Current Algebra and Anomalies

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Introduction

• The Physical Currents
  • Electromagnetic Hadron Currents
  • Weak Lepton Currents
  • Weak Hadron Currents
  • Weak Non-Leptonic Interactions

• Current Commutation Relations

• PCAC
  • Neutron β Decay
  • PCAC for Neutrino Reactions
  • The Adler Consistency Condition
  • Discussion

• Current Algebra: Methods

• Some Applications of Current Algebra
  • Cabibbo-Radicati Sum Rule
  • The Adler Sum Rule for Neutrino Reactions
  • The Adler-Weisberger Formula

• The A-W Formula and Generalizations

• Semi-Leptonic Decay Processes
  • K_ℓ3 Decay
  • K_ℓ4 Decay

• Non-Leptonic Decays
  • K → 3π Decays
  • Hyperon Decay

• References

The following sections are included:

• Introduction

• Canonical and Space-Time Constraints in Current Algebra
  • Canonical Theory of Currents
  • Space-Time Constraints on Commutators
  • Space-Time Constraints on Green's Functions
  • Space-Time Constraints on Ward Identities
  • Schwinger Terms
  • Discussion

• The Bjorken-Johnson-Low Limit

• The π ⁰ → 2_γ Problem
  • Preliminaries
  • Sutherland-Veltman Theorem
  • Model Calculation
  • Anomalous Ward Identity
  • Anomalous Commutators
  • Anomalous Divergence of Axial Current
  • Discussion

• Electroproduction Sum Rules
  • Preliminaries
  • Derivation of Sum Rules, Naive Method
  • Derivation of Sum Rules, Dispersive Method
  • Model Calculation
  • Anomalous Commutators
  • Discussion

• Discussion of Anomalies in Current Algebra
  • Miscellaneous Anomalies
  • Non-Perturbative Arguments for Anomalies
  • Models without Anomalies
  • Discussion

• Approximate Scale Symmetry
  • Introduction
  • Canonical Theory of Scale and Conformal Transformations
  • Ward Identities and Trace Identities
  • False Theorems
  • True Theorems

• EXERCISES

• SOLUTIONS

The following sections are included:

• Introduction

• Description of Vector Gauge Theories
  • Action and Equations of Motion
  • Symmetries
  • Couplings to Matter Fields
  • Classical Gauge Fields

• Quantization
  • Constrained Quantum Mechanical Example
  • Quantizing a Yang-Mills Theory
  • θ Angle
  • Functional Integral Formulation
  • Semi-Classical Picture for the Vacuum Angle

• Fermion Interactions and Anomalies
  • Quantized Fermions
  • Gauge Invariance and Fermion Couplings
  • Chiral (or Axial Vector) Anomalies
  • Some Physical Consequences of Anomalies

• Quantization Constraints on Physical Parameters
  • Dirac Point Monopole
  • Three-Dimensional Gauge Theories
  • Four-Dimensional Gauge Theories
  • Discussion

• Conclusions
  • Abelian Functional Gauge Structures in Non-Abelian Theories
  • Cochains, Cocycles and the Coboundary Operation
  • Cocycles in Quantum Mechanics
    • 1-Cocycle
    • 2-Cocycle
    • 3-Cocycle
  • Cocycles in Quantum Field Theory
    • 0-Cocycle
    • 1-Cocycle
    • 2-Cocycle
    • 3-Cocycle
  • Higher-Dimensional Gauge Dynamics Suggested by the Chern - Simons Term
  • The Dirac Sea and Chiral Anomalies
  • Discussion

• Acknowledgement

• Solutions to Selected Exercises

• References

The following sections are included:

• Introduction

• Chiral Anomalies and Differential Forms

• Transformation Properties of the Anomalies

• Identification and Use of the Anomalies

• Normalization of the Anomalies

• Appendix I. A Simple Formula for α_2π–2

• References

The gauge structure of anomalies and the related currents is analyzed in detail. We construct the covariant forms for both the currents and the anomalies for general gauge theories in even-dimensional space-time. The results are then extended to determine the structure of gravitational anomalies. These can always be interpreted as anomalies for local Lorentz transformations.

A new restriction on fermion quantum numbers in gauge theories is derived. For instance, it is shown that an SU(2) gauge theory with an odd number of left-handed fermion doublets (and no other representations) is mathematically inconsistent.

A new mathematical framework for the Wess-Zumino chiral effective action is described. It is shown that this action obeys an a priori quantization law, analogous to Dime's quantization of magnetic change. It incorporates in current algebra both perturbative and non-perturbative anomalies.

It is shown that in certain parity-violating theories in 4k + 2 dimensions, general covariance is spoiled by anomalies at the one-loop level. This occurs when Weyl fermions of spin-½ or or self-dual antisymmetric tensor fields are coupled to gravity. (For Dirac fermions there is no trouble.) The conditions for anomaly cancellation between fields of different spin is investigated. In six dimensions this occurs in certain theories with a fairly elaborate field content. In ten dimensions there is a unique theory with anomaly cancellation between fields of different spin. It is the chiral n = 2 supergravity theory, which is the low-energy limit of one of the superstring theories. Beyond ten dimensions there is no way to cancel anomalies between fields of different spin.

It is shown that ordinary baryons can be understood as soli tons in current algebra effective lagrangians. The formation of color flux tubes can also be seen in current algebra, under certain conditions.

The following sections are included:

• The H

• The Callan-Rubakov Effect

• Baryon Resonances

• References