Quantum Field Theory

This is an introductory course on gauge theories for students of the fourth year having already attended Theoretical Physics. Although most of the subject is worked out in detail starting from a review of the classical theory of fields, I assume that students already know the properties of relativistic equations for particles of spin 0, 1/2 and 1. The course is divided into two parts. Part A deals with Quantum Electrodynamics and part B with non-abelian gauge theories. The students are allowed to attend either part A or part B alone.

Recommended Textbooks

• An Introduction to Quantum Field Theory
  by M. E. Peskin and D. V. Schroeder
  1995, Addison Wesley
• The Quantum Theory of Fields, vol I and II,
  by S. Weinberg,
  1995 Cambridge University Press

Reference Program (A.Y. 1998-99)

PART A

Review of Classical Electrodynamics

Symmetries and Conservation Laws

The Dirac Monopole

Path Integrals and Perturbation Theory

Beyond the Tree Approximation

Renormalization Theory for QED

PART B

Non-abelian Gauge Theories

Renormalization Group and Asymptotic Freedom

Spontaneously Broken Global Symmetries. The Goldstone theorem

Spontaneously Broken Local Symmetries: The Higgs mechanism

Anomalies

Effective Program (A.Y. 1998-99)

[2/3/99:] General discussion about the plan of the lectures

Part A

Review of Classical Electromagnetism

• [3/3/99:] *Maxwell's equations in covariant notation [IZ/1-1-2]
  *Solution of the Bianchi identities: the electromagnetic field [IZ/1-1-2]
  *Exercise 1: Explicit expression of the electromagnetic field in terms of the field strength
  *Exercise 2: Construction of the most general action reproducing the Maxwell's equations
• [4/3/99:] *End of exercise 1
  *Action for a system of point charges coupled to the electromagnetic field [LL/III,23]
• [5/3/99:] *Exercise 3: Electromagnetic field radiated by a classical accelerating particle [PS/6.1]
• [9/3/99:] *End of Exercise 3
• [10/3/99:] Energy and number of photons emitted form an accelerating particle: the infrared catastrophe [PS/6.1(IZ/1-3-2)]
• [11/3/99:] Gauge choice for a classical system; generalization of the Lorentz gauge
  Gauge invariant coupling between the electromagnetic and the Dirac fields

Symmetries and Conservation Laws

• [12/3/99:] Noether's theorem for internal and space-time continuous symmetries
• [16/3/99:] Exercise 4: infinitesimal variation of a classical action related to general space-time and field transformation
• [17/3/99:] Energy-momentum tensor for the free electromagnetic field; improved energy-momentum tensor; Lorentz invariance, angolar momentum and boost
  

A little digression about duality

• [17/3/99] Duality rotation on electric and magnetic fields in vacuum
• [18/3/99:] The magnetic monopole: singularity of the electromagnetic field for a monopole at rest; the Dirac's solution; quantization of the electric charge
• [19/3/99:] *The Schwinger-Zwanziger quantization condition; the lattice of charges and the theta parameter; SL(2,Z)

Path Integrals

• [23/3/99:] Path integral in Quantum Mechanics [PS/9.1]:
  N=1 degree of freedom; classical limit; generalization to N>1
• [24/3/99:] Exercise 5: Double-slit electron diffraction [PS/9.1]
  The Aharonov-Bohm effect [FLS/15-5]
  Matrix element of a time-ordered product of Heisenberg operators
  Path integral in field theory: the real scalar field [PS/9.2]
• [25/3/99:] Green functions from a path integral [PS/9.2]
  Generating functional of all Green function Z[J] ; the free case
• [26/3/99:] Short discussion about the evaluation of Z[J] in the interacting case (lattice; expansion of the Euclidean action around stationary/finite-action solutions; perturbative expansion)
  Perturbative expansion of Z[J]; Feynman rules in coordinate space
• [30/3/99:] Relation among Green functions and S-matrix elements: the reduction formula [IZ/5-1-2,5-1-3]
• [31/3/99:] Reduction formula in momentum space; poles of Green functions; examples for the scalar field
  Demonstration of the reduction formula
• [8/4/99:] Path integral for the free electromagnetic field; gauge fixing; photon propagator [PS/9.4]
• [9/4/99:] Path integral for the Dirac field; integration over Grassmann variables [PS/9.5]
• [13/4/99:] Exercise 6: Gaussian integral over Grassmann variables
  Feynman rules for Quantum Electrodynamics [PS/9.5(see also 4.7,4.8)]
  
• [14/4/99] Cross section for muon pair production from electron-positron scattering at lowest order; hadron production from electron-positron collisions; the ratio R and the color degree of freedom [PS/5.1].

Beyond the tree level approximation

• [15/4/99] Dimensional regularization [PS/7.5]
  Vacuum Polarization [PS/7.5]
• [16/4/99] Wave function renormalization of the electromagnetic field [PS/7.5]
• [20/4/99] Renormalization conditions and renormalization schemes [PS/10.3]
• [21/4/99] Symmetries of the quantum system; conservation of the electromagnetic current between physical states
  Generating functional W[J] of connected Green functions[PS/11.3,PS/11.5]
• [22/4/99] One-particle irreducible Green functions and the effective action [PS/11.3,PS/11.5]; the loop expansion
• [23/4/99] Functional identity for the effective action in QED; transversality of photon 1PI functions
• [27/4/99] General parametrization of the electron-photon-electron vertex when the electron is on-shell [W/10.6 (also PS/6.2)]
  Normalization of the form factors at q²=0: F(0)+G(0)=1
  Physical meaning of the form factors F and G
• [28/4/99] One-loop contributions to F and G from the vertex correction [W/11.3 (also PS/6.3)]
  The anomalous magnetic moment of the electron: the Schwinger term
• [29/4/99] One-loop contribution to F from the wave function renormalization of [PS/7.1 (also W/11.4)] the Dirac field
  The Ward-Takahashi identity and the universality of the electric charge [PS/7.4,PS/10.3]
• [4/5/99] Exercise 7: Evaluation of the electron wave function renormalization constant at one-loop order in the Feynman gauge [PS/7.1]
• [5/5/99] One-loop cross-section for elastic electron-muon scattering
  Infrared divergence; soft photon radiation [PS/6.1]
• [6/5/99] Cancellation of infrared divergences in the limit -q² much larger than m_e²: the Sudakov double logarithm and the Sudakov form factor [PS/6.4]

General Renormalization Theory

• [7/5/99] Review of on-shell renormalization for QED [PS/10.3]
  Superficial degree of divergence of a one-particle-irreducible diagram in a general theory [W/12.1(PS/10.1)]
• [11/5/99] Renormalizable theories; polynomial character of divergences in one-loop vertex functions [W/12.2(PS/10.1)]
• [12/5/99] List of one-loop divergent 1PI functions in QED [W/12.2(PS/10.1)]
  Non-polynomial character of divergences in higher orders; two-loop vacuum polarization in words [W/12.1,12.2(PS/10.4)]
  Compatibility between renormalization and Ward identities: unitarity of QED and independence of physical results on the gauge parameter
  Definition of gauge anomaly

End of Part A

Part B

Non-abelian gauge theories

• [13/5/99] Classical action for a non-abelian gauge theory [PS/15.1,15.2,15.4]
• [18/5/99] Path integral for non-abelian gauge theories [PS/16.2]
  Gauge fixing and Faddeev-Popov terms in the Lorentz gauge [PS/16.2]
  BRS symmetry [PS/16.4]
• [19/5/99] Nilpotency of BRS transformations [PS/16.4]
  Action for a general gauge choice; ghost number conservation
  Definition of the space of physical states via the BRS condition in the operator formalism [PS/16.4]
• [20/5/99] Slavnov-Taylor identities for the generating functional of 1PI functions [IZ/12-4-2].
• [25/5/99] Short discussion about one-loop renormalization of a non-abelian gauge theory [IZ/12-4-3]
  

The Renormalization Group

• [25/5/99] Motivations and aims of the renormalization group
• [26/5/99] The Callan-Symanzik equation [PS/12.2]
  Asymptotic behaviour of proper Green functions [PS/12.3]
  
• [27/5/99] Fixed points of the renormalization group equations [PS/12.3]
  The origin in the space of coupling constants and the role of perturbation theory
  Computation of the beta function for the scalar theory at one-loop order.
• [1/6/99] Beta function of QED in the Minimal Subtraction (MS) scheme of dimensional regularization [W/18.6]
  Beta function of QED in the one-loop approximation [W/18.6]
  Quantum Cromo-Dynamics (QCD): its action and relation between bare and renormalized coupling constant [PS/16.1].
• [2/6/99] Short survey of the computation of the beta function of QCD in the one-loop approximation; asymptotic freedom of QCD [PS/16.5].

Anomalies

• [2/6/99] Global and local symmetry of a Quantum Field Theory; definition of anomaly
• [3/6/99] The axial current in QED: Ward identity for the generating functional of proper Green functions
  Violation of the Ward identity by the triangle diagram; evaluation of the triangle diagram in dimensional regularization [PS/19.2]
  Comments: Adler-Bardeen no-renormalization theorem

References

• [IZ] Quantum Field Theory
  by C. Itzykson and J.-B. Zuber
  1980, McGraw-Hill
  
• [LL] Fisica Teorica, Teoria dei Campi
  L. D. Landau and E. M. Lifsits
  Ed. Riuniti
  
• [PS] An Introduction to Quantum Field Theory
  by M. E. Peskin and D. V. Schroeder
  1995, Addison Wesley
  
• [W] The Quantum Theory of Fields, vol I and II
  by S. Weinberg
  1995 Cambridge University Press
  
• [FLS] Electromagnetism, vol I
  by R. Feynman, Leighton and Sands
  

Exercises

To absorbe the subject of this course you also need to practice it. You are strongly invited to solve the exercises proposed below.

Path Integrals and Perturbation Theory

• Exercise 1

Beyond the Tree-Level Approximation

• Exercise 1
• Exercise 2

I'm sorry, but I have no time to complete this section, at the moment.

How to prepare the examination

How can you judge that you have reached a good level of understanding and that you are "ready to go"? I tried to figure out a number of questions that you should be able to answer. If you have doubts on any of these points, please ask me to clarify them during the class. Do not esitate. To begin with, have a look to what you should know about the chapter Beyond the tree level approximation. If you have been able to answer to all questions, you can move to General renormalization theory . If you are interested in the second part of the course, you may check your understanding by inspecting Non-abelian gauge theories first. Then go to The renormalization group . Have you passed the previous tests? Very good! You have almost finished. There are few remaining questions on the fascinating subject of Anomalies : answer them carefully. Are you reasonably happy with your answers to the previous questions? If this is the case, you are ready for the examination.

Last modified: May 12 1999 - (feruglio@padova.infn.it)

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