 c fab2 c , as (2a)

A2

= : Abν11 (x1 )Abν22 (x2 ) : [∂ ν1 ∂ ν2 Δ μ1 μ2 (ξ )Δ μ1 μ2 (ξ ) − ∂ ν1 Δ μ1 μ2 (ξ )∂ ν2 Δ μ1 μ2 (ξ ) + : Abν11 (x1 )∂μ2 Abν22 (x2 ) : [∂ ν1 Δ μ1 μ2 (ξ )Δ μν12 (ξ ) − ∂ ν1 Δ μν12 (ξ )Δ μ1 μ2 (ξ ) + : Abν11 (x1 )Abν22 (x2 ) :

[∂ ν1 Δ μ1 μ2 (ξ )∂μ2 Δ μν12 (ξ ) − ∂ ν1 ∂ μ2 Δ μ1 ν2 (ξ )Δ μ1 μ2 (ξ )

− : ∂ μ1 Abν11 (x1 )Abν22 (x2 ) :

[∂ ν2 Δ μ1 μ2 (ξ )Δ ν1 μ2 (ξ ) − Δ μ1 μ2 (ξ )∂ ν2 Δ ν1 μ2 (ξ )

− : ∂ μ1 Abν11 (x1 )∂ μ2 Abν22 (x2 ) : [Δ μ1 μ2 (ξ )Δ ν1 ν2 (ξ ) − Δ μ1 ν2 (ξ )Δ ν1 μ2 (ξ ) − : ∂ μ1 Abν11 (x1 )Abν22 (x2 ) : [Δ μ1 μ2 (ξ )∂ μ2 Δ ν1 ν2 (ξ ) − ∂ μ2 Δ μν12 (ξ )Δ μν21 (ξ )

184

7 Field Theory Without Infinities

+ : Abν11 (x1 )Abν22 (x2 ) :

[∂ ν2 Δ μ1 μ2 (ξ )∂ μ1 Δ ν1 μ2 (ξ ) − Δ μ1 μ2 (ξ )∂ μ1 ∂ ν2 Δ ν1 μ2 (ξ )

+ : Abν11 (x1 )∂ μ2 Abν22 (x2 ) : [Δ μ1 μ2 (ξ )∂ μ1 Δ ν1 ν2 (ξ ) − Δ μν12 (ξ )∂ μ1 Δ μν21 (ξ ) + : Abν11 (x1 )Abν22 (x2 ) :

[Δ μ1 μ2 (ξ )∂ μ1 ∂ μ2 Δ ν1 ν2 (ξ ) − ∂μ2 Δ μ1 ν2 (ξ )∂μ1 Δ ν1 μ2 (ξ ).

(7.89)

7.4.2.7 The Distributions and Their Splitting in the Fermi Gauge The result is the simplest in the case of the Fermi gauge where

Δ μ1 μ2 (ξ ) = −igμ1 μ2 D(−) (ξ ) with D(−) (ξ ) = 0. Limiting ourselves to this case in this subsection, it reads2 again up to the factor g2 fab1 c fab2 c ,  (2a) A2 = : Abμ1 (x1 )Abν2 (x2 ) : 2D(−) (ξ )∂ μ ∂ ν D(−) (ξ ) − 3∂ μ D(−) (ξ )∂ ν D(−) (ξ ) b1 (x1 )∂ μ Aνb2 (x2 ) : D(−) (ξ )D(−) (ξ ) − : Fμν b1 b2 (x1 )Aνb2 (x2 ) − Aνb1 (x1 )Fμν (x2 ) : D(−) (ξ )∂ μ D(−) (ξ ). − : Fμν

(7.90)

All the tensor distributions can be calculated as in the case of the ghost contribution. The results are  1  gμν + 2∂μ ∂ν HA (x), 12   1 D(−) (x)∂μ ∂ν D(−) (x) = gμν − 4∂μ ∂ν HA (x), 12 1 D(−) (x)∂μ D(−) (x) = − ∂μ HA (x). 2

∂μ D(−) (x)∂ν D(−) (x) = −

(7.91) (7.92) (7.93)

Combining all these results, again up to the factor −g2 fab1 c fab2 c , (2a)

A2

= : Abμ1 (x1 )Abν2 (x2 ) : (5gμν − 2∂ μ ∂ ν ) HA (x1 − x2 ) b1 + : Fμν (x1 )∂ μ Aνb2 (x2 ) : HA (x1 − x2 ) b1 b2 (x1 )Aνb2 (x2 ) − Aνb1 (x1 )Fμν (x2 ) : + : Fμν

1 μ ∂ HA (x1 − x2 ) . (7.94) 2

The construction of the causal distribution and its splitting as well as the construction of the S-matrix contribution can be realized as above. Adding the ghost 2 This result needs a lot of systematic tensor calculations which can be realized with the help of a computer algebra program.

7.4 Application to Yang-Mills Theory

185

contribution, up to a factor g2 fab1 c fab2 c /(2π )4 , the one-loop exchange contribution to the gluon self-energy reads, up to local terms, (2)

T2

 2 1 1 (gμν − ∂ μ ∂ ν ) 3 ξ 2 + i  2 1 b1 + : Fμν (x1 )∂ μ Aνb2 (x2 ) : 2 ξ + i  2 1 1 b1 b2 + : Fμν (x1 )Aνb2 (x2 ) − Aνb1 (x1 )Fμν (x2 ) : ∂ μ . 2 ξ 2 + i

= : Abμ1 (x1 )Abν2 (x2 ) :

(7.95)

Adding four-divergences, (2)

T2

=

5 g2 fab1 c fab2 c : Abμ1 (x1 )Abν2 (x2 ) : 3(2π )4    2 1 (gμν − ∂ μ ∂ ν ) + local terms . (7.96) ξ 2 + i

From tensor analysis, the local terms can be written, in general, as (a1 gμν + a2 ∂ μ ∂ ν + bμ ∂ ν + cν ∂ μ + d μν ) δ (4) (x1 − x2 ) where bμ and cν are fixed vectors while d μν is a fixed second-rank tensor. Covariance implies that there are no fixed vectors and no second-rank tensor other than the metric tensor gμν . Dimensional analysis and the absence of a mass parameter implies the vanishing of d μν . Finally, Ward identities imply conservation of the local terms which then restrict to X (gμν − ∂ μ ∂ ν ) δ (4) (x1 − x2 ) where X is a dimensionless parameter.

7.4.2.8 General Relativistic Gauges (2a)

In a general covariant gauge, there are additional terms to A2 factor −g2 fa1 bc fa2 bc , they read

. Again up to the

− (a − 1) : Aaμ11 (x1 )Aaμ21 (x2 ) : ∂ μ2 ∂ ν1 D(−) (ξ )∂μ2 ∂ν1 E (−) (ξ ) − (a − 1) : Aaμ11 (x1 )Aaμ22 (x2 ) : D(−) (ξ )∂μ2 ∂μ1 D(−) (ξ ) + (a − 1) : Aaμ11 (x1 )Aaμ22 (x2 ) :  ∂μ1 ∂ ν1 D(−) (ξ )∂μ2 ∂ν1 E (−) (ξ ) + ∂μ2 ∂ ν1 D(−) (ξ )∂μ1 ∂ν1 E (−) (ξ ) + (a − 1) : Aaμ11 (x2 )Fμa22μ1 (x1 ) + Aaμ11 (x1 )Fμa12μ2 (x2 ) :

∂ ν1 D(−) (ξ )∂μ2 ∂ν1 E (−) (ξ )

186

7 Field Theory Without Infinities

+ (a − 1) : Faμ12 μ1 (x1 )Fμν21 (x2 ) : D(−) (ξ )∂μ1 ∂ν1 E (−) (ξ ) +

(a − 1)2 : Faμ12 μ1 (x1 )F ν2 ν1 (x2 ) : ∂μ1 ∂ν1 E (−) (ξ )∂μ2 ∂ν2 E (−) (ξ ). 2

Because of the presence of the E functions, new distributions must be calculated. In order to simplify this calculation, one defines, as usual,the E function by a derivative of the D function with a squared mass parameter κ

d (−) (−) E (ξ ) = − D (ξ ; κ )

(7.97) dκ κ =0 where D(−) (ξ ; κ ) = −

i (2π )3



d 4 p θ (p0 )δ (p2 − κ )eip·ξ .

(7.98)

It is then clear that all the calculations of new distributions and their splitting will reduce to (7.99) I1 (ξ ; κ ) = −(2π )2 D(−) (ξ )D(−) (ξ ; κ ) and

I2 (ξ ; κ1 , κ2 ) = −(2π )2 D(−) (ξ ; κ1 )D(−) (ξ ; κ2 ).

(7.100)

7.4.3 Calculation of I1 (ξ ; κ ) Let us define the Fourier transform Iˆ1 (k; κ ) =



d 4 ξ I1 (ξ ; κ )eik·ξ .

(7.101)

As for the κ = 0 case, 

    (7.102) d 4 q θ (q0 )θ (k0 − q0 ) δ q2 − κ δ k2 − 2q · k + κ        d3q = & θ k0 − |q|2 + κ δ k2 − 2q0 |q|2 + κ + 2q.k + κ . (7.103) |q|2 + κ

Iˆ1 (k; κ ) =

The calculation is simplified if it is carried out in the particular frame where k = 0. Then, after using spherical coordinates for which angular integration is trivial, Iˆ1 (k0 ; κ ) = 2π Setting

&

 ∞ 0

q2 + κ = z,

Iˆ1 (k0 ; κ ) = 2π

    & & q2 dq & θ k0 − q2 + κ δ k02 − 2q0 q2 + κ + κ . q2 + κ (7.104)  ∞ √

κ

dz

&   z2 − κθ (k0 − z)δ k02 − 2k0 z + κ .

(7.105)

7.4 Application to Yang-Mills Theory

187

Of course the integral vanishes if k0 < 0. Therefore, after the introduction of step functions for taking this fact into account and for allowing the integration interval to be (−∞, +∞), Iˆ1 (k0 ; κ ) = 2πθ (k0 )

 ∞ −∞

dz

&   √ z2 − κθ (k0 − z)δ k02 − 2k0 z + κ θ (z − κ ).

(7.106) Integration over the variable z, using the Dirac-function, gives after some elementary algebra     ˆI1 (k0 ; κ ) = π θ (k0 )θ k02 − κ 1 − κ . (7.107) 2 k02 This result, holding in the particular frame where k = 0, can be analytically continued to an arbitrary frame and  π κ Iˆ1 (k; κ ) = θ (k0 )θ (k2 − κ ) 1 − 2 . (7.108) 2 k Its limit for κ → 0 is, of course, identical with the result previously obtained.

7.4.4 Calculation of Iˆ2 (k; κ 1 , κ 2 ) By definition, Iˆ2 (k; κ1 , κ2 ) =



d 4 qd 4 p θ (q0 )θ (p0 )δ (q2 − κ1 )δ (p2 − κ2 )δ (4) (k − p−q). (7.109)

The result must of course be symmetric in κ1 , κ2 . Performing the integration over q with the help of the δ (4) function and the integration over p0 ,     d3 p & Iˆ2 (k; κ1 , κ2 ) = θ k0 − |p|2 + κ2 2 |p|2 + κ2    2 2 δ k − 2k0 |p| + κ2 + 2p.k + κ2 − κ1 . (7.110) As in the previous case, this integration is performed in the particular frame where k = 0. Spherical coordinates are used again. Then Iˆ2 (k0 ; κ1 , κ2 ) = 2π

 ∞

&

p2 d p

  & θ k0 − p2 + κ2

p2 + κ2  & δ k02 − 2k0 p2 + κ2 + κ2 − κ1 . 

The change of variable z =

&

0

p2 + κ2 is again performed

(7.111)

188

7 Field Theory Without Infinities

Iˆ2 (k0 ; κ1 , κ2 ) = 2π

 ∞ √ κ2

&   dz z2 − κ θ (k0 − z)δ k02 − 2k0 z + κ2 − κ1 .

(7.112)

Integration using the δ function and some elementary algebra lead, in the same way as in the above cases, to  π Iˆ2 (k0 ; κ1 , κ2 ) = 2 θ (k0 ) k04 − 2k02 (κ2 + κ1 ) + (κ2 − κ1 )2 2k0 √ θ (k02 − κ2 + κ1 )θ [(k0 − κ2 )2 − κ1 ]. (7.113) Handling the last two step functions is now necessary. Let us assume, for simplicity, that κ2 ≤ κ1 . Then the step function θ (k02 − κ2 + κ1 ) is equal to 1 and can be dropped out. On another hand, the argument of the second step function can be decomposed into factors   √ 2 √ √  √ √  k0 − κ2 − κ1 = k0 − κ2 − κ1 k0 − κ2 + κ1 . The second of these factors is, by the fact that k0 is positive and the assumption κ2 ≤ κ1 , always positive so that it may be dropped out of the step function which √ √ reduces to θ [k0 − ( κ2 + κ1 ). Introduction of a positive factor in its argument √ √ 2 allows to use instead θ [k0 − ( κ2 + κ1 )2 . After analytic continuation to an arbitrary frame, the final result is  π Iˆ2 (k; κ1 , κ2 ) = 2 θ (k0 ) [k2 − (κ2 + κ1 )2 ] [k2 − (κ2 − κ1 )2 ] 2k   √ √ (7.114) θ k2 − ( κ2 + κ1 )2 . It is symmetric on κ1 and κ2 and holds for any value of these parameters. It can be checked that it reduces to the previous result for κ1 → 0 as well as to the known result ' 2 ˆI2 (k; m2 ) = π θ (k0 ) 1 − 4m θ (k2 − 4m2 ) (7.115) 2 k2 for κ1 = κ2 = m2 .

7.4.5 Calculation of the Tensor Distributions The distributions computed in the previous subsection do not occur directly in the course of the calculations but all the used distributions can be related to them. As an illustrative example,let us consider 

Pˆμν (k) = Let us begin with

d 4 x e−ik·x ∂μ ∂λ D(−) (x)∂ν ∂ λ E (−) (x).

7.4 Application to Yang-Mills Theory

Pˆμν (k; κ ) =



189

d 4 x e−ik·x ∂μ ∂λ D(−) (x)∂ν ∂ λ D(−) (x; κ ).

(7.116)

As in the previous calculations, the expressions of the generalized functions are introduced and the integrations over x and one of the momenta are carried out. This gives (k − κ ) Pˆμν (k; κ ) = − 2(2π )2 2



d 4 q θ (q0 )θ (k0 −q0 )δ (q2 − κ )δ (k2 −2k ·q+ κ )(k −q)μ qν . (7.117)

Because it is a second-rank tensor depending only on the vector k, the tensor method allows to set     Pˆμν (k; κ ) = A k2 gμν + B k2 kμ kν . (7.118) As previouly, the functions A and B can be obtained by taking the trace and multiplying by k μ kν . Using the expression (7.108) of Iˆ1 (k; κ ), one gets, after calculations in the same lines as here above and left to the reader, Pˆμν (k; κ ) = −

1 192π (k2 )3

  3 θ (k0 )θ k2 − κ k2 − κ   gμν k2 (k2 − κ ) + 2kμ kν (k2 + 2κ ) . (7.119)

Taking the derivative with respect to κ , setting κ = 0 and changing the sign lead to 

Pˆμν (k) = =−

d 4 x e−ik·x ∂μ ∂λ D(−) (x)∂ν ∂ λ E (−) (x)   1 θ (k0 )θ k2 (2gμν k2 + kμ kν ). 96π

(7.120)

Let us now quote a few results that can be obtained in the same way 

  1 (7.121) gμν θ (k0 )θ k2 , 32π    i (7.122) d 4 x e−ik·x ∂ ν D(−) (x)∂μ ∂ν E (−) (x) = kμ θ (k0 )θ k2 , 32π   1  gμν gλ τ + gμτ gλ ν + gμλ gντ d 4 x e−ik·x ∂μ ∂ν E (−) (x)∂λ ∂τ E (−) (x) = − 192π   (7.123) θ (k0 )θ k2 . d 4 x e−ik·x D(−) (x)∂μ ∂ν E (−) (x) = −

The derivation by hand of the last result is somewhat heavy. Use of computer algebraic programs as the packages CANTENS or HEPHYS of REDUCE allows one to get the result very quickly.

190

7 Field Theory Without Infinities

7.4.6 The Final Result for the Gluon Self-energy The scalar distribution appearing in (a − 1) and (a − 1)2 terms is the same as the one appearing in the Fermi gauge. Therefore, the splitting is, up to local terms, the same as the one occurring in the Fermi gauge. It is then a simple matter of algebraic calculations to get the final result for the gluon self-energy in any covariant gauge. Up to the factor −g2 fab1 c fab2 c /(2π )4 , it reads (2) T2

 2 1 −3a + 1 μν μ ν (g − ∂ ∂ ) =: : 6 ξ 2 + i  2 1 2a + 1 b1 Fμν (x1 )∂ μ Aνb2 (x2 ) : +: 2 3 ξ + i  2 1 2a − 5 μ b1 b2 ν ν + : Fμν (x1 )Ab2 (x2 ) − Ab1 (x1 )Fμν (x2 ) : ∂ . 6 ξ 2 + i (7.124) Abμ1 (x1 )Abν2 (x2 )

There is no contribution from the (a − 1)2 term because the tensor distribution is symmetric and contracted with antisymmetric tensors. Calculation of the remaining contributions to the second order S-matrix is left as an exercise.

7.5 Summary In finite field theory, the nth order of the S-matrix is given by Tn = An − An = Rn − Rn where An and Rn are defined by An (x1 , . . . , xn ) = ∑ T˜m (X) Tn−m (Y, xn ), P

Rn (x1 , . . . , xn )

= ∑ Tn−m (Y, xn ) T˜m (X). P

The difference Dn = Rn − An is a distribution with causal support and for which the retarded and advanced parts are respectively Rn and An . The splitting of Dn into advanced and retarded parts is unique only for distribution of negative singularity order. Otherwise, an arbitrary linear combination of Dirac function and its derivative of order less than or equal to the singularity order occurs. As in standard theory, the nth order Tn is not univoquely defined but there are no divergences at all. In class III gauges of Yang-Mills theory, the construction of the second order is realized. Nonuniqueness of T2 at the level of tree graphs is removed by the BRST

References

191

invariance of this contribution. It gives rise to the usual quartic term in gauge fields. At the level of loops, only the gluon self-energy is explicitly computed. It involves the distribution   1 d 1 HT (x) = − . (2π )4 dx2 x2 + i After partial integrations, it reads (2)

T2 (x1 , x2 ) =

−11a + 9 2 g fab1 c fab2 c : Abμ1 (x1 )Abν2 (x2 ) : 6 (gμν − ∂ μ ∂ ν ) HT (x1 − x2 ).

References 1. Epstein, H., Glaser, V.: Ann. Inst. Poincar´e 29, 211 (1973) 157 2. Scharf, G.: Finite Quantum Electrodynamics. The causal approach.Texts and Mono-graphs in Physics, Springer, Berlin (1995) 157, 164, 174 3. Caprasse, H.: unpublished 157 4. Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions, NorthHolland Math. Studies, vol. 84 (1984) 157

Chapter 8

Gauges with a Singular C Matrix

8.1 Introduction Two class III gauges with singular C matrix and, consequently, lack of isotropic power counting are of interest: the Coulomb gauge and the family of Leibbrandt gauges. The Coulomb gauge is of historical importance. Often considered as a gauge with no unphysical degree of freedom (class I gauge), it plays an important role in the search for classical solutions. However, its class I property holds only in singular coordinate frames where quantization is cumbersome. In a nonsingular coordinate frame, it is a class III gauge characterized by the tensor Cμν = nμ nν − gμν with n2 = 1. It is clear that det C = 0. The static temporal gauge with Cμν = nμ nν also shows this feature. Leibbrandt gauges are of more recent interest. First considered as resulting from a prescription for the unphysical pole in class II axial gauges, their class III character is nevertheless obvious through the C-matrix Cμν = n∗μ nν . Their interest in perturbative calculations comes from the fact that, with class II gauges, no Faddeev-Popov ghost is required. This ghost-free character is assumed to be conserved in their class III partners because “ghost loops vanish in dimensional regularization”. However, Faddeev-Popov ghosts are required in the BRST formalism associated with class III gauges. Their contribution cannot be computed as it is done in Chap. 6 for other class III gauges because the C-matrix is singular. The present chapter will deal with a method of computing the ghost and other singular contributions by regulating the C-matrix through an interpolation with covariant gauges. Such an interpolation restores the usual power-counting. The singular case is obtained as a limit on the interpolating parameter. For the ghost contribution, this limit exhibits a pole which explains why the det C = 0 case is singular. However, because all the requirements of a consistent gauge i.e. a consistent free theory and the BRST invariance are satisfied, it

Burnel, A.: Gauges with a Singular C Matrix. Lect. Notes Phys. 761, 193–208 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 8 

194

8 Gauges with a Singular C Matrix

is expected that the S-matrix will remain well-defined in the limit. A cancellation of the various singular contributions is expected. We give here an example of this cancellation in the case of the gluon self-energy. The reader will be quickly aware that the calculations are heavier than in a covariant gauge; there is also no ghost decoupling so that there is no particular advantage to use these gauges in practical calculations. They should be considered only as a particularly tedious exercice in noncovariant quantization.1 The chapter is organized as follows. First, the ghost-loop integral is discussed in Leibbrandt gauges and shown to be not defined. Interpolating between such gauges and relativistic ones allow to show how these singular integrals can be regulated. In a second time, the loop integrals are computed in a way as general as possible at the beginning. In order to carry out the entire calculation, a gauge must be chosen. One takes an interpolating planar-type gauge and computes all the divergent parts of the contributing integrals. This task is almost impossible by hand and the results were obtained with the help of REDUCE which was also used to show that the singularities in the interpolating parameter cancel out in the self-energy, as expected. Most parts of this chapter are highly technical and some of the calculations are only sketched.

8.2 The Ghost Loop Contribution 8.2.1 The Leibbrandt Gauges The ghost loop contribution is the simplest example of the singularities that appear in gauges with a singular C-matrix. In order to study such singularities, let us first recall that, in covariant gauges, this contribution can easily be computed either in the usual dimensional regularization or in the finite theory. The result is not ambiguous and is independent of the gauge parameter a. In the Leibbrandt gauges, it is usually assumed to vanish, at least in the dimensional regularization approach, although we have shown that its calculation involves det C, which is here vanishing, at the denominator. It is of pedagogical interest to see why the usual arguments for the vanishing cannot work. In Minkowski space, Leibbrandt gauge calculations involve the integral 

IF =

d 2ω p

n∗ · p n∗ · (p + q) F(p, q) [n∗ · p n · p + i][n∗ · (p + q) n · (p + q) + i]

(8.1)

with various F functions. It is easy to show that it can be written IF = IF,1 + IF,2 1

(8.2)

This result is in contradiction with a popular calculation without ghost contribution but nonlocal counter-terms [1]. Of course, this nonlocality is in contradiction with all the standards of renormalization theory. This point will not be considered further here.

8.2 The Ghost Loop Contribution

with IF,1 =

1 n·q

IF,2 = −i



d 2ω p



195

 n∗ · p n∗ · (p + q) − F(p, q), n∗ · p n · p + i n∗ · (p + q) n · (p + q) + i (8.3)

n∗ · q IF . n·q

(8.4)

This formula reflects the historical way of considering (wrongly) the Leibbrandt gauges as a prescription for the poles occurring in axial gauges. The last term IF,2 is generally overlooked because it involves the same integral as IF but multiplied by a factor  which is taken equal to zero at the end of the calculation. If the change of variable p + q = p is made in the second integral occurring in IF,1 when F = 1, the same integral as the first one is obtained and, finally, I1 = I1,1 = 0. However, both the neglect of IF,2 and the vanishing of I1,1 assume that, after dimensional regularization, the involved integrals make sense. They will only if the F function is sufficiently regular. They are not defined when F = 1 or for another function which does not vanish at infinity. The same kind of wrong results can be obtained in the finite theory. From (7.65), in a general gauge, the ghost contribution involves the distribution Pμ1 μ2 (x) =

1 ˆ μ2 (−) (−) ∂ DC (x) ∂ˆ μ1 DC (x). 2 C00

(8.5)

Its Fourier transform Pˆ μν (k), given by Pˆ μ1 μ2 (k) =

1 (2π )2



ˆ pˆ μ2 (kˆ − p) d 4 p θ (p0 ) δ (pC2 ) θ (k0 − p0 ) δ (kC2 − k˜ · p − p · k) ˆ μ1 ,

(8.6) ˜ In the case is a second-rank tensor depending on the momenta k, kˆ and k− = kˆ − k. of the Leibbrandt gauges, characterized by Cμν = n∗μ nν , it becomes −1 Pˆμν (k) = nμ nν (2π )2



Let us write it 

Ii (k) =

kˆ μ = n∗ · k nμ ,

(8.7)

d 4 p δ (pC2 ) θ (p0 ) δ ((k − p)C2 ) θ (k0 − p0 ) p · n∗ n∗ · (k − p). 1 nμ nν [I2 (k) − k · n∗ I1 (k)] (2π )2

(8.8)

d 4 p δ (pC2 ) θ (p0 ) δ ((k − p)C2 ) θ (k0 − p0 ) (p · n∗ )i .

(8.9)

Pˆμν (k) = where

kC2 = n∗ · k n · k,

196

8 Gauges with a Singular C Matrix

A superficial inspection of these integrals, using (p · n∗ )i δ (p · n∗ p · n) = 0,

i > 0,

(8.10)

would lead to their vanishing. This is however erroneous because such distributionlike relations hold only when the rest of the integrant is a test-function with a regular behaviour. Here, the rest of the integrant is a distribution so that (8.10) cannot be used. One is here faced with erroneus results obtained from a uncareful use of distribution-like relations.

8.2.1.1 The Case n2 = n∗2 Let us now show how to get the correct behaviour for the integrals occurring in the finite theory, first in the case n2 = n∗2 . In this case, a preferred frame in which n = (n0 , n),

n∗ = (n0 , −n),

n0 > 0

(8.11)

can be found. Then 

d 4 p δ [(p0 n0 )2 − (p.n)2 ] θ (p0 ) δ (kC2 − 2n0 k0 n0 p0 + 2n.k n.p)

Ii (k) =

θ (k0 − p0 ) (p0 n0 + n.p)i .

(8.12)

Integrating over p0 , 

d3 p δ (kC2 − 2k0 n0 |n.p| + 2n.p n.k)θ (n0 k0 − |n.p|) (|n.p| + n.p)i . |p.n| (8.13) Because, for i > 0, one obviously has Ii (k) =

1 2n0

(|n.p| + n.p)i = (2 n.p)i θ (n.p), these integrals become Ii (k) =

2i−1 n0



d 3 p θ (n.p)(n.p)i−1 δ (k · n k · n∗ − 2k · n n.p)θ (k0 n0 − n.p) (8.14)

Use of various properties of the Dirac δ -function leads to 2i−2 Ii (k) = θ (k · n∗ ) n0 (n · k)



k · n∗ 2

i−1 

d 3 p θ (n.p) δ (n.p − k · n∗ /2).

(8.15)

Using spherical coordinates for which the integration over azimutal angle is trivial and the one over the radius is performed with the help of the Dirac function, one gets

8.2 The Ghost Loop Contribution



197

d 3 p θ (n.p) δ (n.p − k · n∗ /2) =

 1 π du ∗ 2 ∗ (k · n ) θ (k · n ) . 3 3

2|n|

0

u

(8.16)

The last integral is obviously divergent. This result confirms that the ghost contribution is infinite in Leibbrandt gauges. It is here obtained in an approach which is free of assumptions about ultraviolet regularization. The divergence is then an intrinsic property of the gauge itself and not a defect of a particular regularization scheme. 8.2.1.2 The Case n2 = −n∗2 Let us now consider the case of a vector, for instance n∗ , time-like and the other one space-like. The integrals (8.9) are computed in a frame in which n∗ = (1, 0, 0, 0),

n = (0, 0, 0, 1).

It is easy to obtain

θ (k0 ) Ii (k) = k0 k3



   i d q δ (k3 − q3 ) dq0 q0 δ (q0 ) θ (q0 ) . 3

(8.17)

The second integral vanishes while the first one involves integration of the factor 1 from −∞ to which +∞ which is infinite. The result writes 0 × ∞ and is thus not defined. In this case, the ghost contribution can again not be computed directly in Leibbrandt gauges.

8.2.2 Interpolating Between Leibbrandt and Relativistic Gauges 8.2.2.1 Light-Like n and n∗ Because the ghost-loop cannot be computed directly in the Leibbrandt gauges, let us regulate the involved integrals by an interpolation between these gauges and relativisic gauges. With Leibbrandt gauges, the asymptotic behaviour for large momenta is not the same in all directions of space. The interpolation restores the isotropy and is therefore called here power counting regularization. Moreover, for simplicity of the calculations, this interpolation is made in such a way that it leads to a planar-type gauge. To this aim, one takes Cμν = n∗μ nν −

α nμ n∗ν + α gμν . α +1

Restriction to light-like vectors n2 = n∗2 = 0, n · n∗ = 1 is further made in order to simplify the expressions and the Leibbrandt gauge is considered as the limit for α → 0 of this interpolating gauge. In the preferred frame

198

8 Gauges with a Singular C Matrix

n∗k = −nk ,

n∗0 = n0 ,

one gets, after elementary calculations, (n.p)2 , pk pl C = −α |p| − α +1 kl

2

  n.kn.p pk kl (C +C ) = −2 α k.p + . (8.18) α +1 kl

lk

With respect to the integration variable p, the integrant involves the scalar product with two fixed vectors k and n. In spherical coordinates, elliptic integrals are thus involved. Because the final goal is not to perform a complete calculation but only to study the behaviour near the value α = 0, a particular case, for instance k = 0, can be taken. With respect to covariant gauges, this is not a particular choice of coordinate frame because such a frame is already chosen to be a preferred one. This choice amounts to working only with a particular four-momentum. In this particular case, the integrals involved in the ghost contribution become

  

C00 k02 1 (n.k)2 − α |p|2 − (8.19) θ (k0 ) d 3 p δ IF (k0 , 0) = F(k, p)

2k0 |C00 | 4 α +1 sub where “sub” means now the substitution p0 = k0 /2, k = 0 in the various functions F(k, p) which can occur in the calculation. Use of spherical coordinates and of n20 = |n|2 = 1/2, C00 = 1/2 implies 2π IF (k0 , 0) = θ (k0 ) k0

 1

 ∞

du p2 d p 0

  2

k0 p2 u2 . (8.20) − [α (α + 1) + ] F(k, p)

δ 8 α +1 2 sub

−1

Integration over p can be carried out with the help of the δ -function. The result is

 1

π du 3/2

√ (8.21) θ (k0 )(α + 1) F(k, p) IF (k0 , 0) =

−1 N(u)3/2 2 2 sub where N(u) = α (α + 1) +

u2 2

(8.22)

and “sub” means now the substitutions p0 = k0 /2, k = 0, |p|2 = k02 (α + 1)/8N(u) in F(k, p). As an example, let us compute IF (k0 , 0) when F = p2 =

k02 [u2 + (α + 1)(2α − 1)]. 8N(u)

Then I p2 (k0 , 0) = 2πθ (k0 )

6α 3 + 9α 2 + 6α + 1 3α 2 (α + 1)(2α 2 + 2α + 1)3/2

.

(8.23)

8.3 Singularities Generated by the Lack of Power Counting

199

The result of such an integration is singular at α = 0. In the calculation of the ghost contribution, it is however multiplied by a coefficient α 2 so that the contribution of this integral is finite. All the integrations involved in the calculation of the ghost contribution can be performed in this way. This calculation is somewhat long but the final result can /2. Therefore, in order to get the easily be guessed. Indeed, when α = 0, N(u) = u2& singularity at α = 0, it is sufficient to replace u by 2N(u) in the expression (8.16). Integration over u can be carried out √ du 2 2 √ . = −1 N(u)3/2 α (α + 1) 2α 2 + 2α + 1

 1

This shows how the ghost contribution becomes singular in the Leibbrandt gauges when they are regulated by interpolating with relativistic gauges. The singularity is a pole at α = 0. 8.2.2.2 The Case n2 = −n∗2 In this case, regularization of power counting must also be used. Here, there is no possibility to regulate the power counting with planar type gauges so that the gluon contribution will be more complicated but the ghost contribution is still given by (8.6). Here the regularization is made through Cμν = n∗μ nν + α gμν .

(8.24)

All the calculations can be carried out in a way similar to the previous case. To show that the lack of power counting appears through the presence of singularities in α −1 , it is again sufficient to carry out the calculation in the simpler case q = 0 ˆ Straightforward manipulations lead to with F(q, k) = kˆ · k. I≈

π 2 q θ (q0 ) 3α 0

(8.25)

in the limit α → 0. Again this singularity shows how, in finite theory, the ghost contribution is divergent in the limit α → 0.

8.3 Singularities Generated by the Lack of Power Counting in Ultra-Violet Divergent Perturbative Theory It is of interest to see how the singularities generated by the lack of power counting, which are present as poles in the interpolating parameter, occur in various Feynman

200

8 Gauges with a Singular C Matrix

graphs of the standard perturbative approach with ultra-violet dimensionally regulated integrals and in S-matrix elements. The calculations become quickly very heavy so that one restricts oneself to a case as simple as possible and limits the developments to essential features. The result is that power counting singularities are present in distinct Feyman graphs but cancel out in the S-matrix. The calculations are carried out in the general case as far as possible. When a particular gauge is used, it is chosen as interpolating between relativistic and Leibbrandt gauges in such a way to simplify at most the calculations.

8.3.1 The Loop Integration in a General Gauge A general method to compute dimensionally regularized integrals in any noncovariant gauge is developped in Chap. 5. In Euclidean space, a general integral reads 2n Eμa11 ···a ...μm =



p μ1 . . . p μm d4 p (2π )4 ∏ni=1 [(p + qi )2 ]ai [(p + qi )C2 ]ai+n m

= (−1)

m/2

πω (2π )2ω ∏2n i=1 Γ (ai )

or (m−1)/2

∑

−k

2 Γ

 1 2n ai −1 d ξi ∏i=1 ξi 0



det(B−1 0 ) !

2n

∑ ai − k − ω

δ

2n

!

1 − ∑ ξi i=1

ω +k−∑2n i=1 ai

A0

i=1

k=0

 B0μm μm−1 · · · B0μm−2k+2 μm−2k+1



m−2k

∏ (B0 Q0 )μi + sym.

(8.26)

i=0

where B0 is the inverse of the matrix B−1 0 and B−1 0 =

n

n

i=1

i=1

∑ (ξi 1| + ξn+iC) = 1| − (1| −C) ∑ ξn+i ,

(8.27)

n

Q0 =

∑ (ξi qi + ξn+iCqi ),

(8.28)

i=1 n

A0 =

∑ (qi , (ξi 1| + ξn+iC)qi ) − (Q0 , B0 Q0 ).

(8.29)

i=1

Obviously, this is horrifying in the general case so that one will restrict oneself to a simpler case.

8.3 Singularities Generated by the Lack of Power Counting

201

8.3.2 Restriction to Loops with Two External Particles In the case of loops with two external particles, e.g. the self-energy, the value of the parameter n is equal to 2 and the values of q1 and q2 can be taken, from energymomentum conservation at each vertex, as q1 = 0,

q2 = q.

With such momenta, the expressions of the different parameters B−1 0 , Q0 and A0 become B−1 0 = 1| − (1| −C)(ξ3 + ξ4 ),

(8.30)

Q0 = (ξ2 1| + ξ4C)q,

(8.31)

A0 = (q, (ξ2 1| + ξ4C)(ξ1 1| + ξ3C)B0 q).

(8.32)

It is important to note for the following that B−1 0 depends only on the sum of the Feynman parameters ξ3 + ξ4 .

8.3.3 Further Restriction for m ≤ 6 After reduction to the same denominator, all integrals in the calculation of the selfenergy have denominators of the form p2 (p + q)2 pC2 (p + q)C2 and the numerator is a second-rank tensor involving always six momenta which are either p or q. Therefore with m ≤ 6 must be computed. As a typical example let us expressions like Eμ1111 1 ...μm 1111 consider Eμ1 ...μ6 explicitly: Eμ1111 1 ...μ6

=π

ω



 1 4

4

∏ d ξi δ (1 − ∑ ξi )(det B0 )1/2

0 i=1

i=1

Γ (4 − ω )A0ω −4 (B0 Q0 )μ1 · · · (B0 Q0 )μ6

(8.33)

  1 Γ (3 − ω )A0ω −3 B0μ6 μ5 (B0 Q0 )μ1 · · · (B0 Q0 )μ4 + sym 2   1 + Γ (2 − ω )A0ω −2 B0μ6 μ5 B0μ4 μ3 (B0 Q0 )μ1 (B0 Q0 )μ2 + sym 4    1 ω −1 B0μ6 μ5 B0μ4 μ3 B0μ2 μ1 + sym + Γ (1 − ω )A0 (8.34) 8 +

where “sym” means the convenient symmetrization on the indices. When power counting holds, the divergent part of this integral is regularized as a pole at ω = 2

202

8 Gauges with a Singular C Matrix

which is present in the Γ -functions Γ (2 − ω ) and Γ (1 − ω ) = Γ (2 − ω )/(1 − ω ). Therefore, the divergent part can be written as ! 4 2 Γ (2 − ω )  1 4 π = Eμ1111 ∏ d ξi δ 1 − ∑ ξi (det B0 )1/2 1 ...μ6 div 4 0 i=1 i=1    B0μ6 μ5 B0μ4 μ3 (B0 Q0 )μ1 (B0 Q0 )μ2 + sym   1  − A0 B0μ6 μ5 B0μ4 μ3 B0μ2 μ1 + sym . 2 (8.35) Let us now perform as many integrals as possible. 8.3.3.1 Integration Over ξ 1 The presence of the δ -function makes the integration over the Feynman parameter ξ1 very easy. Care must however be taken with the limits of integration. To this aim, one transforms finite integrals into infinite ones by using step functions and performs the integrals using the properties of the Dirac function. This leads to ! !   1

0

d ξ1 δ

4

∞

i=1

−∞

1 − ∑ ξi F(ξi ) =

=θ

d ξ1 δ

4

4

1 − ∑ ξi F(ξi )θ (1 − ξ1 )θ (ξ1 ) !

i=1

1 − ∑ ξi θ i=2

4

∑ ξi

i=2

! F(ξi )|ξ1 =1−ξ2 −ξ3 −ξ4 .

The variable ξ1 occurs only in A0 not in B0 so that integration over it gives !  4 π 2Γ (2 − ω ) 1 4 1111 Eμ ...μ div = ∏ d ξi θ 1 − ∑ ξi (det B0 )1/2 1 6 4 0 i=2 i=2    B0μ6 μ5 B0μ4 μ3 (B0 Q0 )μ1 (B0 Q0 )μ2 + sym   1  − A0 B0μ6 μ5 B0μ4 μ3 B0μ2 μ1 + sym 2

(8.36)

where now A0 = (q, (ξ2 1| + ξ4C)[(1 − ξ2 − ξ3 − ξ4 )1| + ξ3C]B0 q). Let us anticipate a little on a forthcoming change of variables by setting

ξ = ξ3 + ξ4 .

(8.37)

8.3 Singularities Generated by the Lack of Power Counting

203

8.3.3.2 Integration Over ξ2 Integration over ξ2 is limited by the θ -function to an integral from 0 to (1 − ξ ). The upper limit must be positive so that a factor θ (1 − ξ ) is occurring. The parameter ξ2 occurs only in A0 and Q0 . After straightforward calculations, the integrals occurring in (8.36) give the results A0 (ξ3 , ξ4 ) =

 1−ξ

A0 d ξ2     (1 − ξ )2 ξ (1 − ξ ) 2 1| + C + ξ3 ξ4C B0 q ,(8.38) = (1 − ξ ) q, 6 2

Bμ1 μ2 (ξ3 , ξ4 ) =

0

 1−ξ 0

(B0 Q0 )μ1 (B0 Q0 )μ2 d ξ2

(1 − ξ )3 (B0 q)μ1 (B0 q)μ2 + ξ42 (1 − ξ )(B0Cq)μ1 (B0Cq)μ2 3  ξ4 (1 − ξ )2  (B0 q)μ1 (B0Cq)μ2 + (B0Cq)μ1 (B0 q)μ2 . (8.39) + 2 =

8.3.3.3 Integration Over a Further Feynman Parameter A further integration over one of the remaining variables ξ3 and ξ4 can still be carried out. It is useful to perform the change of variables

ξ3 = ξ − ξ  ,

ξ4 = ξ 

and to integrate over ξ  which is occurring only in A0 and Bμ1 μ2 . The result of this integration is      ξ (1 − ξ )2 ξ (1 − ξ ) ξ2 1| + C + C2 B0 q , A0 (ξ , ξ  ) d ξ  = ξ (1 − ξ ) q, 6 2 6 0 (8.40)   ξ 2 (1 − ξ ) (B0 q)μ1 (B0 q)μ2 Bμ1 μ2 (ξ , ξ  ) d ξ  = ξ (1 − ξ ) 3 0 +

ξ2 (B0Cq)μ1 (B0Cq)μ2 3

  ξ (1 − ξ )  (B0 q)μ1 (B0Cq)μ2 + (B0Cq)μ1 (B0 q)μ2 . + 4 Setting

I¯ = π 2Γ (2 − ω ),

(8.41)

(8.42)

and performing these three integrations over Feynman parameters, one gets the following expression for this particular one-loop integral with two external legs

204

8 Gauges with a Singular C Matrix



 I¯ 1 Eμ1111 d ξ ξ (1 − ξ ) [det B0 (ξ )]1/2 B0μ6 μ5 (ξ )B0μ4 μ3 (ξ ) = 1 ...μ6 div 4 0   (1 − ξ )2 ξ2  (B0 q)μ1 (B0 q)μ2 + (B0Cq)μ1 (B0Cq)μ2 + sym 3 3   ξ (1 − ξ )  (B0 q)μ1 (B0Cq)μ2 + (B0Cq)μ1 (B0 q)μ2 + 4  1 − B0μ6 μ5 (ξ )B0μ4 μ3 (ξ )B0μ2 μ1 (ξ ) + sym 2     ξ (1 − ξ ) ξ2 2 (1 − ξ )2 1| + C + C B0 q . q, 6 2 6

(8.43)

8.3.3.4 The Last Integration Integration over the last parameter ξ can be realized only if an explicit expression for B0 is given. In other words, the gauge must be chosen. Before the gauge choice and in order to simplify the writing, let us make the change of integration variable

ξ=

1 . 1+z

This leads to

  I¯ ∞ 1/2 L μ1 μ2 L μ3 μ4 zdz (det L) = 4 0  2  z z (Lq)μ5 (Lq)μ6 + (Lq)μ5 (CLq)μ6 + (CLq)μ5 (Lq)μ6 3 4  1 + (CLq)μ5 (CLq)μ6 + sym 3    1 2 1 2 1 1 − Lμ1 μ2 Lμ3 μ4 Lμ5 μ6 + sym (q, [ z 1| + zC + C ]Lq) (8.44) 2 6 2 6

Eμ1111 1 ...μ6 div

where the matrix L is given by L = L(z) = (z1| +C)−1 .

(8.45)

This integral is only one particular case of those occuring in the self-energy. Let us give, without details, the results for other integrals with nonvanishing divergent part. They can be computed as above and are    I¯ ∞ 1/2 Eμ1111 L zdz (det L) L + sym , = μ μ μ μ 1 2 3 4 1 ...μ4 div 4 0

(8.46)

8.3 Singularities Generated by the Lack of Power Counting

 I¯ ∞

Eμ1111 =− 1 ...μ5 div 8

0

205

  zdz (det L)1/2 Lμ1 μ2 Lμ3 μ4 qμ5 + sym .

(8.47)

These results hold for any covariant or noncovariant class III gauge provided a preferred frame can be found, a necessary condition for performing the Wick rotation.

8.3.4 Interpolating Between Leibbrandt and Relativistic Gauges In order to proceed further, one needs the explicit expression of the C matrix. Because our aim is to study the singularity of Leibbrandt gauges, this matrix is chosen as Cμν = n∗μ nν −

α nμ n∗ν + α gμν , 2 α + n + n · n∗

Cμ = n∗μ +

α (n2 )2 nμ . (8.48) α + n2 + n · n∗

As already stated here above, it is a planar-type gauge interpolating between Leibbrandt and relativistic gauges. A planar-type gauge is only possible for n2 = n∗2 , a case to which one was already restricted by the condition of finding a preferred frame. This frame can be chosen in such a way that n = (n0 , |n|, 0, 0),

n∗ = (n0 , −|n|, 0, 0),

It is useful to introduce the vectors n± =

n ± n∗ , 2

n+ = (n0 , 0, 0, 0),

n− = (0, |n|, 0, 0).

Then, with these choices and notations, elementary calculations give    2 2  2n− n+ 2(n2+ )2 2 2 2 pCM = + α p0 − p1 + α − α p22 − α p23 , 2n2+ + α 2n2+ + α    2 2  2n− n+ 2(n2+ )2 2 2 2 = + α + p + α −pCE p + α p22 + α p23 0 1 2n2+ + α 2n2+ + α and the C matrix in Euclidean space is   2n2− n2+ 2(n2+ )2 C = diag α + 2 ,α + 2 , α, α . 2n+ + α 2n+ + α

(8.49)

The matrix L is then easily obtained as  1 1 L = diag , , z + α + 2(n2+ )2 /(2n2+ + α ) z + α + 2n2− n2+ /(2n2+ + α )  1 1 , . (8.50) z+α z+α

206

8 Gauges with a Singular C Matrix

They can be written in a tensor-like way in the Euclidean space as C μν = αδ μν + L

μν

2n2+ μ μ (n nν n nν ), 2n2+ + α + + − −

(8.51)

 μ n+ nν+ 1 = δ μν − z+α z + α + 2(n2+ )2 /(2n2+ + α ) −

 μ n− nν− . z + α + 2n2− n2+ /(2n2+ + α )

(8.52)

It is clear that the integrals over z are all convergent for α > 0 but they become divergent when α = 0. As long as only the problem of convergence or divergence near α = 0 is concerned, α can be neglected with respect to 2n2+ and the approximation z + α + 2(n2+ )2 /(2n2+ + α ) −→ z + α + n2+

(8.53)

can be done. Note that α is not neglected further in this expression. These approximations amount to using a C matrix corresponding to the gauge in which Cμν = n∗μ nν + α gμν .

(8.54)

The reader can ask the question why this gauge was not chosen from the beginning. The reason is that it is not a planar-type gauge and its propagator involves then much more terms inducing heavier calculations. Here, we have shown that, when only the behaviour near α = 0 is taken into account, both gauges are equivalent so that, to study this behaviour, one can use (8.54) with a planar-type propagator. In the gauge (8.54), 1  . (8.55) (det L)1/2 = & (z + α ) z + α + n20 z + α + |n|2 When n2 = 0, integrals are more complicated than when n2 = 0 so that, in order to make a further simplification, light-like vectors n and n∗ are used in the following: n2 = n∗2 = 0,

n20 = |n|2 = n2+ = n2− =

n · n∗ . 2

With these restrictions, all the integrals are of the type  ∞

[a, b, c] = 0

za dz (z + α )b (z + α + n20 )c

(8.56)

where a, b, c are integers. They are given in any table of integrals. We have now all the elements allowing the computation of all the integrals involved in the self-energy but the task is extremely heavy and almost impossible by hand. Use of REDUCE in a very elaborate way was needed.2 Of course, we only report the results. Setting 2

The way REDUCE was used in all these calculations is described in [2]

8.4 Cancellation of Divergences at α = 0 in the Self-Energy

I¯ = iπ 2Γ (2 − ω ),

207

r = p + q,

the integral involved in the ghost contribution gives     1 n∗ · p n∗ · r 8 ∗ ¯ d 2ω p 2 − I(q · n ) + ≈ . α →0 3α n · n∗ (n · n∗ )2 (rC + i)(pC2 + i) div (8.57) It confirms that the lack of isotropic power counting in the Leibbrandt gauge manifests itself by a divergence at α = 0. Such a divergence can also be found in   pμ1 · · · pμ4 n∗ · p n∗ · r 2ω . d p 2 (r + i)(p2 + i)(rC2 + i)(pC2 + i) div On the other hand, some final integrals are convergent at α = 0. For instance,   n∗ n∗ pμ1 pμ2 n∗ · p n∗ · r ¯ μ1 μ2 , (8.58) = − I d 2ω p 2 (n · n∗ )2 (r + i)(p2 + i)(rC2 + i)(pC2 + i) div   pμ1 pμ2 pμ3 n∗ · p n∗ · r I¯ = d 2ω p 2 2(n · n∗ )2 (r + i)(p2 + i)(rC2 + i)(pC2 + i) div   ∗ ∗ qμ1 nμ2 nμ3 + sym . (8.59) Let us note that corrections of order α are expected when the approximation (8.53) is not made. In both cases, terms of order 1/α are the same but finite terms can be different. Therefore, using the gauge Cμν = n∗μ nν + α gμν as a planar-type gauge instead of the exact planar-type gauge Cμν = n∗μ nν −

α nμ n∗ν 2 α + n + n · n∗

+ α gμν

will not affect the singular behaviour near α = 0 but only the regular one. In front of the large number of terms involved in the calculation, such an approximation is useful for getting the divergent part at α = 0.

8.4 Cancellation of Divergences at α = 0 in the Self-Energy In the case of the ultraviolet divergent contribution to the gluon self-energy, all the results of integration have been gathered using again REDUCE in a very elaborate way. The final expression of the divergent part of the self-energy is free of any singularity at α = 0. Such a remarkable cancellation [3] can look like a miracle in front of the high number of divergent terms which is handled (the way the calculation is carried out involves singularities α −n with n ≤ 6). However, because Leibbrandt

208

8 Gauges with a Singular C Matrix

gauges are true gauges for Yang-Mills theory, it is expected in any expression occurring in the S-matrix, both for ultra-violet divergent and for finite parts, as well as in the finite theory. Both in the finite theory and in the ultra-violet divergent contribution, the ghost contribution and the gluon one exhibit the same kind of divergence at α = 0. Both contributions can be built with Feynman rules with a difference in the contractions of fields only. A similar structure of terms occurs in both quantities. Therefore, the final cancellation of divergences at α = 0 will also occur in the finite theory. In this case, an explicit check is still heavier than the check performed for the ultraviolet divergent part. Indeed, about 60 tensor valued distributions must be computed. Some of them are tensors of rank 4 depending on 3 momenta, involving therefore 138 scalar valued distributions which can be computed only by hand. The calculation of the divergent part of the gluon self-energy is an illustration of what is happening when isotropic power counting does not hold. After regularization of power counting, some Feynman integrals show further singularities at the critical value of the regularization parameter. These singularities cancel out when all the terms contributing to the S-matrix are taken into account. This cancellation must work at any level provided the gauge with lack of isotropic power counting is really a gauge, a fact which must be checked by the existence of a consistent free field theory and BRST invariance of the Lagrangian.

References 1. Leibbrandt, G.: Noncovariant gauges. Quantization of Yang-Mills and Chern-Simons Theory in Axial-Type Gauges, World Scientific, Singapore (1994) 194 2. Burnel, A., Caprasse, H.:Int. J. Mod. Phys. C3, 321 (1992) 206 3. Burnel, A., Caprasse, H.: Phys. Lett. 265B, 355 (1991) 207

Chapter 9

Conclusion

The study of Yang-Mills theory from quantization to renormalization in the canonical framework shows features that cannot be found with other quantization procedures. Noncovariant gauges of Yang-Mills theory are, from this point of view, particularly instructive. If a consistent quantization procedure following the same pattern as covariant gauges is required, a noncovariant gauge condition must be written as Cμν ∂ μ Aν = 0. Like in the covariant case, an inhomogeneous part is also allowed. The fixed tensor Cμν must be such that the differential operator Cμν ∂ μ ∂ ν be hyperbolic. Moreover, the coordinate frame in which quantization is carried out must be such that C00 = 0. Then, the gauge condition is introduced into the Lagrangian through a Lagrange multiplier and BRST invariance is imposed in order to guarantee unitarity of the theory. Such restrictions do not appear when path-integral methods are used to quantize the theory. Of course, if they are taken into account, path-integral methods can be applied in the same way as in covariant gauges. Let us quote some of the troubles appearing with careless path-integral methods and their solution. 1. Gribov ambiguities. They occur in the Faddeev-Popov quantization procedure when the chosen gauge condition does not restrict the phase space to the physical one. In canonical formalism with class III gauge conditions, the Fock space is an indefinite metric space including an unphysical part. Physical states are cohomology classes of the BRST operator for covariant or noncovariant gauge fixings. There is no ambiguity in such a procedure. 2. Operator ordering problems in the Coulomb gauge. They occur because the Gauss law is solved. Solution of the Gauss law is required if and only if a singular frame C00 = 0 is chosen for the quantization procedure. In nonsingular frames, such a difficulty does not occur at all. 3. Presence of ghosts in Coulomb gauge. In its usual formulation, the Coulomb gauge is class I and should then involve only physical degrees of freedom. Why should then ghosts be required? Actually, the Coulomb gauge is class III in a general coordinate frame. With class III gauges, Faddeev-Popov ghosts are always present. 4. Prescription ambiguity for the unphysical pole at n · k = 0 in axial gauges. In path-integral quantization, such an ambiguity is related to an incomplete gauge fixing. The canonical study of axial gauges as class II gauges shows features that

Burnel, A.: Conclusion. Lect. Notes Phys. 761, 209–210 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 9 

210

9 Conclusion

are completely different from those found in a usual field theory. Unphysical degrees of freedom are governed by a first-order differential evolution equation. The relation between an antiparticle and a particle moving backward in spacetime is lost. Moreover, the Hamiltonian is not bounded from below so that no ground state exists. The solution of this question remains open but it is not limited to find a particular prescription for the unphysical pole. A way to tackle consistently the unphysical degree of freedom must be found and a mechanism restricting the whole Fock space to physical states must be built. Such a mechanism is completely different from the BRST formalism. Let us note that the “Leibbrandt-Mandelstam prescription” is actually not a prescription for handling the unphysical pole but rather a new family of gauges characterized by a class III gauge condition n∗ · ∂ n · A = 0. Though generally overlooked, ghosts are present with a singular contribution related to the singularity of the C-matrix. When perturbative calculations are involved, an additional restriction on the C-tensor must be imposed. Indeed, if det C = 0, new singularities related to the lack of isotropic power counting are generated in some Feynman integrals. These singularities do not depend on the ultraviolet regularization procedure. They can be regulated through an interpolation between the gauge with singular C and a covariant gauge. If the chosen gauge is a true gauge, the singularities coming from det C = 0 cancel out in S-matrix elements. This phenomenon occurs for both Coulomb and Leibbrandt gauges. In both cases, the gluon self-energy for instance is the sum of two singular contributions coming, the first one from the ghost loop, the other one from the gluon loop. The sum is regular. Calculations in the Leibbrandt gauges neglecting the ghost contribution as well as the singular part of the gluon contribution are leading to results which are aberrant in field theory. For instance, nonlocal counterterms are required. Unfortunately, no consistent way to deal with the singularities generated by singular C-matrices other than the regularization by interpolating with covariant gauges is known at the present time. If all these points are taken into account, noncovariant gauges are on the same level of consistency as covariant ones. Perturbative calculations are however much heavier and there is no hope to avoid Faddeev-Popov ghosts in consistent class III gauges.

Appendix A

Notations

Our objects are defined in Minkowski space-time, which is a vector space equipped with a metric. The choice of the metric is the usual one in particle physics gμν = gμν = diag (1, −1, −1, −1). According to this convention and the usual choice of units such that h¯ = c = 1, time is t = x0 = x0 . The scalar product of two vectors is given by x · y = xμ yμ = xμ yμ = xμ gμν xν = x0 y0 − x.y. Usual convention of summation on dummy indices is understood. The metric is not positive defined since x2 = x · x can be positive, negative or zero. Vectors such that x2 = x02 − |x|2 > 0 are called time-like. Those for which x2 < 0 are called space-like while nonzero vectors such that x2 = 0 are called light-like. . A particular symbol is used here when constraints are taken into account: =⇒ means “implies by time derivation”. In other words, the second relation is obtained by taking the time-derivative of the first one. Some particular functions are also often used. The step function is noted and defined by ⎧ ⎨ 1 if x > 0 θ (x) = ⎩ 0 if x < 0. The sign function is noted and defined by ⎧ ⎨ 1 if x > 0 (x) = ⎩ −1 if x < 0. The Dirac function is not a function but the limit of functions. Its main properties are

Burnel, A.: Notations. Lect. Notes Phys. 761, 211–212 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 10 

212

A Notations

 ∞

1.

−∞

f (x)δ (x − y) dx = f (y);

2.

δ (ax) =

1 δ (x). |a|

Here the integral is not a usual integral but satisfies most of the properties of usual integrals. The principal value distribution is defined as   1 x . P = lim 2 →0 x +  2 x Another used distribution is

  1 1 1 x ∓ i = lim = lim 2 = P ∓ iπδ (x). x ± i →0 x ± i →0 x +  2 x

Fourier transform of functions and distributions are often used. Here is a table of the most useful results. f (x)

∞

−∞

f (x) e−ikx dx = F(k)

f (x)

∞

−∞

f (x) e−ikx dx = F(k)

f (−x)

F(−k)

f (ax)

(1/|a|)F(k/a)

f (x + x0 )

eikx0 F(k)

eik0 x f (x)

F(k − k0 )

x f (x)

iF  (k)

f  (x)

ikF(k)

( f ∗ g)(x)

F(k)G(k) & 2 π /ae−k /4a

f (x)g(x)

1/(2π )(F ∗ G)(k)

a/(x2 + a2 )

π e−a|k|

e−a|x|

2a/(k2 + a2 )

θ (x + a)θ (a − x)

2 sin(ka)/k

(sin ax)/x

πθ (k + a)θ (a − k)

e−ax θ (x)

−i/(k − ia)

1/(x + ia)

−2iπ e−ak θ (k)

θ (x)

−i/(k − i)

1/(x − i)

2iπ θ (−k)

δ (x)

1

1

2π δ (k)

sign(x)

−2iP(1/k)

P(1/x)

−iπ sign(k)

cos ax

π [δ (k − a) + δ (k + a)]

sin ax

−iπ [δ (k − a) − δ (k + a)]

−ax2

e

Appendix B

A Useful Fourier Transform

Let us proceed to the calculation of the Fourier transform of (p0 )θ (p2 ), a function appearing with its transform in covariant gauges. Let us begin with 

I(x) =

d 4 p θ (p0 )θ (p2 )eip·x .

(B.1)

Using the property

θ (p20 − |p|2 ) = θ (p0 − |p|) + θ (−p0 − |p|)

(B.2)

and the definition of the step functions, it reduces to 

I(x) =

d 3 p e−ip.x

 ∞ −∞

d p0 θ (p0 − |p|)eip0 x0 .

(B.3)

Making a translation of the p0 variable and using the Fourier transform of the step function given in the table, one gets I(x) =

i x0 + i



d 3 p e−ip.x ei|p|x0 .

(B.4)

Introducing spherical coordinates, carrying out the integration over the angles and d −ipx using p e−ipx = i dx e ,  ∞  2π θ (p) pd p eip(x0 −|x|) − eip(x0 +|x|) |x|(x0 + i) −∞  ∞  ∂ 2iπ θ (p) d p eip(x0 −|x|) + eip(x0 +|x|) . =− |x|(x0 + i) ∂ |x| −∞

I(x) = −

(B.5)

Using again the Fourier transform of the step function,   1 1 ∂ 2π + |x|(x0 + i) ∂ |x| x0 − |x| + i x0 + |x| + i   1 4π ∂ = . |x| ∂ |x| x2 + ix0

I(x) =

(B.6)

Burnel, A.: A Useful Fourier Transform. Lect. Notes Phys. 761, 213–214 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 11 

214

B A Useful Fourier Transform

Using now the property   1 1 = P 2 ∓ iπ (x0 )δ (x2 ) x2 ± ix0 x and checking that

∂ f (x2 ) = 2|x| f  (x2 ) ∂ |x|

where the prime denotes the derivative, the final result is     1 I(x) = 8π P 2 − iπ (x0 )δ  (x2 ) . x

(B.7)

In order to get the Fourier transform of θ (−p0 )θ (p2 ), let us substitute p by −p in (B.1). We get  I(x) =

d 4 p θ (−p0 )θ (p2 )e−ip·x

(B.8)

so that the Fourier transform of θ (−p0 )θ (p2 ) is obtained simply by replacing x by −x in (B.7). Therefore, by substracting both results, 

d 4 p (p0 )θ (p2 )eip·x = −16iπ 2 (x0 )δ  (x2 ).

(B.9)

Application of the Fourier theorem allows to get 

d 4 x (x0 )δ  (x2 )e−ip·x = iπ 2 (p0 )θ (p2 ).

(B.10)

Appendix C

Generalized Functions

C.1 Elementary Solutions of the Klein-Gordon Equation The aim of this appendix consists of developing elementary solutions associated with differential operators occuring in field equations of noncovariant class III gauges. One will extend the method used in the covariant case to noncovariant ones. It is therefore useful to first briefly recall the main functions used in covariant field theory and their properties. Most calculations are only sketched because these covariant functions appear as particular cases of those obtained in the general case for which details are given explicitly.

C.1.1 The Δ Function The Δ function is an elementary solution of the Klein-Gordon equation for which the support is causal. It is given by

Δ (x; m2 ) = −

i (2π )3



d 4 k (k0 ) δ (k2 − m2 ) e−ik·x

(C.1)

or, after integration over k0 , by

Δ (x; m2 ) = −

1 (2π )3



√

k0 =

|k|2 +m2

d 3 k ik·x e sin k0 x0 . k0

(C.2)

This last expression allows to get easily its zero-time properties

Δ (x; m2 ) x =0 = 0, ∂0 Δ (x; m2 ) x =0 = −δ (3) (x). 0

0

In case of vanishing mass, the usual notation is D(x) = Δ (x; 0).

Burnel, A.: Generalized Functions. Lect. Notes Phys. 761, 215–233 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-69921-7 12 

216

C Generalized Functions

C.1.2 The Δ F Function The Δ F function is the Feynman propagator. It propagates positive energy solutions forward in time and negative energy solutions backward in time. It is given by

Δ F (x; m2 ) =

1 (2π )4



d 4 k e−ik·x k2 − m2 + i

.

Again, if the mass is vanishing, the notation is DF (x) = Δ F (x; 0)

C.2 The E Function C.2.1 Definition In a general covariant gauge, one is faced with the problem of defining solutions for the equation E(x) = D(x). (C.3) Although it must be defined with care, such a solution is not described in usual textbooks on field theory. At a first look, one could think that a formal solution like the convolution product E = DH ∗ D of D with particular solution DH of DH (x) = δ (4) (x), for instance, DH (x) = −

1 (2π )4



d4k

eik·x k2 + i

(C.4)

could work. However, the Fourier transform of this convolution leads to the product 1 δ (k2 ) k2 + i which is not defined.

C.2.2 Derivation of the Elementary Solution In order to proceed consistently, let us apply the  operator to Eq. (C.3). The E-function satisfies the homogeneous equation

C.2 The E Function

217

2 E = 0.

(C.5)

Instead of applying twice the same operator, let us replace 2 by ( + κ ) where κ is a parameter with the dimension of a squared mass. Of course, the limit κ → 0 must be taken in order to recover the original equation. In this way, one can identify ( + κ )E with the elementary solution associated with the  operator and E with the elementary solution associated with the ( + κ ) operator. Both solutions are assumed to have causal support so that

and

E(x) = Δ (x; κ )

(C.6)

( + κ )E(x; κ ) = D(x).

(C.7)

Both equations reduce to (C.3) in the limit κ → 0. Subtraction of (C.6) from (C.7) leads to κ E(x) = D(x) − Δ (x; κ ). Dividing by κ ,

D(x) − Δ (x; κ ) . κ The limit for κ = 0 is given by the l’Hospital theorem

d E(x) = − Δ (x; κ )

. dκ κ =0 E(x) =

Using

Δ (x; κ ) = −

i (2π )3



(C.8)

(C.9)

d 4 k (k0 ) δ (k2 − κ ) e−ik·x ,

the expression of the desired function reads E(x) = −

i (2π )3



d 4 k (k0 ) δ  (k2 ) e−ik·x

(C.10)

where δ  is the derivative of the δ -function. According to Eq. (B.1) of appendix B, one can write 1 (x0 )θ (x2 ). (C.11) E(x) = 8π It obviously vanishes for x2 < 0 so that its support is causal.

C.2.3 Zero-Time Properties Zero-time properties of the E function so defined are easily obtained from (C.3) and (C.9) by commuting the time derivative and the limit κ → 0. They are

218

C Generalized Functions

E(x)|x0 =0 = ∂0 E(x)|x0 =0 = ∂02 E(x)|x0 =0 = 0,

∂03 E(x)|x0 =0

= ∂0 D(x)|x0 =0 = −δ

(3)

(C.12)

(x).

(C.13)

C.2.4 Integration Over k0 Integration over k0 can be realised using (C.9) and

Δ (x; κ ) =

i 2(2π )3





d 3 k  ik·x e − e−ik·x

√ . k0 k0 = |k|2 +κ

(C.14)

The dependence on κ occurs only through k0 so that d dk0 d 1 d = = . dκ d κ dk0 2k0 dk0 Therefore i E(x) = − 2(2π )3 i =− 4(2π )3





d3k d 2k0 dk0



 eik·x − e−ik·x

k0 k0 =|k|



d3k  ik·x −ik·x (ik0 x0 − 1)e + (ik0 x0 + 1)e

k03 k

(C.15) 0 =|k|

It is useful to rewrite this equation by using the fixed four-vector n which is chosen in such a way that, in the frame in which the integration is carried out, nμ = gμ 0 = (1, 0, 0, 0). Then 

d3k i E(x) = − 4(2π )3 |k|(k · n)2 

(ik · n x · n − 1)eik·x + (ik · n x · n + 1)e−ik·x

k0 =|k|

C.2.5 The Cauchy Problem Associated with 2 The solution of the Cauchy problem associated with the equation 2 A = 0 is again obtained as the limit for κ → 0 of the solution of

.

(C.16)

C.3 Generalized Functions Associated with the C Operator

219

( + κ )A = 0,

(C.17)

once the solution of  f = 0 is known. This last one reads f (x) = −



↔

d 3 y Δ (x − y; m2 ) ∂0y f (y)

(C.18)

where y0 is arbitrary and ↔

a(y) ∂0y b(y) ≡ a(y)∂0y b(y) − ∂0y a(y) b(y). Therefore, because ( + κ )A satisfies a Klein-Gordon equation with mass 0, the solution of the corresponding Cauchy problem is ( + κ )A(x) = −



↔

d 3 y D(x − y) ∂0y ( + κ )A(y).

(C.19)

In the same way, A satisfy a Klein-Gordon equation with squared mass κ . Therefore, the solution of this Cauchy problem is A(x) = −



↔

d 3 y Δ (x − y; κ ) ∂0y A(y).

(C.20)

By subtracting (C.20) from (C.19),

κ A(x) = −



↔

d 3 y [D(x − y) − Δ (x − y; κ )] ∂0y A(y) − κ



↔

d 3 y D(x − y) ∂0y A(y).

By dividing by κ and taking the limit κ → 0, the solution of the Cauchy problem associated with the homogeneous equation 2 A = 0 is A(x) = −



↔

d 3 y D(x − y) ∂0y A(y) −



↔

d 3 y E(x − y) ∂0y A(y).

(C.21)

As usual, y0 is arbitrary and initial values of ∂0n A, n = 0, . . . 3 are assumed to be given.

C.3 Generalized Functions Associated with the C Operator As in the case of the d’Alembert operator, generalized functions can be associated with the noncovariant operator C defined by C = Cμν ∂ μ ∂ ν for various Cμν fixed tensors . They can be constructed in the same way as the various Δ functions. This construction is made hereunder. It includes the covariant case for which Cμν = gμν .

220

C Generalized Functions

(1)

C.3.1 Elementary Solution DC The equation

C S = 0

(C.22)

can be solved by Fourier transform by setting 

S=

˜ d 4 k eik·x S(k).

(C.23)

˜ = 0 where k2 = C μν kμ kν . Therefore, In momentum space, the equation reads kC2 S(k) C its general solution is ˜ = s(k) δ (kC2 ) (C.24) S(k) where s(k) is arbitrary but regular at kC2 = 0. In analogy with the covariant case, the elementary (s = 1) even solution is written as (1)

DC (x) = N



d 4 k eik·x δ (kC2 )

(C.25)

where N is a normalization factor that will be fixed later on from equal-time properties of the causal solution. As in covariant case, it is useful to decompose this even solution into parts with fixed sign of frequencies. One is thus lead to solve the equation kC2 = 0. In order to simplify the writing of the equations, let us set C0l +Cl0 , 2C00 Ckl kk kl . V = V (k) = C00

Zl =

(C.26) (C.27)

Of course, it is assumed that C00 = 0. In other words, the coordinate frame is not singular. In covariant gauges, Zl = 0. This property does, in general, not hold in noncovariant gauges but, in some of them, it is possible to find coordinate frames in which Zl = 0. In such frames, the calculations and the writing of equations is of course simpler. They are called preferred frames. In the covariant case, all coordinate frames are equivalent and this notion of preferred frame does not exist. Actually, they are all preferred. Let us work as far as possible in a general frame where kC2 = C00 (k02 − 2Z.kk0 +V ). The solutions with respect to k0 of kC2 = 0 are  k0± = Z.k ± (Z.k)2 −V .

(C.28)

(C.29)

C.3 Generalized Functions Associated with the C Operator

Of course,

221

k0+ + k0− = 2Z.k.

(C.30)



Let us set k = k , k0 = k0 + Z.k  in (C.25). Elementary calculations using the known property of the Dirac-function

δ (ax) =

1 δ (x) |a|

allow to write (1)



  N d 4 k δ (k02 −Y 2 ) e−ik.x eik0 x0 eix0 Z.k |C00 |  N = d 4 k δ (k02 −Y 2 )eik+ ·x |C00 |

DC (x) =

(C.31)

where the formal four-vector k+ is defined by k+ = (k0 + Z.k, k) and Y = Y (k) =

k0+ − k0− = 2



(Z.k)2 −V .

(C.32) (1)

Inserting 1 = θ (−k0 )+ θ (k0 ) inside (C.31), a decomposition of DC (x) into parts with positive and negative frequencies can be realized  (1) (+) (−) DC (x) = i DC (x) − DC (x) (C.33) with (±)

DC (x) = ∓

iN |C00 |



d 4 k θ (∓k0 ) δ (k02 −Y 2 ) eik+ ·x .

(C.34)

It is important to note that the degenerate case k0+ = k0− leading to Y = 0 cannot be treated in the same way as hereunder and is therefore excluded from the discussion. A corresponding gauge is the static temporal gauge.

C.3.2 Positive Frequency Part (1)

Other useful expressions for the positive frequency part of the DC function can be obtained. First, by integrating over k0 its expression given by (C.34), one gets (+)

DC (x) = − By changing k into −k in (C.34),

iN |C00 |



d 3 k −ik.x ik− x0 e e 0 . 2Y

(C.35)

222

C Generalized Functions (+)

DC (x) = −

iN |C00 |



d 4 k θ (k0 ) δ (k02 −Y 2 ) e−ik+ ·x

(C.36)

and by changing also k0 into k0 − Z.k (+)

DC (x) = −iN = −iN



d 4 k θ (−k0 + Z.k) δ (kC2 ) eik·x



d 4 k θ (k0 − Z.k) δ (kC2 ) e−ik·x .

(C.37)

C.3.3 Negative Frequency Part (1)

In the same way, the negative frequency part of the DC function is iN |C00 |

(−)

DC (x) =



= iN



  d 4 k θ (k0 ) δ k02 −Y 2 eik+ ·x

d 4 k θ (k0 − Z.k) δ (kC2 ) eik·x .

(C.38)

Integration over k0 leads to (−)

DC (x) =

iN |C00 |



d 3 k −ik.x ik+ x0 e e 0 . 2Y

(C.39)

C.3.4 Elementary Solution with Causal Support The DC function with causal support can now be defined in analogy with the covariant case (+)

(−)

DC (x) = DC (x) + DC (x) =−

iN |C00 |



+ d 3 k −ik.x  ik− x0 e 0 − eik0 x0 . e 2Y

(C.40)

(±)

Equivalent expressions obtained from those of DC (x) are 

DC (x) = iN

d 4 k  (k0 − Z.k) δ (kC2 ) eik·x

(C.41)



iN d 4 k (k0 ) δ (k02 −Y 2 ) eik+ ·x |C00 |    iN = d 4 k θ (k0 ) δ (k02 −Y 2 ) eik+ ·x − e−ik+ ·x |C00 | =

(C.42) (C.43)

C.3 Generalized Functions Associated with the C Operator



= iN

223

 d 4 k θ (k0 − Z.k) δ (kC2 ) eik·x − e−ik·x .

(C.44)

From (C.40), its zero-time properties are obvious. DC (x)|x0 =0 = 0,

∂0 DC (x)|x0 =0 = −

(C.45) N(2π )3 |C00 |

δ (3) (x).

(C.46)

The normalization factor N is now chosen in such a way that these zero-time properties become identical with those of the relativistic function so that

∂0 DC (x)|x0 =0 = −δ (3) (x). Therefore N=

|C00 | . (2π )3

(C.47)

(C.48)

Zero-time properties for higher time-derivative of DC (x) are now derived. Setting ∂Z = 2Z.∂ , the equation C DC = 0 can indeed be written (C00 ∂02 − ∂0 ∂Z +Ckl ∂k ∂l )DC = 0. Using a nonsingular frame C00 = 0 and setting W= one gets

Ckl ∂k ∂l , C00

(C.49)

∂02 DC = ∂Z ∂0 DC −W DC

and, at zero-time,

∂02 DC (x) x

In the same way,

∂03 DC (x) x =0 = (W − ∂Z2 )δ (3) (x),

0 =0

= −∂Z δ (3) (x).

∂04 DC (x) x

(C.50) (C.51)

= ∂Z (2W − ∂Z2 )δ (3) (x). (C.52) In the relativistic case, as well as in preferred frames, ∂Z vanishes. 0

0 =0

C.3.5 Generalization with a Mass Parameter Because some field equations involve the operator C2 and in view of obtaining its solutions in the same way as in relativistic case, it is useful to introduce a parameter

224

C Generalized Functions

κ with the dimension of a squared mass by replacing kC2 by kC2 − κ . All the above equations are identical except that the definition of V is now V=

Ckl kk kl − κ . C00

The generalized functions are simply obtained by the substitution kC2 → kC2 − κ 

−i|C00 | d 4 k θ (k0 − Z.k) δ (kC2 − κ ) e−ik·x , (C.53) (2π )3  i|C00 | (−) ΔC (x; κ ) = d 4 k θ (k0 − Z.k) δ (kC2 − κ ) eik·x , (C.54) (2π )3   −i|C00 | ΔC (x; κ ) = d 4 k θ (k0 − Z.k) δ (kC2 − κ ) e−ik·x − eik·x . (C.55) 3 (2π ) (+)

ΔC (x; κ ) =

From the substitution k → −k in one of the integrals occurring in the last expression, one can also write

ΔC (x; κ ) =

−i|C00 | (2π )3



d 4 k  (k0 − Z.k) δ (kC2 − κ ) e−ik·x .

(C.56)

It is straightforward to see that zero-time properties are identical to those of DC provided W is redefined as Ckl ∂k ∂l + κ W= . (C.57) C00

C.3.6 Analogous of the Feynman Propagator The function ΔCF (x; κ ) is defined as the analogous of the Feynman propagator by (−)

ΔCF (x; κ ) = θ (x0 )ΔC (x; κ ) − ΔC (x; κ ).

(C.58)

Starting from (C.40) and using ±

θ (x0 )eik0 x0 = −

i 2π

 ∞

dk0 eik0 x0 ± −∞ k0 − k0 − i

which results from properties of Fourier transforms, one can write    4 1 d k ik·x 1 1 e θ (x0 )ΔC (x; κ ) = − . − (2π )4 2Y k0 − k0− − i k0 − k0+ − i

(C.59)

Reduction to a common denominator and use of the conventions (C.26), (C.32) lead to

C.3 Generalized Functions Associated with the C Operator

1 eik·x d4k 2 4 (2π ) (kC − κ )/C00 − i(k0 − Z.k)   2   2   kC − κ kC − κ 1 4 ik·x = d ke P + iπ (k0 − Z.k)δ . (2π )4 C00 C00

θ (x0 )ΔC (x; κ ) =

(−)

Substraction of ΔC

225



(C.60)

gives

ΔCF (x; κ ) =

1 (2π )4

=

C00 (2π )4

 

  2   2  k −κ k −κ d 4 k eik·x P C − iπδ C C00 C00 d 4 k eik·x kC2 − κ + iC00

.

(C.61)

It obviously satisfies (C + κ ) ΔCF (x; κ ) = −C00 δ (4) (x).

(C.62)

This is the propagator associated with the operator C + κ . In order that the prescription allows Wick rotation, the coordinate frame will be chosen in such a way that C00 > 0.

C.3.7 Solution of the Cauchy Problem The solution of the Cauchy problem for the homogeneous equation (C + κ )S(x) = 0 is obtained in the same way as in the relativistic case. The general solution of the homogeneous equation is the sum of the convolutions of arbitrary functions with two independent elementary solutions. These ones are taken as ΔC and ∂0 ΔC so that the solution reads 

S(x) =

d 3 y ΔC (x − y; κ ) α (y) +



d 3 y ∂0 ΔC (x − y; κ ) β (y).

(C.63)

Because S(x) and ∂0 S(x) are given at x0 = y0 , zero-time properties of ΔC and its time derivatives allow to write S(x) = −



↔

d 3 y ΔC (x − y; κ ) ∂0y S(y) + ∂Z



d 3 y ΔC (x − y; κ )S(y)

(C.64)

where y0 is, as usual, arbitrary. This solution looks like the one of the relativistic case but there is an additional term in non-preferred frames. Rearranging the various terms, another useful equivalent expression is

226

S(x) = −

C Generalized Functions



d 3 y ∂0 ΔC (x − y; κ ) S(y) −



d 3 y ΔC (x − y; κ )(∂0 − ∂Z )S(y).

(C.65)

C.3.8 Remarks 1. It is obviously essential for the previous construction that C00 = 0. One must never work in a singular frame. 2. It is also essential that the equation kC2 = 0 has two distinct real solutions. Otherwise the parallelism with the relativistic case cannot be realized. 3. If one can find a frame in which C0k +Ck0 = 0, the calculations are much simpler. Such a frame, called a preferred frame, implies k0+ + k0− = 0. It is then sufficient to take the relativistic formulae, to multiply by the factor |C00 | and to replace k2 by kC2 to obtain the desired results.

C.4 Generalized Functions Associated with 2C As in relativistic gauges and even more generally, field equations involve successive applications of second-order differential operators. We have considered 2 in the relativistic case. Here, one is led to study C2 , C and even C2 . Let us begin with the simplest of these complicated operators. It allows a similar treatment as in the relativistic case.

C.4.1 The Elementary Function with Causal Support Let us indeed consider the equation C2 A = 0

(C.66)

C (C + κ )A = 0.

(C.67)

as the κ → 0 limit of the equation

Reproducing the same reasoning as in the relativistic case, the solution reads



d i|C00 |

ΔC (x; κ ) EC (x) = − =− d 4 k  (k0 − Z.k) δ  (kC2 ) e−ik·x . dκ (2π )3 κ →0

(C.68)

C.4 Generalized Functions Associated with C2

227

C.4.2 Zero Time Properties Zero-time properties of this function are easy to obtain. Permutation of the derivative on κ and the limit x0 → 0 and use of zero-time properties of ΔC immediately give

∂0p EC (x) x

= 0,

0 =0

0 ≤ p ≤ 2.

(C.69)

Using C00 ∂02 = C + ∂0 ∂Z −Ckl ∂k ∂l and C EC (x) = DC (x), one gets

C00 ∂02 EC (x) = DC (x) + (∂0 ∂Z −Ckl ∂k ∂l )EC (x).

Therefore, by derivating once more with respect to time and using (C.69),

∂03 EC (x) x =0 0

1 1 (3) = ∂0 DC (x)

=− δ (x). C00 C00 x0 =0

(C.70)

These properties are, of course, similar to those of the E-function and reduce to them in the covariant case.

C.4.3 Integration Over k0 Integration over k0 can be realized using

ΔC (x; κ ) =

i (2π )3



d 3 k  −ik+ ·x e − eik+ ·x 2k0

(C.71)

where k0 = Yκ (k) and noting that the dependence on κ occurs only through k0 . Therefore, as in the relativistic case, d dk0 d 1 d = = dκ d κ dk0 2k0 dk0 and a straightforward calculation gives i EC (x) = − 4(2π )3





d3k  ik+ ·x −ik+ ·x (ik0 x0 − 1)e + (ik0 x0 + 1)e

k03 k

. (C.72) 0 =Y (k)

228

C Generalized Functions

C.4.4 The Cauchy Problem The Cauchy problem is again solved as in the relativistic case. One considers the equation (C.67) and writes successively the solution for both second-order equations (C + κ )A(x) = −



↔

d 3 y DC (x − y) ∂0y (C + κ )A(y)

+ ∂Z C A(x) = −





d 3 y DC (x − y)(C + κ )A(y),

↔

d 3 y ΔC (x − y; κ ) ∂0y C A(y) + ∂Z



(C.73)

d 3 y ΔC (x − y; κ )C A(y).

(C.74) By subtracting these equations, dividing by κ and taking the limit for κ → 0, A(x) = −



+ ∂Z

↔

d 3 y DC (x − y) ∂0y A(y) − 



↔

d 3 y EC (x − y) ∂0y C A(y)

d 3 y [EC (x − y)C A(y) + DC (x − y)A(y)]

(C.75)

where, as usual, y0 is arbitrary. The last term disappears in preferred frames.

C.5 Generalized Functions Associated with C In noncovariant gauges, one encounters field equations f = g where g satisfies C g = 0. As for the previous operators, one would associate elementary functions to the operator C and solve the associated Cauchy problem. The solution of the problem will be easily generalized to a squared mass parameter κ by considering the operator (C + κ ). The above notations are kept in mind.

C.5.1 Elementary Solution with Causal Support One is seeking a generalized function FC (x) which has zero time properties similar to those of the E function. In particular, it must vanish at zero time. One can first require FC (x) = DC (x) and obtain thus the solution as the four-dimensional space convolution FC = DH ∗ DC

(C.76)

C.5 Generalized Functions Associated with C

229

where DH is a particular solution of DH (x) = δ (4) (x). However, such a solution will not satisfy the required zero-time properties and it violates also the symmetric role played by the operators  and C . Therefore, one adds a symmetric part and starts with the obvious solution (a)

(b)

FC = DH ∗ DC + α DCH ∗ D = FC + FC where

(C.77)

DH (x) = C DCH (x) = δ (4) (x)

and α is a parameter that will be fixed later on in order to satisfy zero-time properties and fully respect the symmetry between both operators. There is, a priori, some arbitrariness in the choice of DH and DCH but the final result should not depend on the particular choice made for these functions. Let us take the usual Feynman solution for DH and start thus with (a)

FC (x) =

i|C00 | (2π )3



d 4 k e−ik·x  (k0 − Z.k)

δ (kC2 ) . k2 + i

(C.78)

Setting k0 = k0 + Z.k and reproducing the same kind of calculations leading to (C.35), one obtains   − +  ik.x e−ik0 x0 i e−ik0 x0 (a) 3 e FC (x) = d k − . (C.79) (2π )3 2Y k0+2 − |k|2 + i k0−2 − |k|2 + i At zero time, (a) FC (x, 0) =

i (2π )3



  1 eik.x 1 d k − . 2Y k0+2 − |k|2 + i k0−2 − |k|2 + i 3

(C.80)

Reduction to a common denominator gives, after straightforward calculations, (a)

FC (x, 0) = −

i (2π )3



d 3 k eik.x

2Z.k . (C.81) (V + |k|2 )2 − 4(Z.k)2 |k|2 + i[2(Z.k)2 −V − |k|2 ] In the same way, one takes (b)

FC (x) =

iα (2π )3



d 4 k e−ik·x (k0 )

δ (k2 ) kC2 + iτ C00

.

(C.82)

230

C Generalized Functions

Here, the prescription for the pole at kC2 = 0 is left arbitrary through the factor τ . It (a) will soon appear that τ is fixed by the choice made in FC . Integration over the k0 variable leads to (b)

FC (x) =

α i C00 (2π )3 



d3k

eik.x 2|k|

 e−i|k|x0 e−i|k|x0 − . (C.83) |k|2 − 2Z.k|k| +V + iτ  |k|2 + 2Z.k|k| +V + iτ 

At zero-time and after reduction to the same denominator, (b) FC (x, 0)

α i = C00 (2π )3



d 3 k eik.x

2Z.k . (V + |k|2 )2 − 4(Z.k)2 |k|2 − iτ [2(Z.k)2 −V − |k|2 ]

(C.84)

The vanishing of FC (x, 0) implies that τ must be taken equal to −1 and that α = C00 . Therefore,    2 (C00 ) (k0 ) δ (k2 ) i|C00 | 4 −ik·x  (k0 − Z.k) δ (kC ) + FC (x) = d k e . (C.85) (2π )3 k2 + i kC2 − iC00 This is the required elementary causal solution. It can be generalized to the operator (C + κ ) by simply replacing kC2 by kC2 − κ . The limit for the case where Cμν = gμν is not trivial from this result.

C.5.2 Zero-Time Properties of FC and its Time Derivatives The function FC is defined above in such a way that FC (x, 0) = 0. It is left as an exercise to check that ∂0 FC (x, 0) = 0. For higher derivatives,

∂02 FC = ( + Δ )FC = DC + Δ FC is used. Then, from zero-time properties of DC and previous ones of FC ,

∂02 FC (x, 0) = 0, ∂04 FC (x, 0)

= −∂Z δ

∂03 FC (x, 0) = −δ (3) (x), (3)

(x),

∂05 FC (x, 0) = (W

(C.86) − ∂Z2 − Δ )δ (3) (x).

(C.87)

Such properties also hold for the corresponding function generalized with the introduction of a mass parameter provided that this parameter is included in the definition of W .

C.5 Generalized Functions Associated with C

231

C.5.3 Preferred Frame If a preferred frame can be found, Z.k = 0 in such a frame. Taking also C00 > 0 which is not a restriction, the calculations and the result are considerably simplified. Using   1 1 1 − δ (x) = − (C.88) 2iπ x + i x − i and generalizing by introducing a mass parameter, the FC -function can be rewritten in such a frame as 

|C00 | d 4 k e−ik·x (k0 ) FC (x; κ ) = − (2π )4   1 1 1 1 − . k2 + i kC2 − κ + i k2 − i kC2 − κ − i

(C.89)

Again, it is not clear that this function will reduce to the E function in the relativistic case because, in such a limit, care must be taken with the product of distributions. In order to get the analogous of the Feynman propagator, it is sufficient to compare this case with the reasoning made in the case of the DCF function by replacing 1 1 1 −→ 2 . k ± i kC2 − κ ± i kC2 ± i One immediately gets FCF (x; κ ) = −

C00 (2π )4



d 4 k e−ik·x

1

1

k2 + i

kC2 − κ + i

.

(C.90)

Its generalization to an arbitrary frame and to any value of C00 is FCF (x; κ ) = −

C00 (2π )4



d 4 k e−ik·x

1 1 . k2 + i kC2 − κ + iC00

(C.91)

C.5.4 The Cauchy Problem Solution of the Cauchy problem associated with the equation A = B can be obtained by considering that A = A − DH ∗ B satisfies the massless homogeneous Klein-Gordon equation. Then the solution for a general B can be obviously obtained from the solution of the homogeneous case as

232

C Generalized Functions

A(x) = (DH ∗ B)(x) −



↔ d 3 y D(x − y) ∂0y

[A(y) − (DH ∗ B)(y)]

(C.92)

whatever DH satisfying DH = δ (4) (x) is. As usual, y0 is arbitrary. A particular case of interest for the following occurs when the B-function is proportional to the ΔC - function or to derivatives of this function B(x) = Lx ΔC (x; κ ) where Lx is a derivation operator with respect to the variable x. It may also contain derivatives with respect to κ . In this case, from properties of derivatives of a convolution product, the solution reads A(x) = Lx (D ∗ ΔC )(x) −



H

+ Because



↔

d 3 y D(x − y) ∂0y A(y) ↔

d 3 y D(x − y) ∂0y Ly (DH ∗ ΔC )(y).

(C.93)

Lx (D ∗ ΔC )(x) = 0,

one can write Lx (D ∗ ΔC )(x) as the solution of the Cauchy problem associated with the d’Alembert operator 0 = Lx (D ∗ ΔCH )(x) +



↔

d 3 y D(x − y) ∂0y Ly (D ∗ ΔCH )(y).

(C.94)

Adding (C.93) and (C.94) multiplied by C00 in order to find back the function FC , the solution of the Cauchy problem associated with A(x) = Lx ΔC (x; κ ) is A(x) = Lx FC (x; κ ) +



↔

d 3 y D(x − y) ∂0y Ly FC (y; κ ) −



↔

d 3 y D(x − y) ∂0y A(y). (C.95)

C.6 The G-Functions One last generalized function which is necessary in the study of commutation relations and the search of the propagator remains to be defined. It satisfies GC = EC . Because EC = −

d DC , dκ

(C.96)

C.6 The G-Functions

233

this function GC (x) can be easily obtained from FC (x; κ ) by derivation with respect to κ

d GC (x) = − FC (x; κ )

. (C.97) dκ κ =0 All its properties are then easily derived. In particular, the propagator is given by



1 1 d C00 F 4 −ik·x

GC (x) = − d ke . (C.98) d κ (2π )4 k2 + i kC2 − κ + iC00 κ =0 The solution of the Cauchy problem associated with A(x) = Lx FC (x; κ )

(C.99)

can also be obtained from (C.95) by writing Lx = −Lx

d dκ

in (C.95). It writes A(x) = Lx GC (x) +



↔

d 3 y D(x − y) ∂0y Ly GC (y) −



↔

d 3 y D(x − y) ∂0y A(y). (C.100)

We have thus at our disposal all the generalized functions which are necessary in the study of noncovariant class III gauges.

Index

BRST charge, 95–96 quantization, 13 transformation, 92–93 Causal distribution, 159 advanced part, 159 order of singularity of, 160 retarded part, 159 Causality, 102–104 Constraints first-class, 10 primary, 8 secondary, 9 second-class, 10 Counter-term, 133 Degree of freedom, 12 effective, 141 superficial, 119 Dirac brackets, 11 quantization, 13 Electromagnetic potential, 7 Electromagnetic field, 6 Euler-Lagrange equations, 1–5 Feynman propagator, 76 rules, 111–113 Gauge class I, 12

class II, 13 class III, 13 degree of freedom, 12 fixing, 12 invariance, 7 Leibbrandt, 31 light-cone, 62 parameter, 17 planar-type, 32–63 static temporal, 30 temporal, 62 Ghost-number, 97 Gribov ambiguity, 12 Gupta-Bleuler formalism, 22–56 Hamiltonian, 4 density, 4 formalism, 1 Heisenberg equation, 3 Klein-Gordon equation, 5 Lagrangian, 3 density, 3–5 formalism, 1 renormalized, 133 Multiplicative renormalization, 134 Normal product, 106 One-particle irreducible, 138 Operator

235

236 annihilation, 6–21 creation, 6–21 Poisson brackets, 2–4 Power counting, 119–120 regularization, 130 Propagator, 21 Proper vertex function, 138

Index Singular frame, 33 Unitarity perturbative, 104 of the S-matrix, 102 Weak equality, 9 Wick’s theorem, 109