P231 Weekly

Week 1: Linear Algebra Review

This week we reviewed dimensional analysis, allometry, and the basics of linear algebra. We roughly covered sections 1 - 3.1 in our notes.

Assignments

• Video Introduction

• Pre-Class Survey

• HW1a (submit link): Due 9/27 (Wed)

• HW1b (submit link): Due 10/11 (Mon)

Suggested Reading

Please feel free to read as much or as little of the following references as you feel appropriate for what you'd like to get out of this course.

• "Dimensional analysis, falling bodies, and the fine art of not solving differential equations," Craig Bohren. American Journal of Physics 72, 534 (2004); https://doi.org/10.1119/1.1574042 (access through UCR VPN)This article really captures the spirit of this course and is a non-trivial demonstration of the power of dimensional analysis. It gives us a reason to pause and think about what it means for our idealized models of nature to be reasonable approximations to the complex reality around us.

• "Natural Units and the Scales of Fundamental Physics," Robert Jaffe, Supplementary Notes for MIT’s Quantum Theory Sequence, Feb 2017. Jaffe's notes have plenty of examples of dimensional analysis as well as a thorough introduction to natural units.

• "Introduction to Tensor Calculus," Kees Dullemond & Kasper Peeters (booklet) 2010. This is a reasonable introduction to the "indexology" that we use in physics. The booklet is much more rigorous than we will be. It may help motivate the types of rules that we will build into our index notation. If you enjoy this reference, you can find more in the first few chapters of most general relativity textbooks, for example Sean Carrol's Spacetime and Geometry.

• "Tensors: A guide for undergraduate students," Franco Battaglia, American Journal of Physics 81, 498 (2013); https://doi.org/10.1119/1.4802811 (accessible through UCR VPN)This is an excellent and readable introduction to to tensors. In the last half of the article (beyond our scope) one starts to get a taste of how linear algebra is a local approximation to calculus on curved spaces.

• "Section 11: The Method of Similarity," in Mathematical Methods of Classical Mechanics by V.I. Arnold. This is a delightful book about differential geometry that is disguised about a book about mechanics. In section 11 Arnold describes how to use scaling relations to relate relate different orbits of a central potential (we mention this example in our class notes). Much more fun are the two problems at the end of the section.

• "Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit?," Matta, Massa, Gubskaya, and Knoll, Journal of Chemical Education 88 (2011). Via Fermat's Library. 

Week 2: Linear Algebra Review

This week we reviewed the basics of linear algebra as needed for this course. Our goal is to get used to index notation. We roughly covered section 3 in our notes.

Assignments

• HW1b (submit link): Due 10/11 (Mon), carried over from last week.

Suggested Reading

Read as much or as little as you feel necessary for your our goals in this course.

• See the tensor references from Week 1. 

• See Section 2 and Section 3.1-3.2 of the Manchester notes.

• Section 4.1 of Stone & Goldbart distinguishes between formal and concrete differential operators. For our purposes, a formal differential operator is what one would typically call "a differential operator," examples include the d'Alembertian or the Laplacian. A concrete differential operator comes with a specification of the domain and the boundary conditions. 

• Chapter 4.1-4.8 of Byron & Fuller is a nice summary of finite dimensional vector spaces. Chapter 5 introduces function spaces. 

Week 3: Linear Algebra Review

Differential equations as linear algebra. Eigenfunctions, Green's functions. We roughly covered section 4 in our notes.

Assignments

• HW2a (submit link): Due 10/13 (Wed)

• HmW2b (submit link): Due Nov 1 (extended one week)

• Peer Review (submit link); four total. Assignments on our internal page. Due 10/18

Suggested Reading

• Section 4.1 of the Manchester notes.

• In our lectures we discuss why there there are really only a few differential operators that always keep showing up in physics. Part of this has to do with the symmetries that our physical models satisfy, and part of it has to do with the idea that terms with more derivatives are typically small in the limit where our model is valid. This falls under the idea of an "effective theory." This year there's a virtual seminar series about effective field theories (all talks are recorded) that I recommend for those who are curious.

• Extra: "Dimensional Analysis in Field Theory," Stevenson, Annals of Physics 132, 383 (1981). A non-trivial example of dimensional analysis.

Week 4: Complex Analysis

Complex analysis review. Complex functions. Integration. We roughly covered section 5 in our notes.

Assignments

• HW2b (submit link): Due Nov 1 (extended one week)

Suggested Reading

Please review complex analysis. Our goal is to be able to perform complex contour integrals and to use the residue theorem. 

• Byron & Fuller, Chapter 6

• Boas, Chapter 14 (friendly introduction)

• Matthews & Walker, 3-3 and Appendix A

• Stone & Goldbart, Ch. 17 (succinct with lots of discussions of advanced topics; suggested for those who are already familiar with complex analysis)

• Appel, Chapter 4.1-4.6

• Cahill, Chapter 5.1-5.14

• Peter Oliver's notes (UMN) on complex analysis and conformal mapping

• Extra: "Why i?" Baylis, Huschilt, and Wei. American Journal of Physics 60, 788 (1992); While not quite the direction we're going, students often bring up quaternions and the Pauli matrices when we discuss complex numbers in physics. This article is a nice introduction.

• Extra: Alexandre Eremenko's notes on "Why Airplanes Fly and Ships Sail," an introduction to conformal mapping and fluid dynamics

  • Some background information in an SPS article by Dwight Neuenschwander, "How Airplanes Fly: Lift and Circulation"

• Extra: for those interested in the geometric ideas presented this week, you may enjoy the following Aleph 0 videos:

  • Stokes' Theorem on Manifolds

  • The derivative isn't what you think it is

Week 5: Complex Analysis

Complex analysis continued. Complex functions. Integration. Cauchy's integral theorem and the residue theorem. We roughly covered section 5 in our notes.

Assignments

• HW2b (submit link): Due Nov 1 (extended one week)

Suggested Reading

• While this has nothing to do with our course material, every time I teach this class I am reminded of something called "analytic continuation into superspace" where the idea (or at least the spirit) of analytic continuation is applied to a particular complexification of spacetime called superspace. 

The links below are the same from last week:

• Byron & Fuller, Chapter 6

• Boas, Chapter 14 (friendly introduction)

• Matthews & Walker, 3-3 and Appendix A

• Stone & Goldbart, Ch. 17 (succinct with lots of discussions of advanced topics; suggested for those who are already familiar with complex analysis)

• Appel, Chapter 4.1-4.6

• Cahill, Chapter 5.1-5.14

• Peter Oliver's notes (UMN) on complex analysis and conformal mapping

• Extra: "Why i?" Baylis, Huschilt, and Wei. American Journal of Physics 60, 788 (1992); While not quite the direction we're going, students often bring up quaternions and the Pauli matrices when we discuss complex numbers in physics. This article is a nice introduction.

• Extra: Alexandre Eremenko's notes on "Why Airplanes Fly and Ships Sail," an introduction to conformal mapping and fluid dynamics

  • Some background information in an SPS article by Dwight Neuenschwander, "How Airplanes Fly: Lift and Circulation"

• Extra: for those interested in the geometric ideas presented this week, you may enjoy the following Aleph 0 videos:

  • Stokes' Theorem on Manifolds

  • The derivative isn't what you think it is

Week 6: The Harmonic Oscillator

Finally, we're back to solving for Green's functions of operators that pop up in physics. We roughly covered section 6 in our notes.

Assignments

• HW3a (submit link): Due Wed, Nov 3 

• HW2b review (submit link): Due Mon, Nov 8. Links to your assignments.

  • If your reviewee has not uploaded their assignment, politely email them to remind them to do so.

• HW3b (submit link): Due Mon, Nov 15. Links to your assignment.

Suggested Reading

• The Green's function for the Harmonic Oscillator is "standard textbook" material. You may want to consult your favorite mathematical physics text for further examples and discussions.

Week 7: Kramers-Kronig, Higher-Dimensions

How do the analytic properties of our Green's functions show up in physical phenomena? What does the harmonic oscillator look like in more than 0+1 dimensions? We cover section 7 and 8 of our notes.

Announcements

• No class on Friday, Nov 12; I will try to record some lectures to make up for this.

• No class on Monday, Nov 15

Assignments

• HW3b (submit link): Due Mon, Nov 15. Links to your assignment.

Suggested Reading

• "Kramers–Kronig in two lines," Ben Hu American Journal of Physics 57, 821 (1989); https://doi.org/10.1119/1.15901 

• "What did Kramers and Kronig do and how did they do it?," Craig F Bohren 2010 Eur. J. Phys. 31 573

• "Understanding the Kramers-Kronig Relation Using A Pictorial Proof," Colin Warwick

Week 8: Higher-Dimensions

The harmonic oscillator generalizes to the wave equation in higher dimensions. We cover section 7 and 8 of our notes.

Announcements

• No class on Monday, Nov 15

Assignments

• No homework 4a

• HW4b (submit link) : Due Fri. Dec 3 (last day of the quarter). Links to your assignment.

• No peer reviews.

Suggested Reading

• Fly By Night Physics, Tony Zee: See Appendix G

• Dimensional reduction is discussed in the textbook by Appel.

Week 9: Dynamics & Distributions

Where did our differential equations all come from? Why focus on linear operators?

We cover 9 of our notes.

Lecture Notes: Monday

Announcements

• No class on Wednesday , Nov 24: Thanksgiving break.

• No class on Friday, Nov 26: Thanksgiving break.

Assignments

• No homework 4a

• HW4b (submit link) : Due Fri. Dec 3 (last day of the quarter). Links to your assignment.

• No peer reviews.

Suggested Reading

• Review your favorite reference on random variables and distributions. 

• A really good place for this is to look up modern machine learning references. One recent textbook geared towards physics and along the lines of the philosophy of the course is The Principles of Deep Learning Theory.

Week 10: Feynman Diagrams and Effective Theories

Where did our differential equations all come from? Why focus on linear operators?

We cover 9 of our notes, but the discussion comes from the following hand-written notes:

Lecture Notes

• Monday: introduction to Feynman diagrams

• Wednesday: Feynman diagrams and effective theory

• Friday: Some "big picture" ideas in probability

Assignments

• No homework 4a

• HW4b (submit link) : Due Fri. Dec 3 (last day of the quarter). Links to your assignment.

• No peer reviews.

Suggested Reading

• Review your favorite reference on random variables and distributions. 

• A really good place for this is to look up modern machine learning references. One recent textbook geared towards physics and along the lines of the philosophy of the course is The Principles of Deep Learning Theory.

• Discussion of the Monty Hall problem and metrics on probability distribution space come from Deep Learning in Physics by Tanaka, Tomiya, and Hashimoto

• For some popular science writing about probabilities:

  • How Not to be Wrong, Jordan EllenbergExamples of statistical paradoxes including some of our discussion of whether a diety exists. Ellenberg's popular writing is some of my favorite public-level writing on mathematics.

  • "The Riemann Zeta Conjecture and the Laughter of the Primes," Jim Holt in When Einstein Walked with GödelThis is my favorite application of Gott's Copernican Principle. This is an excellent volume of short, contemplative "connect the dots" essays across mathematical topics.

  • The Doomsday Calculation, William Poundstone. The first couple of chapters go over Gott's estimate, though rest of the book is largely tedious for those with a physics background

  • "A Mini-Introduction to Information Theory," Ed. Witten (arXiv:1805.11965)

  • "A Short Introduction to Entropy, Cross-Entropy and KL-Divergence," Aurélien Géron

  • Introduction to information theory using Wordle, 3Blue1Brown.

• In class we briefly mentioned the idea of renormalization in the context of effective theories. During this last week we lost one of the physicists who developed this idea, Michael Fisher (see also this celebration of Fisher's life). Fisher wrote an excellent review of renormalization in condensed matter physics that I refer you to.

• Just three years ago we lost another theorist who helped shape how we understand renormalization, Joe Polchinski (who you may know more for his two volume set on superstring theory). Polchinski's "Renormalization and effective Lagrangians" is on the must-read list for any theorist. His memoirs from 2017 are on the must-read list for any graduate student in physics.