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
Original posting year of site: 2005
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
Major Sections:
Introduction-Background Chapter 1 of text entitled “Bird’s Eye View”. Intro to, background for, and simplified overview of QFT.
Latest version: March 9, 2010, incorporating a spelling correction, 3 added words in Sect 1.5.2, and three additional rows at the end of Wholeness Chart 1-2 to September 18, 2009 version
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 has corrections on pg 34 plus other minor changes/corrections to Oct 1, 2009 version.
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 July 29, 2010.
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. More QFT to come.
August 21, 2010
Free Fields Wholeness Chart Part 1 of QFT. Free fields summary of theory derivation and key relations. Ranges from 2nd quantization and field equations to observables and propagators. Overviews scalars (Chap. 3), spinors (Chap. 4) and vectors (Chap. 5.)
Latest version August 21, 2010.
Interacting Fields Wholeness Chart Part 2 of QFT. Interacting fields summary of theory derivation and key relations. Ranges from operator observables to Feynman rules.
Latest version 2005.
Renormalization Wholeness Chart Part 3 of QFT. Not completed at present. Renormalization summary of theory derivation and key relations.
Scattering and Decay Wholeness Chart. Part 4 of QFT. Not completed at present. Scattering and decay summary of theory derivation and key relations.
Path
Integrals in Quantum Theories: A Pedagogic 1st Step pdf
vers Feynman’s many paths approach:
the other way besides canonical quantization
to do QFT.
Added April 30, 2009. Modified
May 6, 2009.
Auxiliary Sections:
Probability Density in Relativistic QM and QFT Updated version is now part of Chap. 3 above. 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 Simplified and extensive [i.e., NOT concise], and easier to follow than almost any text. Summary near end of Free Field Wholeness Chart, link above. Now included as part of Chap. 3 above.
Revised and improved: June 26, 2010.
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.
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. Added on October 19, 2007.
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: March 13, 2010
More to come in the future.
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Copyright
Copyright © 2005, 2007, 2009, 2010 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.