quantum field theory

Sabine Hossenfelder’s “Lost in Math”

December 27, 2020 Incoherent ramblings , , , , , , ,

“Lost in Math” is a book that I’ve been curious to read, as I’ve been a subscriber to Sabine’s blog and youtube channel for quite a while.  On her blog and channel, she provides overviews of many topics in physics that are well articulated, as well as what appear to be very well reasoned and researched criticisms of a number of topics (mostly physics related.)  Within the small population of people interested in theoretical physics, I think that she is also very well known for her completely fearlessness, as she appears to have none of the usual social resistance to offending somebody should her statements not be aligned with popular consensus.

This book has a few aspects:

  • Interviews with a number of interesting and prominent physicists
  • A brutal take on the failures of string theory, supersymmetry, theories of everything, and other research programs that have consumed significant research budgets, but are detached from experimental and observational constraints.
  • An argument against the use of beauty, naturalness, and fine tuning avoidance in the constructions of physical theory.  Through the many interviews, we get a glimpse of the specific meanings of these words in the context of modern high level physical theories.
  • Some arguments against bigger colliders, given that the current ones have not delivered on their promises of producing new physics.
  • A considerable history of modern physics, and background for those wondering what the problems that string theory and supersymmetry have been trying to solve in the first place.
  • Some going-forward recommendations.

While there were no equations in this book, it is not a terribly easy read.  I felt that reading this requires considerable physics sophistication.  To level set, while I haven’t studied particle physics or the standard model, I have studied special relativity, electromagnetism, quantum mechanics, and even some introductory QFT, but still found this book fairly difficult (and I admit to nodding off a few times as a result.)  I don’t think this is really a book that aimed at the general public.

If you do have the background to attempt this book, you will probably learn a fair amount, on topics that include, for example: the standard model, general relativity, symmetry breaking, coupling constants, and the cosmological constant.  An example was her nice illustration of symmetry breaking.  We remember touching on this briefly in QFT I, but it was presented in an algebraic and abstract fashion.  At the time I didn’t get the high level view of what this meant (something with higher energy can have symmetries that are impossible at lower energies.)  In this book, this concept is illustrated by a spinning top, which when spinning fast is stable and has rotational symmetry, but once frictional effects start to slow it down, it will start to precess and wobble, and the symmetry that is evident at higher spin rates weakens.  This was a particularly apt justification for the title of the book, as her description of symmetry breaking did not require any mathematics!

Deep in the book, it was pointed out that the equations of the standard model cannot generally be solved, but have to be dealt with using perturbation methods.  In retrospect, this shouldn’t have surprised me, since we generally can’t solve non-harmonic oscillator problems in closed form, and have to resort to numerical methods for most interesting problems.

There were a number of biting statements that triggered laughs while reading this book.  I wish that I’d made notes of more of of those while I read it, but here are two to whet your appetite:

  • If you’d been sucking away on a giant jawbreaker for a century, wouldn’t you hope to finally get close to the gum?
  • It’s easy enough for us to discard philosophy as useless — because it is useless.

On the picture above.

I like reading in the big living room chair behind my desk that our dog Tessa has claimed as her own, so as soon as I get up for coffee (or anything else), she will usually come and plop herself in the chair so that it’s no longer available to me.  If she was lying on the floor, and my wife sits on “her” chair, she will almost always occupy it once Sofia gets up.  Ironically, the picture above was taken just after I had gotten to the section where she was interviewing Chad Orzel, of “How to Teach Quantum Mechanics to your Dog” fame.

Notes for Quantum Field Theory I (phy2403) now available in paper on amazon

May 2, 2019 phy2403 , ,

My notes (423 pages, 6″x9″) from the fall 2018 session of the University of Toronto Quantum Field Theory I course (PHY2403), taught by Prof. Erich Poppitz, are now available on amazon.com (through kindle-direct-publishing, formerly createspace).

These notes are available in three forms, two free, and one paper:

  • On amazon (kindle-direct-publishing) for $11 USD,
  • As a free PDF,
  • As latex sources (, makefiles, figures, …) to build/modify yourself.

This book is dedicated to dad.

Warning to students

These notes are no longer redacted and include whatever portions of the problem set 1-4 solutions I completed, errors and all.  In the event that any of the problem sets are recycled for future iterations of the course, students who are taking the course (all mature grad students pursuing science for the love of it, not for grades) are expected to act responsibly, and produce their own solutions, within the constraints provided by the professor.


The official course outline included:

  1. Introduction: Energy and distance scales; units and conventions. Uncertainty relations in the relativistic domain and the need for multiple particle description.
  2. Canonical quantization. Free scalar field theory.
  3. Symmetries and conservation laws.
  4. Interacting fields: Feynman diagrams and the S matrix; decay widths and phase space.
  5. Spin 1/2 fields: Spinor representations, Dirac and Weyl spinors, Dirac equation. Quantizing fermi fields and statistics.
  6. Vector fields and Quantum electrodynamics.



Final first draft of complete notes for UofT PHY2403, QFT I .

December 27, 2018 phy2403 , ,

I’ve now uploaded a new version of my class notes for PHY2403, the UofT Quantum Field Theory I course, taught this year by Prof. Erich Poppitz.

This update adds notes for all remaining lectures (up to and including lecture 23.)  I’ve made a pass with a spellchecker to correct some of the aggregious spelling erorss, and also redrawn three figures, replacing photos, which cuts the size in half!

I’ve posted the redacted version (316 pages).  The full version, with my problem set solutions (including errors) is 409 pages.

Feel free to contact me for the complete version (i.e. including my problem set solutions, with errors) of any of these notes, provided you are not asking because you are taking or planning to take this course.


  • Preface
  • Contents
  • List of Figures
  • 1 Fields, units, and scales.
  • 1.1 What is a field?
  • 1.2 Scales.
  • 1.2.1 Bohr radius.
  • 1.2.2 Compton wavelength.
  • 1.2.3 Relations.
  • 1.3 Natural units.
  • 1.4 Gravity.
  • 1.5 Cross section.
  • 1.6 Problems.
  • 2 Lorentz transformations.
  • 2.1 Lorentz transformations.
  • 2.2 Determinant of Lorentz transformations.
  • 2.3 Problems.
  • 3 Classical field theory.
  • 3.1 Field theory.
  • 3.2 Actions.
  • 3.3 Principles determining the form of the action.
  • 3.4 Principles (cont.)
  • 3.4.1 d = 2.
  • 3.4.2 d = 3.
  • 3.4.3 d = 4.
  • 3.4.4 d = 5.
  • 3.5 Least action principle.
  • 3.6 Problems.
  • 4 Canonical quantization, Klein-Gordon equation, SHOs, momentum space representation, raising and lowering operators.
  • 4.1 Canonical quantization.
  • 4.2 Canonical quantization (cont.)
  • 4.3 Momentum space representation.
  • 4.4 Quantization of Field Theory.
  • 4.5 Free Hamiltonian.
  • 4.6 QM SHO review.
  • 4.7 Discussion.
  • 4.8 Problems.
  • 5 Symmetries.
  • 5.1 Switching gears: Symmetries.
  • 5.2 Symmetries.
  • 5.3 Spacetime translation.
  • 5.4 1st Noether theorem.
  • 5.5 Unitary operators.
  • 5.6 Continuous symmetries.
  • 5.7 Classical scalar theory.
  • 5.8 Last time.
  • 5.9 Examples of symmetries.
  • 5.10 Scale invariance.
  • 5.11 Lorentz invariance.
  • 5.12 Problems.
  • 6 Lorentz boosts, generators, Lorentz invariance, microcausality.
  • 6.1 Lorentz transform symmetries.
  • 6.2 Transformation of momentum states.
  • 6.3 Relativistic normalization.
  • 6.4 Spacelike surfaces.
  • 6.5 Condition on microcausality.
  • 7 External sources.
  • 7.1 Harmonic oscillator.
  • 7.2 Field theory (where we are going).
  • 7.3 Green’s functions for the forced Klein-Gordon equation.
  • 7.4 Pole shifting.
  • 7.5 Matrix element representation of the Wightman function.
  • 7.6 Retarded Green’s function.
  • 7.7 Review: “particle creation problem”.
  • 7.8 Digression: coherent states.
  • 7.9 Problems.
  • 8 Perturbation theory.
  • 8.1 Feynman’s Green’s function.
  • 8.2 Interacting field theory: perturbation theory in QFT.
  • 8.3 Perturbation theory, interaction representation and Dyson formula.
  • 8.4 Next time.
  • 8.5 Review.
  • 8.6 Perturbation.
  • 8.7 Review.
  • 8.8 Unpacking it.
  • 8.9 Calculating perturbation.
  • 8.10 Wick contractions.
  • 8.11 Simplest Feynman diagrams.
  • 8.12 Phi fourth interaction.
  • 8.13 Tree level diagrams.
  • 8.14 Problems.
  • 9 Scattering and decay.
  • 9.1 Additional resources.
  • 9.2 Definitions and motivation.
  • 9.3 Calculating interactions.
  • 9.4 Example diagrams.
  • 9.5 The recipe.
  • 9.6 Back to our scalar theory.
  • 9.7 Review: S-matrix.
  • 9.8 Scattering in a scalar theory.
  • 9.9 Decay rates.
  • 9.10 Cross section.
  • 9.11 More on cross section.
  • 9.12 d(LIPS)_2.
  • 9.13 Problems.
  • 10 Fermions, and spinors.
  • 10.1 Fermions: R3 rotations.
  • 10.2 Lorentz group.
  • 10.3 Weyl spinors.
  • 10.4 Lorentz symmetry.
  • 10.5 Dirac matrices.
  • 10.6 Dirac Lagrangian.
  • 10.7 Review.
  • 10.8 Dirac equation.
  • 10.9 Helicity.
  • 10.10 Next time.
  • 10.11 Review.
  • 10.12 Normalization.
  • 10.13 Other solution.
  • 10.14 Lagrangian.
  • 10.15 General solution and Hamiltonian.
  • 10.16 Review.
  • 10.17 Hamiltonian action on single particle states.
  • 10.18 Spacetime translation symmetries.
  • 10.19 Rotation symmetries: angular momentum operator.
  • 10.20 U(1)_V symmetry: charge!
  • 10.21 U(1)_A symmetry: what was the charge for this one called?
  • 10.22 CPT symmetries.
  • 10.23 Review.
  • 10.24 Photon.
  • 10.25 Propagator.
  • 10.26 Feynman rules.
  • 10.27 Example: muon pair production
  • 10.28 Measurement of intermediate quark scattering processes.
  • 10.29 Problems.
  • A Useful formulas and review.
  • A.1 Review of old material.
  • A.2 Useful results from new material.
  • B Momentum of scalar field.
  • B.1 Expansion of the field momentum.
  • B.2 Conservation of the field momentum.
  • C Reflection using Pauli matrices.
  • D Explicit expansion of the Dirac u,v spinors.
  • D.1 Compact representation of
  • E Mathematica notebooks
  • Bibliography

my course evaluation comments for PHY2403 — Quantum Field Theory I.

November 20, 2018 phy2403 , , ,

Here are my evaluation comments for QFT I. The university provides an anonymous facility to submit course feedback, but since I have no conflicts that require anonymization, I’m posting my commentary (and rationale for some of my list selections) publicly.

Q) Please comment on the overall quality of the instruction in this course.

Professor Poppitz’s knowledge of the subject matter is impressive and thorough. I expect that this is a particularly difficult course to teach and think that he has done an admirable job trying to work through the maximum amount of material in the limited time available in this course.

It is challenging but fun game (albeit a slightly masochistic one) to keep up with Prof Poppitz’s blistering pace through the course material. Poppitz often says “Phew!!” at the end of the race to complete a long derivation in the allotted time, and I’ll be saying the same thing at the end of this course.

The barrage of abstract material covered in a lecture is often sufficient to leave me with a headache, and it takes a few hours to recover from each class. It takes a few more hours after that to digest the material at a human pace.

This course would strongly benefit from video recorded lectures that some of the more educationally progressive academic institutions currently provide (i.e. MIT’s OCW and Yale’s “Open Yale” courses). With the exception of the UofT SciNet group (phy1610 — Scientific computing for Physicists), I’m not aware of any UofT physics courses that provide such recordings. For phy2403, video recordings would be particularly valuable, as it would allow the student to “pause” the Professor and work through the material presented at an individually suitable pace.

Q) Please comment on any assistance that was available to support your learning in the course.

Prof Poppitz was available continually on the course forum, after class briefly, and in weekly office hours. I received a great deal of helpful assistance from him during the course.

Q) Compared to other courses, the workload for this course was…

I picked Heavy (not Very Heavy), but I’m not in a good position to evaluate since I’m only taking one course.

Q) I would recommend this course to other students.

I picked Mostly (not Strongly). I wouldn’t recommend this course to anybody who was not adequately prepared. I’m not sure that I was. It is a very tough course. I was continually impressed with the other students in the class. I’ve worked slowly for years to gradually build up the background required to take this course, and all the rest of these younglings are downing the material with seeming ease. There are a lot of exceptionally smart students enrolled on this course.

New aggregate notes collection for UofT phy2403 Quantum Field Theory I

October 21, 2018 phy2403 , ,

I’ve uploaded a new aggregate notes collection of my UofT phy2403 Quantum Field Theory I class notes (taught by Prof. Erich Poppitz), which now includes up to Wed Oct 17th’s lecture 11 (but doesn’t have my problem set I solution)

  • 1 Introduction
  • 1.1 What is a field?
  • 1.2 Scales.
  • 1.2.1 Bohr radius
  • 1.2.2 Compton wavelength.
  • 1.2.3 Relations.
  • 2 Units, scales, and Lorentz transformations.
  • 2.1 Natural units.
  • 2.2 Gravity.
  • 2.3 Cross section.
  • 2.4 Lorentz transformations.
  • 3 Lorentz transformations and a scalar action.
  • 3.1 Determinant of Lorentz transformations.
  • 3.2 Field theory.
  • 3.3 Actions.
  • 3.4 Problems.
  • 4 Scalar action, least action principle, Euler-Lagrange equations for a field, canonical quantization.
  • 4.1 Principles cont.
  • 4.2 d = 2 .
  • 4.3 d = 3 .
  • 4.4 d = 4 .
  • 4.5 d = 5 .
  • 4.6 Least action principle (classical field theory)
  • 4.7 Canonical quantization.
  • 5 Klein-Gordon equation, SHOs, momentum space representation, raising and lowering operators.
  • 5.1 Canonical quantization.
  • 5.2 Momentum space representation.
  • 6 Canonical quantization, Simple Harmonic Oscillators, Symmetries
  • 6.1 Quantization of Field Theory.
  • 6.2 Free Hamiltonian.
  • 6.3 QM SHO review.
  • 6.4 Discussion.
  • 6.5 Switching gears: Symmetries.
  • 7 Symmetries, translation currents, energy momentum tensor.
  • 7.1 Symmetries.
  • 7.2 Spacetime translation.
  • 8 1st Noether theorem, spacetime translation current, energy momentum tensor, dilatation current.
  • 8.1 1st Noether theorem.
  • 8.2 Unitary operators.
  • 8.3 Continuous symmetries.
  • 8.4 Classical scalar theory.
  • 9 Unbroken and spontaneously broken symmetries, Higgs Lagrangian, scale invariance, Lorentz invariance, angular momentum quantization
  • 9.1 Last time.
  • 9.2 Examples of symmetries.
  • 9.3 Scale invariance.
  • 9.4 Lorentz invariance.
  • 10 Lorentz boosts, generator of spacetime translation, Lorentz invariant field representation.
  • 10.1 Lorentz transform symmetries.
  • 10.2 Transformation of momentum states.
  • 11 Microcausality, Lorentz invariant measure, retarded time SHO Green’s function.
  • 11.1 Relativistic normalization.
  • 11.2 Spacelike surfaces.
  • 11.3 Condition on microcausality.
  • 11.4 Harmonic oscillator.
  • 11.5 Field theory (where we are going).
  • 12 Independent study problems
  • Appendices
  • A Useful formulas and review
  • Index
  • Bibliography