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 Wednesday 10/4.

  • Submission link

  • Recording tips. 

  • Link to Flip Tanedo's video introduction

• Pre-Class Survey (confidential), please complete by Wednesday 10/4

• Short Homework 1: due Wednesday 10/4. (submission link)

• Long Homework 1: due Wednesday 10/11. (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)

• "Cursed Units," Joseph Newton (YouTube). A collection of cursed scientific units. 

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.  A useful reference is this set of undergraduate-level notes on linear algebra.

• Mon 10/2: vectors, matrices, and tensors. Sketchy notes. (Please refer to our course notes or the undergraduate level notes for details.)

• Wed 10/4: Three types of nice matrix: rotations, invertible transformations, projections. Transformation of tensors. Inner products and what they buy us. Orthogonality and completeness. Discretized function space. [notes]

  • Update: thanks, Alex C., for correcting me.  In my "taxonomy" of nice matrices, I claimed that all invertible square matrices could be diagonalized as M = R-1 M’ R. While any invertible matrix may be diagonalized, only symmetric (Hermitian, self-adjoint) matrices admit a basis of orthogonal eigenvector. For non-self-adjoint matrices, M = R-1 M’ S, where R and S are different rotations.

• Fri 10/6: The inner product. Definition of the adjoint with respect to the adjoint.  See the suggested reading below for some references on the question of why the moment of inertia tensor is not a "moment of inertia matrix." [notes]

Assignments

• From last week: Video Introduction, due Wed

• From last week: Pre-Class Survey (confidential), due Wed

• Short Homework 1: due Wednesday 10/4. (submission link) [source]

• Long Homework 1: due Wednesday 10/11. (submission link) [source]

• 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.

• "Physics 17: Linear Algebra For Physicists" lecture ntoes from my undergraduate linear algebra course. This may be a helpful (if basic) reference for notation and the main unifying ideas of the subject.

• "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.

• One of my favorite examples of a vector space is color space. The three types of cone cells in our eyes  are sensitive to different pieces of the visual spectrum. They define a basis for a three-dimensional vector space corresponding to neurological responses to the electromagnetic spectrum. Other animals can have more types of cone cells, in which case their color spaces are higher dimensional. Here are a couple of nice videos discussing the concept. There are references to more academic papers in our course notes.

  • • "RGB to XYZ: The Science and History of Color" by John Austin" at the Strange Loop Conference, 2019

    • "Color Spaces: Explained from the Ground Up" via- Video Tech Explained

  • We posed the question of why the moment of inertia tensor is a tensor and not a matrix. Many textbooks get this wrong. One way to see this is that the moment of inertia tensor properly has two lower indices. It may help to write the cross prodcut with respect to the Levi-Civita tensor in 3D. Some references:

    • This answer on physics.stackexchange. The presentation is a little technical. It is similar to what one finds in V.I. Arnold's Mathematical Methods of Classical Mechanics in the section on rigid body motion.

    • A really great reference is this article on bivectors: "Teaching Rotational Physics with Bivectors," Steuard Jensen, Jack Poling. (arXiv: 2207.03560) The bivector construction may seem a little unfamiliar, but they do a good job of introducing it. In 3D, the bivector is a nicer way to represent the cross product. However, bivectors (and their cousins, k-forms) generalize to higher dimensions. The appendix describes the moment of inertia tensor in higher dimensions.

    • Along the lines of the above, here's a more systematic introduction to geometric algebra, including a nice discussion of the moment of inertia tensor: "Unifying the inertia and Riemann curvature tensors through geometric algebra" by M.. Berrondo, J. Greenwald, and C. Verhaaren (Am. J. Phys. 80, 905–912).

    • If you want to really dive deeper into this field, check out "Spacetime algebra as a powerful tool for electromagnetism" (arXiv:1411.5002) by Dressel, Bliokh, and Nori. 

    • A nice introduction to the ideas of geometric algebra on YouTube: "A Swift Introduction to Geometric Algebra."

    • A slightly more sophisticated set of videos: "QED Prerequisites Geometric Algebra: Introduction and Motivation"

    • A YouTube explainer on rigid body dynamics using geometric algebra.

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?

• Mon 10/9: Complex vector spaces. Introduction to historgram space and the metric on histogram space. Preview of function space. Integration by parts. [Notes]

• Wed 10/11: Locality and why we primarily care about differential operators in function space. Hermiticity in function space. Eigenfunctions and a rediscovery of Fourier analysis. [Notes]

  • In class we discussed the notion of an effective theory and why higher-order terms in an action are typically small. For example, the Euler-Heisenberg Lagrangian describes electrodynamics including the quantum effects of electrons.

• Fri 10/13: [Notes]

Assignments

• Long Homework 1: due Wednesday 10/11. (submission link) [source]

• Explainer Video: due MONDAY next week, due date extended because I was late to post. (submission link)

  • Assigned problems are on our internal page. Please prepare an explainer video for  the assigned problem from Long Homework 1. 

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)

• Monday 10/16: Complex functions, Riemann sheets (branch cuts), review of "ordinary" complex integrals, complex differentiation, analyticity and the Cauchy–Riemann equations. [notes]

• Wednesday 10/18: Differentiability = analyticity, the geometric perspective. [notes]

• Friday: no class today. 

Assignments

• Short Homework 2: due Wednesday 10/18. (submission link) [source]

• Long Homework 2: due Wednesday 10/25. (submission link) [source]

• Explainer Video: due Friday 10/27. Assignments (from Long HW2). (submission link)

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

Geometric Algebra (for your own edification)

These include the references from week 1

• • • A really great reference is this article on bivectors: "Teaching Rotational Physics with Bivectors," Steuard Jensen, Jack Poling. (arXiv: 2207.03560) The bivector construction may seem a little unfamiliar, but they do a good job of introducing it. In 3D, the bivector is a nicer way to represent the cross product. However, bivectors (and their cousins, k-forms) generalize to higher dimensions. The appendix describes the moment of inertia tensor in higher dimensions.

    • Along the lines of the above, here's a more systematic introduction to geometric algebra, including a nice discussion of the moment of inertia tensor: "Unifying the inertia and Riemann curvature tensors through geometric algebra" by M.. Berrondo, J. Greenwald, and C. Verhaaren (Am. J. Phys. 80, 905–912).

    • If you want to really dive deeper into this field, check out "Spacetime algebra as a powerful tool for electromagnetism" (arXiv:1411.5002) by Dressel, Bliokh, and Nori. 

    • "Imaginary numbers are not real—The geometric algebra of spacetime," Stephen Gull, Anthony Lasenby & Chris Doran  Foundations of Physics 23, 1175–1201 (1993)

    • "Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics," David Hestenes. Am. J. Phys. 39, 1013–1027 (1971).

    • "Oersted Medal Lecture 2002: Reforming the mathematical language of physics" David Hestenes Am. J. Phys. 71, 104–121 (2003).

    • A great historical introduction from Jack Rusher at the Strange Loop Conference in 2023. 

    • A nice introduction to the ideas of geometric algebra on YouTube: "A Swift Introduction to Geometric Algebra."

    • A slightly more sophisticated set of videos: "QED Prerequisites Geometric Algebra: Introduction and Motivation"

    • A YouTube explainer on rigid body dynamics using geometric algebra.

• Monday, Oct 23: Integration of analytic functions along closed contours. We showed that any closed contour integral of a function in a domain where the function is analytic everywhere always vanishes. [notes]

• Wed, Oct 25: Residue theorem. We introduced meromorphic functions: functions that are almost analytic in a region except for isolated point-like singularities. These functions can be expanded about each point in a Laurent expansion, which is a generalization of a Taylor expansion that includes negative powers of (z-z0) to account for the singularity. The least singular singular term in the Laurent expansion of f about a point z0 has a coefficient that is called the residue of f at the point z0. We showed that the closed contour integral of a meromorphic function is proportional to the sum of residues enclosed by that contour. We showed how this can be used to solve real integrals, and motivated how such integrals may show up in the Green's function problem. [notes]

• Fri, Oct 27: How to find residues and pick contours. [notes]

Assignments

• Long Homework 2: due Wednesday 10/25. (submission link) [source]

• Explainer Video: due Friday 10/27. Assignments (from Long HW2). (submission link)

Suggested Reading

• Pick your favorite chapter on complex contour integration from the standard mathematical methods textbooks. I would not recommend starting from a mathematics textbook.

• I really enjoy the first half of Roger Penrose's Road to Reality for this material. Chapter 9 is especially nice for connecting Fourier analysis to the complex plane. The real joy of the book, as with most textbooks, is doing the exercises. 

• Needham's Visual Complex Analysis goes a bit deeper than we need, but delivers on its title by having very helpful visualizations of everything going on. The "amplitwist" term that pervades the book is simply the notion that multiplying by a complex number is a rescaling and a rotation.   

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

• Monday: The Green's function for the simple harmonic oscillator, done by "ducking the poles." [notes]

• Wednesday: Causality and different pole perscriptions. The retarded Green's function for the simple harmonic oscillator. The damped harmonic oscillator. The Feynman prescription for the quantum harmonic oscillator; a hint of the path integral in quantum mechanics and why the Feynman prescription helps with convergence. [notes]

• Friday: passage from a single simple harmonic oscillator to a lattice of coupled harmonic oscillators. Derivation of the wave equation. Intuitive guess for the solution for the wave equation Green's function from the Poisson equation Green's function. [notes]

Assignments

• Short Homework 3: due Wednesday 11/1. (submission link) [source]

• Long Homework 3: due Wednesday 11/8. (submission link) [source]

• Explainer Video: due Friday 11/10. Assignments to be posted (from Long HW3). (submission link)

• Peer Review: due Friday 11/3. Assignments to be posted.

Suggested Reading

• There are some classic textbooks for this: Byron and Fuller, Butkov, Felder and Felder, etc. Pick your favorite. 

• Chapter 9 of Penrose's Road to Reality is not directly applicable, but offers some nice connections. Some of the extra credit problems draw from this source.

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

• Monday: The Green's function for the wave equation in 3+1 dimensions, derived carefully. [notes, be careful with minus signs]

• Wednesday: we were going to talk about Kramers-Kronig, but we ended up talking about the method of images, curvature, what U(1) means in electromagnetism, and a little bit about the Riemann sphere. 

• Friday: university holiday, no lecture.

Suggested Reading

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

• Dimensional reduction is discussed in the textbook by Appel.

• "Wakes and waves in N dimensions," Harry Soodak;  Martin S. Tiersten, Am. J. Phys. 61, 395–401 (1993)

Lectures

• Monday: Kramers-Kronig [notes, from 2017]

• Wed: Basics of probability. The Monty Hall Problem. The Doomsday clock. [notes]

• Fri:  Monty Hall, Entropy  [notes]

Assignments

Highlighted: due this week.

• Short Homework 4: due Wednesday 11/15. (submission link) [source]

• Long Homework 4: due Wednesday 11/22. (submission link) [source]

• Explainer Video 3: due Friday 11/17. Assignments now posted (from Long HW3). (submission link) Sorry for the delay on this.

• Explainer Video 4: due Friday 12/1. Assignments (from Long HW4) (submission link)

• Peer Review schedule: these are finally up. For the next three weeks, you will watch 3 videos and give feedback to your classmates. Please send a copy of your written feedback to your classmate.

  • Peer Review 1: due Friday 11/17. Assignments (total of three). Submission link.

  • Peer Review 2: due Friday 11/24. Assignments (total of three). Submission link.

  • Peer Review 3: due Friday 12/1. Assignments (total of three). Submission link.

  • Peer Review 4: due Friday, 12/8 (only one review) Submission link.

• Essay: please prepare a brief essay (less than 10 pages) that explains how to solve for the Green's function of a differential operator using a Fourier decomposition and imposing causality. You should write this pedagogically: your intended audience is you in 5 years. Imagine that you'd like to explain the main part of this course long after you've forgotten it. I suggest typesetting using LaTeX. (Submission link)

  • I suggest using LaTeX to typeset your notes. If you need some hints to get started:

    • Overleaf is an online LaTeX editor that makes it relatively easy to get started.

    • Flip's paper/notes template with examples. This is a little more involved (multiple files), but may be a useful starting point as you develop your own paper/notes templates.

Suggested Reading

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

• Dimensional reduction is discussed in the textbook by Appel.

• "The Riemann Zeta Conjefture and the Laughter of the Primes" in When Einstein Walked wtih Godel, Holt.

• "A math equation that predicts the end of humanity," William Poundstone. This article draws from Poundstone's book, The Doomsday Calculation, but I found that book rather insufferable. 

• "How to Select Observers," Robert Garisto. Includes analysis of Gott's estimate for the age of the Berlin Wall. Intersting side note: Garisto is now the editor of PRL and, as an undergradaute, caught an error in the Principia Mathematica. 

• Monday: Gaussian integrals and their generalization to probability densities in higher dimensions. [notes]

  • Alex suggested the book Birth of a Theorem by Cedric Villani.

• Wed: class canceled due to travel, my apologies. If you would still like to spend one hour of your life listening to me talk, here's a practice version of the colloquium I was giving at UC Santa Barbara this week.

Assignments

Highlighted: due this week.

• Long Homework 4: due Wednesday 11/22. (submission link) [source]

• Explainer Video 4: due Friday 12/1. Assignments (from Long HW4) (submission link)

• Peer Review schedule: these are finally up. For the next three weeks, you will watch 3 videos and give feedback to your classmates. Please send a copy of your written feedback to your classmate.

  • Peer Review 2: due Friday 11/24. Assignments (total of three). Submission link.

  • Peer Review 3: due Friday 12/1. Assignments (total of three). Submission link.

  • Peer Review 4: due Friday, 12/8 (only one review) Submission link.

• Essay: please prepare a brief essay (less than 10 pages) that explains how to solve for the Green's function of a differential operator using a Fourier decomposition and imposing causality. You should write this pedagogically: your intended audience is you in 5 years. Imagine that you'd like to explain the main part of this course long after you've forgotten it. I suggest typesetting using LaTeX. (Submission link)

  • I suggest using LaTeX to typeset your notes. If you need some hints to get started:

    • Overleaf is an online LaTeX editor that makes it relatively easy to get started.

    • Flip's paper/notes template with examples. This is a little more involved (multiple files), but may be a useful starting point as you develop your own paper/notes templates.

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.

Discussions

This is our last week of in-person instruction.

• Mon: from Gaussian integrals to field theory. What is the partition function? Introduction to interacting field theories. Feynman diagrams. [notes]

• Wed: the path integral in quantum mechanics, reflections on the Legendre transform, constrained functions and Statstical ensembles. [notes]

• Fri: effective theories, dimensional analysis, a little about renormalization. [notes]

Assignments

Highlighted: due this week.

• Peer Review schedule: these are finally up. For the next three weeks, you will watch 3 videos and give feedback to your classmates. Please send a copy of your written feedback to your classmate.

  • Peer Review 3: due Friday 12/1. Assignments (total of three). Submission link.

  • Peer Review 4: due Friday, 12/8 (only one review) Submission link.

• Essay: please prepare a brief essay (less than 10 pages) that explains how to solve for the Green's function of a differential operator using a Fourier decomposition and imposing causality. You should write this pedagogically: your intended audience is you in 5 years. Imagine that you'd like to explain the main part of this course long after you've forgotten it. I suggest typesetting using LaTeX. (Submission link)

  • I suggest using LaTeX to typeset your notes. If you need some hints to get started:

    • Overleaf is an online LaTeX editor that makes it relatively easy to get started.

    • Flip's paper/notes template with examples. This is a little more involved (multiple files), but may be a useful starting point as you develop your own paper/notes templates.

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 a few 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.

• "Don't feed the physicists," Alex Small in Nature magazine (June 2023). This is a charming short science fiction story that captures the spirit of building models of physics.

• "The Renormalization Group," David Skinner, QFT II.

Calvin and Hobbes, by Bill Watterson (6 January 1988)