Schedule

Welcome to grad school. 

• Fri: Course logistics. Review of dimensional analysis, basics of linear algebra. 

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 10/2

• Short Homework 1: due Wednesday 10/2. (submission link)

• Long Homework 1: due Friday 10/11. (submission link)

Suggested Reading

Current lecture notes draft: Sept 24; over the first two weeks of this course we are quickly reviewing the first part of the notes (linear algebra). Most of the material will be familiar to you, but I would like us to get used to the notation.

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, Chaos and Self-Organised Criticality" (Ch. 10) in  Longair, Theoretical Concepts in Physics An Alternative View of Theoretical Reasoning in Physics 

• "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. Current version of the notes: Sept 24.

• Mon 10/2: Overview of the course. Course notes references: Section 1.2, Chapter 18. Please refer to Part I of the notes on linear algebra as needed.

• Wed 10/4: Finite dimensional vector spaces. Upper and lower indices of tensors. Course notes references: See Chapter 5 for indices, Chapter 7 for inner products, Chapter 9.7 on adjoints, Chapter 11 for Eigen-stuff. 

• Fri 10/6: Discussion of SHW1 dimensional analysis and scaling, change of basis, inner products, 

  • Inner products: see Chapter 7 of the notes for all of the details that we pointed to in class. You may also review Chapter 8 which introduces special relativity from the perspective of linear algebra. 

  • Indices of the transpose: see Chapter 9.7 of the notes. 

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 10/2

• Short Homework 1: due Wednesday 10/2. (submission link)

• Long Homework 1: due Friday 10/11. (submission link)

• Explainer Video 1: 

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.

• There's a fantastic history of vectors in mathematics and physics, Vector: a surprising story of space, time, and mathematical transformation by Robyn Arianrhod. It's the kind of history that a physicist would love, connecting some familiar names with a story of the development of an idea that we tend to ignore in our education. 

• 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.

"Infinite dimensional linear algebra" as the linear vector space of functions. Differential operators as linear operators. Complex spaces. And finally, the Green's function. 

• Mon 10/7: Histogram space, derivative operators, and the transition to the continuum. Reading: Chapter 15.10 discusses the limitations of the "histogram basis" picture of function space. There are two challenges: the first is that basis vectors have no well defined normalization, the second is more formal and has to do with the inability to contain all possible limits in this space.

• Mon 10/7 discussion slot: Special relativity. See lecture notes, chapter 8.

  • At Jason's request: an reference for an introduction to fiber bundles: "Topology of Fibre bundles and Global Aspects of Gauge Theories" by Collinucci and Wijns (hep-th/0611201)

• Wed 10/9: Q&A about the homework. Function spaces. Fourier series: example of an interval with Dirichlet boundary conditions. The adjoint of a differential operator as an integration by parts.

• Fri 10/11: Q&A about homework. Fourier series for a periodic function, en route to Fourier transforms. Apologies that we didn't finish the move to Fourier transforms.

  • Flip had an embarassing mistake where he confused active versus passive transformations. When we identify the components of the change of basis <e^i | f_a > as the components of a matrix, this is a passive transformation. The object that  R = <e | f> acts on (through contraction of indices) does not change, only its components change. On the other hand, if we wanted to write R as an "outer product" R ~ Rij | ei > < ej |, then this is an active transform that will move a vector. 

Assignments

• Long Homework 1: due Friday 10/11. (submission link)

• Short Homework 2: due Wednesday 10/16. (submission link)

• Explainer Video 1:  due Friday 10/18 (submission link)

• Peer Review 1: due  Wednesday 10/23 (submission link)

• Long Homework 2: due Friday 10/25. (submission link)

Suggested Reading

• • • Please refer to Part I of our course 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. 

    • 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. 

"Infinite dimensional linear algebra" as the linear vector space of functions. Differential operators as linear operators. Complex spaces. And finally, the Green's function. 

• Mon 10/14: Fourier Transforms

• Wed 10/16: No meeting. Recommendations: take a brief walk to the Botanical Garden, see if you can find the turtles. Or go to Bytes for a coffee, it's one of the few campus eateries that is run by UCR Dining rather than being outsourced to a contractor. Go to Orbach Library and check out the Creat'R Lab. 

• Fri 10/18: No meeting.

Assignments

• Short Homework 2: due Wednesday 10/16. (submission link)

• Long Homework 2: due Friday 10/25. (submission link)

• Explainer Video 1:  due Friday 10/18 (submission link)

  • You have been assigned one problem from Long Homework 1 to present a 5 minute explainer video. Please find your problem on the Internal Sheet.

• Peer Review 1: due  Wednesday 10/23 (submission link)

Suggested Reading

• • • Please refer to Part I of our course 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. 

    • 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. 

"Infinite dimensional linear algebra" as the linear vector space of functions. Differential operators as linear operators. Complex spaces. And finally, the Green's function. 

• Mon 10/21: complex stuff

  • Spherical harmonics as an example of a basis of functions on the sphere.

  • Complex numbers and functions. Complex functions as maps. 

  • Branch cuts and Riemann sheets: what happens when seemingly nice functions aren't obviously single-valued.

  • The Riemann sum as a map.

• Mon 10/21, 3pm: complex calculus

  • "Guide to Cultivating Complex Analysis" by Jiří Lebl has a nice description of holomorphic vs analytic.

    • Holomorphic functions: complex differentiable at every point (in the sense we described in class)

    • Analytic functions: have a convergent power series

  • Thinking about the complex integral as a map. The fundamental theorem of calculus motivates having a 

• Wed 10/23: Laurent series, integrals around little circles and little boxes. What does it take for an integral around a closed path to be nonzero? Comments on generalizations of the fundamental theorem of calculus (Stokes' theroem, Green's theorem), and the generalized Stokes equation for differential forms in n-dimensional space. 

• Fri 10/25: Due to an odd sulfuric smell, we met in the Physics courtyard. We presented the residue theorem, how to find residues, and the main killer app: how to find real integrals using the residue theorem.

Assignments

• Long Homework 2: due Friday 10/25. (submission link) [updated link: fixed typos, thanks Jason Y.] [updated: fixed more typos, thanks Morgan O. and Yuntian L.] There was a mistaken relative sign in the harmonic oscillator operator. No penalties if you did the problems using the incorrect sign. (The correct sign is that the (d/dx)^2 and k^2 have the same relative sign.)

• Peer Review 1: due  Wednesday 10/23 (submission link) Please find your assigned peer reviews on the Internal Sheet.

Suggested Reading

• • • "Guide to Cultivating Complex Analysis" by Jiří Lebl seems to be a nice formal discussion of complex analysis.

    • Visual Complex Analysis by Needham is nice textbook that does not skimp on images. The approach is a little idiosyncratic.

    • Your favorite undergraduate math methods course is a good starting point. I am partial to the textbook by Boas. 

    • What are Riemann sheets good for?

      1. "Resonances and poles in the second Riemann sheet," by Thomas Wolkanowski. A MSc thesis on the significance of the second Riemann sheet in quantum field theory.

      2. "Scattering amplitudes and contour deformations," by Gernot Eichmann, Pedro Duarte, M. T. Peña, Alfred Stadler

Let's solve the Green's function for the one system that really matters. (Why is this the one?)

• Monday: Discussion (from Friday) what if we draw contours that go off the first Riemann sheet? The one complex integral that we should know.

  • Sad news discussed during class: "America’s most beloved bear is dead. Here’s why Grizzly 399 mattered."

• Wednesday: The Green's function for the simple harmonic oscillator, done by "ducking the poles." Causality and different pole perscriptions. The retarded Green's function for the simple harmonic oscillator. The damped harmonic oscillator. 

• 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. 

Assignments

• Short Homework 3: due Wednesday 10/30. (submission link)

• Explainer Video 2:  due Friday 11/1 (submission link); assignments on the internal page

• Peer Review 2: due  Wednesday 11/6 (submission link); assignments on the internal page

• Long Homework 3: due Friday 11/8. (submission link)

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.

      1. "Scattering amplitudes and contour deformations," by Gernot Eichmann, Pedro Duarte, M. T. Peña, Alfred Stadler

This week we treat the wave equation as the limit of a continuum of coupled harmonic oscillators. 

• Monday: No class (no lecture or discussion). 

• Wednesday: Wave Equation

• Friday: Wave Equation

Assignments

• Short Homework 3: due Wednesday 10/30. (submission link)

• Explainer Video 2:  due Friday 11/1 (submission link); assignments on the internal page

• Peer Review 2: due  Wednesday 11/6 (submission link); assignments on the internal page

• Long Homework 3: due Friday 11/8. (submission link)

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)

This week we treat the wave equation as the limit of a continuum of coupled harmonic oscillators. 

• Monday: No class (Veteran's Day). 

• Wednesday: Dimensional Reduction

• Friday: probability, gaussian integrals, generating function of moments, Feynman diagrams

Assignments

• Short Homework 4: due Friday 11/15. (submission link)

• Long Homework 4: due Friday 11/22. (submission link)

• Explainer Video 4:  due Friday 11/15 assignments on the internal page

  • NEW link (11/25): submission link (I got these mixed up, sorry)

• Peer Review 4: due  Wednesday 11/20 (submission link); assignments on the internal page

• Final essay (submission link): write a set of notes for yourself on solving the Green's function for the 3+1 dimensional wave equation. The notes do not have to be long, but the goal is that they should be useful to you in the future. 

Suggested Reading

• • • Your favorite probability textbook. If you do not have one, it should be Probability Theory: The Logic of Science by Jaynes. 

A brief introduction to perturbative field theory. Theories with quadratic actions have linear equations of motion for which the propagation of information is described by linear operators. In th real world, there are non-quadratic "interaction" terms. In the worst case scenario, you must rely on simulations to study these theories. However, when the theory is perturbatively close to a Gaussian fiext point, you may employ a Feynman diagram expansion.

• Monday: field theory, Feynman diagrams as perturbation theory, and a hint of renormalization. Lecture Notes.

• Wednesday: double class, introduction to renormalization group flow. The key insight is that field theories are defined with a cutoff. Divergences in loop integrals remind us that there is a cutoff, but they are not the reason for the cutoff. Renormalization is a transformation in the space of physically equivalent theories with different cutoffs. The key point is that by going to an equivalent theory with a lower cutoff, the logarithmic corrections to your interactions remain small and perturbative. 

  • Introduction to probability. Bayes theorem, Monty-Hall paradox.

• Friday: No class (swapped with QM)

Assignments

• Short Homework 4: due Friday 11/15. (submission link)

• Long Homework 4: due Friday 11/22. (submission link)

• Explainer Video 4:  due Friday 11/15 assignments on the internal page (Explainer 4 assignments now updated, 11/25)

  • NEW link (11/25): submission link (I got these mixed up, sorry)

• Peer Review 4: due  Wednesday 11/20 (submission link); assignments on the internal page

• Final essay (submission link): write a set of notes for yourself on solving the Green's function for the 3+1 dimensional wave equation. The notes do not have to be long, but the goal is that they should be useful to you in the future. 

Suggested Reading

• • • Your favorite probability textbook. If you do not have one, it should be Probability Theory: The Logic of Science by Jaynes. 

    • "Feynman Diagrams," Weinzierl arXiv:2501.08354

This is our last week of in-class meetings. 

• Monday: Monty-Hall from the point of view of Bayes' theorem. Odds and evidence for binary hypothesis testing. (From Jaynes' Probability Theory, Ch. 4) Discussion of how everything gets tricky when we test across a plurality of hypotheses.

• Wednesday: Entropy.

• Friday: No class (swapped with QM)

Assignments

• Short Homework 4: due Friday 11/15. (submission link)

• Long Homework 4: due Friday 11/22. (submission link)

• Explainer Video 4:  due Friday 11/15 assignments on the internal page (Explainer 4 assignments now updated, 11/25)

  • NEW link (11/25): submission link (I got these mixed up, sorry)

• Peer Review 4: due  Wednesday 11/20 (submission link); assignments on the internal page

• Final essay (submission link): write a set of notes for yourself on solving the Green's function for the 3+1 dimensional wave equation. The notes do not have to be long, but the goal is that they should be useful to you in the future. 

Suggested Reading

• • • Your favorite probability textbook. If you do not have one, it should be Probability Theory: The Logic of Science by Jaynes. 

    • "What is Entropy?" arxiv:2409.09232 by John Baez.

    • "Statistical Physics," Jaynes 1962 Brandeis Lectures on Theoretical Physics, Vol 3

    • "Entropy, Cross-Entropy, & KL-Divergence," Aurélien Géron;.

We had planned to miss Wed and Fri, but due to a personal matter I must cancel class on Monday. 

Assignments

• Short Homework 4: due Friday 11/15. (submission link)

• Long Homework 4: due Friday 11/22. (submission link)

• Explainer Video 4:  due Friday 11/15 assignments on the internal page (Explainer 4 assignments now updated, 11/25)

  • NEW link (11/25): submission link (I got these mixed up, sorry)

• Peer Review 4: due  Wednesday 11/20 (submission link); assignments on the internal page

• Final essay (submission link): write a set of notes for yourself on solving the Green's function for the 3+1 dimensional wave equation. The notes do not have to be long, but the goal is that they should be useful to you in the future. 

Suggested Reading

• • • Your favorite probability textbook. If you do not have one, it should be Probability Theory: The Logic of Science by Jaynes. 

    • "What is Entropy?" arxiv:2409.09232 by John Baez.

    • "Statistical Physics," Jaynes 1962 Brandeis Lectures on Theoretical Physics, Vol 3

    • "Entropy, Cross-Entropy, & KL-Divergence," Aurélien Géron;.