Schedule

Course logistics. 

• 1. Fri: A review of dimensional analysis.

Assignments

• Video Introduction: please record a short (~5min) video to introduce yourself to the class. Due next Wed.

  • Submission link

  • Recording tips. 

  • Link to Flip Tanedo's video introduction

• Pre-Class Survey (confidential), please complete ASAP (& by Wed)

• Short Homework 1: due next Wed. (submission link) Corrected 9/26, thanks to Demao for pointing out. 

• Long Homework 1: due Wed in 2 weeks. (submission link)

Suggested Reading

The lecture notes contain the main narrative of the course.  Each week we provide suggested reading for complementary perspectives and deeper dives into the material. You are strongly encouraged to explore the topics as they relate to your interests and needs.

• "Dimensional analysis, falling bodies, and the fine art of not solving differential equations," Craig Bohren. American Journal of Physics 72, 534 (2004); (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.

• "Dimensional analysis as the other language of physics" R. W. Robinett, American Journal of Physics 83, 353 (2015). Very nice introduction to dimensional analysis.

• "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.  (See also John Baez / Don Koks on Physics FAQ)

Finite dimensional linear algebra. We'll wrap up a few ideas in dimensional analysis and remind ourselves of the many ways in which linear algebra shows up.  

This is an odd numbered week: we will have a lecture on Monday afternoon, 3:00 - 3:50pm and the short homework and introductory videos are due this Wednesday.

• 2. Mon 9/26: Dimensional analysis and error estimates. The "deeper" topic is the idea that all of our theories are effective theories whose accuracy depends on how far you are in their domain of validity. Dimensional analysis lets us estimate how big the error is from a simple theory given that we do not calculate the corrections from a more "correct" theory: e.g. how big is the effect of not including air resistance? It would be silly if you had to do the full calculation just to figure out that the correction is small.

• 3. Mon 3pm, 9/26: Vectors, vector spaces, dual vectors, and all that. We went over problem 3 of HW1a. We reviewed (and generalized) our idea of vectors and vector spaces. A good example of a weird vector space is the space of Fibbonacci sequences. We identified the following words as equivalent: vector, column vector, contravariant vector, ket.  We then defined dual vectors as linear maps from vectors to numbers. We defined the following words as equivalent to "dual vectors:" row vectors, covectors, covariant vectors, bra, one-form.

• 4. Wed 9/28: Intro to tensors. 

• No class on Friday, 10/1.

Assignments

• Video Introduction

• Pre-Class Survey (confidential)

• Short Homework: due this Wed. (submission link)

• Long Homework: due Wed next week. (submission link)

• Explainer Video: due Friday next week. (submission link)

  • Assigned problems are on our internal page

Suggested Reading

The lecture notes contain the main narrative of the course.  Each week we provide suggested reading for complementary perspectives and deeper dives into the material. You are strongly encouraged to explore the topics as they relate to your interests and needs.

• "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); (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.

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

Assignments

• Short Homework: due this Wed.

• Long Homework: due Wed next week.

Suggested Reading

A curious foray into "histogram space" as a stepping stone to function spaces: a class of infinite dimensional vector spaces that keep showing up in physics—why?

This is an even numbered week: we will have office hours on Monday afternoon, 3:00 - 3:50pm in Physics 3054 (Flip's office), and the long homework is due this Wednesday.

• 5. Mon 10/3: We reviewed the first homework and focused on the virtues of index notation. Notes. (I suggest the typed up course notes for a more comprehensive discussion, but the hand written notes are a reminder of what we actually talked about in class.)

• 6. Wed 10/5:  Indices are useful for another reason: they tell us about how tensors transform under symmetries. Where do these symmetries come from? They are symmetries of the inner product = dot product = metric. We also introduce the idea of an adjoint = Hermitian conjugate = transpose. Notes.

• 7. Fri 10/7: A bit of a philosophical aside. Why aren't translations part of the symmetries of the Euclidean metric? (Because the "base space" is not the vector space that we usually care about.) Why aren't more of our differential equations linear in spatial derivatives? (Rotational symmetry.) We argued by dimensional analysis that the simplest theories of physics should be second order in derivatives: these respect rotational space(time) symmetry and do not carry any inverse powers of a "microphysics" scale. Notes.

Assignments

• Long Homework 1: due Wed this week. (submission link)

• Explainer Video: due Fri. this week. (submission link)

  • Assigned problems are on our internal page

Suggested Reading

• 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. 

• 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.

• Extra: for more of a discussion about tangent spaces versus base spaces, Arnold's Mathematical Methods of Classical Mechanics is a classic. You can also see Collinucci and Wijn's lectures at the Modave Summer School in 2006 for a brief introduction to the mathematics of fiber bundles. 

"Find out what happens when functions stop being polite and start getting complex" (Apologies to The Real World)

Yep, four lectures this week. 

• 8. Mon 10/10: The joy of eigenfunctions. Finite dimensional "Green's functions" to set things up. Electrostatics as an example. Two ways to find Green's functions: series solution using completeness, patching solution. Getting started with a third method (Fourier transform), and realizing that we need to review complex analysis. Notes.  

• 9. Mon 10/10: Continuing from the morning. Notes (same as above).  

• 10. Wed 10/12: Complex numbers, complex functions, and analyticity = complex differentiability. How is complex space different from two-dimensional Euclidean space? Integrating complex functions. Hint of the tangent space picture of differentiability. Notes.

• 11. Fri 10/14: Complex functions as maps. See the articles below for some discussion of conformal transformations as applied to airplane lift. We talked about multivalued functions by introducing Riemann sheets. The key point is that in order to make sense of a multivalued function, one has to define a branch cut that is the "teleporter" between Riemann sheets . We proved the fundamental theorem of calculus for analytic functions. Notes. There is also  much more discussion in our typed up notes.

Assignments

• Short Homework 2: due this Wed. (submission link)

• Peer Review of Explainer Video 1: due this Friday (submission link), assignments posted on the internal page.

  • What if your peer has not uploaded their video? Please email them to remind them to upload a video.

• Long Homework 2: due Wed next week. (submission link)

• Explainer Video 2: due Fri next week. (submission link), assignments posted on the internal page.

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 is 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

I realized that the link to the course notes is outdated. The 2022 notes (currently being re-written as we go along) are posted here. 

• 12. Mon 10/17:  Integrals around simple poles, the Laurent expansion. Working towards the Cauchy Integral Theorem. Notes.

• 13. Wed 10/19: Finding residues. Notes.

• 14. Fri 10/21: Review of the big picture.  Picking contours to complete real integrals into complex integrals over a loop. The notes contain a fancy example of integrating over a branch cut, but we did not get to it in class. Notes. 

Assignments

• Mid Course Survey, due Fri this week

• Long Homework 2: due Wed this week. (submission link)

• Explainer Video 2: due Fri this week. (submission link)

Suggested Reading

Follow up with your favorite mathematical methods textbook. I like the book by Boas. 

Are we doing physics yet? Plus a foray into the Kramers-Kronig dispersion relation. 

• 15. Mon 10/24: The harmonic oscillator the wrong way: picking the wrong pole prescription. Notes.

• 16. Mon 10/24: The harmonic oscillator the right way: going over the poles. Comparison to the damped harmonic oscillator. Notes.  (same as notes from this morning)

• 17. Wed 10/26: Damped harmonic oscillator. Feynman propagator and the convergence of the path integral. Notes. We'll take it easy today since there is an exam in another class at 11am.

• 18. Fri 10/28: Sketch of the wave operator. Notes.

Assignments

• Short Homework 3: due this Wed. Hint: see last week's notes. (submission link)

• Peer Review of Explainer Video 2: due this Fri. (submission link | Assignments) 

• Long Homework 3: due Wed next week. (submission link)

• Explainer Video 3: due Fri next week. (submission link | Assignments)

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. 

• "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

• "A brief Introduction to Dispersion Relations and Analyticity," Roman Zwicky lecture notes. Highlights the role of analyticity and Kramers-Kronig like relations (dispersion relations) in quantum field theory. These notes do assume some field theory background and so may be relatively advanced in its physics content, but the mathematical content fits well with what we are studying. Theorists may want to save this reference for later.
  
  

"It's like the more dimensions  we come across / The more problems we see." [Apologies to the Notorious B.I.G.]

No lecture on Wednesday or Friday.

• 19. Mon 10/31: No office hours today, no class the rest of this week. The wave operator, i.e. the harmonic oscillator in (3+1)-dimensions. All we're doing is the 4-dimensional momentum-space integral. But let's see it all come together. Remarks on dimensional reduction: see how (3+1)-dimensional wave can be dimensionally reduced into either the 3-dimensional Poisson equation or the (2+1)-dimensional wave equation. Notes.

Assignments

• Long Homework 3: due Wed this week. (submission link)

• Explainer Video 3: due Fri this week. (submission link)

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

No class on Friday in observance of Veteran's Day.

• Meeting 20, Mon 11/7: Dimensional reduction discussion. notes.

• Meeting 21, Mon 11/7: Introduction to Kramers-Kronig and dispersion relations. notes.

• Meeting 22, Wed 11/9: Kramers-Kronig. notes.

Assignments

• Short Homework 4: due this Wed. (submission link)

• Peer Review of Explainer Video 3: due Fri Dec 2. (submission link)

• Long Homework 4: due Fri Dec 2. (submission link) 

• Explainer Video 4: due Fri Dec 2. (submission link)

• Essay prompt: please write a brief instructional note on how to solve for the Green's function of the wave operator in (3+1)-dimensions. You have gone over the steps for this in Problem 1 of Long Homework 4. The target audience is yourself in 5 years. The purpose of the essay is to give a pedagogical explanation for the mathematics and physics of solving for Green's functions.  (submission link)

  • Why are we doing this? As an academic, you will be judged on your ability not only to "do" physics, but also how you communicate it. During this class, we've practiced verbal communication through explainer videos. This essay is meant to give you some practice on academic writing.

  • I recommend (but do not require) using LaTeX, e.g. with Overleaf. Those who are more experienced with LaTeX can use my paper template as a starting point.

  • There is no page requirement or limit. As a benchmark expect something between 2-5 pages. 

Suggested Reading

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

• Dimensional reduction is discussed in the textbook by Appel.

• Mon 11/14:  No meeting or office hours today, apologies. 

• Meeting 23, Wed 11/16: where did fields come from? Where did the wave equation come from? A lattice of harmonic oscillators as an example of microphysics that generates a macroscopic theory described by the wave equation. notes.

• Meeting 24, Fri 11/18: Gaussian Integrals and Feynman Expansions

Assignments

Because they are all inter-related, all assignments are now due the last day of class. I suggest not procrastinating. There will be no peer review of Long HW 4. 

• Peer Review of Explainer Video 3: due Fri Dec 2. (submission link) (assignments)

• Long Homework 4: due Fri Dec 2. (submission link) 

• Explainer Video 4: due Fri Dec 2. (submission link) (assignments)

• Essay prompt: please write a brief instructional note on how to solve for the Green's function of the wave operator in (3+1)-dimensions. You have gone over the steps for this in Problem 1 of Long Homework 4. The target audience is yourself in 5 years. The purpose of the essay is to give a pedagogical explanation for the mathematics and physics of solving for Green's functions.  (submission link)

  • Why are we doing this? As an academic, you will be judged on your ability not only to "do" physics, but also how you communicate it. During this class, we've practiced verbal communication through explainer videos. This essay is meant to give you some practice on academic writing.

  • I recommend (but do not require) using LaTeX, e.g. with Overleaf. Those who are more experienced with LaTeX can use my paper template as a starting point.

  • There is no page requirement or limit. As a benchmark expect something between 2-5 pages. 

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.

No class on Wed/Fri in observance of Thanksgiving/Native American Heritage Day. 

• Meeting 25 Mon 11/21: From coupled harmonic oscillators to fields. Partition functions. notes.

• Meeting 26 Mon 11/21: Correlation functions. 

Assignments

• Short Homework 5: due this Wed. (submission link)

• Peer Review of Explainer Video 4: due this Fri. (submission link)

• Long Homework 5: due Wed next week. (submission link)

Suggested Reading

• 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.

• Meeting 27 Mon 11/28: Two-point correlation functions from a Gaussian N-dimensional probability distribution give Green's functions of linear operators. Nonlinearities may be treated perturbatively with Feynman diagrams. 

• Meeting 28 Wed 11/30: Probability. Examples: Richard Gott's estimate for the lifetime of the Berlin Wall, the Monty Hall Problem. notes.

• Meeting 29 Fri 12/2: Neural networks as almost-linear operators. Big picture of machine learning. (notes)

Assignments

• Long Homework 5: due Wed this week. (submission link)

Suggested Reading

• Gaussian Processes, an interactive visual exploration

• 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.

• "Asymptotics of Wide Networks from Feynman Diagrams," Ethan Dyer, Guy Gur-Ari (physicists at Google)

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