Lattice Quantum Field Theory of the Dirac and Gauge Fields: Selected Topics 9811209693, 9789811209697

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Baaquie, B. E., author. Title: Lattice quantum field theory of the Dirac and gauge fields : selected topics / Belal Ehsan Baaquie. Description: New Jersey : World Scientific Publishing, [2020] | Includes bibliographical references and index. Identifiers: LCCN 2019047773 | ISBN 9789811209697 (hardcover) | ISBN 9789811209703 (ebook for institutions) | ISBN 9789811209710 (ebook for individuals) Subjects: LCSH: Lattice gauge theories. | Quantum field theory. | Lattice field theory. | Dirac equation. | Quantum chromodynamics. Classification: LCC QC793.3.G38 B33 2020 | DDC 530.14/3--dc23 LC record available at https://lccn.loc.gov/2019047773

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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This book is dedicated to the memory of my lifelong friends and companions, Masud Sadique Chullu (1943-2017), Yamin Chowdhury (1951-2018), Syed Muhammad Hashem Raja (1957-2018) and Shahadat Hossain (1953-2020). Their courage, integrity, sincerity, loyalty, spiritual insight and prescient vision continues to inspire all those who knew them – and their memories live on.

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Acknowledgments

I would like to acknowledge and express my thanks to many outstanding scholars and researchers whose work motivated me to study quantum field theory and to grapple with its mathematical formalism. Two peerless scholars who greatly influenced me are Kenneth G. Wilson and Richard P. Feynman. I did my PhD thesis under the guidance of Kenneth G. Wilson at a time when he first introduced the concept of lattice gauge theory. Ken Wilson’s visionary conception of quantum field theory greatly enlightened and inspired me, and continues to do so till today. As an undergraduate I had the privilege of meeting and conversing a number of times with Richard P. Feynman, and which left a lasting impression on me. I thank Du Xin, Muhammad Mahmudul Karim and Nazmi Haskanbancha for valuable input in the preparation of the book. I thank Ali Namazie, Frederick H. Willeboordse, Spenta Wadia, Avinash Dhar, Michael Spalinski, H. R. Krishnamurthy, Michael Peskin, Steve Shenker, Daniel Rosenhouse, Don Lewis, Ivan Todorov and Waseem A. Sayed for many useful discussions. I thank my precious family members Najma, Arzish, Farah and Tazkiah for their love, affection, delightful company and warm encouragement. They have made this book possible.

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Preface

Quantum field theory is undoubtedly one of the most accurate and important scientific theory in the history of science. Relativistic quantum fields are the theoretical backbone of the Standard Model of particles and interactions. Relativistic and nonrelativistic quantum fields are extensively used in a myriad branches of theoretical physics, from superstring theory, high energy physics and solid state physics to condensed matter, quantum optics, nuclear physics, astrophysics and so on. Non-Abelian Yang-Mills gauge fields coupled to fermions are the workhorse of all advanced studies of quantum field theory. The Standard Model, and in particular, QCD (quantum chromodynamics) – which is the theory of quarks coupled to colored gluons – are exemplars of such quantum field theories. Almost all books on quantum field theory study the continuum formulation of the Dirac and Yang-Mills fields [Polyakov (1986), Peskin and Schroeder (1995), Ryder (2001)] and there are many books that offer a comprehensive perspective on quantum field theory (for example, Weinberg (2010), Zinn-Justin (1993)). To provide a different perspective to gauge fields, K. G. Wilson (1974) introduced lattice gauge theory: the formulation of quantum field theory of quarks and gluons defined on a lattice spacetime. The study of lattice Dirac and Yang-Mills fields provides a fresh perspective to the study of non-Abelian gauge field theories. The lattice gauge field has stood the test of time – continuing to provide new results – and is one of Wilson’s lasting legacy to the edifice of theoretical physics [Baaquie et al. (2015)]. Quantum fields, and in particular fermions and gauge fields, defined on a lattice seems to be the only formulation that provides a mathematically rigorous definition of a quantum field theory. The lattice, in addition to providing an ultra-violet cutoff, also allows for the use of computational tools quite distinct from the continuum formulation. It is expected that with the increase in computational power, the lattice formulation will allow for an in-depth study of nonlinear quantum field theories and reveal features and secrets of quantum fields that are not possible to deduce using approximations, such as the weak or strong coupling perturbation theory. There are many books on lattice field theory (e.g. Creutz (1983); Roth (1997); Montvay and Munster (1994)). Given the vast and increasingly complex matheix

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matics of lattice quantum fields, it is virtually impossible for any one book to cover the entire terrain of lattice quantum field theory. The purpose of this book is to provide a primer and an introduction, to the mathematical foundations of the vast and ever changing landscape of lattice quantum fields. A primer with a limited selection of topics can in itself be very interesting and useful. A primer focusing on a few selected aspects is easier for the reader to comprehend as compared with of an exhaustive book that serves as a reference tome meant for the specialists. The topics and calculations in this primer have been chosen to provide such an introduction.

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Contents

Acknowledgments

vii

Preface

ix

1.

Synopsis 1.1 1.2 1.3 1.4

1

Mathematical Background . . . . . Lattice Gauge Field . . . . . . . . Lattice Dirac Field . . . . . . . . . Lattice Gauge Theory Hamiltonian

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2 3 4 4

Mathematical Background

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SU (N ) Compact Lie Groups 2.1 2.2 2.3 2.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vielbein of SU (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . Metric on SU (N ) Group Space . . . . . . . . . . . . . . . . . . . 2.4.1 SU (2) metric . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Invariant Haar Measure and Delta Function . . . . . . . . . 2.5.1 Delta function on group space . . . . . . . . . . . . . . . 2.5.2 SU (2) measure . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 SU (2) measure: invariant . . . . . . . . . . . . . . . . . . 2.6 Campbell-Baker-Hausdorff (CBH) Formula . . . . . . . . . . . . 2.7 Irreducible Representations of SU (N ) . . . . . . . . . . . . . . . 2.8 Peter-Weyl Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Lie Algebra Generators: Differential Operators . . . . . . . . . . 2.10 Vielbein eab and fab for SU (2) . . . . . . . . . . . . . . . . . . . 2.11 Left and Right Invariant Generators . . . . . . . . . . . . . . . . 2.11.1 Differential realization of SO(3) generators on state space 2.12 SU (N ) Character Function Expansion . . . . . . . . . . . . . . . xi

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2.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SU (N ) Kac-Moody Algebra

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3.1 3.2 3.3

Introduction . . . . . . . . . . . . . . . . . . . . Kac-Moody Algebras . . . . . . . . . . . . . . . Functional Differentiation . . . . . . . . . . . . 3.3.1 Chain rule . . . . . . . . . . . . . . . . 3.4 Kac-Moody Generators . . . . . . . . . . . . . . 3.4.1 Kac-Moody generator and the 2-cocycle 3.5 Chiral Field . . . . . . . . . . . . . . . . . . . . 3.6 The WZW Lagrangian . . . . . . . . . . . . . . 3.7 SU (2) Loop Group . . . . . . . . . . . . . . . . 3.8 SU (2) Kac-Moody Algebra . . . . . . . . . . . 3.9 Kac-Moody Commutation Equations . . . . . . 3.10 Virasoro Generator: Point-Split Regularization 3.11 Summary . . . . . . . . . . . . . . . . . . . . . 3.12 Appendix . . . . . . . . . . . . . . . . . . . . . 4.

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SU (3) Character Functions 5.1 5.2 5.3 5.4

Casimir Operator for SU (3) . Evolution Kernel for SU (3) . Character Functions of SU (3) Summary . . . . . . . . . . .

29 30 31 32 33 36 38 40 43 44 46 49 53 54 55

Introduction . . . . . . . . . . . . . . . . . . . . . . . Hamiltonian Operator . . . . . . . . . . . . . . . . . 4.2.1 U (1) and SU (2) evolution kernel . . . . . . . 4.3 Lattice Action for SU (N ) . . . . . . . . . . . . . . . 4.4 Classical Paths and Winding Number . . . . . . . . 4.5 U (1) Path Integral . . . . . . . . . . . . . . . . . . . 4.6 U (1) Path Integral: Classical Paths . . . . . . . . . . 4.7 SU (N ) Continuum Action . . . . . . . . . . . . . . . 4.8 SU (2) Path Integration . . . . . . . . . . . . . . . . 4.8.1 SU (3) path integration . . . . . . . . . . . . 4.8.2 SU (N ) path integration . . . . . . . . . . . . 4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Appendix: Continuum Limit of SU (N ) Path Integral

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SU (N ) Path Integrals 4.1 4.2

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81 83 84 86

Fermion Calculus

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Fermionic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermion Integration . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.3

6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

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Fermion Hilbert Space . . . . . . . . . . . . . . . . 6.3.1 Fermionic completeness equation . . . . . . 6.3.2 Fermionic momentum operator . . . . . . . Antifermion State Space . . . . . . . . . . . . . . . Fermion and Antifermion Hilbert Space . . . . . . Real and Complex Fermions: Gaussian Integration 6.6.1 Complex Gaussian fermions . . . . . . . . . Fermionic Path Integral and Hamiltonian . . . . . Fermionic Operators . . . . . . . . . . . . . . . . . Fermion-Antifermion Hamiltonians . . . . . . . . . A Quadratic Hamiltonian . . . . . . . . . . . . . . 6.10.1 Orthogonality and completeness . . . . . . Fermion-Antifermion Lagrangian . . . . . . . . . . Fermionic Transition Probability Amplitude . . . . Quark Confinement . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . .

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Lattice Gauge Field 7.

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Non-Abelian Lattice Gauge Field 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

113

Introduction . . . . . . . . . . . . . . . . . . . . . . The Weak Coupling Approximation . . . . . . . . . Gauge-Fixing the Lagrangian . . . . . . . . . . . . Zero Mode . . . . . . . . . . . . . . . . . . . . . . . Gauge-Fixed Path Integral . . . . . . . . . . . . . . The Faddeev-Popov Counter-Term . . . . . . . . . Abelian Gauge-Fixed Path Integral . . . . . . . . . Lattice Faddeev-Popov Non-Abelian Ghost Action Lattice BRST Symmetry . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . .

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Abelian Lattice Gauge Field in d = 3 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Introduction . . . . . . . . . . . . Strong Coupling Representation . Weak Coupling Representation . Gauge Invariance . . . . . . . . . Wilson Loop . . . . . . . . . . . . Phase Transition . . . . . . . . . 8.6.1 Mean field approximation Summary . . . . . . . . . . . . . Appendix A . . . . . . . . . . . .

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8.9 9.

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Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Lattice Gauge Field Mass Renormalization 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

Introduction . . . . . . . . . . . . . . . . . . The Propagator and Mass Renormalization The Computational Scheme . . . . . . . . . Determination of mα and mµ . . . . . . . . Determination of mc for Sc [θ + B] . . . . . Expansion of the Gauge Field Action . . . . Determination of mA for S[θ + B] + Sα [B] . One-Loop Mass Renormalization . . . . . . Slavnov-Taylor Identity . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . .

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10. Gauge Field Block-Spin Renormalization 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . 10.2 Two-Dimensional Lattice Gauge Field . . . . . . 10.3 The Renormalization Group Transformation . . . 10.3.1 Renormalization group and fixed points . 10.4 The Recursion Equation for d Dimensions . . . . 10.5 Strong Coupling Approximation for SU (2) . . . . 10.6 Weak Coupling Expansion for SU (2) . . . . . . . 10.7 Weak Coupling Approximation for d = 4 +  . . . 10.8 Weak Coupling SU (2) Gauge Field: β-Function 10.9 Numerical Solution of the Recursion Equation . . 10.9.1 Change of integration variable . . . . . . 10.9.2 Numerical algorithm . . . . . . . . . . . . 10.9.3 Total grid size S(I) . . . . . . . . . . . . 10.10 Numerical Results . . . . . . . . . . . . . . . . . 10.11 Summary: Confinement and Asymptotic Freedom 10.12 Appendix . . . . . . . . . . . . . . . . . . . . . .

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11. Lattice Gauge Field Hamiltonian 11.1 11.2 11.3 11.4 11.5 11.6

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Lattice Gauge Field Hamiltonian . . . . . . . . . . Gauge-Fixed Chromoelectric Operator . . . . . . . Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . Gauge-Fixed Lattice Gauge Field Hamiltonian . . Hamiltonian and Covariant Gauge: Faddeev-Popov Ghost State Space and Hamiltonian . . . . . . . . 11.6.1 BRST cohomology: state space . . . . . . . 11.7 BRST Charge QB . . . . . . . . . . . . . . . . . . 11.8 QB and State Space . . . . . . . . . . . . . . . . .

157 158 159 165 165 169 170 174 175 176 176 178 179 182 184 185 187

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11.8.1 Gupta-Bleuler condition . . . . . . . . . . . . . . . . . . . . 206 11.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Lattice Dirac Field

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12. Dirac Lattice Path Integral 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13

Introduction . . . . . . . . . . . . . . . . . Dirac Field Coordinates . . . . . . . . . . Dirac Lattice Lagrangian . . . . . . . . . . Lattice Fermions and Chiral Symmetry . . Dirac Field: Boundary Conditions . . . . Dirac Fermionic State Space . . . . . . . . Hilbert Space Metric and Transfer Matrix Dirac Lattice Hamiltonian . . . . . . . . . Lattice Path Integral . . . . . . . . . . . . 12.9.1 Normalization constant . . . . . . Evolution Kernel . . . . . . . . . . . . . . Energy Eigenfunctions . . . . . . . . . . . Propagator . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . .

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13. Dirac Hamiltonian 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11

Fermionic Operators . . . . . . . . . . . . Lattice Dirac Hamiltonian . . . . . . . . . Continuum Hilbert Space . . . . . . . . . Continuum Hamiltonian . . . . . . . . . . Dirac Field’s Energy Eigenfunctionals . . Dirac Charge Operator . . . . . . . . . . . 13.6.1 Momentum and spin operators . . Finite Time Dirac Action . . . . . . . . . Continuum Evolution Kernel . . . . . . . Evolution Kernel: General Quadratic Case Evolution Kernel: Dirac Hamiltonian . . . 13.10.1 Chiral charge operator . . . . . . Summary . . . . . . . . . . . . . . . . . .

Lattice Gauge Theory Hamiltonian 14. Lattice Gauge Theory Hamiltonian

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14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13

Axial Gauge: Un0 = I . . . . . . . . . . . . . . . . . Noncanonical Fermion Anticommutation Equations . Lattice Gauge Theory Hamiltonian . . . . . . . . . . Canonical Fermions . . . . . . . . . . . . . . . . . . . Color Charge Operator and Gauss’s Law . . . . . . . Lattice Action from Lattice Hamiltonian . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Fermion Calculus with Gauge Field . . Appendix B: Matrix M . . . . . . . . . . . . . . . . Appendix C: Lagrangian for an Asymmetric Lattice Appendix D: Classical Continuum Limit . . . . . . .

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275 276 277 279 280 284 286 286 288 289 291

Bibliography

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Index

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Chapter 1

Synopsis

Lattice quantum field theory is quantum field theory defined on a lattice spacetime. Quantum field theory based on the spacetime continuum is a vast subject that has been extensively studied since the 1950’s – and many good books have been written on it. Almost all books on quantum field theory study the continuum formulation of the scalar, Dirac and Yang-Mills gauge fields [Polyakov (1986), ZinnJustin (1993), Peskin and Schroeder (1995), Ryder (2001)], and there is not much that is gained by re-deriving well known results. Lattice quantum field theory has come to the forefront of theoretical physics since the 1970’s, largely due to the work of Ken Wilson. It is natural to start one’s study of lattice quantum field theory with scalar quantum fields. This path has been eschewed since the primary result of the study of scalar lattice quantum fields is in developing numerical algorithms and applications based on these algorithms. Instead, the more complicated Dirac and gauge fields are studied to examine and study the mathematical structures and novel features that emerge from their formulation on a lattice spacetime. Lattice gauge theory is the quantum field theory of the Dirac and gauge fields on a lattice spacetime. Lattice gauge theory was introduced by Wilson (1974) to study the theory of strong interactions, and in particular the confinement of quarks. Both the lattice Dirac and lattice gauge fields have features that are quite distinct from their continuum formulation and many new ideas have to be introduced to study their generalization to a lattice spacetime. The book is grouped into the following Four Parts. • • • •

Mathematical Background: Chapter 2 to Chapter 6. Lattice Gauge Field: Chapter 7 to Chapter 11. Lattice Dirac Field: Chapter 12 to Chapter 13. Lattice Gauge Theory Hamiltonian: Chapter 14.

1

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6. Fermion Calculus

2.SU(N) Compact Lie Group

3.SU(N) Kac-Moody Algebra

5.SU(3) Charcter Function

8. Abelian Lattice Gauge Field

4. SU(N) Path Integrals

12. Dirac Lattice Path Integral

7. Non-Abelian Lattice Gauge Field

13.Dirac Hamiltonian

9. Lattice Gauge Field Mass Renormalization

10. Gauge Field Block-Spin 11. Lattice Gauge Field Renormalization Hamiltonian

14. Lattice Gauge Theory Hamiltonian Fig. 1.1

1.1

Flow of the chapters.

Mathematical Background

This part is focused on the necessary mathematical background to the study of lattice gauge theory. The fundamental degrees of freedom of lattice gauge theory are the finite elements of the SU (N ) Lie group and fermionic variables; the lattice gauge field is based on the calculus on compact group manifolds and the Dirac field requires fermionic variables. Chapter 2 is a brief review of compact Lie groups. Finite Lie group elements are nonlinear variables. Integration over the compact group manifold is studied as the Feynman path integral is based on this theory of integration. The differential realization of the Lie group’s generators is analyzed as it is required in defining the lattice gauge field’s Hamiltonian. Chapter 3 is on Kac-Moody algebras and is not required for any of the other Chapters. It has been included as it shows how Lie groups have an infinite dimensional generalization by enlarging the Lie algebra to include the Heisenberg algebra. Kac-Moody algebras can be used for defining new varieties of (lattice)

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3

quantum fields [Baaquie and Parwani (1996)] and also play an important role in superstring theory. Chapter 4 studies the quantum mechanics of the compact SU (N ) degree of freedom and applies the calculus of compact Lie groups to define a path integral for a chiral Lagrangian. The quantum mechanics of the Lie group’s nonlinear variables is a precursor to the more complex lattice quantum gauge field. The Hamiltonian studied in this Chapter will be seen to be the kinetic operator of the lattice gauge field. Chapter 5 applies the result of Chapter 4 to derive the exact expression for the character functions of SU (3) to illustrate the utility of the path integral. The character functions of SU (3) are obtained by techniques that are independent of algebraic techniques. Chapter 6 is on fermion calculus. Fermion calculus is unlike the calculus of real variables and this Chapter provides a brief introduction to fermion calculus from first principles. In particular, the distinctive feature of fermions is the concept of fermions and antifermions and this is discussed for the simplest case. The Dirac field is based on infinitely many independent complex fermionic variables, and this Chapter is a precursor to the discussion on the Dirac field.

1.2

Lattice Gauge Field

The chapters on lattice gauge field study the nonlinear Yang-Mills field defined on a Euclidean lattice spacetime. The lattice quantum gauge field is studied using both the path integral and Hamiltonian formulations. The lattice provides an ultra-violet cutoff for the quantum field and many interesting features of the lattice quantum fields are developed using the concept of the lattice cutoff. The Lagrangian is defined for a four-dimensional lattice embedded in Euclidean spacetime and analyzed following the treatment given by Baaquie (1977b). In Chapter 7 the lattice formulation of Yang-Mills gauge fields is reviewed and is shown to have many features quite different from the continuum formulation. The key idea of gauge-fixing is discussed and it is shown that, since Lie group variables are compact, the lattice gauge field does not need gauge-fixing. The lattice gauge field is a generalization of the continuum formulation; to illustrate that the lattice theory has all the not-so-apparent symmetries, BRST symmetry is derived for the lattice theory. In Chapter 8 the three-dimensional Abelian gauge field is studied as a warm-up to the more complex non-Abelian gauge field. A number of exact results are obtained, including the equivalence of the Abelian gauge field to the discrete Gaussian quantum field. In Chapter 9 mass renormalization for the lattice gauge field is studied to one-loop using the technique of gauge-fixing and Feynman perturbation theory. Quadratic mass divergence occurs for many diagrams, which are expressed in terms

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of lattice constants. Nontrivial lattice constant identities lead to an exact cancellation of the quadratic divergence. Zero mass renormalization is essential for the consistency and renormalizability of the lattice gauge field, and the one-loop calculation verifies this. The block-spin renormalization of the lattice gauge field, discussed for the Migdal model in Chapter 10, is a procedure that encodes Wilson’s formulation of renormalization – and brings out very clearly the infinitely many (coupled) length scales that are inherent in every quantum field theory. A simplified model proposed by Migdal for numerically renormalizing the gauge field is discussed for the SU (2) lattice gauge field. The Migdal recursion equation is solved numerically and one can study the cross-over from weak to strong coupling. The Migdal model shows how the coupling constant of an asymptotically free nonAbelian gauge field continuously becomes larger and larger and smoothly goes over to the strong coupling confining phase. The numerical solution of Migdal’s model provided the first evidence of the confinement of quarks [Baaquie (1977c)]. The lattice gauge field Hamiltonian is studied in Chapter 11. Gauge-fixing for the Hamiltonian requires a formalism based on constraining differential field operators and is quite distinct from the path integral approach based on the action. The exact gauge field Hamiltonian is obtained. 1.3

Lattice Dirac Field

The Dirac field is studied using fermionic variables. The well-known ‘doubling’ of the fermionic states is solved using Wilson fermions – which are shown to arise from the state space interpretation of the path integral. The lattice formulation allows one to study the finite time path integral with boundary conditions fixed by the lattice theory [Baaquie (1983b)]. The evolution kernel is exactly solved for the lattice using the classical solution of the Dirac field. The Ginsparg-Wilson result for lattice fermions is briefly reviewed. The continuum limit is taken and greatly simplifies the evolution kernel. Using the properties of fermionic variables, an explicit representation is obtained for all the eigenfunctions of the Dirac field. Given the importance of the Dirac field, the Hamiltonian of the Dirac field is defined using fermionic variables, directly for the continuum theory. All the eigenfunctions are then given an independent derivation based on the fermionic coordinates of the Dirac state space. 1.4

Lattice Gauge Theory Hamiltonian

The lattice Dirac and gauge fields have been discussed separately, without the two being coupled. In continuum gauge theories, the coupling of the gauge field to the charge carrying fermion field is done via minimal coupling. Minimal coupling leads to a gauge-invariant Lagrangian.

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Chapter 14 discusses the coupling of the gauge field to the Dirac field. In lattice gauge theory, the coupling of the gauge field to the Dirac field is determined by the necessity of preserving local lattice gauge invariance. One starts with the free Dirac field’s Lagrangian: the coupling the Dirac field to the gauge field is realized by placing appropriate products of gauge field links between any two fermion and antifermion field variables that are defined at distinct spacetime points. To obtain the lattice Hamiltonian for the coupled theory, one needs to define the Lagrangian on an asymmetric lattice. The time lattice  is taken to zero while the space lattice spacing a is held fixed. The asymmetric lattice Lagrangian needs to be gauge-invariant; there is also the additional constraint that one obtains the continuum Dirac-Yang-Mills Lagrangian by taking the limit of  → 0, followed by the limit of lattice spacing a → 0. The result obtained for the lattice Hamiltonian is fairly intuitive: the fermion sector consists of the free Dirac lattice Hamiltonian rendered gauge invariant with the appropriate links between nearest neighbor fermionic field variables. The lattice gauge field Hamiltonian is added to the fermion sector. A nontrivial gauge-invariant metric on the state space of the lattice gauge theory results from the lattice Lagrangian and action functional, and is required to make the lattice Hamiltonian Hermitian.

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Mathematical Background

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Chapter 2

SU (N ) Compact Lie Groups

2.1

Introduction

Group theory is a vast subject of mathematics and forms one of the mathematical cornerstones of theoretical physics. Lie group theory plays a major role in various quantum field theories. The representation theory of SU (2) Lie algebra is well known to most physics undergraduates in the study of angular momentum. The quark model of particle physics is based on SU (3) Lie algebra, and Yang-Mills gauge fields introduced SU (N ) Lie algebras to the study of gauge fields. As mentioned earlier, lattice gauge theory is based on degrees of freedom that are the finite element of the SU (N ) group, which are the nonlinear variables required for defining the gauge field theories on a lattice spacetime. The review of Lie groups in this chapter is a necessary preparation for the study of lattice gauge fields. A few of the key concepts of Lie group are reviewed and the discussion is limited to only SU (N ). Good references for the material in this Chapter are the following: [Zelobenko (1973); Haber (2017); Fadin and Fiore (2005); Cutler and Sivers (1978)]. A group G is a collection of elements g ∈ G that obey the following four axioms. • A group multiplication ∗ is defined such that, for g1 , g2 ∈ G, we have g1 ∗ g2 ∈ G. • Group multiplication ∗ is associative so that, for g1 , g2 , g3 ∈ G, (g1 ∗g2 )∗g3 = g1 ∗ (g2 ∗ g3 ). • The identity element I exists such that for g1 , I ∈ G, I ∗ g = g ∗ I = g • The inverse exists, namely for g ∈ G, there exists a g −1 ∈ G such that g ∗ g −1 = g −1 ∗ g = I A Lie group is a group with the additional property that it is constituted by elements that form a continuous differentiable space, with group multiplication having continuity for elements that are continuously varied. The most familiar Lie groups in physics are the unitary (matrix) groups, denoted by U (N ), which are the set of N ×N matrices with complex elements that satisfy the condition that U † U = I, where U † is Hermitian conjugation (defined as the complex conjugation and transposition of the matrix elements). Group multiplication ∗ is 9

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the ordinary matrix multiplication. The group SU (N ) is the special unitary group that has the additional constraint that det U = 1. The Lie groups U (1), SU (2), SU (3), SU (5), SO(10), E6 and E8 × E8 appear in the study of the Standard Model of particle physics and in superstring theory. 2.2

Lie Algebras

Lie groups are a special case of groups that are generated by a Lie algebra. Let U ∈ SU (N ) be an element of the group; it can be represented by the generalization of the exponential function to a matrix function. Repeated indices are always summed unless stated. The Lie group element is given by the following U = exp{iB a X a }

; a = 1, 2, ..., N 2 − 1

(2.2.1)

where X a are the generators of the Lie algebra that can be finite dimensional matrices. The variables B a are the canonical coordinates of the Lie group element. Note that the exponential function, which is given by exp(iθ), θ ∈ [−π, +π], is an element of the U (1) group; the Abelian phase θ is generalized by the Lie group to non-Abelian phases B a . Let C abc be the totally antisymmetric structure constants; for all compact groups abc C are totally antisymmetric in the three indices.1 Defining the commutator of two matrices by [A, B] ≡ AB − BA the Lie algebra is defined by the commutation equation [X a , X b ] = iC abc X c ; X a = Xa = Xa† : Hermitian Tr(X a X b ) = δ ab /s2 (p) The finite dimensional matrices X a = X a (p) – called the generators of the Lie algebra – define the representation of the Lie algebra, are denoted by p, and discussed further in Section 2.7. The structure constants C abc for (finite dimensional) compact Lie groups can be completely classified and lead to the A, B, D and E series of Lie groups [Zelobenko (1973), Varadarajan (1974)]. Noteworthy 2.1: Notation In all the derivations, no distinction is made between upper and lower indices; rather, the choice of upper or lower indices is based on notational convenience. The index for the generators and structure constants will be denoted by both subscripts and superscripts, which are equivalent for compact Lie groups. In particular C abc ≡ Cabc ; X a ≡ Xa ; B a ≡ Ba 1 This

is not the case for non-compact Lie groups such as the Lorentz group.

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Repeated indices are to be summed over; the summation is sometimes explicitly indicated when required for greater clarity. The commutators follow the rule for differentiation; similar to the chain rule of differentiation, one obtains the Jacobi identity for finite matrices A, B, C given by [A, [B, C]] = [[A, B], C] + [B, [A, C]]

(2.2.2)

The Jacobi identity being obeyed by the generators of the Lie algebras is a consequence of group multiplication being associative. The Jacobi identity yields, for the structure constants [X A , [X B , X C ]] = [[X A , X B ], X C ] + [X B , [X A , X C ]] ⇒ C AαI C BCα = C ABα C αCI + C ACα C BαI

(2.2.3)

The Jacobi identity leads to constraints on the allowed set of structure constants C abc . The structure constants themselves provide the adjoint representation of the Lie algebra. Consider the following relabeling in Eq. 2.2.3 I=i ; C=j Using the complete antisymmetry of C abc under the interchange of any two indices, the Jacobi identity can be rewritten as follows C iAα C αBj − C BαI C αAj = C ABα C iαj

(2.2.4)

Using the matrix notation for writing the ij components of the matrices yields, from Eq. 2.2.4 A Cij ≡ C iAj

⇒ [C a , C b ] = C abc C c

(2.2.5)

The generators of the adjoint representation are defined by (F a )ij = iCiaj It follows from Eq. 2.2.5 that the structure constants provide the adjoint representation of the Lie algebra. The adjoint representation of the SU (N ) Lie algebra is defined by N 2 − 1 generators and is given by N 2 − 1 × N 2 − 1 matrices. The generators of the adjoint representation yield the following representation of the group elements (F a )ij = iC iaj

⇒ ρ = eiF

a

Ba

= e−ω

where the adjoint matrix ω is given by X ρ = e−ω ; ωab = C acb Bc = −ωba ⇒ ω T = −ω c

(2.2.6)

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Vielbein of SU (N )

The SU (N ) group space is spanned by all the allowed values of the non-Abelian phases B a ; the space is a compact Riemann space, which is simply connected for N ≥ 2. Define the vielbein of SU (N ) group space, denoted by eab , as follows eab Xb = −iU † ∂a U ; U = exp{iBa Xa }

(2.3.1)

Introduce the homotopy parameter τ by the following U (τ ) = exp{iτ Bα Xα } ; eab (τ )Xb = −iU † (τ )∂a U (τ ) Boundary conditions : eab (1) = eab ; eab (0) = 0

(2.3.2)

Differentiating by τ yields deab (τ ) Xb = −iU † (τ )(−iBα Xα )∂a U (τ ) − iU † (τ )∂a {iBα Xα U (τ )} dτ deab (τ ) Xb = U † (τ )Xa U (τ ) (2.3.3) ⇒ dτ Using ρab as the group element in the adjoint representation defined by U † Xa U = ρab Xb

(2.3.4)

yields from Eqs. 2.3.3 deab (τ ) Xb = U † (τ )Xa U (τ ) = ρaα (τ )Xα ⇒ dτ For Xa in the fundamental representation

deab (τ ) = ρab (τ ) dτ

(2.3.5)

1 δab (2.3.6) 2 The generators of SU (N ) in the fundamental representation Xa obey the following identity [Haber (2017)]   X 1 1 Tr(AB) − Tr(A) Tr(B) Tr(AXa ) Tr(BXa ) = 2 N a ρab = 2 Tr(Xa U Xb U † ) :

Hence X

ρac (U )ρcb (V ) = 4

c

X

Tr(Xa Xb ) =

Tr(Xa U Xc U † ) Tr(Xc U V Xb V † ) = ρab (U V )

(2.3.7)

c

It is shown that definition of the adjoint representation given in Eq. 2.3.4 reproduces the result stated earlier in Eq. 2.2.6. Consider the following U † (τ )Xa U (τ ) = ρaα (τ )Xα ; ρaα (1) = ρaα : ρaα (0) = δa−α This yields dρaα (τ ) Xα = −iBβ U † (τ )[Xβ , Xa ]U (τ ) = −i2 Bβ C βac U (τ )Xc U † (τ ) dτ = −ωac ρcd (τ )Xd ; ωac = Bβ C aβc

(2.3.8)

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Hence, integrating the differential equation below yields, in matrix notation dρaα (τ ) = −ωac ρcα (τ ) ⇒ ρ(τ ) = e−τ ω dτ and yields the result given in Eq. 2.2.6. From Eqs. 2.3.5 and 2.3.9, in matrix notation

⇒ ρ(1) = ρ = e−ω

de(τ ) = ρ(τ ) = e−τ ω dτ and hence, using boundary conditions Eq. 2.3.2, yields Z 1 ∂e(τ ) 1 − e−ω dτ e= = ∂τ ω 0

(2.3.9)

(2.3.10)

One has the following expansion 1 − e−ω 1 (2.3.11) ' 1 − ω + O(ω 2 ) ω 2 The definition of the vielbein given in Eq. 2.3.1 yields the following result e=

eab Xb = −iU † ∂a U ; U = exp{iBα Xα } n o 1 ⇒ eab Xb = −i U † (B) U (Bα + δaα ) − U (B)  † ⇒ U (B)U (Bα + δaα ) = 1 + ieab Xb '= exp{ieab (B)Xb }

(2.3.12)

The result obtained above in Eq. 3.4.20 will prove to be useful in studying the differential representation of the generators. 2.4

Metric on SU (N ) Group Space

The group manifold SU (N ) is a Riemann metric space with constant curvature. The metric is defined using the properties of the group elements. Define the matrix differential of the group element U by dU (dU )αβ = d(exp iB a X a )αβ

(2.4.1)

The metric g ab (B) is defined by [Zelobenko (1973)] g ab (B)dB a dB b =

1 Tr(dU dU † ) s2

(2.4.2)

g ab is invariant under group multiplication; to see this, let V, W ∈ SU (N ) be numerical matrices. Then U 0 = V U W ⇒ dU 0 = V (dU )W ⇒ Tr(dU 0 dU 0† ) = Tr(dU dU † ) From Eqs. 2.4.2 and 2.3.1 1 −i2 † † Tr(dU U U dU ) = eab dBb ea0 b0 dBb0 Tr(Xa Xa0 ) s2 s2 = eab dBb eab0 dBb0 = (eT e)ab dB a dB b

gab (B)dB a dB b =

(2.4.3)

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and the metric is given by  cosh(ω) − 1  (2.4.4) gab = (eT e)ab = eTac ecb = 2 ω2 ab Note that for Lie groups the existence of the underlying generators leads to the factorization of the metric tensor. In matrix notation, Eqs. 2.3.11 and 2.4.4 yield the following 1 − eω 1 − e−ω 1 cosh(ω) − 1 ' 1 + ω 2 + O(ω 4 ) · =2 (2.4.5) ω ω ω2 12 The metric tensor gab defines a space with constant curvature. The space is an Einstein space with the Ricci tensor Rab being proportional to the metric tensor and the Ricci curvature scalar R is a constant. More precisely, as shown by Marinov and Terentyev (1979) 1 Rab = 4gab ; R = N 4 Note the remarkable fact that the constant curvature R of the Lie group depends only on the groups dimensionality. The metric tensor gab yields the following geodesic, denoted by U (t), between group elements I and U g = eT e = −

U (t) = (U )t ; U (0) = I ; U (1) = U

: t ∈ [0, 1]

The group elements U (t) trace out a geodesic in the group manifold. Since U (t)U (t0 ) = U (t + t0 ), the geodesic U (t) forms an Abelian one parameter subgroup of G. 2.4.1

SU (2) metric

As an example, the metric for SU (2) is calculated; the fundamental representation (discussed in Section 2.7) yields, for σ a =Pauli matrices, the following U = exp{iB a σ a /2} ! B 1 + ina σ a sin = cos 2

B 2

! (2.4.6)

where B2 =

X

B a B a , na =

a

Ba B

(2.4.7)

Note − − − d(→ n 2 ) = 2→ n · d→ n =0

(2.4.8)

A straightforward calculation gives 1 Tr(dU dU ) = (dB)2 + 2 sin2 2 †

= g ab dB a dB b

! B − (d→ n )2 2 (2.4.9)

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Change from Cartesian coordinates (B 1 , B 2 , B 3 ) to polar coordinates (B, θ, φ). Then − d(→ n )2 = dφ2 + sin2 θdθ2

(2.4.10)

Let B = a1 , θ = a2 , φ = a3 ; then g ab dB a dB b = g ij dai daj where 

g ij

1/2 = 0 0

 0  0 2 B 2 sin ( 2 )

0 2 sin2 ( B2 ) sin2 θ 0

(2.4.11)

SU (2) is isomorphic to S 3 , and the metric given by 2.4.11 is the metric on S 3 . 2.5

The Invariant Haar Measure and Delta Function

The symbol dU denotes the invariant measure as well as the infinitesimal matrix. It will be clear from the context of its use what it means. In terms of the canonical coordinates, the measure is written as Z Y YZ a dU = µ(B) dB ⇒ dU f (U ) = dB a µ(B)f (U ) (2.5.1) a

a

The Haar measure µ(B) is defined, as is the case for all curved manifolds, by the following p (2.5.2) µ(B) = g(B) ; g(B) = det(g ab (B)) The measure is invariant under group multiplication in that µ(U ) = µ(V U W ) ; U, V, W ∈ SU (N ) This follows from the invariance of the metric given in Eq. 2.4.3. Since det(exp A) = exp(Tr A), from Eq. 2.4.5, in matrix notation and using Caαβ Cbαβ = N δab r n  1 o p µ(B) = det g = det exp ω2 + O(ω 4 ) 12 n1 o  N  = exp Tr(ω 2 ) = exp − B 2 + 0(B 4 ) (2.5.3) 24 24 The result for the measure term given in Eq. 2.5.3 is required for the mass renormalization calculations in Chapters 4 and 9.

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2.5.1

Delta function on group space

Following the procedure for a general curved manifold, the delta function on group space is defined by 1 Y ˜ a) δ(B a − B (2.5.4) δ[U − V ] = µ(B) a where ˜ aX a} U = exp{iB a X a } ; V = exp{iB δ[U − V ] has the important property that, for σ in the group space, δ[σ(U − V )σ † ] = δ[U − V ]

(2.5.5)

The δ-function for the group elements is similar to the usual δ-function; for V in the group space, we have Z Z dU δ[U − V ] = 1 ; dU f (U )δ[U − V ] = f (V ) (2.5.6) 2.5.2

SU (2) measure

The measure for SU (2), from 2.4.11, is given by   Y B sin θ(dBdφdθ) = c sin2 (B/2)dBdΩ dU = µ(B) dBa = c sin2 2 a where dΩ = sin θdφdθ : solid angle Note that for B ' 0, the group space is isomorphic to the three-dimensional Euclidean space. The range of the variables B, θ, φ are such that each group element occurs once and only once. We can fix this directly from Eq. 2.5.7; the solid angle is to be covered once (for n > 2, the same holds true for the higher dimensional solid angles) and the radial variable B take values such that sin2 (B/2) > 0. The point where the radial factor is zero is taken to be the maximum of B. Hence 0 ≤ B ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π

(2.5.7)

Note the factorization of dU into a solid angle and a radial part is a general feature of SU (N ). For all values of the variables, µ(B) ≥ 0. The points where µ(B) = 0 marks the limit of the variables. The constant c is fixed by the normalization condition Z dU = 1 (2.5.8) G

which, in case of SU (2) gives the final result of dU =

1 sin2 (B/2)dBdΩ 4π 2

(2.5.9)

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SU (2) measure: invariant

To illustrate the invariance of the Haar measure under right- and left-multiplication, the case of SU (2) is studied. Choose a representation of SU (2) group element2   a b U= ⇒ U † U = I ; det U = |a|2 + |b|2 = 1 ; a, b : complex numbers −b∗ a∗ The left- and right-invariant Haar measure is defined by Z Z Z +∞ 2 2 daR daI dbR dbI δ(|a| + |b| − 1) ≡ dadbδ(|a|2 + |b|2 − 1) (2.5.10) dU = −∞

The variables a, b with constraint |a|2 + |b|2 = 1 are coordinates for S 3 ; choosing polar coordinates for a, b yields the SU (2) measure given in Section 2.5.2. Consider another SU (2) group element, with constant elements, given by   c d V= ⇒ V † V = I ; det V = |c|2 + |d|2 = 1 ; c, d : complex numbers −d∗ c∗ Left-multiplication yields     ac − b∗ d bc + a∗ d α β VU = ≡ −(bc + a∗ d)∗ (ac − b∗ d)∗ −β ∗ α∗ The matrix V is equivalent to a rotation of a four-vector in four-dimensional Euclidean space and leads to dαdβ = dadb : invariant Furthermore |α|2 + |β|2 = det(V U ) = det(V ) det(U ) = (|c|2 + |d|2 )(|a|2 + |b|2 ) = |a|2 + |b|2 Hence, from Eq. 2.5.10 the measure is left-invariant and given by Z Z Z Z 2 2 2 2 d(V U ) = dαdβδ(|α| + |β| − 1) = dadbδ(|a| + |b| − 1) = dU A similar derivation shows the measure is right-invariant as well. 2.6

Campbell-Baker-Hausdorff (CBH) Formula

Let ex , ey be finite group elements of Lie group G and consider the multiplication ex ey = ez Then [Zelobenko (1973)] z = x + y + [x, y] +

∞ Y m m X X (−1)m−1 [xp1 y q1 · · · xpm y qm ] Q ·P m i (pi + qi ) i pi !qi ! m=1 i=1 p ,q =0 i

2 The

i

following matrix, used in quantum random walks, is an element of U (2) but not of SU (2)   a b ⇒ U † U = I ; det U = −(|a|2 + |b|2 ) = −1 ; a, b : complex numbers b∗ −a∗

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with pi + qi 6= 0 and [a1 a2 · · · an−1 an ] = [a1 [a2 · · · [an−1 , an ] · · · ]] In particular, the leading terms in the expansion are the following [Varadarajan (1974)] 1 1 1 z = x + y + [x, y] − [[x, y], x] + [[x, y], y] 2 12 12 1 1 − [y, [x, y], x]] − [x, [[x, y], y]] + .. 48 48 2.7

(2.6.1)

Irreducible Representations of SU (N )

A representation of the elements of SU (N ) consist of matrices that obey the multiplication law of the group, which in turn requires that the matrices embodying the generators Xa must obey the Lie algebra of the generators of SU (N ). It is noteworthy that matrices of higher and higher dimensions can all provide a representation of the underlying Lie group SU (N ). A faithful representation is one for which each element of the Lie group corresponds to one element of the representation – in other words, the representation is an isomorphism of the group elements to elements of the representation. One can form block diagonal matrices, with each block corresponding to one representation of the Lie group. An irreducible representation cannot be further reduced to a block diagonal form. The fundamental and the adjoint representations are the two most important representations of the group elements of SU (N ) and are discussed below. (p) Let U be an element of the special unitary group SU (N ), and let Dij (U ) be the matrix elements of the irreducible representation given by the composite representation label p. Note 1 < i, j < dp where dp is the dimension of the pth irreducible representation. The traceless Hermitian generators in the pth representation are given by dp × dp matrices X a (p), a = 1, 2, · · · , N 2 − 1, satisfying the Lie Algebra [X a (p), X b (p)] = iCabc X c (p) ;

Tr(X a (p)X b (p)) =

1

δab s2 (p)

(2.7.1)

where Cabc are the antisymmetric structure constants of SU (N ). In terms of dimensionless canonical coordinates B a of SU (N ) an irreducible representation of SU (N ) is given by (p)

Dij (U ) = (eiB

a

X a (p)

)ij

(2.7.2)

The coordinates B a take value in the SU (N ) group space. The simplest Lie Group is the Abelian group U (1) with Ciaj = 0 , i, a, j = 1. The fundamental representation given by U = eiθ : θ ∈ S 1 . The adjoint representation for U (1) is trivial, equal to 1. All the irreducible representation of U (1) are given by integers p where (p)

p =∈ [−∞, +∞] ; X(p) = p ; Dij (U ) = eipθ ; dp = 1

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The quadratic Casimir operator for the pth irreducible representation for SU (N ) is defined by 2

X (p) ≡

2 NX −1

X a (p)X a (p)

(2.7.3)

a=1

Note that the second Casimir operator commutes with all the generators since hX i X X X X aX a, X b = X a [X a , X b ]+ [X a , X b ]X a = iC abc (X a X c +X c X a ) = 0 a

a

a

a,c

Using Schur’s Lemma, it follows that the second Casimir operator is proportional to the identity matrix and yields X 2 (p) = c2 (p)Ip ;

Tr(Ip ) = dp

(2.7.4)

where c2 (p) is the numerical value of the quadratic Casimir operator in the pth irreducible representation and Ip is a dp × dp unit matrix. The structure constants of the Lie algebra are fixed; other constants, including s(p), c2 (p) and dp , depend on the representation p of the Lie algebra. The two most important representations of the Lie algebra of SU (N ) are the fundamental and the adjoint representations. The fundamental and adjoint representations are given by the following. • Fundamental representation. The smallest size matrices that can provide a faithful representation of SU (N ) are N × N Hermitian and traceless matrices. Given their special importance, the generators of the fundamental representation are denoted by T a . The fundamental representation yields the following i 1h 1 Iδab + (dabc + iC abc )T c ; T r(I) = N (2.7.5) T aT b = 2 N where the constants dabc are completely symmetric under the exchange of any two indices. The fundamental representation has the following normalization 1 Tr(T a T b ) = δab 2 Since there are N 2 − 1 generators of the Lie group, taking the trace of the second Casimir operator yields, from Eq. 2.7.4, for the fundamental representation c2 (F ) the following X  1 1 Tr T a T a = c2 (F )N ⇒ (N 2 −1) = c2 (F )N ⇒ c2 (F ) = (N 2 −1) 2 2N a • Adjoint representation. The Jacobi identity given in Eq. 2.2.3 shows that the structure constants themselves provide a representation of the Lie algebra; from Eq. 2.2.5 (F a )ij = iCiaj ; 1 ≤ i, a, j ≤ N 2 − 1

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The adjoint representation, ρ = exp{−ω} ∈ SU (N ), is defined by the action of the group elements on the Lie group generators; for any representation U ∈ SU (N ), from Eq. 2.3.4 U † X a U = ρaα X α For the adjoint representation [Haber (2017)] Tr(F a F b ) = N δab The equation above is also written as C

abc

C

abd

2

= N δab ⇒ F =

2 NX −1

F a F a = N I ⇒ c2 (A) = N (2.7.6)

a=1

In summary, the quadratic Casimir operators are the following: • Fundamental representation of SU (N ) : c2 (F ) = • Adjoint representation of SU (N ) : c2 (A) = N

1 2N

(N 2 − 1)

Some identities of the structure constants are the following [Haber (2017)] Tr(Fa Fb Fc ) =

i N C ade 2

1 Tr(Fa Fb Fc Fd ) = δad δbc + (δab δcd + δac δbd ) 2 1 + (C ade C bce + dade dbce ) 4N 2.8

(2.7.7)

Peter-Weyl Theorem

Let |U i be the coordinate eigenstate for the state space defined on SU (N ); let hp, ij| be the conjugate eigenstate, which are the tenor product of row vectors of a finite dimension that is determined by the representation p, and with 1 ≤ i, j ≤ dp . Then p (p) hp, ij|U i = dp Dij (U ) ; 1 ≤ i, j ≤ dp (2.8.1) Note that hU |U 0 i = δ(U − U 0 ) 0

0 0

hp, ij|p , i j i = δ

pp0

δii0 δjj 0

(2.8.2) (2.8.3)

where the δ-function is defined in Eq. 2.5.4 using the metric on group space. The completeness equation on Hilbert space given by Z XX I= |p, ijihp, ij| = dU |U ihU | (2.8.4) p

ij

where the sum on p is over all the irreducible representations. For U (1) group, the dual basis is hp| and p (p) hp|U i = dp Dij (U ) = eipθ ; hU |U 0 i = δ(θ − θ0 ) ; hp|p0 i = δp−p0

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21

and completeness equation for U (1) is given by Z +π +∞ X 1 I= dθ|θihθ| |pihp| = 2π −π p=−∞ Let f (U ) be any square-integrable function of SU (N ) Z dU |f (U )|2 < ∞

(2.8.5)

G

The Peter-Weyl theorem for a compact Lie group G states the following [Zelobenko (1973)] Z 0 0 1 (`) (m) dU Dij (U † )Di0 j 0 (U ) = δ `m δ ii δ jj : Peter-Weyl theorem d` G which, for U (1) is given by Z

1 2π

+π

dθe−ipθ eimθ = δp−m

−π

The character functions for SU (N ) is defined by χp (U ) =

dp X

(p)

Dii (U ) = Tr(D(p) (U ))

i=1

The Peter-Weyl theorem for the character functions states that Z dU χp (U † )χp0 (U ) = δpp0

(2.8.6)

G

Let f (U ) be an aribtrary function of the group element U . The Fourier theorem states that X f (U ) = dp Tr(D(p) (U )ap ) (2.8.7) p

Using the Peter-Weyl orthogonality theorem gives Z (p) Tr(D (U)ap ) = dU 0 f (U 0 )Tr(D(p) (UU0† ))

(2.8.8)

Also 2

kf k =

Z

dU |f (U )|2 =

X

dp T r(a†p ap )

p

2.9

Lie Algebra Generators: Differential Operators

The Lie group generators are realized by differential operators acting on functions of the Lie group elements. Consider function f (U ) of group element U ; any faithful representation can be chosen and the representation is explicitly shown only if needed. To foreground the general case, consider U (1); then, from Taylor expansion ∂

f (U ) → f (ΦU ) = f (eiφ eiB ) = eiφ i∂B f (U ) ; U = eiB ; Φ = eiφ

(2.9.1)

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The non-Abelian generalization of Eq. 2.9.1, for the action of the group element of SU (N ), is given by f (U ) → f (ΦU ) = eiφa Ea f (U ) ; U = eiBa Xa ; Φ = eiφa Xa where Ba , a = 1, 2, .., N 2 − 1 are the canonical coordinates of SU (N ). The differential operators Ea obey the Lie algebra commutation equation [Ea , Eb ] = iC abc Ec ; a, b, c = 1 · · · N 2 − 1 (2.9.2) The differential operators Ea , unlike the generators of the Lie algebra Xa , are independent of the representation chosen for Xa , and depend only on the properties of the group manifold. From Eq. 2.3.1 eab Xb = −iU † ∂a U ∂ U (2.9.3) ⇒ U Xa = −ie−1 ab ∂b U ≡ fab i∂Bb where, from Eq. 2.3.10 fab eac = δab ⇒ fab ≡ e−1 ab ω ; ωac = C abc Bb (2.9.4) ⇒ f = e−1 = 1 − e−ω The Lie algebra generators, from Eq. 2.9.3 have the following differential realization ∂ Ea = fab (B) ⇒ Ea U = U Xa (2.9.5) i∂Bb The commutation given in Eq. 2.9.2 is reproduced since Ea Eb U = Ea U Xb = U Xa Xb ⇒ [Ea , Eb ]U = U [Xa , Xb ] = iC abc Ec U ⇒ [Ea , Eb ] = iC abc Ec (2.9.6) Eq. 2.9.2 also follows directly from the definition given in Eq. 2.9.5 due to the Maurer-Cartan equation given by fαi ∂i fβγ − fβi ∂i fαγ = −Cαβi fiγ : ∂i ≡ ∂/∂Bi (2.9.7) It is shown that the differential generators of the Lie group Ei are Hermitian differential operators. Note that  ∂ † 1  ∂ † ∂ =− = : Hermitian (2.9.8) i∂Bj i ∂Bj i∂Bj hence, one has ∂fij ∂ ∂ fij = + fij (2.9.9) Ei† = i∂Bj i∂Bj (σ) i∂Bj (σ) From Eq. 2.9.4 ∞ ∞ X X (−1)n f= ωe−`ω = ω(`ω)n (2.9.10) n! `=0

`,n=0

Note from Eq. 2.2.6 



X ∂ω = Ciαj δjα = 0 (2.9.11) i∂Bj ij P α It follows that, when one evaluates j ∂fij /∂Bj , each term in f (ω) – given in the series expansion in Eq. 2.9.10 – yields zero; hence, from Eq. 2.9.9 ∂ Ei† = fij = Ei : Hermitian (2.9.12) i∂Bj Hence, the generators defined in Eq. 2.9.5 are Hermitian operators.3 3 For

the Abelian case E = ∂/i∂B and is Hermitian due to Eq. 2.9.8.

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2.10

23

Vielbein eab and fab for SU (2)

Consider the adjoint matrix representation of the SU (2) group given by ρ = e−ω ; ωab = εaαb θα : a, α, b = 1, 2, 3 with εaαb being the structure constants of SU (2) and is the three-dimensional completely antisymmetric tensor. In matrix notation ω 1 − e−ω ; f= e= ω 1 − e−ω The identity εaαb εcαd = δac δbd − δad δbc yields, in matrix notation ω 3 = −θ2 ω ; θ2 = θα θα

(2.10.1)

2

From Eq. 2.10.1 only ω, ω are independent matrices since ω 3 = −θ2 ω ; ω 4 = −θ2 ω 2 · · · Hence ω2 ω sin(θ) + 2 (1 − cos(θ)) θ θ Using Eq. 2.10.1, the vielbein is given by sin(θ) ω 1 − e−ω = − 2 (1 − cos(θ)) e= ω θ θ The derivation of fab is more involved. Consider the equation ω 1 ω 1 ω 1 + e−ω ω f− ω= − ω = = coth 2 1 − e−ω 2 2 1 − e−ω 2 2 The coth(x) function has the expansion ρ = e−ω = 1 −

coth(x) =

1 x x3 2x5 x7 + − + − + O(x9 ) x 3 45 945 4725

(2.10.2)

(2.10.3)

From Eqs. 2.10.1 and 2.10.3 ω  ω 2 h 1 (θ/2)2 i ω 2(θ/2)4 (θ/2)6 coth = 1+ + + + + ··· 2 2 2 3 45 945 4725  ω 2 h (θ/2)2 i (θ/2)4 2(θ/2)6 (θ/2)8 + + + + ··· = 1+ θ 3 45 945 4725  ω 2 h i 1 − (θ/2) cot(θ/2) (2.10.4) = 1+ θ where 1 x x3 2x5 x7 cot(x) = − − − − + O(x9 ) x 3 45 945 4725 Hence, Eqs. 2.10.2 and 2.10.4 yield the final result 1 f = 1 + ω + ω 2 A(θ) (2.10.5) 2

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where 

 θ θ 1 − cot 2 2 1 1 ' + θ2 + O(θ4 ) 12 16 · 45 For SU (2), the Maurer-Cartan equation (∂i ≡ ∂/∂θi ) given in Eq. 2.9.7 yields 1 A(θ) = 2 θ

fαi ∂i fβγ − fβi ∂i fαγ = −εαβi fiγ The Maurer-Cartan equation are satisfied by the explicit form for f given by Eq. 2.10.5 since A satisfies the remarkable differential equation ∂A 1 + 3A − θ2 A2 = (2.10.6) ∂θ 4 Eq. 2.10.6 will be essential in obtaining the Kac-Moody commutation equation for SU (2). θ

2.11

Left and Right Invariant Generators

The generators for the p irreducible representation, given by Xa ≡ X a (p), yield the following Ea exp{iB a Xa } = exp{iB a Xa }Xa

(2.11.1)

Ean exp{iB a X a } = exp{iB a Xa }Xan

(2.11.2)

From Eq. 2.11.1

It follows from Eq. 2.11.2 that eiφa Ea eiB

a

Xa

= eiB

a

Xa iφa Xa

e

(2.11.3)

Let V be a constant SU (N ) matrix; then from Eq. 2.11.3 eiφa Ea V U = V U eiφa Xa ; U = eiB

a

Xa

(2.11.4)

Hence, the generator Ea (U ) is invariant under a left-multiplication of U by V and hence Ea (U ) = Ea (V U ) ≡ EaL (U ) : left-invariant and E L , from Eq. 2.9.5, can be written as follows EaL ≡ Ea = fab

∂ i∂Bb

(2.11.5)

A right-invariant generator EaR (U ) is defined by EaR U = Xa U From Eq. 2.11.6 EaR (U ) = EaR (U V ) : right-invariant

(2.11.6)

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Hence [EaR , EbR ]U = [Xb , Xa ]U = iC bac EcR ⇒ [EaR , EbR ] = −iC abc EcR From Eq. 2.11.6 R

eiφa Ea U = eiφa Xa (p) U ; U = eiB

a

X a (p)

(2.11.7)

It follows from the definitions that [EaL , EbR ] = 0 In summary, the left- and right-invariant generators of the Lie algebra have the following commutation equations [EaL , EbL ] = iC abc EcL ; [EaR , EbR ] = −iC abc EcR ; [EaR , EbL ] = 0 (2.11.8) Hence, from Eqs. 2.3.4 and 2.11.6 EaR U = Xa U = U U † Xa U = ρab U Xb = ρab EbL U ∂ ∂ T = fab (2.11.9) ⇒ EaR = ρab EbL = ρab fac i∂Bc i∂Bb since, in matrix notation and from Eq. 2.9.4 ω ω f= ; ω T = −ω ⇒ f T = e−ω f = ω (2.11.10) 1 − e−ω e −1 Putting together the left- and right-invariant generators yields, from Eqs. 2.11.4 and 2.11.7, the following L

R

eiφa Ea eiλa Ea U = eiλa Xa U eiφa Xa ; U = eiB

a

Xa

(2.11.11)

The Laplacian on the group space is useful for the Hamiltonian approach to the lattice gauge field, as discussed in Chapter 11. As for any curved space, the Laplacian for functions defined on the group space is given by 1 ∂ √ ab ∂  (2.11.12) gg −∇2 = g ∂B a ∂B b It can be shown that the generators of the Lie algebra Ea yield the following factorization of the Laplacian [Zelobenko (1973)] − ∇2 = EaL EaL = EaR EaR = E 2

(2.11.13)

2

The Laplacian ∇ is a dimensionless second order differential operator defined in Eq. 11.4.5 for the Hamiltonian of the SU (N ) lattice gauge field. From Eq. 2.11.2 the Casimir operator is given by4 E 2 exp iB a X a (p) = {X b (p)X b (p)} exp iB a X a (p) = c2 (p) exp iB a X a (p) ⇒ E 2 exp iB a X a (p) = c2 (p) exp iB a X a (p) : eigenfunctions

(2.11.14)

Hence, from Eq. 2.11.14 (p)

(p)

(p)

−∇2 Dij (U ) = E 2 Dij (U ) = c2 (p)Dij (U )

(2.11.15)

X 2 (p)|p, iji = c2 (p)|p, iji

(2.11.16)

Note that and for U (1), c2 (p) = p2 . 4 Using

P

a

Eq. 2.11.14.

EaR EaR = E 2 yields E 2 exp iB a X a (p) = exp iB a X a (p) · c2 (p) which is equivalent to

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2.11.1

Differential realization of SO(3) generators on state space

The action of group elements define linear transformations on an underlying linear vector space. Let v be column vector of an dp -dimensional linear vector space. The action of the group element is defined by matrix multiplication and yields v → w = Uv

(2.11.17)

For simplicity, consider the special case of the adjoint representation of SU (2), as this plays a special role in angular momentum theory. The generators and the group element in the adjoint representation of SU (2) are given by (Xi )ab = iaib ; ρ(θ) = e−ω ; (ω)ab = θi aib : a, i, b = 1, 2, 3 The group element acts on three-dimensional Euclidean space E3 with (real) three component vectors denoted by v; the group rotates the vector as follows v → ρv Let v = (x1 , x2 , x3 ) and let f (x) be functions of E3 ; the generators on functions of E3 , denoted by La , are given by La = −iaij xi

∂ ∂xj

⇒ [La , Lb ] = iabc Lc

The generators La are the angular momentum operators of quantum mechanics. The generators provide the realization of Eq. 2.11.17 for the specific case of the rotation x into y. Using the identity valid for O being an operator or a matrix  1 N lim 1 + O = eO N →∞ N yields, in matrix notation x → y = eiθa La x = lim

N →∞



1+i

θ a N La x N

Note (iθa La )xk = −i2 θa aik xi = −θa kai xi = −(ωx)k

⇒ (iθa La )x = −ωx

Hence x → y = lim

N →∞

2.12



1−

ω N x = e−ω x = ρ(ω)x N

SU (N ) Character Function Expansion

For the case when f (U ) is a trace function such that f (U ) = f (V U V † ), Eq. 2.8.7 yields X X f (U ) = dp ap Tr(D(p) (U )) = dp ap χp (U ) (2.12.1) p

p

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The character expansion reduces, for the U (1) case, to the discrete Fourier transform. The irreducible representations of U (1) are given by exp ipθ, p ∈ Z and Eq. 2.12.1 yields dp = 1 ; χp (U ) = eipθ

⇒ f (θ) =

+∞ X

ap eipθ

p=−∞

The Fourier representation is useful for performing convolutions. To define blockspins in Migdal’s model, discussed in Chapter 10, a four-fold convolution has to be performed. Let eA0 be the action function defined for lattice spacing a; the renormalization transformation R generates the action A1 with lattice spacing 2a and is given by the following (four-fold convolution) eA 0 =

X

dp cp χp (U ) ⇒ eA1 = R(eA0 ) =

p

nX

o2d−2 dp c4p χp (U )

p

The above equation is useful for generating a strong coupling expansion of R, since then the coefficients cp rapidly go to zero. However, for the weak coupling sector, too many terms have to kept in the series expansion, and working in the group space becomes necessary. Let X a be the fundamental representation of the generators of SU (N ), and let Tr(X a X b ) =

1 δab 2

(2.12.2)

Define Φ to be the group element in the fundamental representation and let φa be the canonical coordinates for Φ a

Φ = eiφ

Xa

It is shown below that5 n 1 o X g2 † lim dp e− 2 c2 (p) χp (Φ) ' exp Tr(Φ + Φ ) →0 2g 2  p

(2.12.3)

(2.12.4)

The choice of the fundamental representation for Φ is arbitrary; using any other irreducible representation for Φ would only change the effective coupling constant g. To prove 2.12.4, consider the following Fourier expansion o X n 1 † Tr(Φ + Φ ) = dp ap χp (Φ) (2.12.5) exp 2g 2  p The orthogonality theorem given in Eq. 2.8.6 yields ( ) Z 1 1 † † dΦχp (Φ ) exp Tr(Φ + Φ ) ap = dp 2g 2  5 This

identity will be required in the discussion of SU (N ) in Chapter 4.

(2.12.6)

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For  → 0, expanding Φ about unity and using Gaussian integration yields Z P a 1 Y ∞ − 1 φa φa dφa Tr(e−iφ Xa (p) )e 2g2  a ap = dp a −∞ " # Z 1 Y ∞ g2 X a a a b a b = dφ Tr Ip − iφ Xa (p) − X (p)X (p)φ φ dp a −∞ 2 a,b

×e "

− 2g12 

P

a

φa φa

2

+ O( )

# 1 g X a a = Tr Ip −  X (p)X (p) + O(2 ) dp 2 a ! g2 g2 1 Tr[Ip ] × 1 −  c2 (p) ' e− 2 c2 (p) = dp 2 2

where the last line follows from Eq. 2.7.4. Hence Eq. 2.12.4 is verified by Eq. 2.12.5 due to the result that ap ' e− 2.13

g2 2

c2 (p)

+ O(2 )

(2.12.7)

Summary

The emphasis on the review of Lie groups has been on its finite group elements. The exponential mapping of the Lie algebra to the group manifold provides a representation of the finite group elements. The Lie group forms a continuous differentiable space and the properties of this group space were studied in some detail. The Riemannian metric on the group space was derived and the integration measure was defined. Integration theory over the group space was studied as this forms the basis of the path integral for lattice gauge fields. The generators of the Lie algebra were realized as differential operators on the group manifold. This realization is required to define the non-Abelian electric field operator for the lattice gauge field. The special case of SU (2) was studied in some detail.

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Chapter 3

SU (N ) Kac-Moody Algebra

3.1

Introduction

Lie algebras have a natural and nontrivial extension – based on the synthesis of calculus with the generators of the Lie algebras – to infinite dimensional Kac-Moody algebras, also called affine Lie algebras. This synthesis is similar in spirit to that of linear algebra with calculus, which leads to the subject of functional analysis. Compact Lie groups and Lie algebras have a nontrivial generalization to the infinite dimensional Kac-Moody algebras. The Kac-Moody algebra critically hinges on the properties the underlying Lie algebra – as well as the global properties of the underlying Lie group manifold – to form a closed algebra, and this feature will be brought out in the discussion given below. The properties of the compact Lie group turn out to be crucial in algebraic structure of Kac-Moody algebras, and it is for this reason that Kac-Moody algebras have been included. Although lattice field theories have not yet been defined based on the Kac-Moody algebras, this could nevertheless be a useful future avenue of research. Consider a loop S 1 parametrized by a continuous variable denoted by σ ∈ [−π, +π]; at every point of the loop is a compact Lie group Gσ , which for concreteness is taken to be SU (N ); the loop group space is the continuous tensor product of all the Lie groups and is given by LG =

+π O

Gσ

: Loop group

σ=−π

The loop group consists of all maps from S 1 into a compact Lie group G. The continuous ‘tensor product’ needs to be rendered finite to be well defined. It will be seen later, in Section 3.10, that the loop group is not a simple tensor product. This is because there are anomalies that arise due to the continuous tensor product introducing singularities that are rendered finite by appropriately regularizing the loop group generators. The Kac-Moody generators is realized as functional differential operators in Section 3.4. In Section 3.5, the two-dimensional chiral field is analyzed. The WZW 29

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Lagrangian is obtained in Section 3.6 as an application of the functional differential realization of the Kac-Moody algebra. The realization of the SU (2) Kac-Moody algebra using functional differentiation has been obtained by Baaquie (1992). The special case of SU (2) Kac-Moody algebra is worked out explicitly to verify the general result. In Section 3.7, the required SU (2) group theory is discussed. Since the KM algebra is realized as the central extension of the loop group, the loop group algebra is analyzed as a preliminary step. In Section 3.8 the SU (2) KM commutator is explicitly evaluated and the SU (2) realization is shown to be exact. The KM commutation equation is explicitly verified in Section 3.9. The suitably regularized second Casimir of the loop group is constructed and shown to be anomalous. The regularized Virasoro generators are defined essentially as the second Casimir of the KM generators, and in Section 3.10 the Virasoro commutation equations are obtained. All regularizations are done by the technique of point-splitting. KM algebras have important and diverse application in physics [Goddard and Olive (1988)] and are indispensable in string theories defined on group manifolds [Gepner and Witten (1986)]. Kac-Moody algebras can be used for defining spacetime gauge fields, with the dimension coming from the continuous label for the Kac-Moody generators, given by S 1 has an interpretation of extra space dimensions [Baaquie (1991b)]. Supersymmetric gauge fields with the Kac-Moody being the gauge group can be defined [Baaquie (2000)] and the U (1) Kac-Moody gauge field has been shown by Baaquie and Parwani (1996) to be asymptotically free. 3.2

Kac-Moody Algebras

The differential representation Ea of the Lie group generators, discussed in Section 2.9, can be generalized to the loop group generators Ea → Ea (σ) with commutation similar to Eq. 2.9.2 and given by [Ea (σ), Eb (σ 0 )] = iC abc Ec (σ)δ(σ − σ 0 ) ; a, b, c = 1 · · · N 2 − 1 ; σ ∈ S 1

(3.2.1)

The generators Ea (σ) are anomalous due to the δ-function in the commutation equation. Note Ea (σ) from Eq. 2.9.12 is a Hermitian differential operator. The loop group generators Ea (σ) have anomalous properties different from those for the underlying compact Lie group. For example, the operator Ea (σ)Ea (σ) is singular and has to be regularized. Furthermore, unlike the case for a Lie group for which Ea Ea commutes with all the generators, it is shown in Section 3.10 that Ea (σ)Ea (σ) no longer commutes with all the generators of the loop group [Pressley and Segal (1988)]. One would like to couple the different Lie groups Gσ to go beyond the loop group. This is achieved by a non-trivial fibration of the loop group by the Abelian group U (1) leading to twisted tensor product for the Lie groups Gσ . The key step in enhancing Lie algebras to an infinite dimensional algebra is to combine the finite

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dimensional Lie algebra of the compact Lie group with the infinite dimensional operator algebra encoded in Heisenberg commutation equation. The extension is made by adding the (infinite dimensional) unit operator I of the Heisenberg algebra to the generators of the Lie algebra Ea (σ). This gives rise to the infinite dimensional Kac-Moody algebras. The infinitely many generators of the Kac-Moody algebra are given by Ka (σ); I : a = 1 · · · N 2 − 1 ; σ ∈ S 1 where I is the unit operator on the Kac-Moody group. The SU (N ) Kac-Moody algebra is given by [Ka (σ), Kb (σ 0 )] = iC abc Kc (σ)δ(σ − σ 0 ) + ikδab δ 0 (σ − σ 0 )I

(3.2.2)

and h0 (σ) ≡

∂h ; [Ka (σ), I] = 0 ∂σ

To have a well defined path integral for the WZW model, which is based on the Kac-Moody group, one needs to have k = n/4π, n = integer. Let the Lie group coordinates for group Gσ be given by θa (σ); the loop group generators are extended by a function to the Kac-Moody generators in the following manner: Ea (σ) → Ka (σ) = Ea (σ) + kFab (σ)θb0 (σ)I

(3.2.3)

Note for the Abelian case, Cabc = 0 and the Kac-Moody algebra reduces to the Heisenberg algebra. An element of the infinite dimensional Kac-Moody group is given by I n o U[θ] = exp ikI + i dσθa (σ)Qa (σ) [Qa (σ), Qb (σ 0 )] = iC abc Qc (σ)δ(σ − σ 0 ) + ikδab δ 0 (σ − σ 0 )I

(3.2.4)

where Qa (σ) are infinite dimensional constant matrices. The term kI is usually ignored in U[θ] since it gives an Abelian phase that is accounted for only if necessary, as in the discussion of the 2-cocycle in Section 3.4.1. 3.3

Functional Differentiation

The Kac-Moody group has coordinates given by θa (σ); partial derivatives by the coordinates of the compact Lie group are replaced by functional differentiation by θa (σ). The generators of the Kac-Moody algebra are defined in terms of functional differential operators, which is briefly reviewed. Consider variables fn , n = 0, ±1, ±2 . . . ± N that satisfy ∂fn = δn−m ∂fm

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Let t = n, with N → ∞. The limit  → 0 yields ∂fn δf (t) 1 ∂fn → ≡ lim 0 ∂fm δf (t ) →0  ∂fm δf (t) 1 ⇒ = lim δn−m → δ(t − t0 ) 0 →0  δf (t )

(3.3.1)

In general, the functional derivative of Ω[f ] – an arbitrary functional of the path f (t) – is denoted by δ/δf (t) and is defined by Ω[f (t0 ) + δ(t − t0 )] − Ω[f ] δΩ[f ] = lim →0 δf (t) 

(3.3.2)

In the notation of state space one has D δ E δ δΩ[f ] f hf |Ωi = Ω = δf (t) δf (t) δf (t) Note that  has the dimensions of [f ] × [t]. Examples • Consider the linear function Ω[f ] = f (t0 ); then, from Eq. 3.3.2 δΩ[f ] δf (t0 ) f (t0 ) + δ(t − t0 ) − f (t0 ) = = lim = δ(t − t0 ) →0 δf (t) δf (t)  R • Let Ω[f ] = dτ f n (τ ); from above Z δΩ[f ] δf (τ ) = dτ nf n−1 (τ ) (3.3.3) δf (t) δf (t) Z = dτ nf n−1 (τ )δ(t − τ ) = nf n−1 (t) (3.3.4) 3.3.1

Chain rule

The chain rule for the calculus of many variables has a generalization to functional calculus. Consider a change of variables from fn to gn ; the chain rule of calculus yields N X ∂ ∂gm ∂ = ∂fn ∂fn ∂gm m=1

As before, let t = n, t0 = m; re-write above expression as follows " #" # N X 1 ∂ 1 ∂gm 1 ∂ =  ∂fn  ∂fn  ∂gm m=1 Taking the limit of N → ∞ and  → 0 yields Z 1 ∂ δ δg(t0 ) δ lim → = dt0 : Chain rule →0  ∂fn δf (t) δf (t) δg(t0 )

(3.3.5)

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3.4

33

Kac-Moody Generators

Two well-known realizations of the KM algebras are firstly the algebra of currents bilinear in the free two-dimensional non-Abelian Dirac field, and secondly the fieldtheoretic construction using the vertex operator of Frenkel and Kac [Francesco et al. (2012)]. The shortcomings of both these realizations are (i) they use fermion-like fields and construct the KM generators as bilinears in these fields, and (ii) normal ordering is essential for obtaining the central extension. Normal-ordering is not required to define the KM algebra; the realizations of KM algebra using fermion bilinears or exponential of free boson introduce spurious singularities and hence need regularization. The KM generators are considered to be tangent vector fields on the infinite dimensional KM-group manifold. The KM-group manifold is obtained by a nontrivial fibration of the loop group by a U (1) fiber [Pressley and Segal (1988)]. The Kac-Moody generators are explicitly realized as functional differential operators expressed in terms of the coordinates of the (infinite-dimensional) group manifold. The derivation of the Lie group generator Ea given in Eq. 2.9.3 is repeated for the Kac-Moody group. Using Eq. 3.2.4, let Yaσ ≡ U †

δU iδθa (σ)

(3.4.1)

Introduce the homotopy parameter τ and define Yaσ (τ ) = U †

n δU(τ ) ; U(τ ) = exp iτ iδθa (σ)

I dσθa (σ)Qa (σ)

o

(3.4.2)

with boundary conditions Yaσ (1) = Yaσ ; Yaσ (0) = 0 Repeating the steps taken to obtain Eq. 2.3.3 yields dYaσ (τ ) = U † (τ )Qa (σ)U(τ ) dτ

(3.4.3)

with boundary condition dYaσ (0) = Qa (σ) dτ Repeating the steps in Eq. 2.3.8 and differentiating Eq. 3.4.3 yields, using Eq. 3.2.4   I h i d dYaσ (τ ) = −iU † (τ ) dσ 0 θα (σ 0 ) Qα (σ 0 ), Qa (σ) U(τ ) dτ dτ I n o = −iU † (τ ) dσ 0 θα (σ 0 ) iC aαb Qb (σ 0 )δ 0 (σ 0 − σ) + ikδαa δ 0 (σ 0 − σ) U(τ ) n o = −iU † (τ ) iθα (σ)C aαb Qb (σ) − ikθα0 (σ) U(τ ) = −θab

dYbσ (τ ) − kθa0 (σ) dτ

(3.4.4)

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Writing the non-Abelian indices in Eq. 3.4.4 using matrix and Dirac notation and suppressing the σ index, yields dY (τ ) E E d dY (τ ) E (3.4.5) = −θ − k θ0 ; (θ)ab ≡ θα (σ)C aαb dτ dτ dτ d  τ θ dY (τ ) E ⇒ e−τ θ e = −k |θ0 i (3.4.6) dτ dτ Integrating the homotopy parameter from 0 to τ yields, using Eq. 3.4.3 Z τ Z τ d  τ 0 θ dY (τ 0 ) E dτ 0 eτ θ e = −k |θ0 i dτ 0 dτ dτ 0 0 E E  E 1  τθ τ θ dY (τ ) ⇒e − Q = −k e − 1 θ0 dτ θ dY (τ ) E E  E 1 (3.4.7) ⇒ = e−τ θ Q − k 1 − e−τ θ θ0 dτ θ Eq. 3.4.7 has an important special case of τ = 1. From Eq. 3.4.3 dYaσ (1) = U † (1)Qa (σ)U(1) = U † Qa (σ)U dτ and hence, from Eq. 3.4.7 [Baaquie (1988)] U † Qa (σ)U = ρab Qb (σ) − keab θb0 (σ)

(3.4.8)

with
1 1 − e−θ ; ρ = e−θ θ Integrating Eq. 3.4.7 with τ from 0 to 1, and using dY (0)/dτ = 0 from Eq. 3.4.3, yields the result Z 1 Z 1  E E Z 1  dY (τ ) −τ θ 0 1 = dτ dτ e dτ 1 − e−τ θ Q − k |θ i dτ θ 0 0 0   E 
 E 1 1 1 1 − e−θ Q − k 1 − 1 − e−θ θ0 ⇒ |Y (1)i = θ θ θ
0 1 ⇒ |Y (1)i = e|Qi − k 1 − e |θ i (3.4.9) θ From Eq. 2.9.3 e=

Yaσ (1) = Yaσ = U †

δU iδθa (σ)

Writing out Eq. 3.4.9 in components yields from Eq. 2.9.3 n1
o δU U† = eab (σ)Qb (σ) − k 1−e (σ)θb0 (σ) iδθa (σ) θ ab

(3.4.10)

The definition of the generator of the loop group is taken from the Lie group, and from Eq. 2.9.5 is given by Ea (σ) = fab

δ θ
; f = e−1 = iδθb (σ) 1 − e−θ

(3.4.11)

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From Eqs. 3.4.10 and 3.4.11 h n1 i
o δU (σ)θb0 (σ) = U eab (σ)Qb (σ) − k 1−e iδθa (σ) θ ab h i ⇒ Ea (σ) + kFa U = UQa (σ) = Ka (σ)U

35

(3.4.12)

Hence, the Kac-Moody generator, from the defining Eq. 3.4.12, are given by n1
o (σ)θb0 (σ) (3.4.13) f −1 Ka (σ) = Ea (σ) + kFa (σ) ; Fa (σ) = θ ab Eq. 3.4.13 is a result quoted, without a proof, in Baaquie (1986, 1988, 1991b). In components, Eq. 3.4.13 yields Ka (σ) = faβ (σ)

δ ∂θγ (σ) −1 + kθaβ (σ)[fβγ (σ) − δβγ ] iδθβ (σ) ∂σ

The function Fa (σ) is well behaved for all values of θ; in particular, for θ ' 0
1 1 1 f − 1 ' + θ + O(θ2 ) (3.4.14) θ 2 12 For the Abelian case, Cabc = 0 and yields k 0 θ (σ) 2 Hence the Abelian Kac-Moody algebra, which is isomorphic to the Heisenberg algebra, is given by F (σ) =

K(σ) =

k δ + θ0 (σ) iδθ(σ) 2

(3.4.15)

Noteworthy 3.1: Kac-Moody Central Extension The derivation of the central extension is based on solving Eq. 3.4.5; writing the equation in vector notation yields(θ0 = θa0 ) d2 y dy +θ + kθ0 = 0 2 dτ dτ that yields the solution given in Eq. 3.4.9 Z 1 Z 1 Z τ 0 −τ θ dy(0) −τ θ y(1) = y(0) + dτ e −k dτ e dτ 0 eτ θ θ0 dτ 0 0 0 Z 1 Z 1 Z τ 0 = dτ e−τ θ Q − k dτ e−τ θ dτ 0 eτ θ θ0 0

0

0

The integrations, in matrix notation, yield the following Z 1 1 dτ e−τ θ = (1 − e−τ θ ) θ 0 and Z

1

dτ e 0

−τ θ

Z

τ

0 τ 0θ

dτ e 0

Z =

1

dτ 0

 1
 1 1 1 − e−τ θ = 1 − 1 − e−θ θ θ θ

(3.4.16)

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and we have obtained the result given in Eq. 3.4.10 h1 n1 i
o δU (σ)θb0 (σ) ; eab (σ) = (1 − e−τ θ ) = eab (σ)Qb (σ) − k 1−e U† iδθa (σ) θ θ ab ab Furthermore, the central extension is given by
 1 1 f 1 − 1 − e−θ = f − 1) θ θ θ ˜ α (σ), similar to the case of Lie groups There is another set of KM generators K that have left- and right-invariant generators, that is defined, similar to Eq. 3.4.12, by ˜ α (σ)U = Qα (σ)U K and hence ˜ α (σ), K ˜ β (σ 0 )] = [Qβ (σ 0 ), Qα (σ)]U [K

(3.4.17)

Note that on the right hand side of Eq. 3.4.17 the order of the commutation equation has been reversed; hence, from Eq. 3.2.4, one obtains the commutation equations (note the minus signs) ˜ α (σ), K ˜ β (σ 0 )] = iC βαγ K ˜ γ (σ 0 )δ(σ 0 − σ) + ikδβα δ 0 (σ 0 − σ) [K ˜ γ (σ)δ(σ − σ 0 ) − ikδαβ δ 0 (σ − σ 0 ) = −iC αβγ K ˜ α (σ) have a realSimilar to the derivation for Kα (σ), the associated generators K ization given by T ˜ α (σ) = fαβ K (σ)

δ iδθβ (σ)

T + k(fαβ − δαβ )ξβγ θγ0 (σ)

˜ α (σ) commutes with Kβ (σ 0 ) Furthermore, K ˜ α (σ), Kβ (σ 0 )] = 0 [K The commutation equation Eq. 3.2.2 is valid everywhere on the KM-group manifold. Although explicit coordinates θα (σ) were used in Eq. 3.4.12 to define Kα , the expression for Kα is coordinate independent in the sense that the form of Kα ˜ α is also coordinate independent. is the same in all coordinate patches. Similarly, K 3.4.1

Kac-Moody generator and the 2-cocycle

An exact realization of the Kac-Moody generators was derived in Baaquie (1986). The derivation is based on the theory of projective representation, and the 2-cocycle determines the central extension of the loop group generators [Mickelsson (1985)]. In general, to the infinite set of coordinates of the KM group θα (σ) a phase ϕ is also required to express the KM group element Z   U [ϕ; θ] = exp ikϕ + i dσθα (σ)Qα (σ)

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Note the coordinate ϕ does not appear in Kα since a basis has been chosen for the KM-algebra in which k is proportional to the identity operator. The projective representation of the KM group element is given by [Mickelsson (1985)] ˜ θ]U e [k; θ] = exp{i(k˜ + k + ω2 [θ, e θ]}U [θe ◦ θ] U [k;

(3.4.18)

where θe ◦ θ, for each σ, is composed according to the composition rules of G. Consider multiplying group elements with coordinates θ1 , θ2 , θ3 . KM group multiplication is associative in that the order of multiplication does not matter. Hence     U [θ1 ]U [θ2 ] U [θ3 ] = U [θ1 ] U [θ2 ]U [θ3 ] Associativity implies that the phase factor ω2 satisfies the following 2-cocycle condition [Mickelsson (1985)] ω2 (θ1 , θ2 ) + ω2 (θ1 · θ2 , θ3 ) = ω2 (θ1 , θ2 · θ3 ) + ω2 (θ2 , θ3 )

(3.4.19)

The 2-cocycle can be directly obtained using the Campbell-Hausdorff-Baker formula. It can also be obtained from the torsion tensor of the Lie group given in Eq. 3.6.10 using the descent equation of G-cohomology [Mickelsson (1985); Baaquie (1991a)]. Consider the functional derivative o in δ U[k; θ] = − U † [k; θ]U[k; θ + δaα δ(σ − σ 0 )] − 1 −iU † [k; θ] δθa (σ)  Using the group multiplication with 2-cocycle given in Eq. 3.4.18 and the loop group composition equation given in Eq. 3.4.20 yields from above δ −iU † [k; θ] U[k; θ] δθa (σ) " # n o i 0 = − exp iω2 [θ, θ + δaα δ(σ − σ )] + ieab (θ(σ))Qb (σ) − 1  =

1 ω2 [θ, θ + δaα δ(σ − σ 0 )] + eab (θ(σ))Qb (σ) 

Since ω2 [θ, θ] = 0 with notation anticipating later results ˜ 1 δω2 [θ, θ] 0 ω[θ, θ + δaα δ(σ − σ )] = ≡ keab Fα [θ(σ)]  δ θ˜a (σ) θ=θ e Multiplying both sides of Eq. 3.4.20 with fαβ yields h i Eα (σ) + kFα (σ) U = UQα (σ) = Kα (σ)U ⇒ Kα (σ) = Eα (σ) + kFα (σ)

(3.4.20)

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Hence, from above, the central extension is given by ˜ δω2 [θ, θ] kFα (σ) = fαβ δ θ˜β (σ) θ=θ e For the Abelian U (1) group, the Kac-Moody algebra is given by the Heisenberg algebra [Q(σ), Q(σ 0 )] = ikδ 0 (σ − σ 0 ) Using the Campbell-Hausdorff-Baker formula for the Abelian case 1

ex ey = eX+Y + 2 [X,Y ] yields n I o n I o exp i dσθ(σ)Q(σ) exp i dσφ(σ)Q(σ) I n I o i2 = exp i dσ(θ + φ)(σ)Q(σ) + dσdσ 0 θ(σ)φ(σ 0 )[Q(σ), Q(σ 0 )] 2 I n kI o = exp i dσθ0 (σ)φ(σ)} exp{i dσ(θ + φ)(σ)Q(σ) (3.4.21) 2 Hence, from Eq. 3.4.21, the Abelian 2-cocycle is given by I k dσθ0 (σ)φ(σ) ω2 [θ, φ] = 2 It can be directly verified that the Abelian 2-cocycle satisfies the cocycle condition given in Eq. 3.4.19. Note that, as expected ω[θ, θ] = 0 The Abelian central extesion is given by δω2 [θ, φ] δφ(σ)

φ=θ

=

k 0 θ (σ) 2

Hence the Abelian Kac-Moody generator is given by δ k K(σ) = + θ0 (σ) iδθ(σ) 2

(3.4.22)

and reproduces the result obtained earlier in Eq. 3.4.15. 3.5

Chiral Field

The chiral field describes a two-dimensional quantum field theory with the degree of freedom being the nonlinear Lie Group; it has many applications in mathematical physics and is a first step in discussing the WZW Lagrangian. Its essential properties are reviewed. The chiral field’s degree of freedom is the group element given by g(σ) = exp{Xa θa (σ)} ; [Xa , Xb ] = iC abc Xc

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The Sugawara Hamiltonian for the chiral field is given by [Sugawara (1968)] I X 1 dσ {λ2 Ea2 (σ) + (1/λ2 )A2a (σ)} (3.5.1) HC = 2 a The kinetic term, from Eqs. 3.2.3 and 3.4.11, is given by [Ea (σ), Eb (σ 0 )] = iC abc Ec (σ)δ(σ − σ 0 ) δ θ
Ea (σ) = fab ; f = e−1 = iδθb (σ) 1 − e−θ The potential term is given by ∂θβ (σ) Aα (σ) = eβα (σ) ≡ eTαβ (σ)θβ0 (σ) ∂σ The commutation equation of Ea (σ) with Aα (σ) is derived. From Eq. 2.3.1, consider the following ∂g(σ) ∂θγ † ∂g(σ) g † (σ) = g (σ) = iθγ0 eγα (σ)Xα = iAα (σ)Xα ∂σ ∂σ ∂θγ Using the identity

(3.5.2) (3.5.3)

(3.5.4)

[A, BC] = [A, B]C + B[A, C] yields h  ∂g(σ) i ∂g(σ) ∂  Ea (σ 0 ), g † (σ) = [Ea (σ 0 ), g † (σ)] + g † (σ) [Ea (σ 0 ), g(σ)] (3.5.5) ∂σ ∂σ ∂σ Note that   [Ea (σ 0 ), g(σ)] = Ea (σ 0 )g(σ) = g(σ)Xa δ(σ − σ 0 )

Using     Ea (σ 0 ) g † (σ)g(σ) = 0 ⇒ Ea (σ 0 )g † (σ) = −Xa g † (σ)δ(σ − σ 0 ) yields   [Ea (σ 0 ), g † (σ)] = Ea (σ 0 )g † (σ) = −Xa g † (σ)δ(σ − σ 0 ) Hence, from Eq. 3.5.5 h  ∂g(σ) i ∂g(σ) ∂  Ea (σ 0 ), g † (σ) = −Xa g † (σ) δ(σ − σ 0 ) + g † (σ) g(σ)Xa δ(σ − σ 0 ) ∂σ ∂σ ∂σ h i ∂g(σ) ∂ (3.5.6) = − Xa , g † (σ) δ(σ − σ 0 ) + Xa δ(σ − σ 0 ) ∂σ ∂σ From Eq. 3.5.4 h ∂g(σ) i Xa , g † (σ) = iθγ0 eγα (σ)[Xa , Xα ] = i2 C aαβ Xβ θγ0 eγα (σ) = −C aαβ Xβ Aα (σ) ∂σ Hence, from Eqs. 3.5.4 and 3.5.6, one obtains the commutator ∂ [Ea (σ 0 ), iAα (σ)Xα ] = C aαβ Xβ Aα (σ) + Xa δ(σ − σ 0 ) ∂σ ⇒ [Ea (σ), Ab (σ 0 )] = iC abβ Aβ (σ)δ(σ − σ 0 ) + iδab δ 0 (σ − σ 0 ) (3.5.7) In the Sugawara approach, one postulates the above commutation equation given in Eq. 3.5.7 and then derives the expression for the kinetic operator Ea (σ) given in Eq. 3.5.3.

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3.6

The WZW Lagrangian

As an application of the realization of the Kac-Moody algebra, the two-dimensional Wess-Zumino-Witten (WZW) chiral theory [Witten (1983)] is given a Hamiltonian formulation using the KM algebra. The WZW model has wide applications in physics, including the study of superstring theories as well as black holes. The WZW action is derived by exploiting the exact form of the KM realization [Baaquie (1986)]. In the WZW case, unlike the usual sigma-model, the kinetic operator given in Eq. 3.5.2 is replaced by πα (σ) Eα (σ) → πα (σ) and, instead of Eq. 3.5.2, one postulates anomalous commutation equations, which for d = 2 is given by [πα (σ) , πβ (σ 0 )] = iCαβγ [πγ (σ) − kAγ (σ)]δ(σ − σ 0 )

(3.6.1)

Note for k = 0, one recovers the chiral field’s commutators. Furthermore, as in Eq. 3.5.7 [πα (σ) , Aβ (σ 0 )] = iCαβγ Aγ (σ)δ(σ − σ 0 ) + iδαβ ∂σ δ(σ − σ 0 ) where −1 Aα (σ) = eTαβ (σ)θβ0 (σ) = eβα (σ)∂σ θβ (σ) , eαβ = fαβ

and [Aα (σ) , Aβ (σ 0 )] = 0 The WZW Hamiltonian has the Sugawara form of the non-linear chiral model given in Eq. 3.5.1; for dimensionless coupling constant λ2 , we have [Sugawara (1968)] Z 1 dσ[λ2 πα (σ)πα (σ) + (1/λ2 )Aα (σ)(σ)Aα (σ)] HW ZW = 2 HW ZW together with the anomalous commutation equation for πα yields, for k = n/4π(n = integer) the WZW field equations. A concrete realization for πα (σ) is obtained by choosing 2k for the central extension in the KM algebra. Then it can be shown that, for Kac-Moody generators given by Kα (σ) πα (σ) = Kα (σ) − kAα (σ) = Eα (σ) + 2kFα (σ) − kAα (σ)

(3.6.2)

Eq. 3.6.2 yields, using Eq. 3.5.7, the following commutation equation [πα (σ) , πβ (σ 0 )] = [Kα (σ) , Kβ (σ 0 )] − 2k[Eα (σ) , Aβ (σ 0 )]   = iCαβγ Kγ (σ) + 2kiδαβ ∂σ δ(σ − σ 0 ) − 2k iCαβγ Aβ (γ)δ(σ − σ 0 ) + iδab δ 0 (σ − σ 0 ) = iCαβγ [πγ (σ) − kAγ (σ)]δ(σ − σ 0 )

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and we have recovered the result given in Eq. 3.6.1. Note the Kac-Moody algebra’s central extension is necessary for canceling the central extension in the commutation of Ea with Ab , as given in Eq. 3.5.7. Eq. 3.6.1 yields for the WZW Hamiltonian Z 1 HW ZW = dσ[λ2 (Eα + 2kFα − kAα )2 + (1/λ2 )A2α ] (3.6.3) 2 In effect, the WZW central extension term in the action (whose coefficient is k) acts as a field dependent background field for the kinetic operator Eα . The WZW Lagrangian is given by the Dirac-Feynman formula (N () is a normalization constant) e = hθ| exp(−iHW ZW )|θi e lim N () exp{iLW ZW [θ; θ]}

→0

(3.6.4)

where |θi = ⊗|θ(σ)i is the coordinate eigenstate. For infinitesimal time , plane wave states can be used to evaluate the Lagrangian using Eq. 3.6.4. The fact that θα are curved variables does not enter to leading order. Let plane wave states be given by Z n Z o hθ|pi = exp i dσpα (σ)θα (σ) ≡ eipθ ; Dp|pihp| = I The HamiltonianR operator acts on the bra vector; this yields – from Eq. 3.6.3 and suppressing the dσ integration in the Hamiltonian – the following Z e = Dphθ| exp(−iHW ZW )|pihp|θi e hθ| exp(−iHW ZW )|θi 2 o ∂ λ2  e fαβ + 2kFα (θ) − kAα (θ) eip(θ−θ) 2 i∂θβ o n 2 2 λ2 e ' e−i(1/2λ )Aα (θ) exp − i (fαβ pβ + 2kFα (θ) − kAα (θ))2 eip(θ−θ) (3.6.5) 2 2

= e−i(1/2λ

)A2α (θ)

exp

n

− i

Make the change of variables −1 pβ → fβa pa = eβa pa ; Dp → det(e)Dp

Ignoring the det(e) as it does not affect the result to O() yields, from Eq. 3.6.4 Z −iHW ZW e hθ|e |θi = Dp exp{−i(1/2λ2 )A2α (θ)} × exp{−i(λ2 /2)(pα + 2kFα (θ) − kAα (θ))2 ei(θ−θ)ep} e

Z =

e · (p − 2kF + kA)} Dp exp{−i(1/2λ2 )A2α (θ) − i(λ2 /2)(pα )2 + i(θ − θ)e (3.6.6)

Define ∂θ θ − θe = θ˙ ; θ˙ = = ∂t θ ∂t

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Performing the Gaussian integration in Eq. 3.6.6 over

R

Dp, and from Eq. 3.6.4

e = hθ| exp{−iHW ZW }|θi e lim N () exp{iLW ZW [θ; θ]} n o  ˙ F − 1A = exp − i(1/2λ2 )A2α (θ) + i(1/2λ2 )(eαβ θ˙β )2 − 2ikθe (3.6.7) 2 R Restoring the dσ integration and ignoring the normalization, it follows that Z LW ZW = dσ(L0 + L1 ) →0

where L0 =

1 X 1 X [(eαβ θ˙β )2 − (eβα θβ0 )2 ] = − 2 [(eβα ∂t θβ )2 − (eβα ∂σ θβ )2 ] 2 2λ α 2λ α

and  1  L1 = −2k θ˙α eαβ Fβ − Aβ = −2k∂t θα tαβ ∂σ θβ 2

(3.6.8)

The antisymmetric matrix t is given by n 1 − e−θ 1 o t = e θ−1 (f − 1) − eT = θ−2 (θ − sinh(θ)) ; e = 2 θ

(3.6.9)

The metric on the Lie group manifold and the Minkowski metric is given by gαβ = eαγ eβγ ; η µν = diag(1, −1) where ξ1 = t ; ξ2 = σ Hence LW ZW = (1/2λ2 )η µν ∂µ θα ∂ν θβ gαβ (θ) + kµν ∂µ θα ∂ν θβ tαβ (θ) The first term is the σ-model Lagrangian; the second term is the local expression for the WZW term. Compactifying d = 2 spacetime into a two-sphere S 2 , and using Stokes’ theorem gives the WZW action as [Witten (1983)] Z Z 1 k SW ZW = 2 d2 ξL + d3 ξµνρ ∂µ θα ∂ν θβ ∂ρ θγ Tαβγ (θ) 2λ 3 where the last term is integrated over a 3-ball bounded by S 2 ; the completely antisymmetric torsion tensor Tαβγ =

1 Cabc eαa eβb eγc 2

is given by [Cronstrom and Mickelsson (1983)] (∂α ≡ ∂/∂θα ) Tαβγ = ∂α tβγ + ∂β tγα + ∂γ tαβ

(3.6.10)

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SU (2) Loop Group

To gain further insight into the Kac-Moody generators Ki (σ), the loop generators Ei (σ) are analyzed and then their central extension to the Kac-Moody case is considered. The loop group generators can be constructed directly from the underlying compact SU (2) group and all the equations can be studied in full detail. Furthermore, given the special and important role of the SU (2) KM algebra in conformal field theory, the specific case of G=SU (2) is worked out from first principles and in a completely explicit form [Francesco et al. (2012)]. The SU (2) loop group generators Ei (σ) have the commutation equation [Ei (σ), Ej (σ 0 )] = εijk Ek (σ)δ(σ − σ 0 ) : i, j, k = 1, 2, 3

(3.7.1)

Let θi (σ), i = 1,2,3 be all the possible maps from S 1 into SU (2); this defines the coordinates of the KM group manifold [Baaquie (1988)]. Define the adjoint matrix ω by ωij (σ) = εiαj θα (σ)

(3.7.2)

Then, from Eq. 2.9.5 Ei (σ) =

X

fij (σ)

j

δ iδθj (σ)

(3.7.3)

where, from Eq. 2.9.4, in matrix notation f= Let θ =

√

ω 1 − e−ω

θi θi ; for SU (2), from Eq. 2.10.5

1 ω 3 = −θ2 ω ⇒ f = 1 + ω + A(θ)ω 2 2 where, from Eq. 2.10.6   1 θ θ A(σ) ≡ A[θ(σ)] = 2 1 − cot θ 2 2 1 1 ' + θ2 + O(θ4 ) 12 16 · 45

(3.7.4)

The loop group commutator Eq. 3.7.1 implies the Maurer-Cartan equation (∂i ≡ ∂/∂θi ) that follows from Eq. 2.9.7 fαi ∂i fβγ − fβi ∂i fαγ = −εαβi fiγ

(3.7.5)

The explicit form for f given by Eq. 3.7.4 yields Eq. 3.7.5 since A satisfies the remarkable differential equation ∂A 1 + 3A − θ2 A2 = ∂θ 4 Eq. 3.7.6 will be essential in obtaining the KM commutation equation. θ

(3.7.6)

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3.8

SU (2) Kac-Moody Algebra

The general case of the SU (N ) Kac-Moody Algebra is analyzed by first studying the special case of SU (2). The Kac-Moody algebra can be studied explicitly for SU (2), and in full detail, using the exact results for the SU (2) compact Lie group. The SU (2) case provides independent verification for the results obtained [Baaquie (1992)]. The SU (2) KM algebra is given by [Ki (σ), Kj (σ 0 )] = iεijk Kk (σ)δ(σ − σ 0 ) + ikδij δ 0 (σ − σ 0 )I : i, j, k = 1, 2, 3 where σ parametrizes a loop and the prime denotes ∂/∂σ. The KM generators are realized as the central extension of the loop group generators Ei (σ) given by Ki (σ) = Ei (σ) + kFi (σ)

(3.8.1)

where Fi (σ) are functions of the SU (2) loop group coordinates. The Kac-Moody generators are given in the general case by Baaquie (1986) Ki (σ) = Ei (σ) + kFi (σ)

(3.8.2)

where from Eq. 3.7.3 Ei (σ) = fij (σ)

δ iδθj (σ)

and the central extension is −1 Fi (σ) = ωij (fjk − δjk )θk0 (σ) ; ωij (σ) = εiαj θα (σ)

Hence, for SU (2), from Eqs. 3.7.4 and 3.7.4     1 1 θ θ 0 Fi (σ) = δij + A(σ)ωij (σ) θj (σ) ; A[θ(σ)] = 2 1 − cot (3.8.3) 2 θ 2 2 The Kac-Moody generators, from Eqs. 3.8.2 and 3.7.1, are given by [KI (σ), KJ (σ 0 )] = iCIJK Ek (σ)δ(σ − σ 0 ) + k[EI (σ)FJ (σ 0 ) − EJ (σ 0 )FI (σ)] (3.8.4) Consider the following ΓIJ = EI (σ)FJ (σ 0 ) − EJ (σ 0 )FI (σ)

(3.8.5)

Note 1 iEI (σ)FJ (σ 0 ) = − fIJ (σ)δ 0 (σ − σ 0 ) − fIα (σ)ωJα (ω 0 )A(σ 0 )δ 0 (σ − σ 0 ) 2 +δ(σ − σ 0 )fIα (εJαβ A + ωJβ ∂α A)θβ0 (ω) (3.8.6) Eqs. 3.8.5 and 3.8.6 yield iΓIJ = Γ1 + Γ2 + Γ3

(3.8.7)

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where 1 2 2 Γ1 = δIJ δ 0 (σ − σ 0 ) − [ωIJ (σ)A(σ) + ωJI (σ 0 )A(σ 0 )]δ 0 (σ − σ 0 ) 2 1 0 + ωIJ δ(σ − σ 0 ) 4 Γ2 = [fIi (σ)ωiJ (σ 0 )A(σ 0 ) + fJi (σ 0 )ωiI (σ)A(σ)]δ 0 (σ − σ 0 ) Γ3 = Γα (σ)θα0 δ(σ − σ 0 ) Note that ˜ α (σ)θα0 δ(σ − σ 0 ) Γ1 + Γ2 = −δIJ δ 0 (σ − σ 0 ) + Γ where ˜ α = 1 εIαJ + εIαJ A + ωIJ ∂α A + 1 ωIi εiαJ A − 1 ωiJ εIαi A − ωIi ωjJ εiαj A2 Γ 4 2 2 and which yields ˜ α = − 1 εIJα − 3εIJα A + ωIJ ∂α A + ωJα ∂I A − ωIα ∂J A Γα + Γ 4 +(ωIi εiαJ − ωJi εiαI )A + (ωIα θJ − ωJα θI − ωIJ θα )A2

(3.8.8)

Using the identity ωIα θJ θα0 − ωJα θI θα0 − ωIJ θα θα0 = εIJα θ2 θα0 and ∂α A = (θα /θ)∂A/∂θ yields the following ˜ α )θ0 Γ4 = (Γα + Γ α   1 ∂A 0 − θ 2 A2 = − εIJα θα − εIJα ωαβ Aθβ0 − εIJα θα0 3A + θ 4 ∂θ

(3.8.9)

The term in the bracket above is equal to 1/4 due to Eq. 3.7.6, which is a result that follows from the underlying SU (2) compact Lie group; this fact illustrates the essential role that the underlying compact Lie groups play in the structure and consistency of the Kac-Moody groups and algebras. Hence, from Eqs. 3.8.3 and 3.8.9   1 Γ4 = −εIJα δαβ + Aωαβ θβ0 = −εIJα Fα (σ) (3.8.10) 2 and from Eq. 3.8.7, 3.8.8 and 3.8.10 iΓIJ = −δIJ δ 0 (σ − σ 0 ) − εIJα Fα (α)δ(σ − σ 0 ) which yields from Eqs. 3.8.4 and 3.8.5 the KM commutator [KI (σ), KJ (σ 0 )] = iεIJK Kk (σ)δ(σ − σ 0 ) + ikδIJ δ 0 (σ − σ 0 )

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˜ i (σ), which commute with Ki (σ) There is another set of SU (2) KM generators K and have the algebra (note the negative signs) ˜ i (σ), K ˜ j (σ 0 )] = −iεijk K ˜ k (σ)δ(σ − σ 0 ) − ikδij δ 0 (σ − σ 0 ) [K with the realization [Baaquie (1986)] T ˜ i (σ) = fij K (σ)

  1 δ + − δij + A(σ)ωij (ω) θj0 (σ) iδθj (σ) 2

A number of identities specific to SU (2) had to be used for constructing the KM commutator, and they shed new light on the structure of the KM algebra and showed its essential connection with the algebra of the underlying SU (2) compact Lie group. 3.9

Kac-Moody Commutation Equations

The general result derived in Section 3.4 for the generators of the Kac-Moody algebra is verified by an explicit and long calculation. Given the importance of the Kac-Moody algebra and its novelty, an explicit calculation seems to be in order. The calculation shows, as expected, that the central extension – which is given by the constant k – crucially depends on the underlying compact Lie algebra; in particular, the derivation relies on the complete antisymmetry of the structure constants, which holds for compact Lie groups. The results of this Section are based on Baaquie (1988). The notation being employed is reviewed; θα (σ) is the infinite set of real coordinates of the KM-group. C α are the generators of the adjoint representation of G, and are antisymmetric matrices Cβαγ ; define the antisymmetric matrix θ(σ) = θα (σ)C α ≡ θ ; (C α )βγ ≡ Cβαγ

(3.9.1)

ξ(σ) = θ−1 (σ)

(3.9.2)

f (σ) = θ/[1 − exp(−θ)]

(3.9.3)

and its inverse

and the matrix

To write Kα (σ) compactly, define the operator Eα (σ) = fαβ (σ)

δ iδθβ (σ)

(3.9.4)

where δ/δθα (σ) is the functional derivative, and the function Fα (σ) = [fαβ (σ) − δαβ ]ξβγ (σ)θγ0 (σ)

(3.9.5)

Kα (σ) = Eα (σ) + kFα (σ)

(3.9.6)

Then, from Eq. 3.4.13

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and, from Eq. 3.2.2 [Ka (σ), Kb (σ 0 )] = iC abc Kc (σ)δ(σ − σ 0 ) + ikδab δ 0 (σ − σ 0 )I

(3.9.7)

Kα (σ) is a Hermitian functional differential operator acting on functionals of the KM group elements. The KM generators have the commutator [KI (σ), KJ (σ 0 )] = [EI (σ) , EJ (σ 0 )] + k{(EI (σ)FJ (σ 0 )) − (EJ (σ 0 )FI (σ))} (3.9.8) The matrix fαβ given in Eq. 3.9.3 satisfies the Maurer-Cartan equation ∂α = ∂/∂θα fIβ ∂β fJK − fJβ ∂β fIK = −CIJα fαK ,

(3.9.9)

which yields from Eq. 3.9.4 [EI (σ) , EJ (σ 0 )] = iCIJα Eα (σ)δ(σ − σ 0 )

(3.9.10)

Define the function ΓIJ by ΓIJ (σ − σ 0 ) = (Ei (σ)FJ (σ 0 )) − (EJ (σ 0 )FI (σ))

(3.9.11)

Eqs. 3.9.4 and 3.9.5 yield iΓIJ (σ − σ 0 ) = Γ1 + Γ2 + Γ3 + Γ4 + Γ5 ,

(3.9.12)

where Γ1 = (fIβ ∂β fJγ − fJβ ∂β fIγ )ξγα θα0 δ(σ − σ 0 ) = −CIJα fαγ ξγα θα0 δ(σ − σ 0 ) and Γ2 = fIβ fJγ (∂β ξγα − ∂γ ξβγ )θα0 δ(σ − σ 0 ) , and Γ3 = −(fIβ ∂β ξJα − fIβ ∂β ξIα )θα0 δ(σ − σ 0 ) , and Γ4 = fIβ (σ)fJγ (σ 0 ){ξγβ (σ) − ξγβ (σ 0 )}δ 0 (σ − σ 0 ) , and Γ5 = fIβ (σ)ξJβ (σ 0 )δ 0 (σ − σ 0 ) − fJβ (σ 0 )ξIβ (σ)δ 0 (σ − σ 0 )

(3.9.13)

Consider the identity   x(σ 0 ) − x(σ) δ 0 (σ − σ 0 ) = x0 (σ)δ(σ − σ 0 ) ⇒ x(σ 0 )δ 0 (σ − σ 0 ) = x(σ)δ 0 (σ − σ 0 ) + x0 (σ)δ(σ − σ 0 )

(3.9.14)

This leads to a simplification of Γ4 yields Γ2 + Γ4 = 0

(3.9.15)

∂α ξβγ + ∂γ ξαβ + ∂β ξγα = 0

(3.9.16)

since it can be shown that

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From Eqs. 3.9.13 and 3.9.14 0 0 Γ5 = (fIβ ξJβ + fJβ ξIβ )δ(σ − σ 0 ) − {f ξ + (f ξ)T }IJ (σ)δ 0 (σ − σ 0 )

(3.9.17)

From Eqs. 3.9.2 and 3.9.3 yield f ξ + (f ξ)T =

1 1 + =1 −θ 1−e 1 − eθ

(3.9.18)

This remarkable identity – which comes from the property of the underlying compact Lie group SU (N ) – is the reason why, in spite of the apparently field dependent expression for ΓIJ , one ends up with a field independent central extension k for the KM commutation equations. From Eqs. 3.9.17 and 3.9.18 Γ5 = δ 0 (σ − σ 0 ) + Γ6

(3.9.19)

After some simplifications, Γ3 + Γ6 = {fIβ ∂J ξαβ − fJβ ∂I ξαβ − ∂α (f ξ)JI }θα0 δ(σ − σ 0 )

(3.9.20)

Using the crucial identity given in (A.1) ∂α (f ξ)JI = fβI ∂J ξαβ − fJβ ∂I ξαβ ,

(3.9.21)

Γ3 + Γ6 = (f − f T )Iβ ∂J ξαβ θα0 δ(σ − σ 0 )

(3.9.22)

f − fT = θ

(3.9.23)

Γ3 + Γ6 = CIJβ ξβα θα0 δ(σ − σ 0 )

(3.9.24)

yields

From Eq. 3.9.3

and hence

Collecting Eqs. 3.9.13, 3.9.15, 3.9.19 and 3.9.24 yields iΓIJ (σ − σ 0 ) = −CIJα (fαγ − δαγ )ξγα θα0 δ(σ − σ 0 ) − δIJ δ 0 (σ − σ 0 ) ,

(3.9.25)

or more succinctly, using Eq. 3.9.11 (EI (σ)FJ (σ 0 )) − (EJ (σ 0 )FI (σ)) = iCIJα Fα (σ)δ(σ − σ 0 ) + iδIJ δ 0 (σ − σ 0 ) (3.9.26) The expected commutation equation of the KM-generators, from Eqs. 3.9.8, 3.9.10 and 3.9.26 is given [KI (σ) , KJ (σ 0 )] = iCIJα Kα (σ)δ(σ − σ 0 ) + ikδIJ δ 0 (σ − σ 0 )

(3.9.27)

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Virasoro Generator: Point-Split Regularization

The Virasoro generator is realized as a second order Hermitian functional differential operator L(σ) and is essentially the KM algebra’s second Casimir operator K 2 (σ). K 2 (σ) is singular due to the singularity arising from δ 2 /δθ2 (σ) and hence needs to be regularized. The Virasoro generator is regularized using the point-splitting method [Baaquie (1988)]. Noteworthy 3.2: Representation of the δ function Consider the following representation of the Dirac-delta function  1 h 1 1 i η 1 0 : x 6= 0 δ(x) = lim = = − η→0 2πi x − iη ∞:x=0 x + iη π x2 + η 2

(3.10.1)

A proof for the representation, using a Fourier transform, is given by Z Z Z η η dx dp dx f (x) = eipx fp π x2 + η 2 π 2π x2 + η 2 Z Z Z η 1 dp = dp e−η|p| fp → fp = f (0) = dxδ(x)f (x) π 2η 2π which is the expected result. Furthermore (note the signs on the iη’s) i 1 h 1 1 δ 0 (x) = − 2πi (x + iη)2 (x − iη)2

(3.10.2)

The full content of the commutation equation given in Eq. 3.9.10 is realized by the short distance operator product expansion, which for σ ≈ σ 0 is given by 1 Cαβγ Eγ (σ) + finite terms (3.10.3) Eα (σ)Eβ (σ 0 ) ∼ = 2π σ − σ 0 − iη To see that Eq. 3.10.3 yields the expected commutation equation, consider [Eα (σ), Eβ (σ 0 )] = Eα (σ)Eβ (σ 0 ) − Eβ (σ 0 )Eα (σ) 1 n Cαβγ Eγ (σ) Cβαγ Eγ (σ 0 ) o = − 0 2π σ − σ 0 − iη σ − σ − iη n o 1 1 1 = Cαβγ Eγ (σ) − + O(η) 0 0 2π σ − σ − iη σ − σ + iη = iCαβγ Eγ (σ)δ(σ − σ 0 ) Similarly, the other commutation equations can be written in terms of the short distance expansion as follows (note the signs of the iη terms below) δαβ 1 1 Cαβγ Fγ (σ) (Eα (σ)Fβ (σ 0 )) ∼ + finite terms + = 0 2 2π (σ − σ + iη) 2π σ − σ 0 − iη

(3.10.4)

Eqs. 3.10.3 and 3.10.4 yield1 Kα (σ)Kβ (σ 0 ) ∼ = 1 Note

kδαβ 1 Cαβγ Kγ (σ) + + finite terms 2π (σ − σ 0 + iη)2 σ − σ 0 − iη

that Eq. 3.10.5 also follows directly form Eq. 3.9.27.

(3.10.5)

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The iη’s in the short distance expansion can be ignored if an explicit point-splitting method is used for regularizing singular operators – for example, by introducing an infinitesimal ε. This is because the limit of η → 0 is always taken before the limit of ε → 0 is taken. The leading term for expansions given in Eqs. 3.10.4 and 3.10.5 have a positive sign since, for σ = σ 0 +ε, the singularity for the regularized positive valued operator should have a positive value – as given in Eq. 3.10.9. Consider the regularized operator using the point-splitting method 2 Ereg (σ) = lim

ε→ 0

1 {Ea (σ)Ea (σ + ε) + Ea (σ + ε)Ea (σ)} + const 2

(3.10.6)

Using the identity [AB, C] = A[B, C] + [A, C]C yields [Ea (σ)Ea (σ + ε), Eα (σ 0 )] = Ea (σ)[Ea (σ + ε), Eα (σ 0 )] + [Ea (σ), Eα (σ 0 )]Ea (σ + ε)  = iC aαb Ea (σ)Eb (σ + ε)δ(σ + ε − σ 0 ) − Ea (σ)Eb (σ + ε)δ(σ − σ 0 ) The short distance expansion given in Eq. 3.10.3 yields Ea (σ)Eb (σ + ε) = −

1 abc C Ec (σ) 2πε

The identity2 C aαb C abc = −c2 (A)δαc where c2 (A) is the second Casimir of the adjoint representation, yields the result o c2 (A) Eα (σ) n 2 [Ereg (σ), Eα (σ 0 )] = i δ(σ + ε − σ 0 ) − δ(σ − σ 0 ) 2π ε c2 (A) 0 =i δ (σ − σ 0 )Eα (σ) (3.10.7) 2π Note Eq. 3.10.7 shows that the loop group is not simply a tensor product of the underlying compact Lie group since the second Casimir does not commute with all the generators. It can be shown that for σ parametrizing a loop, Eα (σ) are the generators of the loop group [Pressley and Segal (1988)]. The naive commutation of E 2 (σ) with Eα (σ 0 ) yields zero, and the anomaly in Eq. 3.10.7 arises from the short distance singularity in Eq. 3.10.3. Eq. 3.10.7 is the anomaly of E 2 (σ) of the loop group [Pressley and Segal (1988)]. Define the regularized Hermitian operator using the point-splitting method 2 Kreg (σ) = lim

→ 0

2 For

1 a {K(σ)K(σ + ) + K(σ + )K(σ)} + 2 + b 2 

SU (N ), from Eq. 2.7.6 c2 (A) = N .

(3.10.8)

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To fix the constant a, note from operator product expansion given in Eq. 3.10.5 ∼ kdimG · 1 K(0)K() = 2π 2 and hence a kdimG 1 2 (3.10.9) Kreg (0) = · 2 + 2 +b 2π   2 To render Kreg (0) finite choose kdimG (3.10.10) 2π 2 2 All matrix elements of Kreg (σ) are finite; the constant b fixes the value of Kreg (0), which is left undetermined due to the regularization. In complete analogy to calculation for obtaining Eq. 3.10.7, one obtains a=−

2 [Kreg (σ), Kα (σ 0 )] = iβδ 0 (σ − σ 0 )Kα (σ)

(3.10.11)

where c2 (A) (3.10.12) 2π In the approach of Knizhnik and Zamolodchikov (1984) the constant β is indirectly evaluated using properties of the null vector and vacuum state of the semidirect product of the KM and Virasoro algebras. Eq. 3.10.11, however, shows that β results directly from the anomaly in E 2 (σ) and is independent of the properties of the null and vacuum states. For the Virasoro commutator, using Eqs. 3.10.8 and 3.10.11 yields 1 2 2 [Kreg (σ), Kreg (σ 0 )] = iβ(Kσ Kσ0 + + Kσ0 + Kσ )δ 0 (σ − σ 0 ) 2 1 + iβ(Kσ Kσ0 + Kσ0 Kσ )δ 0 (σ − σ 0 − ) (3.10.13) 2 All the operators at σ and σ 0 have to be isolated, with no cross-terms. To do this Eq. 3.9.14 is used; in particular, it yields the following β = 2k +

0 Kσ0 + δ 0 (σ − σ 0 ) = Kσ+ δ 0 (σ − σ 0 ) + Kσ+ δ(σ − σ 0 )

From the short distance expansion given in Eq. 3.10.5, the leading singularity of the following operator product is given by  2 2a ∂  kdimG 0 = 3 Kσ Kσ+ = Kσ Kλ = ∂λ 2π (σ − λ)3 λ=σ+  λ=σ+ Hence ∼ Kσ Kσ+ δ 0 (σ − σ 0 ) + 2a δ(σ − σ 0 ) + O() (3.10.14) Kσ Kσ0 + δ 0 (σ − σ 0 ) = 3 From Eqs. 3.10.8, 3.10.13 and 3.10.14 n o 2 2 2 2 β −1 [Kreg (σ), Kreg (σ 0 )] = i Kreg (σ) + Kreg (σ 0 ) δ 0 (σ − σ 0 ) − 2ibδ 0 (σ − σ 0 ) ia 0 2ia {δ (σ − σ 0 ) + δ 0 (σ − σ 0 − )} + 3 {δ(σ − σ 0 ) − δ(σ − σ 0 − )} (3.10.15) 2   1 2 2 0 0 0 = i{Kreg (σ) + Kreg (σ )}δ (σ − σ ) − 2ibδ 0 (σ − σ 0 ) − iaδ 000 (σ − σ 0 ) (3.10.16) 6 −

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The singular terms of O(−2 ) and O(−3 ) in Eq. 3.10.15 arising from Eqs. 3.10.8 and 3.10.14 cancel – as is necessary if point-split regularization is to make sense. Define the Virasoro generator by 1 2 L(σ) = Kreg (σ) (3.10.17) β Then, from Eq. 3.10.11 [L(σ), Kα (σ 0 )] = iδ 0 (σ − σ 0 )Kα (σ)

(3.10.18)

The requirement that L(σ) generates reparametrizations fixes the normalization in Eq. 3.10.17. The Virasoro algebra is given, from Eq. 3.10.16, by the following  2a  1 n 12b o [L(σ), L(σ 0 )] = i(L(σ)+L(σ 0 ))δ 0 (σ −σ 0 )−i δ 0 (σ −σ 0 )+δ 000 (σ −σ 0 ) β 12 a (3.10.19) For the Virasoro algebra to contain the subalgebra SU(1,1) of Mobius transformations, b is fixed to be kdimG 1 a=− (3.10.20) b= 12 24π and hence i c 0 [L(σ), L(σ 0 )] = i(L(σ) + L(σ 0 ))δ 0 (σ − σ 0 ) + {δ (σ − σ 0 ) + δ 000 (σ − σ 0 )} (3.10.21) 2π 12 The central charge c is given by   ˜ a 2kdimG ; k˜ = 2πk c = −2π 2 = β 2k˜ + c2 (A) Eq. 3.10.21 agrees with the known result [Goddard and Olive (1988)]. Note that in the realization of Kα the anomaly in E 2 (σ) is the source of c2 (A) in the central charge, and without this factor c would have been a trivial extension of U (1) factors. So far, only local properties of Kα (σ) have been used to obtain the Virasoro algebra; we now further assume that σ parametrizes a (periodic) loop, with ∈ [−π, π]; then, due to periodicity +∞ X

Kα (σ) =

α einσ K−n

n=−∞

and from Eq. 3.9.6 γ β ˜ n+m,0 [Knα , Km ] = iCαβγ Kn+m + iknδ

For the Virasoro operator, let L(σ) =

X

einσ Ln

n

Taking the ε → 0 limit yields from Eqs. 3.10.17 and 3.10.22 Ln =

+∞ 1 X Kn−m Km , n 6= 0, β m=−∞

(3.10.22)

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53

and L0 =

∞ o X 1n 2 1 K0 + 2 K−m Km − c β 24 m=1

Note that Ln and L0 are the usual ‘normal ordered’ expressions for the Virasoro generators except for the constant in L0 . In fact, the constant c/24 is the trace anomaly [Goddard and Olive (1988)]. This expression for L0 is the correct one for computing the character functions of the KM-group and for proving modular invariance of the KM-partition function [Francesco et al. (2012)]. Eq. 3.10.21 yields 1 [Ln , Lm ] = (n − m)Ln+m + cn(n2 − 1)δn+m,0 (3.10.23) 12 The SU (1, 1) closed subalgebra is generated by L−1 , L0 , L1 , and fixing b in Eq. 3.10.20 is equivalent to redefining L0 so as to obtain L±1 , L0 as the generators obeying the SU (1, 1) algera3 [L1 , L−1 ] = 2L0 3.11

Summary

Kac-Moody algebra is the synthesis of the concept algebra with that of calculus, and is achieved by combining the finite dimensional Lie algebra with the infinite dimensional Heisenberg algebra. In the realization considered here, the infinite dimensional KM-group manifold is the starting point of the analysis. The KM commutation equations are a result of the specific properties of the underlying compact group G and of the local properties of the KM group; the KM generators clearly reveal the inner structure of the KM group, including its relation with the loop group. Kac-Moody generators were realized, for any value of the central extension, as functional differential operators expressed in terms of the coordinates of the KM group manifold. The KM generators do not need any regularization, and the central extension is due to underlying properties of the KM-group manifold. The derivation of WZW action from the Hamiltonian, using the exact form of the KM realization, showed the connection of the 0-cocycle topological WZW terms with the 2-cocycle, which is algebraic in character since it results from the KM commutation equations [Mickelsson (1985)]. Kac-Moody generators can be defined on the complex plane: the analysis of Sections 3.9 and 3.10 can be carried out on the complex plane by replacing σ by a complex variable z and ∂/∂σ by ∂/∂z. Periodicity in σ of Kα (σ) is replaced by assuming Kα (z) is analytic in z. The realization of Kac-Moody generators was used in defining the Virasoro algebra in parameter space using point splitting and does not require the use of 3 By redefining L to L + const, we can always adjust the central extension such that L 0 0 ±1 , L0 forms a closed SU(1,1) sub algebra (see, for example Francesco et al. (2012)).

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normal ordering. The Virasoro operator L(σ) has a short distance singularity. The derivation given in Section 3.10 shows that point-split regularization of the Virasoro operator is adequate for obtaining the correct result for the central extension [Baaquie (1988)]. 3.12

Appendix

A proof is given of ∂α (f ξ)JI = fβI δJ ξαβ − fJβ δI ξαβ Recall (C α )ij = Ciαj ; θ = C α θα ; ξ = θ−1 Hence f = θ/[1 − exp(−θ)] ; e = f −1 Let g = f ξ = 1/[1 − exp(−θ)] ⇒ h = g −1 = 1 − exp(−θ) The definitions above yield ∂α (f ξ) = ∂α (g) = −g∂α (h)g = −ge−θ C β eαβ = −g(1 − h)C β eαβ Using the identity [C α , g] = (1 − h)C β hαβ yields, from the two equations above [C α , g] = −g[C α , h]g = −g(1 − h)C β eαβ = (hf )αβ ∂β g = θαβ ∂β g The final result is given by ∂α (f ξ)JI = ξαβ [C α , g]JI = (ξC J ξ)αβ fβI = fβI ∂J ξαβ − fJβ ∂I ξαβ

(A.1)

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Chapter 4

SU (N ) Path Integrals

4.1

Introduction

The study of SU (N ) non-Abelian degrees of freedom is central to the study of YangMills gauge fields. Furthermore, lattice gauge fields are defined directly in terms of the finite group elements of SU (N ). To gain some insight into the behavior of the non-Abelian degree of freedom and of the lattice gauge field, the Feynman path integral for the SU (N ) degrees of freedom is studied. This Chapter is based on the formalism of quantum mechanics, since a system constituted by one nonlinear (curved) degree of freedom is analyzed. The degree of freedom is the unitary matrix, and this Chapter is a precursor to the study of lattice gauge fields, in which infinitely many unitary matrices (degrees of freedom) are coupled. In quantum mechanics, the degree of freedom is often considered to be the position of the particle, which is a vector of flat Euclidean space. This Chapter studies the quantum mechanics of the SU (N ) degree of freedom that takes values in the SU (N ) group space. The SU (N ) group space is a simply connected compact Riemannian curved metric space – having constant positive curvature. Classical paths consist of trajectories in SU (N ) space. Let H be the Hamilton operator defined by the Laplace-Beltrami operator on SU (N ) space. The evolution kernel is the matrix elements of the finite evolution operator exp(−T H), where T is Euclidean time. The evolution kernel is computed using two different methods, first by using the eigenfunctions of H and then by using the finite time path integral. Comparing these two methods will allow us to calculate the eigenfunctions of H using the finite time path integral and give new insights into the relationship between the eigenfunctions of H and the classical paths given by the Lagrangian L. Many of the results obtained in this Chapter have direct application to nonAbelian lattice gauge theory. The action of this Chapter is the limit of zerodimensional space for the lattice gauge field, which is discussed in Chapter 7. The Hamiltonian H discussed in this Chapter is the pure kinetic term of the lattice gauge field Hamiltonian studied in Chapter 11. 55

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The Laplace-Beltrami operator on SU (N ) has been studied by Wadia (1980) and Menotti and Onofri (1981) using differential techniques. Some of the results derived here coincide with their results and provide a useful check to the path integral calculations. An exact continuum path integral expression of exp(−T H) is obtained for the SU (N ) using lattice gauge theory methods. The path integrals for U (1), SU (2) and SU (3) are evaluated exactly and it is seen that the SU (N ) case is completely reduced to solving the SU (2) path integral. Hence the main focus in doing path integration is on the SU (2) path integral. By comparing the result of the SU (3) path integral for exp(−T H) with its eigenfunction expansion, an explicit expression is obtained for the character functions of SU (3). It has been hitherto an intractable problem to exactly evaluate all the character functions of SU (3) using properties of SU (3) Lie algebra. The path integral approach yields this new result. 4.2

Hamiltonian Operator

Define the Hamiltonian operator to be purely kinetic and is given for curved space by the Laplace-Beltrami operator as 2 ˆ = − g ∇2 (4.2.1) H 2 U where g 2 is the coupling constant and carries the dimension of mass. From the results of Section 2.11, the Hamiltonian can be written in terms of the generators of the Lie algebra Ea and yields 2 X ˆ =g H E2 (4.2.2) 2 a a The Laplace-Beltrami operator is given in Eqs. 11.4.5 and 2.11.13. Schr¨odinger’s equation for Euclidean time t is given by ∂ψ ˆ Hψ(U )=− (4.2.3) ∂t The Euclidean time Schr¨ odinger’s equation is the heat diffusion equation on the SU (N ) manifold. The wave functions ψ are elements of the Hilbert space defined on the SU (N ) group manifold. (p) It can be seen from Eq. 2.11.15 that the energy eigenfunctions are Dij (U ) with eigenenergy equal to (1/2)g 2 c2 (p), where c2 (p) is the second Casimir eigenvalue and (p) having d2p degeneracy, with dp being the dimension of the irrep Dij (U ). The probability amplitude or the evolution kernel for finite Euclidean time T , from Eq. 4.2.1, is defined by n g2 T o ˆ K(T ) = hV |e−T H |W i = hV | exp ∇2V |W i (4.2.4) 2 For the case of H being the Laplacian, K(T ) is also called the heat kernel. K(T ) is the probability amplitude that degree of freedom starting from the point W in

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SU (N ) will be at the point V in SU (N ) after time T , and is the matrix element of ˆ the time evolution operator e−T H . The completeness equation given in Eq. 2.8.4 yields XX ˆ K(T ) = dp hV |e−T H |p, ijihp, ij|W i (4.2.5) p

= =

ij

XX p

ij

X

e−

(p)

dp Dji (V † )e−

g2 T 2

c2 (p)

g2 T 2

c2 (p)

(p)

Dij (W )

dp Tr D(p) (V † W )

(4.2.6) (4.2.7)

p (p)

since, from Eq. 4.2.2, hV |E 2 |p, iji = c2 (p)Dji (V † ). The character function of SU (N ) is defined by χp (U ) = Tr D(p) (U )

(4.2.8)

and yields K(T ) = hV | exp

n g2 T 2

o X g2 T ∇2V |W i = dp e− 2 c2 (p) χp (U ) ; U = V † W (4.2.9) p

Equation above can be used for the identifying the character functions. If one 2 does an expansion of K(T ) in powers of e−g T /2 , the coefficients are the character functions modulo the dimension of the representation. 4.2.1

U (1) and SU (2) evolution kernel

For the U(1) Abelian group, χp (U ) = U p = eipB and hence ∞ X g2 T 2 ˆ iB −T H K(T ) = h1|e |e i = e− 2 p eipB ; − π < B < π

(4.2.10)

p=−∞

For SU (2), we use polar coordinates given in Section 2.5.2. Let n ˆ be the unit norm vector in three dimensions and σi (p)/2 the SU (2) generators in the pth representation; this yields D(p) (U ) = eiB nˆ ·˜σ(p) ; 0 < B < π

(4.2.11)

with c2 (p) = p(p + 1) ; dp = 2p + 1 ; p = 0, 1/2, 1, 3/2 . . . For the fundamental representation, with p = 1/2, the generators are the Pauli matrices. The SU (2) characters are χp (B) = Tr D(p) (U ) =

p X

eimB =

m=−p

1 sin[(2p + 1)B] sin B

(4.2.12)

and hence for SU (2) K2 =

X p=0,1/2,1...

(2p + 1)e−

g2 T 2

p(p+1) sin[(2p

+ 1)B] sin B

(4.2.13)

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Let dp = 2p + 1 = m : m = 1, 2, 3, . . .

⇒ c2 (p) = p(p + 1) =

1 2 m −1 4

Up to an overall constant, Eq. 4.2.13 yields ∞ X g2 T 2 sin(mB) K2 = me− 8 m sin B m=1 =

(4.2.14)

∞ g2 T 2 −i X me− 8 m eimB sin B m=−∞

(4.2.15)

The character functions for SU (N ) are formally given by a formula by Weyl. But the explicit evaluation of the character functions for SU (3) and higher order groups, unlike the SU (2) case, has not been tractable using algebraic methods. 4.3

Lattice Action for SU (N )

A lattice path integral for the probability amplitude K(T ) is derived. Let N be a large integer. Then, using the completeness equation N − 1 times, yields the following  T   ˆ |W i H (4.3.1) K(T ) = hV | exp(−T H)|W i = hV | exp − N · N ) ( Z N −1 N −1 Y Y T ˆ (4.3.2) dUn hUn+1 |e− N H |Un i = n=1

n=0

with the boundary conditions U0 = W ; UN = V

|w>

(4.3.3)

R

Note that X |n|>1

Dn2 '

Z 1

N

dd n ' n2d−4

Z

N

nd−1

1

dn ' N 4−d n2d−4

Hence, for N → ∞   N 4−d , 

1 ln Z ∼ ln N,  Nd  constant,

2 ε

2-1

>

I

2 a0

I

Fig. 10.11 The ultra-violet fixed points is twice unstable. The infra-red fixed point leads to either a strongly coupled theory or a free massless theory. The two phases are separated by a critical phase with G∗ = 2 − 1.

The fixed point G∗ = 2 − 1 is twice unstable. For G0 < G∗ , GI → 0 as I → ∞; for G0 > G∗ , GI → ∞ as I → ∞. See Figure 10.11. Both the fixed points G∗ = 0 or G∗ = ∞ are now stable, with the fixed point at G∗ = 2 − 1 being twice unstable. The system undergoes a phase transition at G∗ = 2 − 1, and the behavior of the gauge field at the G∗ = 2 − 1 fixed point is described by a scale invariant theory. For d = 4 +  > 4, the asymptotically free domain of G ' 0 is separated by a phase transition from the strongly coupled domain of G ' ∞, and the gauge field does not continuously go from the asymptotically free theory to the strongly coupled theory; that is, for d > 4, the gauge-field cannot simultaneously exhibit freelike behavior at short distances, and strong coupling behavior at large distance. The UV fixed point with G∗ = 2 − 1 is a strongly coupled theory. The renormalization group trajectories are shown in Figure 10.11. Furthermore, as  → 0, the G∗ = 2 − 1 fixed point coalesces with G∗ = 0 fixed point – making it once unstable – and leading to the gauge field having no phase transition separating the weak from the strong coupling sector.

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10.8

175

Weak Coupling SU (2) Gauge Field: β-Function

From above, the weak coupling recursion equation for d = 4 +  > 4 is given by GI+1 = 24−d GI (1 + GI ) = 2− GI (1 + GI )

(10.8.1)

To convert to the original coupling constant recall that GI =

1 2 g ; a ≡ 2I a0 2 I

Using aI+1 = 2aI

⇒ aI+1 = a + a ⇒ GI+1 =

1 2 g (aI + aI ) 2 I+1

Eq. 10.8.1 yields   1 1 1 2 g (a + a) = 2− g 2 (a) 1 + g 2 (a) 2 2 2 In terms of the coupling constant g(a), Eq. 10.8.1 yields g 2 (a + a) = g 2 (a) + a

  ∂g 2 (a) 1 + · · · = 2− g 2 (a) 1 + g 2 (a) ∂a 2

(10.8.2)

Let a momentum scale µ be defined by µ=

1 a

⇒ a

∂g 2 (a) ∂g = −2gµ ≡ −2gβ ∂a ∂µ

where the β-function is defined by β=µ

∂g ∂µ

Hence, from Eq. 10.8.2, the Migdal β-function is given by 1 βM = (1 − 2− )g − 2− g 3 2

(10.8.3)

Taking the limit of  → 0 yields, from Eq. 10.8.3, the following βM = −0.5g 3 The result we have obtained for the gauge field β-function is a special case of the SU (Nc ) gauge field. The following exact result, for SU (2)(Nc = 2) and g ≈ 0 is given, from Peskin and Schroeder (1995) βExact = −

11 3 11Nc 3 g =− g ' −0.05g 3 48π 2 24π 2

The Migdal approximation yields a β-function that is qualitatively similar to the exact result. It is not clear how one can improve the Migdal approximation or whether it can be used to make any empirical predictions.

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Numerical Solution of the Recursion Equation

Recall from Eq. 10.5.5 we have V˜ (I, τ ) =

Z

dσV (I, Tr(σ))V (I, Tr(σ † τ ))

Z =

dσV (I, Tr(σ))V (I, Tr(στ ))

(10.9.1)

In obtaining Eq. 10.9.1, the following was used: (a) that V (I, τ ) and V˜ (I, τ ) are trace functions of τ and (b) for SU (2), Tr(σ † ) = Tr(σ) and d(σ † ) = d(σ −1 ) = dσ. From Eq. 10.5.4 nZ o2d−2 V (I + 1, τ ) = dW V˜ (I, Tr(W ))V˜ (I, Tr(W τ )) (10.9.2) Since the invariant measure dW is always ≥ 0, it follows that V (I + l, τ ) ≥ 0 if V (I, τ ) ≥ 0. The positivity of V (I, τ ) ensures that the effective action is always real, given that V (I, τ ) ≥ 0 – and yields results for numerical calculations that are rapidly convergent. Eqs. 10.9.1 and 10.9.2 are studied numerically. Since both equations are essentially the same, except for the additional step of introducing dimension, for now let us focus on Eq. 10.9.1. For notational convenience, V (I, τ ) is considered to be a function of 21 Tr(τ ). From Eq. 10.6.10 Z Z Z +1 1 π 2 dσ = sin σdσ dx π 0 −1 where x = cos θ; choosing the spherical coordinate system for σ such that τˆ is in the 3-direction, yields 1 1 Tr(σ) = cos σ ; Tr(στ ) = cos σ cos τ − x sin σ sin τ 2 2 Hence 1 V˜ (I, τ ) = π 10.9.1

Z 0

π

dσ sin2 σV (I, cos σ)

Z

+1

dxV (I, cos σ cos τ − x sin σ sin τ ) (10.9.3) −1

Change of integration variable

Recall, from Eq. 10.6.9, x = cos(θ). Make a change of variable from x to ξ; let cos ξ = cos σ cos τ − x sin σ sin τ ⇒ − sin ξdξ = − sin σ sin τ dx sin ξ dξ dx = sin σ sin τ

(10.9.4)

The upper and lower limits of integration for variable ξ are given as follows4 4 A careless treatment of the limits of ξ results in the wrong limits and the recursion equation fails to give the correct answer.

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• Upper limit x = +1 : cos ξ = cos(σ + τ ) ⇒

ξ = (σ + τ )mod 2π ≡ min{σ + τ, 2π − σ − τ }

• Lower limit x = −1 : cos ξ = cos(σ − τ ) ⇒

ξ = |σ − τ |

Note ξ ∈ [0, π] ⇒ σ, τ ∈ [0, π] From Eq. 10.9.3 1 sin τ V˜ (I, τ ) = π

Z

π

Z

(σ+τ )mod2π

dσ sin σV (I, cos σ)

dξ sin ξV (I, cos ξ)

(10.9.5)

|σ−τ |

0

Let Z

(σ+τ )mod2π

W (I; σ, τ ) ≡

dξ sin ξV (I, cos ξ)

(10.9.6)

|σ−τ |

Then 1 sin τ V˜ (I, τ ) = π

Z

π

dσ sin σV (I, cos σ)W (I; σ, τ )

(10.9.7)

0

The two integrations to be performed numerically are Eqs. 10.9.5 and 10.9.7. The change of variable made in Eq. 10.9.4 is important for the numerical calculation. As things stand in Eq. 10.9.3, for performing the integration on a grid of points for the x integration the function V (I, σ) is required at points that are not the same as the points necessary to perform the σ-integration. And if some interpolated values of V (I, τ ) are used for performing the x-integration, the errors introduced are large. Hence, if we are to use values of the function V (I, σ) or V˜ (I, σ) at grid points which are computed by the recursion formula, then Eq. 10.9.3 cannot be used. What Eq. 10.9.4 does is that it produces the function V˜ (I, σ) at the same grid points that are to be used for performing the σ- and ξ-integrations. In other words, the change of variable from x to ξ results in Eq. 10.9.5 that uses the value of V (I, τ ) only at the grid points that are fixed and which are the same as the ones used for performing both the ξ, σ-integration. Hence, no interpolation for values of V (I, τ ) are necessary. Also, in computing W (I; σ, τ ), the limits on the integral always fall on the fixed grid points, with no interpolation being made for the values of V (I, τ ). To summarize, the change of variable from x to ξ allows us to perform, for each iteration, both integrations in Eqs. 10.9.5 and 10.9.7 on the same fixed grid points for which the recursion formula gives values of the computed function.

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Note after the change of variable in Eq. 10.9.4, the function that is required is generically f (I, θ) = sin θV˜ (I, cos θ) This appears in the integrand and is also the results of the numerical integration, as can be seen from Eqs. 10.9.5 and 10.9.7. In fact, Eq. 10.9.5 can be written as 1 f˜(I, τ ) = π

Z

π

Z

(σ+τ )mod2π

dσf (I, σ) 0

dξf (I, ξ) |σ−τ |

The functions f˜(I, τ ), f (I, θ) are defined on the same grid points, with the variation in the limits of integration on the given grid, as seen in Eq. 10.9.5, generating the result sin τ V (I, τ ). 10.9.2

Numerical algorithm

Eqs. 10.9.5 and 10.9.7 are ideally suited for a numerical solution. Each step in the recursion is identical as far as the structure of the integrals goes – with the input function changing from V (I, σ) to V (I +1, σ). The function V (I +1, σ) is computed from V (I, σ), and in turn is the input for calculating V (I + 2, σ), and so on. The initial function V (1, σ) for the recursion is the exponential of the bare action given by V (1, σ) = exp

n 1 o (cos σ − 1) G1

The input coupling constant G1 is a parameter that can be varied. For the purpose of performing the integrations numerically, the variables σ and ξ are discretized. The range of integration, for the first step in the recursion, is divided into N number of grid points. The total number of grid points may vary with each step in the recursion. The number N is an input, whereas N (I) is fixed by the computer program; the total number of grid points for the Ith step in the recursion is denoted by N (I), with maximum value of N (I) being 2N number of grid points. The only other input for the computer program is the maximum number of iterations to be performed, denoted by M . In summary, a computer program based on the numerical algorithm discussed above has the following four parameters as input • • • •

The The The The

initial plaquette action with coupling constant G1 dimension of spacetime d maximum number of grid points for the integration 2N maximum number of allowed iterations M .

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Total grid size S(I)

Consider the case of G1 ' 10−3 ; then n 1 o n 1 2o V (1, σ) = exp (cos σ − 1) ∼ exp − σ G1 2G1

√ From above, it can be seen that V (1, σ) rapidly goes to zero when σ  1/ G1 . The size of the grid S(1) is chosen such that V (1, S(1)) ≥ G21 on all points inside √ this space; for G1 small, S(1) ∼ G1 . The choice of V (1, N ) ∼ G21 is necessary to do the second order calculation of GI for d = 4.5 With this in mind, for each iteration I, the range of integration of variable σ is given by the total grid size S(I) such that V (I, S(I)) > G2I

For reasons to be explained later, the total grid size S(I) must always be of the form π/2n−m (n, m = integers); n is chosen from the equation p π S(1) = n ∼ G1 2 Given S(1) and N , the initial grid spacing (1) is initially chosen so that (1) =

S(1) N

In general6 S(I) =

π 2n−m

; (I) =

S(I) N (I)

Qualitatively, S(I) < π/2 for the weak coupling region and π/2 < S(1) < π for the cross-over region. As discussed above, the initial total grid size is fixed such that V (1, N ) = G21 . The computed function’s normalization is chosen such that V (I, 1) = V˜ (I, 1) = 1

(10.9.8)

As the iteration is performed to compute V (2, σ), V (3, σ), and so on, the interval of σ on which these functions are > G22 , G23 and so on also increases – since for d = 4 the coupling constants GI are increasing. Figures 10.12(a) and (b) show the action and total grid size as the one iterates and the running (effective) coupling constant increases. Eq. 10.9.7 is used to compute the function V˜ (I, τ ) for values of τ until the point when V˜ (I, S(I)) = G2I ; N (I) is defined using the equation (I)N (I) = S(I) ; N ≤ N (I) ≤ 2N This procedure of increasing the total number of lattice points is continued until N (I) > 2N ; when this happens, all the odd lattice points are dropped from the calculation, and the lattice spacing is consequently doubled. Figure 10.12(a) shows 5 For

d < 4, G21 can be replaced by say a fixed value equal to 0.001 without any large errors. computer program is slightly more complicated and S(I) may not be of the form π/2n−m for the first few steps. 6 The

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V(σ) GI 0.01

(a)

2

V(S(I))=GI’

ε

-π/2

-π

S(I)

π/2

π

V(σ)

(b)

GI’ 0.4

2

V(S(I’))=GI’

-π

-π/2

ε’

π/2

S(I’)

π

σ

Fig. 10.12 (a)Weak coupling regime: 0 < S(I) < π/2. The action and total grid size for GI ' 0.01 and with  = S(I)/N (I). (b) Cross-over regime: π/2 < S(I) < π. The action and total grid size for I 0 > I and G0I ' 0.1 and with 2 = S(I 0 )/N (I 0 ).

grid spacing  and, for example, Figure 10.12(b) in which the grid spacing 0 has doubled compared to previous recursions. Furthermore, the doubling of the total number of lattice points m times implies that π S(I) = n−m ; m = 0, 1, · · · , n 2 The procedure of increasing the total number of lattice points is continued until S(I) = π; when S(I) = π, the number of lattice points is permanently fixed, and the S(I) no longer is allowed to increase. This is the reason why S(1) had to have the form π/2n ; because after doubling the lattice spacing n times, we end up with S(I) = π. The value of S(I) = π is reached when the gauge field becomes strongly coupled. The strong coupling regime is shown in Figure 10.13. This method of doubling the lattice spacing is crucial in allowing one to go from the weak to the strong coupling domain. The reason being that if the lattice spacing was kept fixed at its initial value, and the starting is from the weak coupling sector, then one would eventually need about 210 grid points to cover the interval [0, π]. This would make the calculation infeasible.

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V(σ) GI’’ 1.6

-π

π/N

S(I’’)= π

σ

Fig. 10.13 Strong coupling limit: S(I) = π. The total space is fixed to be S(I) = π with fixed grid spacing  = π/N .

The function W (I; σ, τ ) is computed using Simpson’s rule. sin ξV (I, cos ξ); then from Eq. 10.9.6 Z (σ+τ )mod2π W (I; σ, τ ) ≡ f (ξ)dξ = W (I; τ, σ)

Let f (ξ) =

(10.9.9)

|σ−τ |

Since W (I; σ, τ ) is symmetric under σ ↔ τ , only the case of σ > τ needs to be considered. Hence, the numerical integration that needs to be performed is only the following Z (σ+τ )mod2π W (I; σ, τ ) ≡ f (ξ)dξ σ−τ

and yields the the following interval for ξ-integration: • For σ + τ < 2π ; σ + τ − (σ − τ ) = 2τ • For σ + τ > 2π ; 2π − (σ + τ ) − (σ − τ ) = 2(π − σ) In other words, the interval for the ξ-integration is always even. The total number of grid points N (I) is always arranged to give an even number of intervals so that Simpson’s rule, given below, can be used for the ξ-integration7 Z 3 f (ξ)dξ = const[f1 + 4f2 + f3 ] (10.9.10) 1

A straightforward extension can be made for arbitrary number of even intervals. The trapezoidal rule is used for evaluating Z π dσ sin σV (I, cos σ)W (I; σ, τ ) 0

7 The

overall constant is removed by the normalization given in Eq. 10.9.8.

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since the trapezoidal rule is more accurate for evaluating the integral of a periodic function over its period than is Simpson’s rule. However, the gains of using the trapezoidal rule are minor, and it is preferred mostly due to its simplicity. The reason for using Simpson’s rule for W (1; σ, τ ) is that if the trapezoidal rule is used, then there are large systematic errors in going from the weak to the strong coupling domain. The definition of the running coupling constant for the entire renormalization trajectory is made by assuming that the functional form of the effective action is given by o n 1 2HI (10.9.11) (cos α − 1) + (cos α − 1)2 VE (I, cos α) = exp GI GI The coupling constant GI is determined by using the above ansatz and can be evaluated by comparing the numerical value of the computed function V (I, cos α) at three lattice points. The ansatz is exact for the weak and strong coupling limits, and is a natural interpolation for the intermediate cross-over region, for which GI is evaluated for a number of adjacent points. It is found that the variation in using different definitions for the coupling constants in the cross-over region is about 10%. The coupling constant GI is taken to be the one evaluated using V (I, 2), V (I, 3) and V (I, 4). The numerical accuracy of the computer program is checked with the weak and strong coupling analytic results. The program is accurate, for each iteration, to 1% when calculations are performed with ∼ 100 grid points. The computer program produces the sequence of functions V (I, σ) from which GI can be computed. In Figure 10.14 the results are plotted. 10.10

Numerical Results

The sequence of coupling constants for different initial values of G1 lie on the same trajectory; changing the value of G1 simply shifts the sequence of coupling constants along the trajectory on which the other sequences lie. Taking the limit of G1 → 0 yields the renormalized coupling constant as a function of the effective lattice spacing. The renormalization group trajectory for the sequence of coupling constants {GI }∞ I=0 has three distinct regions, namely the weak coupling regime, the strong coupling regime and the cross-over region. These three are smoothly connected and can be identified only qualitatively. By convention, the intermediate region is identified to be such that the deviation from the weak and strong approximations is > 10%. The numerical result, for d = 4 (the numbers depend on d) is shown in Figure 10.14, yields the following. • Weak coupling regime is for 0 ≤ GI ≤ 0.3. The coupling constant increases gradually.

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2.0

11.4 1.8 1.6

GI

Strong Coupling(~5%)

1.4 1.2 1.0 0.8

Intermediate Coupling

0.6 0.4 0.2

Weak Coupling(~10%) 4

8

12

16

20 24

28

I Fig. 10.14 Graph of coupling constant renormalization trajectory, with GI vs. I, where I = number of iterations, and GI is the effective coupling constant for the gauge field.

• Intermediate regime is for 0.3 ≤ GI ≤ 1.5. The coupling constant abruptly increases over a small range. • Strong coupling regime is for 1.5 ≤ GI ≤ ∞. The coupling constant increases rapidly. The sequence of effective actions are monotonically increasing functions; recall the effective actions are normalized such that V (I, 1) ≡ 1; that is V (I + 1, α) ≥ V (I, α), α > 1. Since V (I = ∞, α) = V∗ (α) = 1, we see that the sequence of effective actions converge uniformly to the stable strong coupling fixed point action. In fact, it is found numerically that, to ∼ 15% accuracy, the form of the initial action is valid for all I. V (I, α) = exp

n 1 o (cos α − 1) GI

The strong coupling fixed point reached by the numerical calculation depends on the total range of integration used for the intermediate and strong coupling regimes.

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The exact theory gives 0 ≤ σ ≤ π. In the numerical calculation, if the range 0 ≤ σ ≤ a : a > π or a < π is used, then a fixed point was reached which was different from V∗ (σ) = 1. This is the reason why S(1) needs to have the form π/2n ; because after doubling the lattice spacing n times, the total grid space ends up with S(I) = π – and which results in the correct strong coupling fixed point. The range 0 ≤ σ ≤ π means that each group element of SU (2) is covered once and only once in the numerical integration: the strong coupling fixed point is very sensitive to the entire structure of the non-Abelian gauge group. This fact is also apparent in the exact lattice gauge theory, where the strong coupling expansion involves the entire gauge group. Summary: Confinement and Asymptotic Freedom





>




>

>

>

>

>

>




a

> 2a Fig. 10.15 Gauge field renormalization: the link representing the gauge field changes with the increase in the size of the lattice.

Migdal’s recursion formula is, in essence, a scheme for gauge field renormalization, with the recursion equation being an expression of the renormalization group transformation. The procedure for renormalization encodes the intuitive formulation of Wilson (1983) that a quantum field theory consists of infinitely many coupled length scales. Each lattice size has its own length scale, which is its lattice spacing, and there is a separate effective coupling constant GI for the effective action and gauge field degrees of freedom for that length scale. For example, Figure 10.15 shows that under renormalization, the link variable representation of the lattice gauge field becomes longer and longer for increasing lattice spacing, providing an intuitive realization of gauge field renormalization. Figure 10.15 shows that the renormalization of the gauge field leads to the intuitive result that classical gauge fields are represented by very long link variables and hence

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their variation over space is smooth. It is for this reason that classical gauge fields can be approximated by differentiable functions – as is the case for the Maxwell Abelian gauge field. The physical interpretation of GI is that it is the strength of the gauge field felt by particles which couple to it, say quarks, when these quarks are separated by a distance of 2I a0 (a0 = original spacetime lattice spacing). Asymptotic freedom implies that G1 → 0 as a0 → 0; for the sake of discussion, let G1 ' 10−3 for a0 ' 10−16 cm. The recursion equation shows that as the quarks are separated to larger distances, the strength of gauge field increases continuously. The fact that, for d = 4, the model has an unstable weak coupling fixed point and a stable strong coupling fixed point implies that the quarks behave almost like free particles at short distances, and become strongly coupled at large distances. The absence of any other fixed points shows that the weak and strong coupling behavior of the gauge field is not separated by any phase transition, and the quarks go continuously from their short distance weak coupling behavior to their strongly coupled behavior, and which gives rise to the bound states of the quarks. Since the coupling constant becomes arbitrarily large for an arbitrarily large distance, the quark-antiquark separate to a definite distance, after which the quarkgauge field system produces quark-antiquark pairs – since this becomes energetically more favorable than any further separation of the quark-antiquark in question. This explains why quarks in a bound state cannot be separated to macroscopic distances – and results in the permanent confinement of quarks. Of course, in the above discussion, it has been assumed that the qualitative behavior of the pure gauge field is not destroyed when the quark field is coupled to it. Given this assumption, Migdal’s model provides a theoretical example of the physical idea of quark confinement, and gives a tractable example of an asymptotically free field theory going over to a strongly coupled system. As is well known, this rather remarkable behavior is unique to non-Abelian gauge fields and is shown by no other known quantum field. 10.12

Appendix

Eq. 10.6.12 yields 2 Z n 2(cos u cos σ − 1) o eG π du sin2 u exp V˜ (I, τ ) = const π 0 G h i 4H 4H (cos σ cos u − 1)2 + (sin σ sin u)2 + O(G2 ) (10.12.1) × 1+ G 3G The computation of the Appendix will give the result stated in Eq. 10.6.14. Recall σ = τ /2. Let cos σ a= : 0(1/G) (10.12.2) G √ Since σ, u = 0( G) the following approximations are made:

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• •

2 2 (cos σ cos u − 1) = (cos σ − 1) − au2 + O(G) G G sin2 u = u2 + O(G) = u2 [1 + O(G2 )]

(10.12.3)

• (cos u cos σ − 1)2 = (cos σ − 1)2 − (cos σ − 1)u2 + O(G2 ) • sin2 σ sin2 u = (sin2 σ)u2 + O(G2 ) Let o 4H 4H n 1 (cos σ − 1)2 ; Q = sin2 σ − (cos σ − 1) G G 3 Then above equations and Eq. 10.12.1 yield (up to constants) Z π 2 V˜ (I, τ ) = e2(cos σ−1)/G duu2 e−au [P + Qu2 ] (10.12.4) 0 Z 2 1 ∞ duu2 e−au [P + Qu2 ] ' e2(cos σ−1)/G 2 −∞ n1 o i 2(cos σ−1)/G h 4H e = (cos σ − 1)2 + 6H sin2 σ − (cos σ − 1) + O(G2 ) 1+ 3/2 G 3 cos σ From Eq. 10.6.13 V˜ (I, τ ) = exp{C(τ )} P =1+

Therefore, up to a constant, from Eq. 10.12.4 2 3 4H C(τ ) = (cos σ − 1) − ln cos σ + (cos σ − 1)2 G 2 G n1 o + 6H sin2 σ − (cos σ − 1) + O(G2 ) 3 Since σ = τ /2, we have r 1 1 2 sin σ = − (cos τ − 1) ; cos σ = 1 + (cos τ − 1) 2 2 1 2 ln cos σ = (cos τ − 1) + O(G ) 4 Therefore  h 1 3 5H i 1 1 C(τ ) = − (cos τ − 1) + (cos τ − 1)2 H − 2G 8 2 4G 4 But 1 2H 0 C(τ ) = 0 (cos τ − 1) + 0 (cos τ − 1)2 G G Hence, the final result is given by 3 5H 1 1 − = 0 G 2G 8 2 2H 0 1 1  H− = 0 G 4G 4

(10.12.5)

(10.12.6)

(10.12.7) (10.12.8)

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Chapter 11

Lattice Gauge Field Hamiltonian

The Hamiltonian for QCD (quantum chromodynamics) has been widely studied using the lattice and continuum formulations. The mathematical treatment of gauge fixing the Yang-Mills Hamiltonian goes back to Schwinger (1962). The paper by Christ and Lee (1980) gives a clear and complete treatment of gauge-fixing the continuum non-Abelian gauge field Hamiltonian. The lattice gauge field Hamiltonian has been derived by Kogut and Susskind (1975). The lattice gauge field Hamiltonian can be regulated to all orders, and can be used for calculations involving two loops or higher. To use the lattice Hamiltonian for the weak-coupling approximation, it is necessary to choose a gauge, for example, the Coulomb gauge. Gauge-fixing the lattice gauge theory Hamiltonian essentially involves only the lattice gauge field and the quarks (Dirac field) enter only through the quark-color-charge operator. For this reason, gauge-fixing is analyzed before the complete derivation of the color-charge-operator, discussed later in Chapter 14. Gauge-fixing the lattice action is similar to the continuum case, as can be seen from the path integral formulation discussed in Section 7.5. Similarly, gauge-fixing the lattice Hamiltonian is very similar in spirit to gauge-fixing the non-Abelian continuum Hamiltonian. The discussion on gauge-fixing the lattice gauge-field Hamiltonian is based on the treatment of Baaquie (1985), and is similar to the continuum treatment of Christ and Lee (1980). The Coulomb gauge for the continuum Abelian gauge field Hamiltonian is discussed at length in Baaquie (2018) and it is recommended that the Abelian case be reviewed to facilitate navigating the more complex non-Abelian case. The lattice gauge field is defined using finite-group elements of SU (N ) as the fundamental degrees of freedom, whereas the continuum uses only the Lie algebra of SU (N ). This difference introduces a lot of extra complications leading to significant differences between the lattice and continuum gauge field Hamiltonians – both for the kinetic operator and the potential term. In particular, the lattice gauge field has an additional complication that is absent in the continuum case, which is the distinction between the left- and right-group multiplication required for defining lattice gauge transformation. Given an appropriate generalized interpretation of the basic symbols, it will turn 187

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out that the form of the gauge-fixed continuum and lattice Hamiltonians are very similar. 11.1

Lattice Gauge Field Hamiltonian

Consider a three-dimensional Euclidean lattice with spatial lattice spacing given by a; Uni , i = 1, 2, 3 is the SU (N ) link degree of freedom from lattice site n to n + ˆi (ˆi is the unit lattice vector in the i th direction) and ψn , ψ¯n , is the lattice Dirac field. A detailed derivation of the full lattice gauge theory Hamiltonian H is given in Chapter 14. In Section 14.3, the Hamiltonian H for SU (N ) is obtained in the axial gauge, specified by Un0 = I ; n : four-dimensional lattice From Eq. 14.5.5, Hamiltonian H is given by ¯ ψ, U ] H = HG [U ] + HD [ψ, where HD is the quark-gauge-field part. The derivation of the lattice gauge field Hamiltonian is given based on a summary of Section 14.3; it is obtained from the asymmetric lattice gauge-field Lagrangian by taking the limit of  → 0. The asymmetric Lagrangian, from Eq. 7.1.3, is L(n; ) =

3 3 1  X 1 aX † Tr(W + W ) + Tr(Wn,ij ) ; i, j = 1, 2, 3 n,0i n,0i 4g 2  i=1 4g 2 a i6=j=1

The asymmetric lattice Lagrangian in the axial gauge, given in Eq. 7.1.3, yields from the Dirac-Feynman formula, the gauge field Hamiltonian 0 Y 0 lim hU 0 |e−HG |U i = e−HG (U ) δ(Uni − Uni ) = N exp{L(n; )} (11.1.1) →0

ni

( = lim exp →0

1 a 4g 2 

3 X i=1

† 0 Tr(Un,i Un,i

) 3 1  X 0 + h.c.) + 2 Tr(Wn,ij + Wn,ij ) 4g a i6=j=1

For  → 0 Eq. 4.3.5 yields ( )
a 1 X † 0 0 † exp Tr Uni Uni + Uni (U )ni  4g 2 ni ) ( Y g2  X 2 0 0 ∇ (Uni )) δ(Uni − Uni ) = exp + a n,i ni where ∇2 is the Laplace-Beltrami operator for SU (N ) discussed in Section 2.9. The gauge-field Hamiltonian in the axial gauge, from Eqs. 11.1.1 and 14.5.4, is given by g2 X 2 1 X † HG = − ∇ (Uni ) − Tr(Uni Un+ˆi,j Un+ U† ) ˆ j,i nj a n,i 2ag 2 n,ij =

g2 X X 2 1 X † Ea (Uni ) − Tr(Uni Un+ˆi,j Un+ U† ) ˆ j,i nj a n,i a 2ag 2 n,ij

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where Ea (Uni ) is the chromoelectric field operator (see Eqs. 11.1.4 and 11.1.13). Gauge-transformations are given by Uni → Uni (Φ) ≡ Φn Uni Φ†n+bi

(11.1.2)

The Hamiltonian acts only on gauge-invariant wave functionals Ψ. As given in Eq. 14.3.5, the wave functional Ψ is invariant under time-independent gaugetransformations given in (11.1.2); that is, Ψ[U ] = Ψ[U (Φ)] Choose canonical coordinates link variable is given by

a Bni ;

(11.1.3)

suppressing the lattice and vector indices, the

Uni = exp(iB a Xa ) The differential operator Ea (U ) is discussed in Section 2.9. From Eqs. 2.11.4 and 2.11.7 (summing on repeated non-Abelian indices) yields1 EaL (U ) = fab (U )

δ ∂ δ ∂ T ≡ ; EaR (U ) = fab (U ) ≡ i∂B b iδL B a i∂B b iδR B a

(11.1.4)

As discussed in Eq. 2.11.4, EaL (U ) is left-invariant in the sense that EaL (V U ) = EaL (U ) ; V ∈ SU (N ) and EaL (U ) is right-invariant in the sense that EaL (U V ) = EaL (U ) ; V ∈ SU (N ) From Eq. 2.11.8, the operators EaR and EaL are the lattice chromoelectric field operators, which are first-order Hermitian differential operators with the commutation equation, from Eq. 2.11.8, given by [EaL , EbL ] = iC abc EcL ; [EaR , EbR ] = −iC abc EcR ; [EaR , EbL ] = 0 EaR (U ) = Rab (U )EbL (U ),

Rab (U ) = Tr(Xa U Xb U † )

where Rab is the adjoint representation given by2 Rab (U ) = Tr(Xa U Xb U † )

(11.1.5)

Recall Xa are the generators and C abc the structure constants of SU (N ). The choromelectric operators are the non-Abelian quantum field generalization of the electric field of the Abelian gauge field. In canonical coordinates the gauge transformation is given by Φn = exp(iφan Xa ) and from Eq. 2.11.11 a

R

a

a

L

a

eiφn Ea (U ) U = eiφn Xa U = Φn U ; eiφn Ea (U ) U = U eiφn Xa = U Φn 1 Explicit

(11.1.6)

expression for fab is given in Eqs. 2.9.4, but will not be required for the derivations of this Chapter. 2R ab is the notation for the adjoint representation, which is different from the one used in Chapter 2, is chosen so as to not clash with the definition of the quark charge operator.

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The gauge transformation, from Eq. 11.1.6, iso hence given by  Xn    a R L exp i φn E (Uni ) − Ea (Un−bi,i ) Ψ[U ] = Ψ U (Φ)

(11.1.7)

ni

Since, from Eq. 11.1.3 Ψ[U ] = Ψ[U (Φ)] Eq. 11.1.7 yields Gauss’s " law # o Xn EaR (Uni ) − EaL (Un−bi,i ) |Ψi = 0

(11.1.8)

i

Gauge invariance of the Hamiltonian under time independent gauge transformations, as expressed in hEq. 14.2.14, yields the following i X HG , EaR (Uni ) − EaL (Un−bi,i ) = 0 i

Hence, the constraint equation given in Eq. 11.1.8 holds for all values of time. Gauss’s law confirms the identification EaL (Uni ) as the chromoelectric operator of the gauge field corresponding to the link variable Uni . ¯ ψ, U ) is discussed in Section 14.7 The lattice quark-color-charge operator ρna (ψ, and is defined in Eq. 14.7.14; it satisfies [ρna , ρmb ] = iCabc ρnc δnm Hence, in the presence of the chromoelectric charge operator ρna , from (14.7.9) Gauss’s law yields h i P L L 0= i [Rab (Uni )Eb (Uni ) − Ea (Un−b i,i )] − ρna |Ψi hP i ab L ≡ D E (U ) − ρ ] |Ψi (11.1.9) mi na m,i nmi b ab where Dnmi is the lattice covariant backward derivative. Let |n, ai be a ket vector of lattice site n and non-Abelian index a; then, from Eq. 11.1.9, the real matrix Di given by ab Dnmi = hn, a|Di |m, bi = Rab (Uni )δnm − δab δn−bim (11.1.10) It is seen from above that Di performs a finite rotation Rab on the ket vector and then displaces it in the backward direction. The full Hamiltonian of the gauge field coupled to fermions is a sum of the kinetic and potential energy and is given by ¯ ψ, U ] H = K(U ) + P [ψ, (11.1.11) where, from Eq. 11.4.5 g2 X 2 ∇ (Uni ) (11.1.12) K=− 2a n,i

and from Eq. 2.11.133 −∇2 (U ) =

X

EaL (U )EaL (U )

a 3 As

shown in Eq. 2.11.13, the Laplacian can also be written as X −∇2 (U ) = EaR (U )EaR (U ). a

(11.1.13)

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11.2

191

Gauge-Fixed Chromoelectric Operator

We can see from Gauss’s law that all the Uni ’s are not required to describe the gaugeinvariant wave functional Ψ. We gauge transform Uni to a new set of variables Vni which are constrained; the constrained variables Vni will decouple from Gauss’s law. Consider the change of variables from {Uni } to {Φ, Vni } given by Uni = Φn Vni Φ†n+bi

(11.2.1)

The new gauge degrees of freedom {Vni } are constrained, with {Vni } having one constraint for each n; the Coulomb gauge for the lattice is given by X Xna (V ) ≡ Im Tr Xa (Vni − Vn−bi,i ) = 0 (11.2.2) i

The explicit form of the gauge constraint will not be required for the derivations below. In canonical coordinates we have Vni = exp(iAani Xa ),

Φn = exp(iφan Xa ),

a Uni = exp(iBni Xa )

A small variation Aa + dAa yields   ∂V (A) a † V (A + dA) = V (A) 1 + V (A) dA ∂Aa R = V (A)[1 + iXa eab (A)dAb ]

(11.2.3)

where, as in Eq. 2.3.1 eab (A) ≡

R eab (A)

  † ∂V L Xb = eba = −iTr V (A) ∂Aa

Define L R δL Aa = eab (A)dAb ; δR Aa = eab (A)dAb

(11.2.4)

From Eq. 2.9.4 R L T fab = fab ; fab = fab = fba

Then V (A + dA) = V (A)(1 + iXa δR Aa ) = (1 + iXa δL Aa )V (A) As shown in Eq. 2.9.4 L R R eL aα fab = δab ; eaα fab = δab

(11.2.5)

and hence matrix eaα is the inverse of faα . Under the change of variables from Uni to Vni , given in Eqs. 11.2.1 and 11.2.2, ¯ ψ, U ) in Eq. (11.1.11) can be expressed as a function of the potential energy P(ψ, only Vni . For the kinetic energy K the expression for EaL is required. Noteworthy 11.1: Non-Abelian change of variables Recall from Eq. 2.9.5, the left-invariant chromoelectric operator is given by ∂ L Ea = fab (B) ⇒ EaL U = U Xa i∂B b

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Consider a change of variables from B a to Aa ; the chain rule yields X ∂Aa ∂ X ∂Aa ∂ ∂ L = = eL aβ (A)fβα (A) b b a b ∂B ∂B ∂A ∂B ∂Aα a a,α,β

a X eR X δ R Aβ δ ∂ βa (A)∂A L = f = (A) βα ∂B b ∂Aα ∂B b δL Aβ a,α,β

β

Hence L Ea = fab (B)

X δ R Aβ δ ∂ = i∂B b δL B b iδL Aβ

(11.2.6)

β

Consider h(B), a function of the B a variables; then, similar to the derivation above (summing over repeated non-Abelian index) dh(B) =

∂h δh dB a = δR B a a ∂B δL B a

(11.2.7)

Vni = Φ†n Uni Φn+bi

(11.2.8)

Consider from Eq. 11.2.1

The left hand side of Eq. 11.2.8, from Eq. 11.2.7, yields dVni =

δVni δR BAa = iVni Xa δR Aa δ L Aa

and the right hand side of Eq. 11.2.8 yields   a d Φ†n Uni Φn+bi = −i(Xa Φ†n Uni Φn+bi )δR φan + i(Φ†n Uni Xa Φn+bi )δR Bni a +i(Φ†n Uni Φn+bi Xa )δR φn+bi

Recall from Eq. 11.1.5 Rab (U ) = Tr(Xa U Xb U † ) From above, after some simplifications † b δR Aani = δR Φan+bi − Rab (Vni )δR φbn + Rab (Φ†n+bi )δR Bni

(11.2.9)

Using the chain rule and Eq. 11.2.6 yields X ∂Aa X ∂φa ∂ ∂ ∂ n ni = + a b b b ∂φa ∂A ∂Bmj ∂B ∂B n ni mj mj n,a n,i,a =

X n,i,a,α,β

R eaα (Ani )

∂Aα ∂ ni L faβ (Ani ) β + . . . b ∂Bmj ∂Ani

Therefore, from Eqs. (11.1.4) and (11.2.10) X δ R Aa X δ R φa δ δ δ n ni EbL (Umj ) = = + a b b b iδ φa iδ A iδL Bmj δ B δ B L L L L n ni mj mj n n,i

(11.2.10)

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b The coefficient functions δR Aani /δL Bmj of the above equation are evaluated. The constraint given in Eq. 11.2.2 is valid under variations of Aani to Aani + dAani and yields

0 = Xna (A) = Xna (A + dA) Hence, from Eqs. 11.2.7 and 11.2.11 X b Γab nmi (A)δR Ami = 0

(11.2.11)

(11.2.12)

m,i

For constraint given in Eqs. 11.2.2 and 11.2.12 yields a b ab ab Γab δ i,m nmi = hn, a|Γi |m, bi = δXn /δL Ami = ωni δnm − ωn−b i,i n−b

where, from Eqs. 11.2.2 and 11.2.12 † ab ωni = Tr(Xa Vni Xb + Xb Vni Xa )

The constraint Eq. 11.2.12 on δAani determines δφ/δB. Consider from Eq. 11.2.2, the following variation: Vni (A + dA) = Φ†n (φ + dφ)Uni (B + dB)Φn+bi (φ + dφ) and which yields, from Eq. 11.2.9 † b δR Aani = δR φan+bi − Rab (Vni )δR φbn + Rab (Φ†n+bi )δR Bni X ab b ≡ Dnmi δR φbm + Rab (Φ†n+bi )δR Bni

(11.2.13) (11.2.14)

m

From Eqs. 11.2.13 and 11.2.14, the lattice covariant forward derivative operator Di is given by † ab Dnmi = hn, a|Di |m, bi = δab δn+bi,m − Rab (Vni )δnm

Eqs. 11.2.12 and 11.2.14 yield X X b hn, a|Γi Di |m, biδR φbm + hn, a|Γi RiT |m, biδR Bmi =0 m,i,b

(11.2.15)

m,i,b

where T stands for transpose and hn, a|Ri |m, bi = δnm Rab (Φn+bi )

(11.2.16)

Hence, from (11.2.15) we have      1  δR φan T  = − n, a  Γj Rj  m, b (11.2.17) b Γ·D δL Bmj P where (Γ · D)−1 is the inverse of operator i Γi Di . From Eqs. 11.2.14 and 11.2.17       δR Aani 1 T T   = − n, a  Di Γj Rj − Rj δij  m, b (11.2.18) b Γ·D δL Bmj

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Hence, from Eqs. 11.2.11, 11.2.17, and 11.2.18     X   1 δ δ T  δ − D = n, a m, b Γ R ij i j j   b Γ·D δL Aani δL Bmj n,i    X  1  δ T − n, a  m, b Γ R Γ · D j j  δ φan L n

(11.2.19)

Eq. (11.2.19) provides the solution for expressing the unconstrained chromoelectric operator δ/δL B in terms of the new constrained gauge-fixed chromoelectric operator δ/δL A and the gauge transformation δ/δL φ. In essence, this solves the problem of gauge fixing the lattice Hamiltonian. Note Eq. 11.2.18 yields the following identity X δR Aani hl, c |Γi | n, ai =0 b δL Bmj n,i as expected from Eq. 11.2.12. Also, from Eq. 11.2.12 the gauge-fixed chromoelectric operator is transverse since X δ =0 (11.2.20) hn, a |Γi | m, bi δL Abmi m,i Hence, from Eqs. 11.1.4 and 11.2.20          T 1 δ b   m, c eL , A Γ Γ = δ δ δ − n, a mm ij ac cb (Amj )  i Γ · ΓT j  δL Abni mj 11.3

Gauss’s Law

To check that constrained variables Vni decouple from Gauss’s law, recall from Eqs. 11.1.10 and 11.2.19, we have X m,j

X

   δ δ = m, b DjT  l, c b b δL Bmj δL Bmj m,j     X   1 δ T T = n, a  δ − D Γ R D j i j j  l, c a  ij Γ · D δ A L ni n,ij    X  1  δ T T  − n, a  Γj Rj Dj  l, c a Γ · D δ L φn n,ij

hl, c |Dj | m, bi

(11.3.1)

From the definitions of Di and Di given in Eqs. 11.1.10 and 11.2.15, respectively, one has the crucial operator identity RjT DjT = −Dj RT

(11.3.2)

hn, a |R| m, bi = δnm Rab (Φn )

(11.3.3)

where

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Noteworthy 11.2: Proof of RjT DjT = −Dj RT From Eq. 11.1.10, the real matrix Di is given by ab Dnmi = hn, a|Di |m, bi = Rab (Uni )δnm − δab δn−bim

and from Eq. 11.2.15 Di is given by † ab Dnmi = hn, a|Di |m, bi = δab δn+bi,m − Rab (Vni )δnm

Furthermore, from Eqs. 11.2.16 and 11.3.3 hn, a|Ri |m, bi = δnm Rab (Φn+bi ) ; hn, a |R| m, bi = δnm Rab (Φn ) Eq. 2.3.7 gives the following multiplication law for the adjoint representation Rac (U )Rcb (V ) = Rab (U V ) Note that Rab (U † ) = Rab (U ) To prove the identity, the left hand side yields, using the definition of Di given in Eq. 11.1.10       

 X

 n, a RjT DjT  m, b = n, a RjT  l, c l, c DjT  m, b l,c

=

X

 δnl Rca (Φl+j ) Rbc (Unj )δml − δcb δm−j,l

l,c

= δnm Rba (Unj Φn+j ) − δm−j,n Rba (Φn+j ) = δnm Rba (Φn Vnj ) − δm−j,n Rba (Φn+j )

(11.3.4)

The right hand side, using the definition of Di given in Eq. 11.2.15, yields  

 X

   n, a Di RT  m, b = hn, a |Di | l, ci l, c RT  m, b l,c

X † = δac δn+j,l − Rac (Vnj )δnl δml Rbc (Φm ) l,c

= δn+j,m Rba (Φm ) − δnm Rca (Vnj )Rbc (Φm ) = δn+j,m Rba (Φn+j ) − δm−j,n Rba (Φm Vnj )

(11.3.5)

Hence, from Eqs. 11.3.4 and 11.3.5, the following identity has been proven RjT DjT = −Dj RT Hence, from Eqs. 11.3.1 and 11.3.2 the first term in Eq. 11.3.1 is zero and X δ δ δ cb Dlmj = Rcb (Φl ) = (11.3.6) b b δ φcl δ B δ φ R L L mj l m,j

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Recall from Eq. 11.2.1 that ψn = Φn ζn ; ψ¯n = ζ¯n ; Φ†n = Uni = Φn Vni Φ†n+bi From Eq. 11.1.9, Gauss’s constraint is given by   X ab ¯ ψ, U )i = 0  Dnmi EbL (Umi ) − ρna  |Ψ(ψ, m,i

and from Eq. 11.3.6 X

ab Dnmi EbL (Umi ) =

m,i

δ iδR φan

Hence, Vni has decoupled from Gauss’s constraint, and from Eqs. 11.1.9 and 11.3.6   δ ¯ ζ, V )i = 0 − ρna |Ψ(ζ, (11.3.7) iδR φan Using Eq. 11.1.9 δ exp(iφα ρα ) = ρa exp(iφα ρα ) (11.3.8) iδR φa ¯ ψ, U ) is gauge invariant yields and solving Eq. 11.3.7 using that Ψ(ψ,  X  ¯ ψ, U ) = exp i ¯ ζ, V ) Ψ(ψ, ρna φan Ψ(ζ, (11.3.9) n

Eq. 11.3.9 shows that the state space of the lattice gauge theory is gauge-invariant ¯ ψ, U ) given in Eq. 11.3.9 obeys Gauss’s law, note up to a phase. To verify Ψ(ψ, from Eqs. 11.3.6 and 11.3.8  X ab ¯ ψ, U )i  Dnmi EbL (Umi ) − ρna  |Ψ(ψ, m,i

!  X δ a ¯ ζ, V )i = 0 − ρna exp[i ρna φn ] |Ψ(ζ, = iδR φan n The change of variables from {Uni } to {Vni , Φn } has a Jacobian given by the Faddeev-Popov determinant. Similar to the evaluation of the ghost Faddeev-Popov in gauge-fixing path integral, it canZ be shown that the Jacobian is equal to Y Y J −1 [V ] = dΦn δ[Xna (Φn Vni Φ†n+bi )] (11.3.10) 

n

n,a

For weak coupling, J [V ] has been evaluated to O(A2 ) in Section 7.8. Hence, suppressing the fermion field variables, for a gauge-invariant operator G and gaugeinvariant state |Ψi, from Eqs. 11.2.1 and 11.3.9 YZ hΨ |G| Ψi = dUni Ψ∗ [U ]G[U, δ/δU ]Ψ[U ] n,i

=

YZ

" dVni

Y

δ[Xna (V )] Ψ∗ [V ]J 1/2 [V ] exp i

n,a

n,i

b δ/δV ]J −1/2 [V ]) × (J [V ]G[V, " " # # X exp i φan ρna J 1/2 [V ]Ψ[V ] 1/2

n

"

## X n

φan ρna

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Hence, the effective wave functional defined by absorbing the Jacobian is given by e ] = J 1/2 [V ]Ψ[V ] Ψ[V and the effective operator is " " # # X X 1/2 a a e G = J [V ] exp i φn ρna G exp −i φn ρna J −1/2 [V ] n

(11.3.11)

n

such that e G| e Ψi e hΨ |G| Ψi = hΨ| 11.4

(11.3.12)

Gauge-Fixed Lattice Gauge Field Hamiltonian

The kinetic operator is given from Eqs. 11.1.4, 11.1.12 and 11.1.13 by the following: K=

δ δ a iδ B a iδL Bni L ni

(11.4.1)

where all repeated indices are summed over. Symbolically write the transformation Eq. 11.2.19 as δ δ = Lpq δ L Bp δL Cq Then from Eqs. 11.4.1 and 11.4.2     δ 1 δ δ δ K = Lpq Lpq0 = LLTqp Lpq0 iδL Cq iδL Cq0 L iδL Cq iδL Cq0

(11.4.2)

(11.4.3)

where L = detkLab k

(11.4.4)

As for any curved space, the Laplacian for functions defined on the group space is given by 1 ∂ √ ab ∂  −∇2 = gg (11.4.5) g ∂B a ∂B b The transformation given by Eq. 11.2.19 yields L = J [V ]

(11.4.6)

and the Jacobian J is given by Eq. 11.3.10. The choice of operator ordering given by Eq. 11.4.3 allows for further simplifications. Recall that from Eq. 11.2.20 δ/δL Aani is ‘transverse’; using this equation and Eq. 11.4.3 yields     1 δ δ 1 δ δ a0 a K= J + + D 0 0 J iδL Aani iδL Aani J iδL φan iδL Aan0 i n ni       1  1 δ δ T T bb0  Γ Γ × J n, a  m, b + D (11.4.7) 0 0 mm k  Γ · D j j D T · ΓT  iδL φbm iδL Abm0 k

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The effective Hamiltonian, using (11.3.11), is given by e = J 1/2 eiφan ρna He−iφan ρna J −1/2 H Note that a

e−iφn ρna

a δ eiφn ρna = ρmb b iδL φm

Hence, the final expression for the gauge-fixed lattice Hamiltonian is given by    2 δ −1/2 e = + g J −1/2 δ H J J 2a iδL Aani iδL Aani         1 1 δ a0 a T   m, b D + J −1/2 − ρ J n, a Γ · Γ 0 na 0 Γ · D D T · ΓT  iδL Aan0 i n ni    δ T bb0 ¯ ζ, V ) (11.4.8) Dmm − ρmb J −1/2 + P (ζ, 0k 0 iδL Abm0 k Note that the redundant gauge degree of freedom Φn has completely decoupled from the Hamiltionian and state space. The quark-color charge ρna has the instantaneous non-local and non-Abelian lattice Coulomb potential given by (Γ · D)−1 Γ · ΓT (D T · ΓT )−1 The state functionals depend on only the constrained variables Vni , that is ¯ ζ, V ) e = Ψ( e ζ, Ψ Recall from Eq. 11.2.21 the commutation equation is given by          T 1 δ b   Γj m, c eL ,A = δnm δij δac − n, a Γi cb (Amj ) δL Aani mj Γ · ΓT 

(11.4.9)

(11.4.10)

Eqs. 11.4.8, 11.4.9 and 11.4.10 completely define the gauge-fixed Hamiltonian for the SU (N ) lattice gauge field. The redundant gauge degrees of freedom {Φn } have been completely decoupled from the system, as expected. e in Eq. 11.4.8 is exact, and is equally valid for strong The expression for H and weak couplings. Comparing Eqs. 11.4.1 and 11.4.7, shows that the coordinates {Uni } are analogous to Cartesian coordinates for the gauge field whereas coordinates {Vni } are analogous to curvilinear coordinates. 11.5

Hamiltonian and Covariant Gauge: Faddeev-Popov Quantization

The Hamiltonian for the lattice gauge field was discussed in Section 11.2 using the Coulomb gauge – which is manifestly non-covariant. The Hamiltonian can be chosen for a covariant gauge, but this can lead to non-physical state vectors that have zero and even negative norm, and which is not allowed by the Born interpretation of the state vector representing probabilities.

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The essential ideas of the Faddeev-Popov formalism are discussed for the simpler case of the photon field. For the Abelian gauge field, the Gupta-Bleuler procedure allows for a covariant gauge-fixing of the Hamiltonian, and is based on constraining state space so as to avoid the non-physical states. The Gupta-Bleuler procedure cannot be applied to non-Abelian gauge fields. However, with the introduction of the Faddeev-Popov ghost fields, a covariant formulation of the Hamiltonian and state space can be defined using the Faddeev-Popov quantization that generalizes to the non-Abelian case. The formalism is based on BRST cohomology and discussed below. All calculations are carried out for the Abelian gauge field defined on Euclidean spacetime. Only the Abelian gauge field is discussed as it illustrates the key features of the covariant state space. The continuum formulation is used due to its notational simplicity, and because the lattice generalization can be read off from the continuum derivation. The discussion follows the derivations given in Baaquie (2018). The full power of the Faddeev-Popov formulation comes to the fore in the study of Yang-Mills non-Abelian gauge fields. The gauge-fixed action, with a covariant gauge-fixing term −α(∂µ Aµ )2 /2, from Eq. 7.7.7, is given by Z Z Z α 1 4 2 4 2 d xFµν − d x(∂µ Aµ ) + d4 x∂µ c¯∂µ c SGF = − 4 2 = S + Sα + SF P (11.5.1) The path integral for the gauge-fixed Abelian theory is given by4 Z Z = DAµ D¯ cDc exp{SGF } 11.6

Ghost State Space and Hamiltonian

The state space of the quantum field is determined by the time derivative terms in the action; the reason being that the time derivative couples the gauge field at two different instants and at each instant, the gauge field is a coordinate of the underlying state space. The ghost action is similar to the action for the complex scalar field [Baaquie (2018)]; the state space has the fermion coordinate eigenstates and is given by Y |¯ c, ci = |¯ ci ⊗ |ci ≡ |¯ c(~x)i ⊗ |c(~x)i ~ x

The completeness equation is given by Z D¯ cDc|¯ c, cih¯ c, c| = I The inner product is given by h¯ c, c|¯ c0 , c0 i = δ(¯ c − c¯0 )δ(c − c0 ) 4 Fermion

integration is reviewed in Chapter 6.

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The rules of fermion calculus yield δ(¯ c − c¯0 )δ(c − c0 ) = −(¯ c − c¯0 )(c − c0 )

(11.6.1)

To prove above statement, consider an arbitrary function f (¯ c, c) of c¯, c with the following Taylor expansion f (¯ c, c) = α + β¯ c + γc + ω¯ cc Using the rules of fermion integration given in Section 6.2 yields Z Z 0 0 d¯ cdcf (¯ c, c)(¯ c − c¯ )(c − c ) = d¯ cdc (α¯ cc − β¯ cc¯0 c − γc¯ c0 + ω¯ cc¯ c0 c0 + · · · ) Z = d¯ cdcc¯ c (α + β¯ c0 + γc0 + ω¯ c0 c0 + · · · ) = −f (¯ c0 , c0 ) where · · · refers to terms that go to zero. Hence Eq. 11.6.1 has been verified. In general, for N complex fermions, one has N Y

N Y

δ(¯ cn − c¯0n )δ(cn − c0n ) = (−1)N

n=1

(¯ cn − c¯0n )(cn − c0n )

n=1

The gauge-fixed action in the Feynman gauge, with α = 1, from Eq. 11.5.1 is given by Z Z 1 d4 x(∂µ Aν )2 + d4 x∂µ c¯∂µ c (11.6.2) SGF = − 2 In the covariant gauge, the gauge field state space requires all four components of the gauge field. The completeness equation is given by 3 YZ 3 Z Y Y I= dAµ (~x)|Aµ (~x)ihAµ (~x)| ≡ DAµ |Aµ ihAµ | µ=0 ~ x

µ=0

The completeness equation for the gauge plus ghost field is given by Z Y I = DAµ D¯ cDc |¯ c, c; Aµ ih¯ c, c; Aµ |

(11.6.3)

µ

The state space of the gauge and ghost field system is given by a tensor product of the two state spaces V = VA ⊗Vc¯,c The connection of the state space and action to the Hamiltonian is similar to the case of the gauge-field given in Eq. 11.1.1. The Dirac-Feynman formula for Euclidean time is given by 0

0

0

h¯ c, c; Aµ |e−H |¯ c0 , c0 ; A0µ i = N ()eL(¯c,¯c ,c,c ;Aµ )

(11.6.4)

where N () is a normalization. The action consists of two decoupled free fields; the Hamiltonian is the sum of the gauge field and ghost Hamiltonians and given by H = HA ⊕ HG

(11.6.5)

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The Hamiltonian of the continuum gauge field is obtained from Eq. 11.1.1 and, for α = 1, is given by (index i summed over) Z Z δ2 1 1 3 d x 2 + d3 x(∂i Aν (~x))2 ; i = 1, 2, 3 HA = − 2 δAµ (~x) 2 Z 1 δ = d3 x[πµ2 + (∂i Aν )2 ] ; πµ (~x) = −i (11.6.6) 2 δAµ (~x) To obtain the ghost field Hamiltonian, note that the ghost Lagrangian density, from Eq. 11.6.2, is given by 1 (¯ c − c¯0 )(c − c0 ) + ∂i c¯∂i c (11.6.7)  Fermion calculus yields the following identity n n δ2 o δ2 o 0 0 |¯ c , c i = exp −  δ(¯ c − c¯0 )δ(c − c0 ) h¯ c, c| exp −  δ¯ cδc δ¯ cδc    δ2  (¯ c − c¯0 )(c − c0 ) = − (¯ c − c¯0 )(c − c0 ) +  =− 1− δ¯ cδc n1 o = − exp (¯ c − c¯0 )(c − c0 ) (11.6.8)  Note the rather unfamiliar result that the factor of  is inverted in the exponent since the  term appears in the numerator on the right hand side, unlike the case for real variables where Gaussian integration results in the  factor being in the denominator. Similar to Eq. 11.1.1, the ghost field Hamiltonian is given, up to irrelevant constants, by the Dirac-Feynman formula LG =

exp{−HG }h¯ c, c|¯ c0 , c0 i = h¯ c, c| exp{−HG }|¯ c0 , c0 i = exp{LG }

(11.6.9)

Hence, from Eqs. 11.6.7, 11.6.8 and 11.6.9, the ghost field Hamiltonian is given by Z i h δ2 − ∂i c¯(~x)∂i c(~x) (11.6.10) HG = d 3 x δ¯ c(~x)δc(~x) 11.6.1

BRST cohomology: state space

The Coulomb gauge, discussed in Section 11.2, is manifestly noncovariant and explicitly breaks Lorentz invariance. The gauge condition is A0 = 0 supplemented by imposing the transversality condition on the state space given by ∂i Ai = 0. The normal mode method of gauge-fixing also breaks Lorentz invariance since only the space components of the gauge field appear in the normal mode expansion, with the time component of the gauge field removed by gauge-fixing. One can also obtain a state space description of the photon field using a covariant gauge that respects Lorentz symmetry. For example, consider the Lorentz gauge defined by ∂µ Aµ = 0; one can consistently quantize the photon field using the Gupta-Bleuler formalism. However, as mentioned earlier the Gupta-Bleuler approach cannot be generalized to Yang-Mills non-Abelian gauge fields.

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The BRST method of quantization is a modern formulation of the Gupta-Bleuler approach that is equally valid for the Yang-Mills fields. The ghost fields are used to remove the extra degrees of freedom for the case of a covariant (Lorentz invariant) gauge, and this formalism can be generalized to the lattice gauge field, as in Chapter 7. The BRST method has the following ingredients. The gauge-fixed action has a BRST symmetry with a fermionic BRST charge operator QB . The BRST conserved charge obeys [H, QB ] = 0, where H is the Hamiltonian of the gauge fixed theory given in Eq. 11.6.5. The state space is enlarged to include the ghost field VA ⊗ Vc¯,c with completeness on this state space given by Eq. 11.6.3. The physical state space is a subspace of VA ⊗ Vc¯,c and is defined for |Φi ∈ VA ⊗ Vc¯,c . The physical gauge invariant states |Φi obey further conditions of BRST cohomology, given below, that defines the physical gauge invariant state subspace. • The operator QB is nilpotent since Q2B = 0. • The exact states are of the form QB |χi that are automatically annihilated by QB . These exact states are states that correspond to pure gauge transformations such that Aµ = ∂µ φ. • Physical states |Φi are constrained to be annihilated by QB and are said to closed under QB and obey QB |Φi = 0. Since charge QB is conserved, the constraint is conserved over time. • Physical states |Φi are not exact, that is |Φi = 6 QB |χi. • States that can be written as |Φi + QB |χi are all equivalent to |Φi and can be shown to differ by only a gauge transformation. • Physical states |Φi are precisely the gauge invariant states of the photon field. The BRST formalism is shown in Section 11.8.1 to reduce to the Gupta-Bleuler scheme for the case of the Abelian gauge field.

11.7

BRST Charge QB

The gauge fixed action given in Eq. 11.6.2 is BRST invariant, which is the result of choosing a gauge and its compensating Faddeev-Popov counter term. The Noether current due to BRST invariance yields the BRST charge QB [Das (2006), ZinnJustin (1993)]. It is convenient to write the gauge-fixed action in terms of an auxiliary field as this simplifies the derivation. From Eq. 11.6.2, in the Feynman gauge with α = 1, the gauge fixed action

SGF

1 =− 4

Z d

4

2 xFµν

1 − 2

Z

4

2

d x(∂µ Aµ ) +

Z

d4 x∂µ c¯∂µ c

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is re-written using an auxiliaryZ field G as followsZ ˜ eSGF = DGeS ; S˜ = d4 xL˜ 1 1 2 − G2 + i∂µ GAµ + ∂µ c¯∂µ c L˜ = − Fµν 4 2 Consider a fermionic parameter λ such that

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(11.7.1) (11.7.2)

λ2 = 0 ; {λ, c¯} = 0 = {λ, c} The BRST transformation consists of an infinitesimal gauge transformation for Aµ together with the following transformation of the ghost and auxiliary fields δAµ = λ∂µ c ; δ¯ c = −iλG ; δc = 0 = δG

(11.7.3)

Note one can define a BRST transformation with δ¯ c = 0 and δc 6= 0; this gives rise to the same BRST charge. The BRST variation of Eq. 11.7.1 is given byZ Z δ S˜ = iλ

∂µ G∂µ c + iλ

G∂ 2 c = 0

BRST invariance yields a conserved fermionic charge QB using the Noether current. The BRST current is given by the varying both Aµ and c¯, c in the Lagrangian. Using  1  ∂ 2 − Fµν = Fνµ = −Fµν ∂(∂µ Aν ) 4 the variation of the Lagrangian in Eq. 11.7.1 under the BRST symmetry given in Eq. 11.7.3 yields ∂L ∂L jµ = δAν + δ¯ c = λ∂ν cFνµ − iλG∂µ c (11.7.4) ∂(∂µ Aν ) ∂(∂µ c¯) Since the BRST current is conserved ∂µ jµ = 0, dropping the parameter λ yields the conserved BRST charge Z Z d3 xj0 =

QB = Z =

d3 x (∂i cFi0 − iG∂0 c)

d3 x (−c∂i Fi0 − iG∂0 c)

The Euler-Lagrange equation for Aµ   ∂L ∂L = ∂µ ∂(∂µ Aν ) ∂Aν leads to the following Z QB =

⇒ ∂µ Fνµ = i∂ν G

d3 x (ic∂0 G − iG∂0 c)

The Euler-Lagrange equation for G   ∂L ∂L = ⇒ i∂µ Aµ = −G ∂µ ∂(∂µ G) ∂G yields the final expression Z for BRST charge  QB = d3 x c(~x)∂0 (∂µ Aµ (~x)) − (∂µ Aµ (~x))∂0 c(~x)

(11.7.5)

Note that all operators are considered to be time dependent Heisenberg operators and the time derivatives are taken on the Heisenberg operators.

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QB and State Space

The conserved charge is an operator QB that acts on the state space of the gauge fixed action. To obtain an explicit representation of the operator QB the operator representation of the quantum fields Aµ and c¯, c are written in terms of creation and destruction operators. Since the auxiliary field has been removed in Eq. 11.7.5, QB follows from the gauge fixed action given by Eq. 11.5.1; fixing α = 1 yields Z Z 1X 4 2 d x(∂µ Aν ) + d4 x∂µ c¯∂µ c (11.8.1) SGF = − 2 µ,ν The gauge field consists of four free scalar fields and the ghost field is the fermionic version of the free complex scalar field. Hence we can use the results of free fields with some minor modifications. The normal mode expansion in Euclidean time t = −iτ of the gauge field, similar to the case for the Euclidean free field, is the following [Baaquie (2018)] Z Z Z d3 p 1 p (11.8.2) (e−Ep~ τ +i~p·~x ap~µ + eEp~ τ −i~p·~x a†p~µ ) ; = Aµ (τ, ~x) = (2π)3 2Ep~ p ~ p ~ and where Ep~ = |~ p|

(11.8.3)

The equal time canonical commutation equations yields [ap~µ , a~† ] = (2π)3 δ 3 (~ p − ~k)δµ−ν kν

with the rest of the commutators being zero. The ghost field are similar to the complex scalar fields and one has the expansion for Euclidean spacetime given by Z 1 p (e−Ep~ τ +i~p·~x Ap~ + eEp~ τ −i~p·~x Bp~† ) c(τ, ~x) = 2Ep~ p ~ Z 1 p c¯(τ, ~x) = (e−Ep~ τ +i~p·~x Bp~ + eEp~ τ −i~p·~x A†p~ ) (11.8.4) 2Ep~ p ~ The Fourier expansion of c, c¯ given in Eq. 11.8.4 above is for Euclidean spacetime. Under charge conjugation Ap~ ↔ Bp~ ⇒ c ↔ c¯ The equal time canonical anticommutation equations n ∂c(τ, ~x) o n ∂¯ o c(τ, ~x) , c(τ, ~x0 ) = δ 3 (~x − ~x0 ) = , c¯(τ, x~0 ) ∂τ ∂τ yield {Ap~ , A†p~0 } = {Bp~ , Bp†~0 } = δ 3 (~ p − p~0 ) ; {Ap~ , Ap~0 } = 0 = {Ap~ , Bp†~0 } = 0 = {Bp~ , Bp~0 } and with all the other anticommutators being zero.

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In the Feynman gauge, the action as given in Eq. 11.8.1 for the gauge field is equivalent to four decoupled scalar fields. Hence, the gauge field Hamiltonian, in Euclidean time (t = −iτ ), as given in Eq. 11.6.6, yields Z Z 3 Z i 1 X h  ∂Aµ (τ, ~x) 2 2 ~ ; HA = + (∇Aµ (τ, ~x)) = d3 x (11.8.5) − 2 µ=0 ~x ∂τ ~ x The ghost Hamiltonian is similar to the complex scalar field and given by Z h i δ2 − ∂i c¯(~x)∂i c(~x) HG = δ¯ c(~x)δc(~x) Z~x h i = ∂0 c¯(τ, ~x)∂0 c(τ, ~x) − ∂i c¯(~x)∂i c(~x) (11.8.6) ~ x

The signs of the kinetic and potential term of HG are fixed by the rules of fermion integration. Note the differing signs for HA , HG for the kinetic and potential terms; this is the reason that the zero point energy of the bosonic and fermionic fields have opposite signs.5 Substituting the Fourier expansion for Aµ , c¯, c given in Eqs. 11.8.2 and 11.8.4 yields the following H = HA + HG Z h i = Ep~ a†p~µ ap~µ + Ap†~ Ap~ + Bp~† Bp~ + E0

(11.8.7)

p ~

Substituting Eqs. 11.8.2 and 11.8.4 into the expression for QB , given in Eq. 11.7.5, and after some algebra, yields the conserved (time independent) BRST charge operator Z 3 Z X † † QB = − |~ p |(ap~0 Ap~ + ap~0 Bp~ ) + i pi (a†p~i Ap~ + ap~i Bp~† ) (11.8.8) p ~

p ~

i=1

To simplify the notation define p0 ≡ i|~ p|

(11.8.9)

Note Eq. 11.8.9 is what one expects from the continuation of Minkowski to Euclidean time. The BRST charge can be written in the following invariant form 3 Z   X (11.8.10) QB = i pµ a†p~µ Ap~ + ap~µ Bp~† µ=0

p ~

To show that QB is conserved, consider the following. Z [QB , HA ] = i (−pµ ap†~µ Ap~µ + pµ ap~µ Bp~† ) p ~ Z [QB , HG ] = i (pµ a†p~µ Ap~µ − pµ ap~µ Bp~† ) = −[QB , HA ] p ~

5 Hence

E0 = ( 12 · 4 − 1)(2π)3 δ (3) (0)

R p ~

Ep~ = (2π)3 δ (3) (0)

R p ~

Ep~ .

(11.8.11) (11.8.12)

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From above and Eq. 11.8.7 [QB , H] = [QB , HA + HG ] = 0 and hence we have confirmed that for the quantized theory BRST charge is conserved. BRST charge is not a Hermitian operator, with its Hermitian conjugate given by Q†B

Z =−

|~ p p ~

|(ap~0 A†p~

+

a†p~0 Bp~ )

−i

3 Z X i=1

p ~

pi (ap~i A†p~ + a†p~i Bp~ )

The significance of Q†B is discussed by Malik (2001). To verify that Q2B = 0, using the anticommuting property of Ap~ , Bp~† yields the following X Z XZ Q2B = i2 pµ kν [ap~µ , a~† ]Bp~† A~k = i2 pµ pµ Bp~† Ap~ (11.8.13) µ,ν=0

kν

p ~,~ k

µ=0

p ~

But from Eq. 11.8.9 pµ pµ = p20 +

3 X

pi pi = −|~ p |2 + (~ p )2 = 0

i=1

Hence, from Eq. 11.8.13 and above Q2B = 0 We have the important result that as an operator QB is nilpotent, namely that Q2B = 0 is an operator equation that is valid both on-shell and off-shell. As discussed in Section 11.6.1, every vector |Φi in the physical gauge invariant state space is annihilated by QB , that is (the physical state is not exact) QB |Φi = 0 11.8.1

(11.8.14)

Gupta-Bleuler condition

It is shown how the definition of state space given by BRST quantization reduces to the Gupta-Bleurel constraint on state space when the gauge field state space is considered by itself, without the presence of the ghost field. From Eqs. 11.8.13 and 11.8.14 Z 0 = QB |Φi = i (pµ a†p~µ Ap~ + pµ ap~µ Bp~† )|Φi p ~

Since the gauge and ghost field are decoupled, the physical state vector is taken to be a tensor product |Φi = |ΦA i|ΦG i

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Ghost number is conserved and |ΦG i is taken to have zero ghost number. From the expression for QB given in Eq. 11.8.13, the ghost state is taken to be the ground state |ΦG i = |ΩG i that is annihilated by Ap~ , Bp~ , defined by6 Ap~ |ΩG i = 0 = Bp~ |ΩG i The BRST constraint reduces to Z QB |Φi = −i p ~

pµ ap~µ |ΦA iBp~† |ΩG i = 0

Since Bp~† |ΩG i = 6 0, to achieve QB |Φi = 0 the following constraint is imposed on the gauge field state vectors QB |Φi = 0 ⇒ pµ ap~µ |ΦA i = 0 This constraint can be written more transparently in real space; from Eq. 11.8.2 ∂µ Aµ (~x) = ∂µ A(−) x) + ∂µ A(+) x) µ (~ µ (~ (+)

(−)

where Aµ has all the annihilation operators and Aµ The BRST constraint yields the following

has all the creation operators.

pµ ap~µ |ΦA i = 0 ⇒ ∂µ A(−) x)|ΦA i = 0 : Gupta-Bleuler condition µ (~ Note that, since hΦA |∂µ A(+) =0 µ the Gupta-Bleuler condition implies that for any physical state |ΦA i   (−) hΦA |∂µ Aµ |ΦA i = hΦA | ∂µ A(+) + ∂ A |ΦA i = 0 µ µ µ

(11.8.15)

In other words, instead of imposing the covariant gauge condition of the operator equation ∂µ Aµ = 0, in the Gupta-Bleuler approach the weaker condition is imposed, as in Eq. 11.8.15, that the expectation value of the operator ∂µ Aµ for any gauge invariant (physical) state has to be zero. In conclusion, the physical gauge invariant state space for both the Abelian and non-Abelian gauge fields, is given by the constraint QB |Φi = 0 ; |Φi = |ΦA i|ΩG i The physics of the BRST constraint on state space for non-Abelian gauge fields has been discussed in Peskin and Schroeder (1995). 6 It

can be verified by using the ghost Hamiltonian HG given in Eq. 11.8.6 that n Z d3 p o HG |ΩG i = 0 ⇒ h¯ c, c|ΩG i = N exp − |~ p |¯ c(~ p )c(~ p) 3 (2π)

where c¯(~ p ); c(~ p ) are the Fourier transform of the coordinate basis c¯(~ x); c(~ x).

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Summary

The two underpinnings of the lattice gauge field – namely that it is defined directly using the finite group element of the gauge group and secondly that it has a lattice cutoff – introduce many new features that are absent in the continuum formulation. For the case of the Hamiltonian, the finite group element leads to the chromoelectric field operators being differential operators on the Lie group manifold. Gauge-fixing necessitates the removal of the extra gauge degrees of freedom from the kinetic operator. The gauge-constraint (Coulomb gauge) requires constraining the differential operators and requires properties of the lattice chromo-electric field operator that do not arise in the continuum formulation. In particular, the change of variables eliminating the redundant gauge degree of freedom from the chromoelectric operator depends on the structure of the Lie group space – and which is unlike the continuum case where only the Lie algebra is required for gauge-fixing. The BRST operator provides an exactly conserved charge that was used for choosing a covariant gauge for the gauge-field Hamiltonian. The closed but not exact states of BRST cohomology precisely pick out the all gauge-invariant states: this procedure works for both Abelian and non-Abelian gauge fields. The case of the Abelian gauge field was exactly solved, and the computation was carried out in continuum spacetime notation. The notation for the continuum case is more convenient than the lattice case, and the continuum formulation – for the Abelian gauge field – can be carried over to the lattice case by inspection.

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Chapter 12

Dirac Lattice Path Integral

The Dirac field, which is a particular case of a field based on fermionic variables, was first proposed by P. A. M. Dirac in 1928 to describe the relativistic electron in four-dimensional spacetime [Dirac (1999)]. Every electron has spin 1/2 and hence needs two independent fermionic variables for its description; its antiparticle, namely the positron is also spin 1/2 and needs another two independent fermionic variables. Hence in total a Dirac fermion needs four fermionic degrees of freedom and is consequently described by a four-component fermionic variable, and which transforms as a spinor under Lorentz transformation. The electron carries electric charge and hence can couple to the Abelian gauge field. The Dirac field has many realizations, with particular varieties of the Dirac field being realized in weak interactions as leptons, and in strong interactions by quarks – both of which are described by Dirac fields that carry non-Abelian charges [Baaquie and Willeboordse (2015)]. The Dirac field is sometimes referred to as a quark field when one is emphasizing its non-Abelian nature. This Chapter is based on the results obtained in Baaquie (1983b). The Dirac field is studied using lattice fermions introduced by Wilson (1974) and further elaborated by Ginsparg and Wilson (1982). In particular, the lattice is used for calculating the evolution kernel of the Dirac field. The continuum limit is taken, and the complete energy eigenfunctionals of the free Dirac field – as well its propagator – are then evaluated using the kernel. 12.1

Introduction

The time evolution kernel, or the probability amplitude, is defined for quantum fields by the matrix elements of exp(−itH/~) between arbitrary initial and final field configurations, where t is the physical time and H the Hamiltonian operator. In Euclidean time, the probability amplitude is given by exp(−τ H). The kernel contains the complete content of the quantum field theory. For the free boson field, the kernel has been evaluated by Feynman and Hibbs (1965). A similar calculation is carried out for the free Dirac field in Baaquie (1983b) There are a number of problems with the fermions which are not present in 211

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the boson case. Note that for the free boson field, the evolution kernel is given by the finite time classical action – with the appropriate boundary conditions on the classical field. However, in the case of the Dirac field the classical Dirac Lagrangian is zero, giving an incorrect result for the kernel. The result stems from the ambiguity in incorporating the boundary conditions for the finite time Dirac action and this short-coming is resolved using the lattice formulation that results in a boundary term for the finite time continuum action. Furthermore, the lattice formulation shows that the appropriate boundary conditions for the continuum Dirac field are the same as for the lattice theory. The ambiguities of the continuum fermion path integral for the Dirac field have been noted by many authors [Creutz (1983); Montvay and Munster (1994); Roth (1997)]. Wilson’s formulation of the lattice fermions incorporates the boundary values in an unambiguous manner and specifies a particular way of defining the fermion initial and final field configurations. With the hindsight gained from the lattice, a derivation is given in Section 12.2 for the coordinates of the Dirac field directly from the continuum formulation. 12.2

Dirac Field Coordinates

What are the field coordinates, namely the degrees of freedom, for the Dirac field’s state vector? For starters, consider the Dirac fermion field directly in the continuum formulation without recourse to the lattice. The Dirac equation was obtained by Dirac by starting from the field equation for a scalar field given by the Klein-Gordon equation and reducing the second derivatives to first derivatives [Baaquie (2018)]. The resulting field equation in Minkowski time is given by   ∂ p · ~r + m ψ = 0 iγ0 + i~ ∂t and the conjugate Dirac equation is   ∂ p · ~r + m = 0 ψ¯ − iγ0 − i~ ∂t ¯ ψ are the Dirac four-component spinors, m is the mass term and γµ where ψ, the gamma matrices. Boosting the spinors to the rest-frame results in setting the three-momentum p~ = 0 in above equations, and yields the classical solution given by     ∂ ∂ iγ0 + m ψc = 0 ; ψ¯c − iγ0 + m = 0 (12.2.1) ∂t ∂t Using the 2 × 2 block representation, and denoting the two component upper and lower doublet spinors u  by subscripts   and` yields
1 0 ψu γ0 = ; ψ= ; ψ¯ = ψ¯u , ψ¯` (12.2.2) 0 −1 ψ` Consider the following ansatz ψ¯c , ψc  eimt ψu (12.2.3) ψc = e−imt ψ`

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and ψ¯c = e−imt ψ¯u , eimt ψ¯`




(12.2.4)

It can be readily seen that the solutions given in Eqs. 12.2.3 and 12.2.4 satisfy the Dirac field equation given in Eq. 12.2.1. The time dependence of a state with energy E is given by e−iEt ; from Eqs. 12.2.3 and 12.2.4, the time evolution of the degrees of freedom is given by the following ψ¯u ; ψ` : evolving forward in time and ψ¯` ; ψu : evolving backward in time Hence, we conclude that • the field coordinates propagating forward in time are the conjugate coordinate eigenstates hψ¯u , ψ` | • and those propagating backward in time are the coordinate eigenstates |ψ¯` , ψu i The same conclusion is reached in Section 12.3 using the Wilson lattice fermions. Under spatial rotations, for the choice of the gamma matrices given in Eq. 12.3.2, the upper and lower components of ψ¯ and ψ transform independently. Hence, each of the upper and lower components ψ¯u , ψ` , provide a two-dimensional spin 1/2 spinor representation of the rotation group in three-dimensional space. 12.3

Dirac Lattice Lagrangian

The reasoning used by Dirac to discover his equation was discussed in Section 12.2. One can use a different line of reasoning developed in Chapter 6 to also arrive at the Dirac equation. Recall from Eq. 6.11.4, fermion calculus and the nature of the fermion and antifermion state space yields the Minkowski time Dirac Lagrangian in one spacetime dimension as given by   ∂ +m ψ LM = −ψ¯ iγ0M ∂tM Extending the time coordinate t to space and time t, xi ; i = 1, 2, 3 requires constraints be put on the variation of the field variables in the space direction, which is realized by having derivative in the space direction, with their own Gamma matrices. Requiring a Lorentz invariant Lagrangian yields   ∂ ∂ LM (x) = −ψ¯ iγ0M − iγiM + m ψ ; i = 1, 2, 3 ∂tM ∂xi M µ ¯ = −ψ(iγµ ∂ + m)ψ ; µ = 0, 1, 2, 3 (12.3.1)

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¯ where ψ(x) and ψ(x) are four-component anticommuting spinors, m is the mass; γµM are the Minkowski space gamma matrices that satisfy the following anticommutation equation {γµM , γνM } = ηµν I where I is a 4 × 4 unit matrix and ηµν is the Lorentz metric given by ηµν = diag(1, −1, −1, −1) Define for Euclidean time τ Euclidean gamma matrices γµ with the following representation (σi are the Pauli matrices)1     1 0 0 −σi t = −iτ, γ0M = γ0 = , −iγiM = γi = i (12.3.2) 0 −1 σi 0 and the Hermitian gamma matrices have the following anticommutation equations {γµ , γν } = δµν I ; 㵆 = γµ Hence, from Eqs. 12.3.1 and 12.3.2 the Dirac Euclidean Lagrangian is given by   ∂
∂ ¯ + γi +m ψ (12.3.3) L(x) = −ψ(x) γµ ∂µ + m ψ(x) = −ψ¯ γ0 ∂τ ∂xi Re-writing the Dirac Lagrangian in a symmetric form (more appropriate for the lattice theory) yields the four-dimensional continuum spacetime action given by Z S = d4 xL(x)  1 ¯ ¯ ¯ ψ(x)γµ ∂µ ψ(x) − ∂µ ψ(x)γ (12.3.4) µ ψ(x) − mψ(x)ψ(x) 2 Discretize spacetime into a four-dimensional lattice, with x = na, where n is a four-dimensional lattice point and with lattice spacing a; let µ ˆ be a unit vector in the µ-direction. More precisely L(x) = −

x = na ; n = (n0 , n1 , n2 , n3 ) : lattice point µ ˆ : unit vector with µ = 0, 1, 2, 3 Using an approximation for the first derivative yields the naive lattice action of the form X S= L˜n (12.3.5) n

¯ L˜n = ma4 ψ(na)ψ(na) o 1 Xn¯ ¯ ¯ −a4 ψ(na)γµ [ψ(na + µ ˆa) − ψ(na)] − [ψ(na +µ ˆa) − ψ(na)]γ µ ψ(na) 2a µ o a3 X n ¯ ¯ ¯ = ma4 ψ(na)ψ(na) − ψ(na)γµ ψ(na + µ ˆa) − ψ(na +µ ˆa)γµ ψ(na) 2 µ 1 Dirac

matrices have infinitely many allowed representations.

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Define dimensionless lattice fermion field variables ψ¯n , ψn by the following (the constant K is defined for simplifying the final result) 2K = (4 + ma)−1  3 1/2 a ψn = ψ(na) 2K  3 1/2 a ¯ ψ¯n = ψ(na) 2K From Eq. 12.3.5, the naive lattice Lagrangian is given by o Xn L˜n = −2Kmaψ¯n ψn + K ψ¯n+ˆµ γµ ψn − ψ¯n γµ ψn+ˆµ

(12.3.6)

(12.3.7)

µ

For zero space and one discrete time dimension, the following Dirac lattice Lagrangian, given in Eq. 6.11.3, was obtained from the underlying fermionic Hilbert space  1 ψ¯n+ˆ0 (1 + γ0 )ψn + ψ¯n (1 − γ0 )ψn+ˆ0 : d = 1 (12.3.8) −ψ¯n ψn + 2(1 + ma) One expects to recover the expression given in Eq. 12.3.8 as the one discrete time dimension limit of the four-dimensional Dirac lattice Lagrangian. To have fourdimensional hyper-cubic symmetry for the lattice Euclidean Lagrangian consistent with Eq. 12.3.8 entails adding the following term to the naive lattice Lagrangian given in Eq. 12.3.7 o Xn K ψ¯n+ˆµ ψn + ψ¯n ψn+ˆµ (12.3.9) µ

Note that, in the a → 0 limit, this term yields an extra contribution to the mass term of the continuum Lagrangian that, from Eq. 12.3.9 – to leading order in the lattice spacing a – is given by 3 n o X K ψ¯n+ˆµ ψn + ψ¯n ψn+ˆµ = 8K ψ¯n ψn + O(a2 ) (12.3.10) µ=0

Adding Eq. 12.3.10 to the naive Lagrangian given in Eq. 12.3.7 does not change the continuum limit. Hence, define the Dirac lattice Lagrangian by the following 3 n o X ˜ Ln = Ln + K ψ¯n+ˆµ ψn + ψ¯n ψn+ˆµ − 8K ψ¯n ψn (12.3.11) µ=0

Using Eq. 12.3.7 for L˜n and with 8K + 2Kma = 1 ⇒ 2K =

1 4 + ma

yields the Dirac lattice Lagrangian given by 3 n o X Ln = −ψ¯n ψn + K ψ¯n (1 − γµ )ψn+ˆµ + ψ¯n+ˆµ (1 + γµ )ψn µ=0

(12.3.12)

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The lattice Dirac field is written in terms of Wilson fermions that consist of introducing the projecting operators 1 ± γµ for defining the nearest neighbor coupling of the Dirac field. The four-dimensional Dirac lattice Lagrangian given in Eq. 12.3.12 has the requisite hyper-cubic symmetry that reduces to the one-dimensional time lattice result given in Eq. 12.3.8; in particular, the hyper-cubic symmetry ensures that the fermionic degrees of freedom that propagate are the ones required by the Lagrangian in Eq. 12.3.8. Although the lattice Lagrangian L in Eq. 12.3.12 has the same a → 0 limit as the earlier naive Lagrangian L˜ given in Eq. 12.3.7, the full a 6= 0 lattice theories for the two Lagrangians are radically different. The lattice Lagrangian L allows the propagation of the correct four degrees of freedom, namely the two spin states each for the particle and the antiparticle as can be seen by the argument that led to the Lagrangian given in Eq. 12.3.8; in the continuum limit, Lagrangian L yields the correct state space for the spin 1/2 Dirac particle. In contrast, there are eight propagating degrees of freedom for the naive La˜ the reason being that the projection operators 1 ± γ0 , which project grangian L; ˜ If one takes conout two degrees of freedom for every step in time, are absent in L. ˜ one obtains the incorrect limit of tinuum limit of the states space generated by L, two distinct species of spin 1/2 Dirac particles for the quantum spectrum of states. Note that the time lattice spacing has been taken to be equal to the spatial lattice spacing, and the symmetric limit of a → 0 is taken for obtaining the continuum limit. One could have equivalently started with continuous three-space and a time lattice, and which is required for obtaining the continuum Dirac Hamiltonian. The symmetric hyper-cubic symmetric spacetime lattice is more appropriate for the path integral formulation. 12.4

Lattice Fermions and Chiral Symmetry

Massless fermions have chiral symmetry. In the Standard Model of particle physics, the neutrino is massless and so it is necessary to have a discretization scheme for the fermions that respects chiral symmetry. An excellent summary of chiral symmetry for lattice fermions is given by Chandrasekharan and Wiese (2004). In this Section, the calculation is done in Euclidean spacetime since it provides for a well-defined path integral. The inversion of spacetime, from x → −x is realized for fermions by the transformation ψ(x) → ψ(−x) = γ5 ψ(x) The gamma five matrix γ5 in Euclidean space is given by γ5 = γ0 γ1 γ2 γ3 ⇒ {γµ , γ5 } = 0 Chiral symmetry is defined by local rotations of the fermion fields and given by ¯ iαγ5 ψ → eiαγ5 ψ ; ψ¯ → ψe

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An action having chiral symmetry yields ¯ iαγ5 , eiαγ5 ψ] = S[ψ, ¯ ψ] S[ψe A mass term in the lattice action, under a chiral transformation, yields the following ψ¯n ψn → ψ¯n e2iαγ5 ψn 6= ψ¯n ψn and breaks chiral symmetry. By this criterion, Wilson fermions break chiral symmetry since the Lagrangian given in Eq. 12.3.12 has a mass-like terms. To obtain chiral symmetry starting with Wilson fermions requires fine-tuning the mass-term in the lattice action, and is not very suitable for numerical simulations. Since one is interested in lattice theories that yield chiral symmetry in the continuum limit, one can ask a separate question: which lattice theories have a (hidden) chiral symmetry coming from the continuum action? In a landmark paper, Ginsparg and Wilson (1982) provided a definition of chiral symmetry for lattice fermions that are obtained by a block-spin renormalization group transformation, starting from continuum fermions having an exact chiral symmetry. It was later realized that the Ginsparg-Wilson lattice fermions define what is termed as perfect lattice fermions: fermions that arise from the block-spin of continuum chiral invariant fermions, and hence do not have any artifacts of the lattice – although the continuum results might not be apparent in the lattice.2 Consider d-dimensional Euclidean spacetime with chiral fermions χ, χ ¯ having an action SI (χ, χ) ¯ that is invariant under chiral rotations SI (eiαγ5 χ, χe ¯ iαγ5 ) = SI (χ, χ) ¯ The continuum fermions χ, χ ¯ are combined into ‘block-spin’ fermions χn , which are defined on a discrete spacetime lattice, denoted by lattice points n. The block-spin fermions are obtained by an averaging procedure over a hypercube E(n), which is centered on n, and weighted by a density function ρ(x). The block-spin fermions are given by the following Z Z χn ≡ dd xρ(x)χ(x) ; χ ¯n ≡ dd xρ∗ (x)χ(x) ¯ E(n)

E(n)

with X

⊕E(n) = Rd

n

Let the fermions on the lattice be given by ψn , ψ¯n with action for lattice fermions ¯ the block-spin renormalization group transformation yields the following be S(ψ, ψ); Z h i ¯ e−S(ψ,ψ) = DχDχ ¯ exp − (ψ¯m − χ ¯m )Amn (ψn − χn ) − SI (χ, χ) ¯ with sum over all repeated lattice indices. Take Amn = aδm−n for simplicity as it does not change the result. Hence Z h i ¯ e−S(ψ,ψ) = DχDχ ¯ exp − a(ψ¯n − χ ¯n )(ψn − χn ) − SI (χ, χ) ¯ 2 Block-spin renormalization for gauge fields is discussed in Chapter 10; a simple one-dimensional case is discussed in Baaquie (2014).

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The kernel of the block-spin transformation h i exp − a(ψ¯n − χ ¯n )(ψn − χn ) explicitly breaks chiral symmetry as this is required for the renormalization group transformation to have a nonsingular fixed point [Ginsparg and Wilson (1982)]. Consider a global chiral transformation on the lattice ψn → eiγ5 ψn ; ψ¯n → ψ¯n eiγ5 Since SI (χ, χ) ¯ is invariant under the chiral transformation, one can do the change of fermion integration variables χn → eiγ5 χn ; χ ¯n → χ ¯n eiγ5 and this yields ¯ iγ5 )} exp{−S(eiγ5 ψ, ψe Z h i = DχDχ ¯ exp − a(ψ¯n − χ ¯n )e2iγ5 (ψn − χn ) − SI (χ, χ) ¯

(12.4.1)

¯ Suppose SI (χ, χ) ¯ is a quadratic function of χ, χ; ¯ this in turn implies that S(ψ, ψ) ¯ is a quadratic function of ψ, ψ given by ¯ = ψD ¯ −1 ψ ≡ ψ¯m D−1 ψn ; D−1 = −D−1 S(ψ, ψ) mn mn nm

(12.4.2)

¯ iγ5 ) = S(ψ, ψ) ¯ + iψ{D ¯ −1 , γ5 }ψ S(eiγ5 ψ, ψe

(12.4.3)

and

Note from above that D−1 breaks chiral symmetry since {D−1 , γ5 } 6= 0. The right hand side of Eq. 12.4.1, using fermion differentiation, yields Z h i DχDχ ¯ exp − a(ψ¯n − χ ¯n )(1 + 2iγ5 )(ψn − χn ) − SI (χ, χ) ¯ Z   h i = DχDχ ¯ 1−2ia(ψ¯n − χ ¯n )γ5 (ψn −χn ) exp − a(ψ¯n − χ ¯n )(ψn −χn )−SI (χ, χ) ¯  i ∂ ∂  ¯ γ5 ¯ e−S(ψ,ψ) = 1+2  a ∂ψ ∂ ψ

(12.4.4)

where the final result has been obtained using Tr(γ5 ) = 0. Furthermore, from Eq. 12.4.2 ∂ ∂ −S(ψ,ψ) ¯ ¯ ¯ −1 γ5 D−1 ψe−S(ψ,ψ) γ5 e = −ψD ∂ψ ∂ ψ¯ The left hand side of Eq. 12.4.1 yields, using Eq. 12.4.3   iγ5 ¯ iγ5 ¯ ¯ −1 , γ5 }ψ e−S(ψ,ψ) e−S(e ψ,ψe ) = 1 − iψ{D Hence, from Eqs. 12.4.5 and 12.4.6 2 ¯ −1 ¯ −1 , γ5 }ψ ψD γ5 D−1 ψ = ψ{D a

(12.4.5)

(12.4.6)

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and which yields the final result 2 Dγ5 + γ5 D = γ5 : Ginsparg-Wilson Relation a The Ginsparg-Wilson result provides a criterion for chiral lattice fermions that apparently break chiral symmetry – but which are guaranteed to yield the correct chiral symmetric continuum limit without the need for any fine tuning. Although it would seem that Ginsparg-Wilson criterion is difficult to obtain, Hasenfratz et al. (2001) have obtained classically perfect lattice gauge theories that obey the Ginsparg-Wilson criterion to a very high accuracy. Fermions obeying the GinspargWilson criterion, in principle, have non-local couplings in spacetime. Luscher (1999) further developed the Ginsparg-Wilson result by defining a modified chiral symmetry transformation for the lattice – which depends on the lattice gauge field – and that reduces to the continuum one for zero lattice spacing. Based on the modified definition of chiral transformations on a lattice, Luscher (2000) obtained a ground-breaking result in formulating an exact nonperturbative theory of lattice chiral gauge theories. 12.5

Dirac Field: Boundary Conditions

To analyze the Dirac field’s state space, and in particular, to calculate the evolution kernel, the boundary conditions for the Dirac lattice action need to be ascertained, and hence the action for finite (Euclidean) lattice time is analyzed. Consider a finite open lattice of N sites in the time direction and a periodic lattice of L3 sites for the spatial dimensions. We are essentially concerned with the coupling in the time direction. Hence, using the periodicity of the spatial lattice, define the Fourier transform in the spatial coordinates by the following L−1 3 h X Y 1 X 2πipi ni /L i e , p = (p1 , p2 , p3 ) (12.5.1) eip·n = L p =0 p i=1 i X X ψ(n,n0 ) = eip·n ψpn0 ; ψ¯(n,n0 ) = e−ip·n ψ¯pn0 (12.5.2) p

p

This gives for the Lagrangian (dropping the subscript 0 on the time coordinates so that n0 → n) n o L = −ψ¯pn Lψpn + K ψ¯pn (1 − γ0 )ψpn+1 + ψ¯pn+1 (1 + γ0 )ψpn (12.5.3) Using the scalar product only in the spatial indices yields   3 3 X X L = 1 − 2K cos pi + i2K γi sin pi i=1

i=1

= α + iγ · β where, from Eq. 12.3.2   3 X 0 σ iγ · β = i2K γi sin pi = 2K −σ 0 i=1

(12.5.4)

: σ=

3 X i=1

sin pi σi

(12.5.5)

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Define the upper and lower two-component spinors, denoted by subscript u and `, respectively, by the following   ψpnu ψpn = , ψpn = (ψ¯pnu , ψ¯pn` ) (12.5.6) ψpn` Note that from the definition of γ0 in (2.2), 12 (1 ± γ0 ) are the protection operators for the upper and lower components. The fermionic degrees of freedom, in the momentum representation, completely factorize, with no coupling between different momenta; henceforth the momentum index is completely suppressed and restored only when necessary. Hence, the coupling in the time direction of the Dirac lattice Lagrangian is given as follows L = −ψ¯n Lψn + 2K ψ¯n` ψn+1` + 2K ψ¯n+1u ψnu

(12.5.7)

The equation above shows that the Lagrangian propagates degrees of freedom (ψ¯n` , ψnu ) at time n to (ψ¯n+1u , ψn+1` ) at time n + 1. It follows from Eq. 12.5.7 that the boundary fermionic coordinates values at time n = 0 is (ψ¯0` , ψ0u ), and the other remaining fermionic coordinates, namely (ψ¯0u , ψ0` ), do not couple to the next time instant at n = 1 and hence are decoupled from the lattice Dirac field. Similarly, the final boundary fermionic coordinates at final time n = N are (ψ¯N u , ψN ` ) and the other remaining fermionic coordinates at n + N , namely (ψ¯N ` , ψN u ) do not couple to earlier times and decouple from the lattice Dirac action.3 Hence choose as the initial fermion field configuration the coordinate eigenstate tensor product defined on the spatial lattice Y ⊗|ψ¯(n,0,`) , ψ(n,0,u) i ≡ |ψ¯` , ψu i : initial coordinate state vector (12.5.8) n

Similarly, choose for the final fermion field configuration the conjugate coordinate eigenstate tensor product Y ⊗hψ¯(n,N,u) , ψ(n,N,`) | ≡ hψ¯u , ψ` | : final dual coordinate state vector (12.5.9) n

The boundary values of the Dirac field are shown in Figure 12.1; for every point in space, the lattice the boundary values are as given in the Figure 12.1. 12.6

Dirac Fermionic State Space

The results of fermion calculus discussed in Chapter 6 are generalized to the case of the Dirac field. Eqs. 12.5.8 and 12.5.9 show that the coordinate basis for the state of the Dirac field factorizes into a tensor product over the lattice sites. Hence, one can analyze each lattice site one by one and compose the field’s basis states using a tensor product and for this reason the index for the lattice site is suppressed. 3 A derivation of the boundary conditions for the Dirac field in the continuum time theory is given in Section 12.2.

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ψ u, ψl

…

n=N

ψl, ψu

n=0 Fig. 12.1

The boundary values of the Dirac field; only the time lattice is displayed.

The Dirac field’s coordinate as well as its dual coordinate for a single lattice site are given by anticommuting ψ¯ and ψ four-component fermionic variables. The Dirac field’s degrees of freedom   ψ ψ = u ; ψ¯ = [ψ¯u , ψ¯` ] ψ` are four-component Dirac (complex) spinors. To define operators and state vectors, choose ψ¯` , ψu as the coordinate variables and ψ¯u , ψ` as the conjugate variables. ψ¯` , ψu is the analogue of the x coordinate and ψ¯u , ψ` that of the p coordinate of single-particle quantum mechanics. ˆ¯ , ψˆ and their eigenstates |ψ¯ , ψ i as well as the The coordinate operators ψ u ` u ` conjugate equations are defined as " #   ˆ¯ ¯ ψ ` |ψ ¯l , ψu i = ψl |ψ¯l , ψu i (12.6.1) ˆ ψ ψu u h i   ˆ¯ , ψˆ = hψ¯u , ψ` | ψ¯u , ψ` hψ¯u , ψl | ψ (12.6.2) ` u ˆ¯ creates spin-up and spin-down particles when applied on the vacuum state and ψ ` ψˆu does the same for antiparticles. The coordinate eigenstates are |ψ¯` , ψu i and the dual coordinate eigenstates are ¯ hψu , ψ` |. The two components of doublets are labeled by subscripts 1 and 2. Let |f i be an element of Hilbert state space. Then, due to the anticommuting property of the fermionic variables ψ¯µ , ψ` , the general coordinate Taylor expansion for the state vector is given by f (ψ¯u , ψ` ) = hψ¯u , ψ` |f i = f0 +

2 X (f1i + f2i ψ`1 ψ`2 )ψ¯ui + f3 ψ¯u1 ψ¯u2 + f6 ψ`1 ψ`2 i=1

+

2 X i=1

(f4i + f5i ψ¯u1 ψ¯u2 )ψ`i +

2 X ij=1

f7ij ψ¯ui ψ`j + f8 ψ¯u1 ψ¯u2 ψ`1 ψ`2

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The state vector is specified by sixteen complex coefficients fα;ij and corresponds to the sixteen fermion states allowed for a given momentum (or at a given position) by the Pauli exclusion principle. Conjugation is defined in Section 6.5 by the following operations: ¯ 0 , ψ¯ → γ0 ψ. • ψ → ψγ • the order of the fermion variables is reversed. • all the coefficients are complex conjugated. Let |f i be a state vector and hg| be the conjugate state vector. Then, denoting conjugation by †, yields f (ψ¯u , ψ` ) = hψ¯u , ψ` |f i ; g † (ψ¯` , ψu ) = hg|ψ¯` , ψu i Recall conjugation is defined by ¯ 0 , ψ¯ → γ0 ψ ψ → ψγ In particular ψu → ψ¯u ; ψ` → −ψ¯` The conjugate state vector is given by f † (ψ¯` , ψu ) = hf |ψ¯` , ψu i = f ∗ (ψu , −ψ¯` )

(12.6.3)

and with the order of the fermion variables being reversed in f (ψu , −ψ¯` ). For a general choice of the metric, the scalar product is X hf |gi = fα∗ tαβ gβ (12.6.4) ∗

αβ

where α, β are the general index. Note that the metric tαβ is chosen to be positive definite. The scalar product can be represented by means of fermionic integration as follows Z † ¯ ¯ ¯ ψ)g(ψ¯u , ψ` ) hf |gi = dψdψf (ψ` , ψu )T (ψ, (12.6.5) ¯ ψ) is determined by the metric tαβ . where the metric T (ψ, The scalar product of the state vector with a conjugate state vector is positive definite, namely (i) hf |f i ≥ 0 and hf |f i = 0 iff |f i = 0; ˆ † |gi = hg|G|f ˆ i∗ (ii) hf |G The resolution of the identity operator acting on the fermionic Hilbert space is given by Z ¯ ψ¯` , ψu iT (ψ, ¯ ψ)hψ¯u , ψ` | I = dψdψ| (12.6.6) For the diagonal metric (which appears in the continuum case) ¯ ψ) = exp(−ψψ) ¯ = exp T (ψ,

n

−

2 X s=1

o (ψ¯us ψus + ψ¯ls ψls )

(12.6.7)

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The metric in Eq. 12.6.7 implies tαβ = δαβ in Eq. 12.6.4. The completeness equation for metric in Eq. 12.6.7 is given by Z Y 4 ¯ I= dψ¯α dψα |ψ¯` , ψu ie−ψψ hψ¯u , ψ` | (12.6.8) α=1

The metric in Eq. 12.6.7, using the completeness equation as a consistency condition, yields nX o ¯ hψ¯u , ψ` |ψ¯` , ψu i = exp (ψ¯us ψus + ψ¯`s ψ`s ) = exp(ψψ) (12.6.9) s

12.7

Hilbert Space Metric and Transfer Matrix

The state space of the (four-dimensional) Dirac field is fixed by the kinetic term in the Lagrangian. The lattice (Euclidean) Dirac Lagrangian, obtained by discretizing the continuum case is given by Eq. 12.3.12 L = −ψ¯n ψn + K

3 X

{ψ¯n (1 − γµ )ψn+ˆu + ψ¯n+ˆu (1 + γµ )ψn }

u=0

= −ψ¯n ψn + 2K(ψ¯n` ψ(n+ˆo)` + ψ¯(n+ˆo)u ψnu ) +K

3 X

{ψ¯n (1 − γi )ψn+ˆi + ψ¯n+ˆi (1 + γi )ψn }

i=1

Hence (ψ¯n` , ψnu ) propagates to (ψ¯(n+ˆo)u , ψ(n+ˆo)` ) yielding the coordinates for the Dirac Hilbert space as ˆ hψ¯(n+ˆo)u , ψ(n+ˆo)` |e−H |ψ¯n` , ψnu i (12.7.1) Recall, from Eqs. 12.5.8 and 12.5.9, the Dirac field’s coordinate eigenstates are given by O hψ¯u , ψ` | = hψ¯~xu , ψ~x` | : Coordinates of the Dirac Hilbert space ~ x

|ψ¯` , ψu i =

O

|ψ¯~x` , ψ~xu i : Coordinates of the dual Dirac Hilbert space

~ x

A state functional of the Dirac field |Φi has the coordinate representation given by   (12.7.2) hψ¯u , ψ` |Φi = Φ ψ¯u , ψ` Consider a d-dimensional Euclidean spacetime lattice with lattice spacing a. Let the (d−1)-dimensional spatial lattice be of infinite size and the time lattice be of finite size N . The finite time Wilson lattice action is given by Eq. 12.3.12 SN = −

N X X

ψ¯n ψn

n0 =1 ~ n

+K

N XX 3 X

{ψ¯n (1 − γµ )ψn+µ + ψ¯n+µ (1 + γµ )ψn }

n0 =0 ~ n µ=0

(12.7.3)

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Note that in the first term in the action above, the sum over the time lattice is from 1 ≤ n0 ≤ N ; in the second term of the lattice action, the sum is over 0 ≤ n0 ≤ N and the boundary conditions at n0 = 0 and n0 = N + 1 are correctly incorporated. The lattice Hamiltonian H is given by the path integral as follows: N YZ Y ¯ ¯ lim hψu , ψ` | exp{−a(N + 1)H}|ψ` , ψu i = dψ¯n dψn exp(SN ) (12.7.4) N →∞

n0 =1 ~ n

Consider the case for N = 1; from Eq. 12.5.7, the action is given by S1 = −

1 X

ψ¯n Lψn + 2K

n=1

1 X

{ψ¯n` ψn+1` + ψ¯n+1u ψnu }

n=0

¯ + 2K ψ¯` ζ` + 2K ζ¯u ψu + 2K ζ¯` ψ` + 2K ψ¯u ζu = −ζLζ

(12.7.5)

where ψ1 = ζ ; ψ¯1 = ζ¯ ; ψ¯0` = ψ¯` ; ψ0u = ψu ; ψ¯2u = ψ¯u ; ψ2` = ψ` Eqs. 12.5.4 and 12.5.5 yield ¯ = −αζζ ¯ − 2K ζ¯u σζ` + 2K ζ¯` σζu −ζLζ Hence S1

e



   ¯ ¯ ¯ ¯ = exp 2K ψu ζu + 2K ζ` ψ` + 2K ζ` σζu exp − αζζ   ¯ ¯ ¯ × exp 2K ψ` ζ` + 2K ζ` ψ` − 2K ζu σζ`

(12.7.6)

and hψ¯u , ψ` | exp(−2aH)|ψ¯` , ψu i =

Z

¯ exp{S1 (ψ, ¯ ψ; ζ, ¯ ζ)} dζdζ

(12.7.7)

Consider the following transfer matrix of the Dirac Hamiltonian ( ) X X −aH ¯ ¯ ¯ ¯ ¯ hψu , ψ` |e |ψ` , ψu i = exp 2K ψ~n ψ~n − K (ψ~n γi ψ~n+i − ψ~n+i γi ψ~n ) ~ n

~ ni

(12.7.8) where ~n is a (d − 1)-dimensional spatial lattice point with i = 1, ...(d − 1). Fourier transforming the Dirac field to momentum space yields the transfer matrix     X X hψ¯u , ψ` |e−aH |ψ¯` , ψu i = exp 2K ψ¯p~ ψp~ − 2iK ψ¯p~ γi sin(pi )ψp~ (12.7.9)   p ~

p ~i

Recall from Eq. 12.3.2 that the gamma matrices have the following representation     X X 0 −σi 0 −σ γi = i ⇒ − 2iK γi sin(pi ) = 2K : σ= σi sin(pi ) σi 0 σ 0 i

i

For notational simplicity suppress the summation over the momentum index. ψp~ = ψ ; ψ¯p~ = ψ¯

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The Dirac spinor’s two component representation yields   ψ ψp~ = u ; ψ¯p~ = [ψ¯u , ψ¯` ] ψ` and hence    0 −σ ψu ¯ ¯ 2K[ψu , ψ` ] = 2K(−ψ¯u σψ` + ψ¯` σψu ) σ 0 ψ` The Fourier representation, using the notation of Eq. 12.5.5, yields  ¯ + 2K ψ¯` σψu − 2K ψ¯u σψ` hψ¯u , ψ` |e−aH |ψ¯` , ψu i = exp 2K ψψ

(12.7.10)

Note the nearest neighbor term, containing the σ matrix, couples only the ket coordinates ψ¯` , ψu and the bra dual coordinates ψ¯u σψ` , with no coupling of the dual bra and ket coordinates. This is reflected in the fact that the boundary terms will not appear in potential terms in the action, as in the example discussed in Noteworthy 12.1. From Eqs. 12.7.5, 12.7.6, 12.7.7 and 12.7.10 Z ¯ exp{S1 (ψ, ¯ ψ; ζ, ¯ ζ)} hψ¯u , ψ` | exp(−2aH)|ψ¯` , ψu i = dζdζ (12.7.11)  = exp 2K ψ¯u σψ` − 2K ψ¯` σψu Z ¯ ψ¯u , ψ` | exp(−aH)|ζ¯` , ζu iT (ζ, ¯ ζ)hζ¯u , ζ` | exp(−aH)|ψ¯` , ψu i (12.7.12) × dζdζh The Hamiltonian given by Eq. 12.7.10 is obtained by comparing Eq. 12.7.12 with Eq. 12.7.6; the comparison is sufficient to obtain the Hamiltonian. The following terms in Eq. 12.7.6 −2K ζ¯u σζ` + 2K ζ¯` σζu are the value of the Hamiltonian for the bra and ket states |ζ¯` , ζu ihζ¯u , ζ` | that are summed over in the path integral. The terms 2K ψ¯` σψu − 2K ψ¯u σψ` are in the expression for the Hamiltonian given in Eq. 12.7.10 but do not appear in S1 , and hence need to be introduced by hand in Eq. 12.7.12 for it to equal Eq. 12.7.11. These terms do not appear inside the path integral in Eq. 12.7.12 as these are boundary terms and do not appear in the potential terms of the action. See discussion in Noteworthy 12.1. It is the remarkable properties of the representation of the gamma matrices chosen for the lattice Dirac field that the gamma matrix γ0 and γi ; i = 1, 2, 3 together ‘conspire’ to yield a well-defined Hamiltonian. It was crucial that all γi ; i = 1, 2, 3 be off-diagonal for the coordinates and the dual coordinates to yield the factorization of the terms in the Hamiltonian and yield the boundary terms that made the equality of Eqs. 12.7.11 and 12.7.12 possible – and which in turn yields the Hamiltonian.

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In fact, if one computes hψ¯u , ψ` | exp(−aN H)|ψ¯` , ψu i, the boundary terms that do not appear in the potential terms of the action go to zero as N → ∞, as is the case for the case discussed in Noteworthy 12.1. Noteworthy 12.1: Boundary Terms in the Action Consider a path integral for x(t) with action given by  Z T  1  dx 2 dt m2 + V (x) S=− 2 dt 0 and with boundary conditions x(0) = x ; x(T ) = x0 On discretizing the action one obtains, for T = N  N −1 N −1 X X 1 S = − m2  (xn+1 − xn )2 −  V (xn ) 2 n=0 n=1

with boundary conditions x0 = x ; xN = x0 Note the boundary values occur only in the kinetic term and the potential term does not have terms V (x0 ), V (xN ) that depend on x0 , xN ; these missing boundary terms go to zero as  → 0 and, like the case for the lattice Dirac Hamiltonian, need to be introduced by hand for a system with two lattice steps in time. Similarly, in deriving the Hamiltonian for the Dirac field, the boundary values do not appear in the potential of the action, as the potential has no coupling with the boundary values; these boundary terms also become zero in the limit of zero lattice spacing. Hence, we conclude that the Hamiltonian given in Eq. 12.7.9 is the correct expression. The metric on Hilbert space is determined by the terms in Eq. 12.7.6 that do not couple to the boundary values and yield  ¯ ζ) = exp − αζζ ¯ T (ζ, (12.7.13) Note that the Wilson metric for the Hilbert space of lattice Dirac field, given in Eq. 12.7.15 depends on the action through the parameter K. In general, the action (Lagrangian) contains both the Hamiltonian and the (kinematical) structure of state space. Recall the completeness equation is given in Eq. 12.6.6 for the general lattice ¯ ψ) metric T (ψ, Z ¯ ψ¯` , ψu iT (ψ, ¯ ψ)hψ¯u , ψ` | I = dψdψ| From Eq. 12.7.13, writing out in full the metric gives  X   X  ¯ ψ) = exp − αψψ ¯ T (ψ, = exp − 1 − 2K cos(pi ) ψ¯p~ ψp~ p ~

i

(12.7.14)

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Note that the metric given above in Eq. 12.7.14 is identical to the metric studied in Eq. 6.3.3, since the Wilson metric has factorized in momentum space. Fourier transforming the Dirac field to the space lattice yields, from Eq. 12.7.14    X X
 ¯ ψ) = exp − ψ¯~n ψ~n + K ψ¯~n ψ~n+i + ψ¯~n+i ψ~n (12.7.15) T (ψ,   ~ ni ~ n ( ) X
¯ ψm ~ ~n)ψ~n = exp − ~ M (m, mn ~

The inner product of the basis states, similar to the result in Eq. 6.3.5, is given by  1 ¯ hψ¯u , ψ` |ψ¯` , ψu i = exp + αψψ α     X X
1 = exp ψ¯~n ψ~n − K ψ¯~n ψ~n+i + ψ¯~n+i ψ~n (12.7.16)   det M ~ n

~ ni

where det M =

Y

α(~ p)

p ~

For the four-dimensional continuum limit, the Hilbert space metric is given by the limit of a → 0 for d = 4 and yields, since 2K = 1/(4 + ma), the following   Z     X ¯ ¯ ¯ ψ) = exp − 1 − 3 a3 ψ(x)ψ(x) → exp − d3 xψ(x)ψ(x) T (ψ, 2K n Hence, for d = 4 the continuum metric is given by  Z   Z ¯ ψ) = exp − d3 xψ¯~x ψ~x = exp − T (ψ,

d3 p ¯ ψp~ ψp~ (2π)3

 (12.7.17)

and the inner product of the basis states with the dual state is given by Z  Z  d3 p ¯ hψ¯u , ψ` |ψ¯` , ψu i = exp d3 xψ¯~x ψ~x = exp ψ ψ (12.7.18) p ~ p ~ (2π)3 12.8

Dirac Lattice Hamiltonian

The transfer matrix is given by the matrix elements of exp{−aH}. The Wilson lattice metric given in Eq. 12.7.14 yields the following coordinate inner product ( ) X X hψ¯u , ψ` |ψ¯` , ψu i = exp ψ¯n ψn − K (ψ¯n ψn+i + ψ¯n+i ψn ) (12.8.1) n

ni

where a normalization constant has been dropped. The (lattice) Hamiltonian H is defined in terms of the transfer matrix by the following ¯ hψ¯u , ψ` |e−aH |ψ¯` , ψu i = e−aH(ψ,ψ) hψ¯u , ψ` |ψ¯` , ψu i (12.8.2)

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The lattice equation for the transfer matrix e−aH is given by Eq. 12.7.9 ( ) X X hψ¯u , ψ` |e−aH |ψ¯` , ψu i = exp 2K ψ¯n ψn − K (ψ¯n γi ψn+i − ψ¯n+i γi ψn ) n

ni

(12.8.3) The left hand side of Eq. 12.7.9, using Eq. 6.8.4, yields n o ¯ ψ) + O(a2 ) lim hψ¯u , ψ` |e−aH |ψ¯` , ψu i ' hψ¯u , ψ` |ψ¯` , ψu i 1 − aH(ψ, a→0 ( ) X X 1 ¯ ¯ ¯ ¯ exp ψn , ψn − K (ψn ψn+i + ψn+i ψn ) − aH(ψ, ψ) (12.8.4) ' det M n ni Hence, from Eqs. 12.8.3 and 12.8.4, the Dirac lattice Hamiltonian is given by X 1 H = − (2K − 1) ψ¯~n ψ~n a ~ n   1 KX ¯ ¯ ψ~n (1 − γi )ψ~n+i − ψ~n+i (1 + γi )ψ~n + ln det M (12.8.5) + a a ~ ni

The results obtained in this Section are further analyzed for the eigenstates of the Dirac field in Section 13.2. Noteworthy 12.2: Dirac Lagrangian and Metric The metric on Hilbert space given in Eq. 12.7.15    X X
 ¯ ψ) = exp − T (ψ, ψ¯~n ψ~n + K ψ¯~n ψ~n+i + ψ¯~n+i ψ~n   ~ n

~ ni

is the result of the hyper-cubic symmetric lattice Lagrangian given in Eq. 12.3.12. An alternative choice for the Dirac Hamiltonian, discussed in Section 14.8, can be obtained by choosing the canonical metric that is given by n X o ¯ ψ) = exp − Tc (ψ, ψ¯~n ψ~n (12.8.6) ~ n

The canonical metric given in Eq. 12.8.6 yields an asymmetric action functional. Transforming from the axial gauge back to a manifestly gauge-invariant expression yields the lattice action (n = (n0 , ~n))  X X 1 † S=− ψ¯n ψn + (1 − m0 ) ψ¯n+ˆ0 (1 + γ0 )Un0 ψn ψ¯n (1 − γ0 )Un0 ψn+ˆ0 2 n n   X 0 hnm − ψ¯n ψm δn0 ,m0 + SG † h nm 0 nm The pure gauge field action SG is not affected by the metric. The lattice gauge theory Hamiltonian, from Eq. 14.8.1, is given by   X X 0 hnm ¯ ψ) = m0 b ψ, ψ¯n ψn + ψ¯n ψ m + HG H( h†nm 0 n nm

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Lattice Path Integral

The probability amplitude, or the evolution kernel, is the amplitude that the field configuration |ψ¯n` , ψnu i to be in the conjugate configuration hψ¯nu , ψn` | after time T . That is, the evolution kernel K is defined by (12.9.1) K(ψ¯u , ψ` ; ψ¯` , ψu ; T ) ≡ hψ¯u , ψ` e−T H ψ¯` , ψu i K(ψ¯u , ψ` ; ψ¯` , ψu ; T ) can be expressed as a Feynman path integral over the anticommuting fermion field variables, with appropriate boundary conditions on the integration variables. Let T = N a, and impose the boundary conditions n = 0; ψ¯0` = ψ¯` , ψ0u = ψu n = N;

ψ¯N u = ψ¯u ,

ψN ` = ψ`

K is obtained by integrating over all the fermion field variables from n = 1 to N − 1, and yields −1 Z Y NY K(ψ¯u , ψ` ; ψ¯` , ψu ; T ) = dψ¯pn dψpn exp(S) p n=1

where (suppressing the momentum summation) S=−

N −1 X

ψ¯n Lψn + 2K

n=1

N −1 X

{ψ¯n` ψn+1` + ψ¯n+1u ψnu }

n=0

S is the action for finite time N a; note that the first term in S has N − 1 terms whereas the second term coupling nearest neighbors in time have N terms. The boundary values appear in S only in the nearest-neighbor time coupling term. Since S is quadratic, the path integral can be performed for K by solving the classical field equation, together with the given boundary conditions. Suppose ξ¯n , ξn satisfy the classical field equation and the given boundary condition, that is ¯ ξ)/δ ψ¯n = 0 ; δS(ξ, ¯ ξ)/δψn = 0, δS(ξ, (12.9.2) ξ¯0N = ψ¯l , ξ0u = ψu ; ξ¯N u = ψ¯u , ξN l = ψl Define the new integration variables ζ¯n , ζn by ψn = ζn + ξn , ψ¯n = ζ¯n + ξ¯n dψn = dζn ,

(12.9.3)

dψ¯n = dζ¯n

ξ¯n , ξn satisfying the boundary conditions yields ζ¯0l = ζ¯N u = 0 ; ζ0u = ζN l = 0 Hence, the new variables ζn , ζ¯n are independent of the boundary values, and form a so-called open chain boundary fermion system. Making the change of variables in S gives N −1 N −1     X X S=− ζ¯n Lζn + K ζ¯nl ζn+1l + ζ¯n+1u ζnu + 2K ξ¯0l ξ1l + ξN¯ u ξN −1u n=1

n=1

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The integration variables ζn , ζ¯n have decoupled from the classical solution ξn , ξ¯n and yield the result ¯ K(ψ¯u , ψl , ψ¯` , ψu : T ) = hψ¯u , ψl |e−aN H |ψ¯` , ψu i = C(N ) exp(H(ξ, ξ))

(12.9.4)

where ¯ = 2K(ξ¯0l ξ1l + ξ¯N u ξN −1u ) H(ξ, ξ) −1 Z Y NY ¯ C(N ) = dζpn dζ¯pn exp S(ζ, ζ)

(12.9.5) (12.9.6)

p n=1

The normalization constant C(T = N a) is evaluated in Section 12.9.1. Note from Eq. 12.9.5 that only the boundary values of ξ¯n appear in the solution. Hence only the time dependence of the classical solution for ξn is required; using the field equation given in Eq. 12.9.2 yields αξnu = −β · σξnl + 2Kξn−1u

(12.9.7)

αξnl = −β · σξnu + 2Kξn+1l   X α = 1 − 2K cos pi ; βi = 2K sin pi

(12.9.8)

i

Make the ansatz ξnu = eλn A + e−λn B ξnl = eλn C + e−λn D Solving the field equations in Eqs. 12.9.7 and 12.9.8 with the boundary conditions given in Eq. 12.9.3 gives the solution [Baaquie (1983b)] " #    Σ(N − n) − shλn β · σ 1 ψu ξnu α (12.9.9) = −n) ψl ξnl Σ(N ) − shλ(N β · σ Σ(n) α with     α 2K β2 2chλ = 1+ + 2 2K α α 2K Σ(n) = shλn − shλ(n − 1) α There is a similar equation for ξ¯n . Substituting the above solution for ξn in Eq. 12.9.5 gives, restoring the momentum index, the result H(ψ¯pu ψpl , ψ¯pl ψpu ; N ) " #  shλ(N −1) X 2K shλ − β · σ ψpu α (ψ¯pu ψ¯pl ) shλ(N −1) = ψpl Σ(N ) β·σ shλ α p   X 2K shλ(N − 1) ¯ = ψp (−iβ · γ)ψp + shλ ψ¯p ψp (12.9.10) Σ(N ) α p

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where ψp =

  ψpu , ψpl

ψ¯p = [ψ¯pu , ψ¯pl ]

The result given in Eq. 12.9.10 is exact, and can be used to study Dirac fermions on a finite lattice. For example, a direct calculation for H with N = 2 verifies the result. The result for d = 2 can be obtained from Eq. 12.9.10 by replacing β · σ by 2K sin p and considering the ψ and ψ¯ to be doublets. The Dirac equation is linear and hence the continuum limit does not need renormalization; taking the continuum limit of a → 0 greatly simplifies the result. Define the following dimensional quantities Z Z 1 X 1 d3 k = lim 3 = T = N a, k = pa ; t = na ; 3 a→0 a (2π) k p p ω = k2 + m2 , µ(t) = ωchωt + mshωt To take the limit a → 0, the fields ξ, ξ¯ are rescaled using Eq. 12.3.6 and only the non-vanishing terms are retained. The continuum solution of the Dirac field’s classical solution is [Baaquie (1983b)]    1 ξku (t) µ(T − t) = ξkl (t) µ(T ) −shω(t − T )k · σ

[ξ¯ku (t) ξ¯kl (t)] = [ψ¯ku ψ¯kl ]



µ(T ) −shωtk · σ

−shωtk · σ µ(t)

  ψku ψkl

 1 −shω(t − T )k · σ µ(T − t) µ(T )

The continuum classical solutions, yield from Eq. 12.9.5, the following Z ¯ F(ξ, ξ) = {ξ¯ku (T )ξku (T ) + ξ¯kl (0)ξkl (0)} k

giving the final result [Baaquie (1983b)] ¯ = F(ξ, ξ)

Z  k

 shωT ω (−ψ¯ku k · σψkl + ψ¯kl k · σψku ) + (ψ¯ku ψku + ψ¯kl ψkl ) µ µ

µ = ωchωT + mshωT

(12.9.11)

The normalization constant C(T ) is evaluated in Subsection 12.9.1 and yields the final result  2 µ(T ) K(ψ¯u , ψl ; ψ¯l , ψ¯u ; T ) = Πp exp(F) = C(T ) exp(F) (12.9.12) ω

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12.9.1

Normalization constant

The normalization constant C(T ) is calculated in the continuum limit. Recall that from Eqs. 12.9.1 and 12.9.4 K ≡ hψ¯u , ψl |e−T H |ψ¯l , ψu i = C(T ) exp F

(12.9.13)

To directly calculate C(T ), requires performing the fermion path integral given in Eq. 12.9.6 and then taking the continuum limit. The path integral can be avoided by, instead, separately evaluating the antiperiodic trace of K and exp(H); this procedure yields C(T ).4 Using standard finite time methods [Kapusta (1993)], the left hand side of Eq. 12.9.13 yields i4 h 1 p ˜ ; ω = p2 + m2 TrK = Πp 2ch ωT 2 ˜ stands for the antiperiodic trace. where Tr The antiperiodic trace of the right hand side of Eq. 12.9.13 yields Z ¯ −ψψ ¯ ˜ exp F = dψdψe Tr exp H(ψ¯l , −ψ¯u ; ψ¯u − ψl )     2 1 4ω 2  1 µ+ω shωT k · σ = Πk det ch ωT = Πk µ −shωT k · σ µ + ω µ 2 Hence, from Eq. 12.9.13 and equations above  2 2  µ ωchωT + mshωT C(T ) = Πk = Πk ω ω

(12.9.14)

The continuum and lattice formulation have the following similarities. • The result for H is dimensionally correct. In particular, the break-up of the spinors into upper and lower components for the lattice survives the continuum limit; that is, the initial and final continuum field configurations are given by {ψ¯l (~x), ψu (~x)} and {ψ¯u (~x), ψl (~x)}, respectively, although the continuum theory does not explicitly have this to start with. It was shown in Section 12.2 that the continuum formulation determines the field coordinates based on the choice of the Dirac γ matrices. ¯ can be derived directly from the • The solutions for the continuum ξ(t) and ξ(t) continuum theory, and discussed in Chapter 13. ¯ given in Eq. 12.9.11 can be directly obtained from the • The result for H(ξ, ξ) continuum theory; one needs to use the fermionic coordinates in continuous space and the evolution kernel can be obtained from the continuum eigenstates of the Dirac field. This is discussed in Chapter 13. • A continuum action with the correct boundary terms is written in Chapter 13 using the lattice results as a guide. 4 Note that C(T ) could equally be obtained by taking the periodic trace of both sides of Eq. 12.9.1.

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Evolution Kernel

The Hamiltonian is defined using the evolution kernel by noting that, from Eq. 12.7.18 ∂ ˆ ψ¯l ψu i − K = hψ¯u ψl |H| ∂T T =0 ¯ ψ)hψ¯u ψl |ψ¯l ψu i = H(ψ, Z  ¯ ψ) exp = H(ψ, ψ¯p ψp p

¯ ψ) is the Hamiltonian. Using Eq. 12.9.12 for the kernel yields and H(ψ,   ∂ ∂C(0) ∂H(0) − K|T =0 = − + exp H(0) ∂T ∂T ∂T Dropping a constant yields the Dirac Hamiltonian Z ¯ H(ψ, ψ) = ψ¯p (iγ · p + m)ψp

(12.10.1)

p

¯ ψ) is a functional of both the field configuration ψ¯l (x), ψu (x) as well as the H(ψ, conjugate coordinates ψ¯u (x), ψl (x), and is analogous to the function H(p, x) defined on the phase space for a single particle. The evolution kernel contains the complete eigenenergies and eigenfunctions of the quantum field [Baaquie (2014)]. Let En and |Φn i be the eigenenergies and eigenfunctions. The completeness equation given in Eq. 12.7.17 defines the matrix elements and scalar products of the wave functions, and yields hΦm |H|Φn i = En δnm The time evolution operator can be written as X e−T H = e−T En |Φn ihΦn |

(12.10.2)

n

where the sum is over all the eigenfunctions, and K(ψ¯u , ψ¯` , ψ¯` , ψu ; T ) ≡ hψ¯u , ψ` |e−T H |ψ¯` , ψu i X = e−T En Φn (ψ¯u , ψ` )Φ†n (ψ¯` , ψu ) n

=

YX p

e−T En (p) Φnp (ψ¯pu , ψp` )Φ†np (ψ¯pl , ψpu )

(12.10.3)

np

where the conjugate Φ†n is defined in Eq. 12.6.3. The last equation above is due to the factorization of the kernel in momentum space. The eigenfunctions will hence be solved for each p separately. To evaluate En and Φn , note that in the kernel K, besides an overall factor of e2ωT in C(T ), only powers of e−ωT appear in H(T ). Hence, expanding K in powers of e−ωT yields (for each p) K = e2ωT (K0 + e−ωT K1 + e−2ωT K2 + e−3ωT K3 + e−4ωT K4 )

(12.10.4)

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Note that the expansion for K terminates due to the anticommutation of the fermions. Hence, comparing Eqs. 12.10.3 and 12.10.4 yields, after some algebra, all the eigenfunctions. The calculation is done in Section 12.11, and the results are summarized here. For each momentum p, there are five eigenenergies and sixteen eigenfunctions, which are orthonormal. The term excitation is used for a particle or antiparticle operator applied on the vacuum state and the term pair stands for applying an particle-antiparticle operator to the vacuum state. The Dirac field has the following eigenenergies and eigenfunctions. (1) E0 = −2ω: a unique vacuum state Ω; (2) E1 = −ω: four eigenfunctions of a single excitation; (3) E2 = 0: six eigenfunctions, four of which are pairs and two are two excitation states; (4) E3 = ω. four eigenfunctions which are a single excitation mixed with excitation and pair; (5) E4 = 2ω: one eigenfunction, which is the superposition of the vacuum state, a single pair and two pairs. These sixteen orthonormal eigenfunctions form a complete basis for the Hilbert space of states; it is a consequence of the exclusion principle that only sixteen states are allowed for each momentum p. For starters, the vacuum state is calculated. Let T → ∞; then from Eqs. 12.9.11 and 12.9.12 K = e−T E0 Ω(ψ¯u ψl )Ω† (ψ¯l ψu ) + O(e−T (E1 −E0 ) ) 2  Z   P 1 ¯ m+ω exp − ψpu · σψpl = eT (2 p ωp ) Πp 2ω p ω+m Z  1 ¯ ψpl p · σψpu + O(e−ωT ) × exp ω + m p From Eqs. 12.10.3 and 12.10.4, the vacuum energy and the vacuum eigenfunction is given by X Xp p2 + m2 E0 = −2 ωp = −2 p

p

Ω(ψ¯u ψl ) = hψ¯u ψl |Ωi  Z  Y m + ω 1 = exp − ψ¯pu p · σψpl 2ω p ω+m p

(12.10.5)

The energy for the vacuum is a negative divergent quantity and so is the constant ¯ ψ). Both these divergent quantities arise dropped in Eq. 12.10.1 in obtaining H(ψ, due to the ordering of operators chosen and can be removed by appropriately reordering (normal ordering) the Hamiltonian. Normal ordering is not done since firstly only energies with E0 subtracted will enter our calculations; and secondly normal ordering the Hamiltonian would alter the Lagrangian – and which is not desirable for the path integral approach.

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For the vacuum state, note that for T → ∞ the coordinates (ψ¯u , ψl ) and ¯ (ψl , ψu ) – referring to the later and earlier times – completely factorize in K, and hence allow for defining the vacuum state Ω. This factorization is a reflection of the uniqueness of the vacuum, and non-factorization would be an indication of the multiplicity of vacuum states, linked with symmetry breaking. The vacuum consists of the superposition of particle-antiparticle pairs of all momentum p. The field in the vacuum state is correlated over spatial distance of m−1 and the spin orientation of the field in the vacuum state is coupled by the helicity operator p · σ/|p|. 12.11

Energy Eigenfunctions

The eigenenergies and eigenfunctions for the free Dirac field are evaluated [Baaquie (1983b)]. The momentum index is suppressed since the field factorizes in this representation. Recall from Eqs. 12.10.4 and 12.10.3  2   ω ¯ shωT µ(T ) ¯ ¯ ¯ (−ψu p · σψ` + ψ` p · σψu ) + (ψu ψu + ψ` ψ` ) exp K= ω µ(T ) µ(T ) X = e−T En Φn (ψ¯u , ψ` )Φ†n (ψ¯` , ψu ) n

To simplify the algebra, consider the so-called helicity basis. Let the Pauli matrices be       0 1 0 −i 1 0 σ1 = σ2 = , σ3 = 1 0 i 0 0 −1 Then 

cos θ p · σ = p −iφ e sin θ

−e−iφ sin θ − cos θ



= pU σ3 U † where p = p(cos φ sin θ, sin φ sin θ, cos θ) ; p = |p|   cos 12 θ −e−iφ sin 21 θ , UU† = 1 U = −iφ cos 12 θ e sin 12 θ The matrix U rotates the spin of the doublets to lie parallel (or antiparallel) to their three-momentum p, the upper component having helicity +1 and the lower component of the doublet having helicity −1. Choose the helicity basis given by    †  U 0 U 0 ψ→ ψ ; ψ¯ → ψ¯ 0 U 0 U† For the doublets, use the notation " # 1 ψu(l) ψu(l) = , 2 ψu(l)

1 2 ψ¯u(l) = (ψ¯u(`) , ψ¯u(`) )

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In the new basis the evolution kernel is given by   2  p µ(T ) ω ¯ exp − K= (ψ¯u σ3 ψl − ψ¯l σ3 ψu )shωT + (ψu ψu + ψ¯l ψl ) ω µ(T ) µ(T ) (12.11.1) To expand K in powers of e−ωT , expand shωT and µ−1 (T ) to e−4ωT and write out the exponential term and the normalization constant to O(e−4ωT ). This yields K = e2ωT (K0 + e−ωT K1 + e−2ωT K2 + e−3ωT K3 + e−4ωT K4 ) Eqs. 12.10.3 and 12.11.1 – and after considerable algebra – yield the results given below. (I) E0 = −2ω: ground state; non-degenerate  2   m+ω p (ψ¯u σ3 ψl − ψ¯l σ3 ψu ) K0 = exp − 2ω m+ω † ¯ ¯ = Ω(ψu , ψ` )Ω (ψu , ψ` )

(12.11.2)

where the vacuum state is given by Φ0 = Ω(ψ¯u , ψ` )     p ¯ m+ω exp − ψu σ3 ψl = 2ω m+ω (II) E0 = −ω: first excited level; four-fold degenerate 4

K1 =

X 2ω ¯ † (ψ¯` , ψu ) = Ω(ψ¯u , ψ` )ψψΩ Φi Φ†i ω+m i=1

where Φ1 = N ψ¯u1 Ω(ψ¯u , ψ` ) ; Φ2 = N ψ¯u2 Ω(ψ¯u , ψ` ) Φ3 = N ψl1 Ω(ψ¯u , ψ` ) ; Φ4 = N ψl2 Ω(ψ¯u , ψ` ) p N = 2ω/(ω + m) The first excited level consists of a single excitation on the vacuum. (III) E2 = 0: second excited level; six-fold degenerate     2  ω−m ω ¯ ω 2 ¯ ¯ K2 = 2 Ω(ψu , ψ` ) 1 + (ψu σ3 ψu ) + (ψψ) Ω† (ψ¯u , ψ` ) ω+m p p =

10 X i=5

Φi Φ†i

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where

237

ω ω Φ5 = N 0 ψ¯u σ1 ψl Ω(ψ¯u , ψ` ) ; Φ6 = N 0 ψ¯u σ2 ψ2 Ω(ψ¯u , ψ` ) p p   ω ω Φ7 = N 0 1 + ψ¯u σ3 ψ` Ω((ψ¯u , ψ` ) ; Φ8 = N 0 ψ¯u ψ` Ω(ψ¯u , ψ` ) p p √ ω 1 2 √ ω 1 2 0 0 ¯ ¯ ¯ Φ9 = N 2 ψu ψu Ω(ψu , ψ` ) ; Φ10 = N 2 ψ` ψ` Ω(ψ¯u , ψ` ) p p p 0 N = 2(ω − m)/(ω + m)

The states Φ5 , Φ6 , and Φ8 are the spin-one pair states and Φ8 is the singlet pair state. State Φ7 is a superposition of the vacuum state and a pair. States Φ9 and Φ10 are the two excitation states. (IV) E3 = ω: third excited level; four-fold degenerate " 2ω(ω − m) ¯ ¯ + 2ω ψψ( ¯ ψ¯u σ3 ψl − ψ¯l σ3 ψu ) Ω(ψu , ψ` ) ψψ K3 = (ω + m)2 p #  2 14 X ω 3 ¯ Φi Φ†i (ψψ) Ω† (ψ¯u , ψ` ) = + p i=11 where   2ω ¯ Φ11 = N 00 ψ¯u1 1 + ψu σ3 ψl Ω(ψ¯u , ψ` ) p   2ω ¯ ψu σ3 ψl Ω(ψ¯u , ψ` ) Φ12 = N 00 ψ¯u2 1 + p   2ω ¯ ψu σ3 ψl Ω(ψ¯u , ψ` ) Φ13 = N 00 ψ¯l1 1 + p   2ω ¯ ψu σ3 ψl Ω(ψ¯u , ψ` ) Φ14 = N 00 ψ¯l2 1 + p p 00 N = 2ω(ω − m)/(ω + m). These states are a superposition of a single excitation and an excitation and a pair. (V) E4 = 2ω: fourth excited level; non-degenerate K4 = Φ15 Φ†15 ,     ω−m p ¯ Φ15 = exp ψu σ3 ψl 2ω ω−m The state Φ15 consists of a mix of no particle, one pair and two pairs. Note the similarity in form of Φ15 with the vacuum state Ω, the two being related by m → −m. In summary, the energy spectrum of the free fermion field, for each momentum p, consists of five energy levels and sixteen eigenfunctions. These eigenfunctions are orthonormal hΦn |Φm i = δnm

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and form a complete basis for the Hilbert space of states. The eigenfunctions for the full field theory are elements of the Hilbert space formed by the tensor product over p, and are given by Y Φn [ψ¯u , ψ` ] = Φnp (ψ¯np , ψ`p ) p

and the eigenenergies given by En =

X

Enp (p)

p

These basis states can be used as a complete basis for doing perturbation theory, and are an alternative to the usual Fock-space approach. 12.12

Propagator

The propagator is calculated using the evolution kernel and the vacuum state. Recall that, for T > 0, the propagator is 0 α ¯β Gαβ pT δ(p − p ) = hΩ|ψpT ψp0 0 |Ωi α where ψpT , ψ¯pβ0 0 are Heisenberg field operators. From standard field theory the Euclidean propagator, in matrix notation, is given by [Peskin and Schroeder (1995)] Z +∞ eip0 T dp0 GpT = −∞ 2π ip0 γ0 + ip · γ + m   e−ωT m + ω −p · σ = (12.12.1) p·σ m−ω 2ω

The Heisenberg operators have the time evolution ψpT = eT H ψp e−T H ψ¯pT = eT H ψ¯p e−T H with ψp ψ¯p being the Schr¨ odinger operators. Ignoring from now on the momentum conserving δ-functions, and using the completeness equation gives T E0 Gαβ hΩ|ψpα e−T H ψ¯pβ |Ωi, pT = e Z R R ¯ ¯ αβ T E0 ¯ ζdζhΩ| ¯ GpT = e dξdξd ξ¯` , ξu ie− p ξξ ξpα hξ¯u , ξ` |e−T H |ζ¯` , ζu ie− p ζζ ζ¯pβ hζ¯u , ζ` |Ωi Z R ¯ ¯ = eT E0 [ξpα ζ¯pβ ] · Ω† (ξ¯` , ξu )K(ξ¯u , ξ` , ζ¯` , ζu ; T )Ω(ζ¯u , ζ` )e− p (ξp ξp +ζp ζp ) ¯ ζζ ¯ ξξ, Z =N ξpα ζ¯pβ exp(F) ¯ ζζ ¯ ξξ,

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The exponential of the effective action F is equal to the product of the vacuum wave functional and its dual, the evolution kernel and the metric; Eqs. 12.10.4 and 12.10.5 yield Z  shωT 1 ¯ ξpl p · σξpu + (−ξ¯pu p · σξpl + ζ¯pl p · σζpu ) F= ω+m µ p  ω 1 ¯ ¯ p ξp − ζ¯p ζp + (ξ¯pu ζpu + ζ¯pl ξpl ) − ζpu p · σζpl − xi (12.12.2) µ ω+m  2  2 2ω −2ωT µ N = Πp e ; µ ≡ µ(T ) ω ω+m The generating functional for F is evaluated as this is useful for evaluating the ¯ p be the four-component fermionic sources; define n-point functions. Let hp , h   Z Z Z ¯ ¯ ¯ W (h, h) = N exp F + hp ξp + ζp hp (12.12.3) ¯ ζζ ¯ ξξ,

p

p

  2 Z    1 ωT ω ¯ iα · p ζ exp − ζ 1 + = N Πp e µ ω+m ¯ ζζ     p·σ ¯ × exp e−ωT h¯u + h¯l ζu + ζh ω+m Z  ¯ = exp hp Mp hp

(12.12.4)

(12.12.5)

p

where Mp =

 e−ωT m + ω p·σ 2ω

−p · σ m−ω

 (12.12.6)

Hence, the propagator is given by Gαβ pT =

δ2

¯ h) W ( h, = Mpαβ ¯ ¯ βp h=0,h=0 δhβp δ h

(12.12.7)

Eqs. 12.12.1 and 12.12.6 show that equation above is the required result. The propagator Gαβ pT can also be evaluated using the eigenfunctions. The notau` is used for the 2 × 2 matrix entries in Eq. 12.12.6. In terms of , G tion of Guu pT pT the eigenfunctions of the Dirac field, the propagator is given by X G = eT E0 hΩ|ψp e−T H ψ¯p |Ωi = e−T (En −E0 ) hΩ|ψp |Φn ihΦn |ψ¯p |Ωi (12.12.8) n

where Eq. 12.10.2 has been used to obtain the equation above. It can be shown that only the first excited level, that is, states with E1 = −ω, contributes to Eq. 12.12.8. Hence G = e−ωT

4 X i=1

hΩ|ψp |Φi ihΦi |ψ¯p |Ωi

(12.12.9)

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where the states Φ1 , . . . , Φ4 are given in Section 12.11. Consider evaluating G`u as an example. From Eq. 12.12.9 −ωT G`u αβ = e

4 X

α β hΩ|ψ`p |Φi ihΦi |ψ¯pu |Ωi

i=1

Using β ψ¯pu |Ωi = N −1 |Φβ i,

β = 1, 2 ; N =

p

2ω/(ω + m)

yields −ωT G`u αβ = e

=N

4 X α hΩ|ψp` |Φi ihΦi |Φβ iN −1

i=1 −1 −ωT

e

α α ¯β hΩ|ψpl |Φβ i = e−ωT hΩ|ψp` ψpu |Ωi

(12.12.10)

where the orthonormality equation has been used. The wave functions in Section 12.11 are given in the helicity basis. To evaluate the propagator the completeness equation Z ¯ ¯ ψ).hψ¯u , ψ` | I = DψDψ| ψ¯` , ψu iT (ψ, is required. To evaluate Eq. 12.12.10, note that the vacuum functionals generate the term given in Eq. 12.11.2; performing the fermion integration using Eq. 6.6.11 yields Z −ωT ¯ ¯ ψ)hψ¯u , ψ` |ψ α ψ¯β |Ωi G`u = e DψDψhΩ| ψ¯` , ψu iT (ψ, αβ p` pu Z n Z o −ωT α ¯β † ¯ ¯ ¯ =e DψDψψp` ψpu Ω(ψu , ψ` )Ω (ψu , ψ` ) exp − ψ¯p ψp p



2

Z

m+ω α ¯β ¯ e−ωT DψDψψ p` ψpu 2ω   Z Z p (ψ¯u σ3 ψl − ψ¯l σ3 ψu ) − ψ¯p ψp × exp − p p m+ω Z Z −ωT e d3 p p(σ3 )αβ ; ≡ = 2ω (2π)3 p =

Transforming back to the original momentum basis yields, in matrix notation, the expected result G`u =

e−ωT p·σ 2ω

Similarly, all the other elements of GpT can be calculated using the states of the first excited level.

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Summary

The free Dirac field’s evolution kernel was evaluated using the lattice formulation. The field coordinates ψ¯u (x), ψ` (x) were used due to the lattice approach. In Chapter 13, the continuum Dirac theory is recast using ψ¯u (x), ψ` (x) as the dynamical degrees of freedom. The main difficulty with the treatment of the lattice Dirac field is the question of chiral symmetry. For the choice of the γµ ’s made, the matrix γ5 interchanges upper and lower components. Hence chiral transformations are non-diagonal on the coordinates chosen for the state space; to describe the state space of a chiral field another set of coordinates need to be used. Other ways of defining the lattice fermions that are non-local in time cannot be used as the state space discussed in this Chapter crucially hinges on the nearest-neighbor coupling in time, and on the use of the projectors (1 ± γ0 ) in the Lagrangian. The calculation performed can be extended to interacting Dirac field and consists of perturbatively calculating with an open boundary interacting fermion system. The boundary values enter via the classical solution that acts as a background field for the quantum variables [Baaquie (1986)]. The evolution kernel can be used to calculate the transition amplitudes for nonlocal field configurations like vortices, strings, and so on. It also provides a different perspective for studying quantum field theories.

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Chapter 13

Dirac Hamiltonian

The Feynman path integral is usually defined using the infinite time action, since one is normally calculating the scattering matrix for which infinite time processes are required. In effect, one evaluates the time ordered vacuum expectation values of the field operators using the infinite time path integral. However, for time independent bound states, the correct way to describe the system is to use the time independent eigenenergies and eigenfunctionals of the quantum field, and one is naturally led to the Hamiltonian formulation. For example, in Baaquie (1983a, 2018) the eigenenergies of the (bound) eigenstates of two-dimensional Quantum Electrodynamics are evaluated, and it is found that the single fermion eigenenergies and eigenstates are eliminated in the interacting theory showing the confinement of the fundamental fermions. Since the mesons and hadrons are relativistic quark confining bound states of the interacting quarkgluon quantum field, the description of such bound states necessitates the study of the Hamiltonian approach to Quantum Chromodynamics. The first step is to study the Hamiltonian formalism for the free quark field. In this Chapter, the Dirac Hamiltonian is extensively studied using the anticommuting fermionic variables, and in particular its close connection with path integration is shown. The discussion in Chapter 12 on the Dirac field is based on the path integral. The eigenfunctions obtained earlier in Section 12.11, using the path integral, are re-derived using the Hamiltonian – illustrating the complimentary nature of the Hamiltonian and path integral formulations. One of the advantages of using fermion calculus is that the Hilbert space of the Dirac field can be given an explicit realization in terms of fermionic variables and many features of the Dirac field state vectors have a transparent representation. The Dirac vacuum, which is the single most important state of the Dirac field, is derived exactly and its correlations and other properties are studied in detail. This is in contrast to the approach taken in the standard canonical approach [Peskin and Schroeder (1995)], for which the state space in general is defined in terms of the action of anticommuting creation and destruction operators. Furthermore, as shown in Section 13.8, the fermionic variables allow an explicit computation of the 243

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evolution kernel of the Dirac field, something that is not clear how to carry out in the canonical formulation. One of the main results of this Chapter is to derive the boundary term for the finite time Dirac action – and show that the evolution kernel is given precisely by the boundary term action. 13.1

Fermionic Operators

The results obtained in Section 6.3 are applied to the Dirac field. The coordinate operators are  op    ψ¯u ψ¯u ¯ hψu , ψ` | ¯op |f i = f (ψ¯u , ψ` ) (13.1.1) ψ` ψ` The conjugate operators are given by the completeness equation.  op    Z ψ¯ ζ¯` ¯ ¯ (ζ¯u , ζ` ) hψ¯u , ψ` | ¯`op |f i = dζdζ hψ¯u , ψ` |ζ¯` , ζu i exp(−ζζ)f (13.1.2) ψu ζu The inner product yields  δ    − δψ` ¯ ζ` (13.1.3) hψ¯u , ψ` |ζ¯` , ζu i =  δ  exp(ψ¯u ζu + ζ¯` ψ` ) ζu ¯ δ ψu

where the fermionic derivatives are definedas in Chapter 6. Hence  δ  op  − δψ` ψ¯` |f i =  δ  f (ψ¯u , ψ` ) hψ¯u , ψ` | op ψu ¯

(13.1.4)

δ ψu

In summary, the following is the operator representation in the coordinate basis !  δ δ  ψ = δψ¯u , ψ¯ = ψ¯u , − (13.1.5) δψ` ψ` and yield the anticommutation equations {ψ¯α , ψ¯β } = {ψα , ψβ } = 0 ; {ψ¯α , ψβ } = γ0αβ (13.1.6) The anticommutators are a result of our definition of the metric, and we will see that for the lattice there is a more general anticommutator than Eq. 13.1.6. Note that in the usual scheme for quantization, ψα is taken as the independent degree of freedom, ψ¯α the conjugate variable and the anticommutation relation is postulated. In the lattice approach, the anticommutator is the result of the definition of the state space and the metric defining the inner product on this space, and can have forms more general than the canonical anticommutation relation. ¯ ψ) of the operLet G be an arbitrary operator. Then the matrix elements G(ψ, ator, as in Eq. 6.8.2, are defined by ¯ ψ)hψ¯u , ψ` |ψ¯` , ψu i = G(ψ, ¯ ψ) exp(ψψ) ¯ hψ¯u , ψ` |G|ψ¯` , ψu i ≡ G(ψ, The left hand side is sometimes called the kernel of the operator and the quantity ¯ ψ) the normal symbol. The conjugate operator G is defined by conjugating G(ψ, ¯ ψ) , and operator G is Hermitian if, for arbitrary |f i and each matrix element G(ψ, hg|, we have hf |G† |gi ≡ hg|G|f i∗ = hf |G|gi : Hermitian (13.1.7)

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13.2

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Lattice Dirac Hamiltonian

The Hamiltonian operator for the Dirac field has been obtained in Section 12.8 starting from the finite time lattice action. To obtain the Hamiltonian from the action, one essentially has to ‘divide’ out the metric and the inner product of the Dirac field coordinate eigenstates from the expression for e−aH . All coordinates – both lattice and continuum as well as space and momentum indicies – in this Chapter are in three dimensions as the Hamiltonian makes no reference to time. For this reason, the vector notation will not be used. The free Dirac Hamiltonian has been derived in Section 12.8 and is given, up to a constant, by Eq. 12.8.5 X 1 ψ¯~n ψ~n H = − (2K − 1) a ~ n   KX ¯ ψ~n (1 − γi )ψ~n+i − ψ¯~n+i (1 + γi )ψ~n (13.2.1) + a ~ ni

The lattice Hamiltonian obtained in Eq. 13.2.1 is Hermitian as defined in Eqs. 6.8.5 and 13.1.7 – and hence has only real eigenvalues. Note that the mass term for H arises partly from both the lattice time derivative term of the action and from the coordinate inner product. Similarly, the nearest neighbor coupling term in H has a contribution from the inner product. Define the following new rescaled lattice variables r r 1 − 2K ad−2 ψn = ψ(x) ; x = na ζn = a 2J r r 1 − 2K ¯ ad−2 ¯ ζ¯n = ψn = ψ(x) (13.2.2) a 2J where 2K =

1 1 ; 2J = (ma + d) (ma + d − 1)

Hence, the Dirac Hamiltonian is given by o X Xn ¯ ζ) = H(ζ, ζ¯n ζn − J ζ¯n (1 − γi )ζn+i + ζ¯n+i (1 + γi )ζn n

(13.2.3)

(13.2.4)

ni

where J is the nearest-neighbor coupling constant given by Eq. 13.2.3. Note that to obtain H from the lattice action, the time coupling terms have been dropped and the nearest-neighbor coupling in the Lagrangian suitably altered. The metric in the rescaled variables is given by o n X X ¯ ζ) = exp − aJ ζ¯n ζn + aJ (ζ¯n ζn+i + ζ¯n+i ζn ) T (ζ, K n ni n Z o = exp − λp ζ¯p ζp (13.2.5) p

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where  X aJ  1 − 2K cos pi ; λp = K i

Z ≡ p

3 Z Y i=1

+π

−π

dpi 2π

(13.2.6)

Note that p~ is the dimensionless periodic momentum. The metric is positive definite only if 0 < 2K