Pedagogic Aids to
Quantum Field Theory
Statement of Web Site Purpose
The goal of this web site is to provide assistance in learning quantum field theory (QFT). It is an outgrowth of my own notes on how one could present the subject in a more student friendly manner, compiled when I was first studying QFT over 20 years ago. In that sense, it is a sort of “peer instruction” (a teaching tool of recognized merit.)
The material herein is part of a textbook I am presently writing, in which I hope to make learning QFT markedly easier, and thus more efficient, than do typical texts. The pedagogic principles to be employed in so doing are summarized in a preliminary preface to the text (see Preface.)
As one example of these principles, the presentations herein are NOT concise, but step-by-small-step, with extensive explanation and elaboration. Conciseness, in my experience the source of great inefficiency in learning, and considerable suffering by new students, is much in fashion among the well practiced. I go to great pains to avoid it.
As another example, material is often overviewed/summarized in what I call a “Wholeness Chart” (called a “Unifying Chart” in my earlier work), so named because it reveals in chart form the underlying connections that unite various aspects of a given theory into a greater whole. Learning a computer program line-by-line is immensely harder than learning it with a block diagram of the program, showing major sections and sub-sections, and how they are all interrelated. The same is true in physics. Wholeness Charts for physics parallel block diagrams for computer programs and, I submit, are just as advantageous for the learning process. A picture is worth a thousand words, and in my own learning experiences, I have found Wholeness Charts to be invaluable aids that coalesce a lot of different information into one central, easy-to-see, easy-to-understand, and easy-to-reference, framework.
This web site is neither static nor comprehensive, as the material in it is continually changing and expanding. Rather than wait until the book is finished, I felt it advantageous to make certain core material available online as it becomes completed.
Regardless of the state of evolution of the site, I hope that whatever students of QFT find here will be helpful.
Robert D. Klauber, PhD
Note: I have been informed that some pages here do not reproduce very well for those who have disabled JavaScripts. Those who have done so may need to turn it back on.
Feedback Welcome
If you find errors herein or have suggestions on how any aspect of QFT might be presented in an easier to understand manner, please let me know via rdklauber###quantumfieldtheory.info (### used to thwart bots and spam - change ### to “at” sign.)
Contents
Initial chapters of book, hopefully to be completed in Spring 2012, titled
Student Friendly Quantum Field TheoryTM
Introduction-Background Chapter 1 of text entitled “Bird’s Eye View”. Intro to, background for, and simplified overview of QFT. Comparison of QFT, non-relativistic quantum mechanics, and relativistic quantum mechanics. Overview of physics and how these three quantum theories are related to each other and to classical theory.
Latest version: November 15, 2011 text inserted into Sect 1.1 on pg 1. Prior version, May 10, 2011 some changes to Sect. 1.5.3.
Chapter 2. Foundations Natural units, notation, summary of classical variational mechanics, Shroedinger and Heisenberg pictures, summary of quantum theories, 2nd quantization. Appendix: Simplified intro to contravariant/covariant components.
Latest version: April 10, 2010.
Part I: Free Fields
Chapter 3. Scalars: Spin 0 Fields Relativistic quantum mechanics (RQM): deducing the wave equation, relativistic probability density, explanation for “funny” relativistic normalization constants, negative energies. Quantum field theory (QFT): deducing the field equation, deriving coefficient commutators, number operator form of Hamiltonian, vacuum energy, creation and destruction operators, normal ordering, 4 currents, observable operators, bosons and commutators, QFT states as harmonic oscillators, step-by-small-step derivation of the propagator.
Latest version: January 20, 2012 version has small addition middle of pg 77 and added Appendix C. July 15, 2011 version added Appendix B plus changes on pg 77 and corrections to Feynman propagator derivation from January 26, 2011 version.
Chapter 4. Spinors: Spin 1/2 Fields Development of the Dirac equation, its solutions, spinors, their properties, probability density, energies of the solutions, spin operator, helicity operator for relativistic quantum mechanics (RQM). QFT Lagrangian, Hamiltonian, anti-commutation relations, number operators, Dirac vacuum energy, positive energy for real Dirac particles, creation and destruction operators, four-current, charge operator, momentum operator, QFT spin and helicity operators, subtleties regarding relativistic spin, spinor Feynman propagator.
Latest version: March 9, 2012 has modifications to Appendix A and minor typos corrected over July 12, 2011 version, which had corrections in signs in spinor propagator derivation of March 2, 2011 version. pgs 119 to 122.
Chapter 5. Vectors: Spin 1 Fields Review of classical electromagnetism, 3D+1 formulation and 4D (covariant) formulation. RQM for photons. QFT for photons. At end is Free Fields Wholeness Chart summarizing Chaps. 3, 4, and 5.
Version date: November 15, 2011 typo corrections. Prior version, Aug 4, 2011, had typo corrections and change to Sect. 5.3.5 pg 149 over July 12, 2011 versions, which had sign corrections in Wholeness Chart 5-4 propagator derivation summary section over May 28, 2011 version.
Chapter 6. Symmetry, Invariance, and Conservation for Free Fields Simplified explanation of symmetry. Symmetry of the laws of nature as discovered by Einstein. Symmetry in classical mechanics. QFT: Lorentz symmetry of form of Lagrangian density. Noether’s theorem. Deriving charge conservation using Noether’s theorem. Intro to gauge theories.
Version date: December 27, 2011 revised Prob. 13 over November 15, 2011 version, which had minor type corrections over November 1, 2011 version. Aug 2, 2011 vers had typo corrections and modified Sect 6.6 over July 27, 2011 version.
Free Fields Wholeness Chart Overview of scalars (Chap. 3), spinors (Chap. 4) and vectors (Chap. 5.) Free fields summary of theory derivation and key relations. Ranges from 2nd quantization and field equations to observables and propagators. This is the same chart found at the end of Chap. 5.
Latest version Aug 4, 2011. Typos corrected plus some revisions to propagator derivation summary Feb 8, 2011.
Part II: Interacting Fields
Chapter 7. Interactions: The Underlying Theory Electromagnetic interactions in relativistic quantum mechanics (RQM). General theory of interactions in QFT, including Interaction Picture, S operator, S matrix, Dyson expansion of the S operator, and Wick’s theorem.
Latest version: Dec 1, 2011 corrected eq 7-39 and associated eq in WhCharts 7-1 and 7-2. Oct 21, 2011 version added new Sect. 7.3.11 on pg 195. Sept 23, 2011 vers had minor corrections/changes to Aug 5, 2011 version. Aug 5 version had Sect 7.3.10 added, typos corrected over July 26, 2011 version.
Chapter 8. QED: Interaction Theory Applied to Electromagnetism Introductory sections.
Latest version: July 26, 2011. Complete chapter in forthcoming book.
Interacting Fields Wholeness Chart Overviews Chaps 7 and 8, the essence of Part 2 of QFT. Interacting fields summary of theory derivation and key relations. Ranges from operator observables to Feynman rules.
Latest version Sept 7, 2011.
The following chapters will be in forthcoming book. Chap 17 is available for viewing now by clicking link below.
Chapter 9. Higher Order Corrections. The source of the notorious infinities in QFT transition amplitude calculations.
Chapter 10. The Vacuum Revisited. The famous lay literature particle/antiparticle pair creation and destruction in the vacuum scenario examined in light of QFT.
Chapter 11. Symmetry, Invariance, and Conservation for Interacting Fields. The role of symmetry in interaction theory.
Part III: Renormalization: Conquering Those Notorious Infinities
Chapter 12. Overview of Renormalization. Simplified overview of renormalization with minimum math. Redefining charge and mass leads to a finite theory.
Chapter 13. Renormalization Toolkit. Assembling some math relations needed in renormalization.
Chapter 14. Putting it all together. Formal renormalization procedure.
Chapter 15. Regularization. Evaluating the unbounded integrals.
Part IV: Scattering
Chapter 16. Scattering in QFT. From transition amplitudes/probabilities to scattering cross sections, anomalous magnetic moment, and Lamb shift.
Addenda
Chapter 17. Path
Integrals in Quantum Theories: A Pedagogic 1st Step This
chapter is available by clicking on this link. Feynman’s many paths approach: the
other way besides canonical quantization
to do QFT.
Latest version October 24, 2011. includes minor additions and typo corrections to June 7, 2011 and May 6, 2009 versions.
Chapter 18. Ontology of QFT. What is really real in QFT.
Summary
Chapter 19. Summary of QFT Basic Principles and QED. An overview of the entire book.
End of Book Material
Auxiliary and Advanced Material:
Probability Density in Relativistic QM and QFT Updated version is now part of Chap. 3 above. See Sects. 3.1.4 and 3.7. The differences between probability density in non-relativistic quantum mechanics and relativistic quantum theories. A confusing topic for students not explained in any text known to the author.
Why we use "funny" normalization factors in the relativistic solutions This is actually the same link as probability density above. These two topics are closely related.
Derivation of the Propagator - Step by Baby Step Scalar propagator derivation now included as part of Chap. 3 above as Section 3.13, near the end of the chapter. Simplified and extensive [i.e., NOT concise] and easier to follow than almost any text. Summary near end of Free Field Wholeness Chart, see link above. Spinor propagator derivation is in Chap 4.
Compton Scattering Transition Amplitude Step by small step derivation showing general methodology for finding any interaction transition amplitude. This is the formal, i.e., long, method, not using Feynman rules. Feynman rules are the shortcut derived from this method.
Chirality and Helicity (Added October 29, 2007)
Chirality vs Helicity summary A summary chart explaining, comparing, and contrasting chirality and helicity, two concepts that are often confused for one another. Modified January 18, 2012.
Chirality and Helicity in Depth Closer look at chirality and helicity using examples, rather than pure theory. Also shows how chirality and helicity become effectively the same thing when v=c.
The Seesaw Mechanism A way in which the low mass of neutrinos may be explained. Advanced material. New material added on January 18, 2012.
Non Eigen States, Wave Packets, and the Hamiltonian in QFT A look at states in QFT which are not eigenstates of three momentum k, and additionally, wave packet states in QFT. The action of the QFT Hamiltonian on these states is investigated. This material does not seem to be available in any QFT texts.
Posted: November 21, 2011. Added Wholeness Chart summary of entire treatment over version of March 13, 2010
More to come in the future.
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Copyright
Copyright © 2005, 2007, 2009, 2010, 2011, 2012 Robert D. Klauber. Until and unless other notice is given, free distribution of paper and electronic copies to students is permitted and encouraged, provided appropriate notification is given of this web site as the source. Free distribution of ten or fewer copies to others is also permitted. To include any original material, or original presentation of known material, contained herein in a publication, written prior approval from the author is required.