Schedule

Google Calendar Link (requires RWeb login)

Welcome to grad school. 

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

  • We talked about how math and physics are different. Some of the ideas that came up were formalism, notion of truth,  a multiplicity of descriptions (and when two physical theories are equivalent versus effectively equivalent), whether one is a tool to the other. 

  • See the references below that discuss whether dx and Δx are mathematically equivalent.

Assignments

• Video Introduction: please record a short video to introduce yourself to the class. Due next Wednesday, Oct. 1. Guidelines: approximately 5 minutes, share your name, research interests, where you are from (both university and hometown), and anything else you would like to share (hobbies, recommendations, aspirations, etc.).

  • Submission link: Introductory Video

  • You may use any recording device (laptop, phone), please show your face to help us get to know each other. You must upload in a way that is accessible to the other members of the class (e.g. share on UCR's Google Drive and whitelist @ucr accounts).

  • Link to Flip Tanedo's video introduction

• Pre-Class Survey, please complete by Wednesday, Sept. 1 

  • Submission link: Pre-Class Survey

• Short Homework 1: due Wednesday, Oct. 1. 

  • Submission link:  Short HW 1

  • Please note that while you may edit your responses in the form, the pdf you upload cannot be replaced. Please double check your file before you upload for this and all of your uploaded pdfs.

• Long Homework 1: due Friday, Oct. 10

  • Submission link:   Long HW 1

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.

• See our collection of dimensional analysis articles on our references page

• Every year we start this class by asking how physics is different from mathematics. One of the topics that regularly comes up is that physicists treat the infinitesimal dx as a very-small-but-finite difference Δx. I do this often.  The intuition gets wacky when you look at second derivatives. The following references dig into when this is valid:

  • Ely, R. "Teaching calculus with infinitesimals and differentials." ZDM Mathematics Education 53, 591–604 (2021). https://doi.org/10.1007/s11858-020-01194-2

  • "Extending the Algebraic Manipulability of Differentials," Jonathan Bartlett, Asatur Zh. Khurshudyan. arXiv:1801.09553

• Further discussion on physics and mathematics:

  • "Why Physics Is Unreasonably Good at Creating New Math," Ananyo Bhattacharya in Nautilus

  • "The importance of stupidity in scientific research" by Marting Schwartz in Journal of the Cell and an informal follow up, "The centrality of stupidity in mathematics" on the Math for Love blog.

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Finite dimensional linear algebra. 

• Mon 9/30: We discussed SHW1 Problem 3. We introduced the Green's function problem as finding the inverse of a differential operator, motivating this by the electrostatic potential being an integral (linear combination) of point charge potentials. We then did a lightning review of linear algebra with our index notation.

  • A puzzle: why is the moment of inertia tensor not called the moment of inertia matrix? (Hint: "Tensors are objects that transform as tensors.") Use the definition of the moment of inertia tensor to motivate why it should have two lower indices.  

  • A trickier puzzle: some textbooks will use the matrix transformation law for the rotation of the moment of inertia tensor. Explain why this turns out to be acceptable for rotations, argue that this is not true for transformations that are not isometries of the Euclidean metric. 

  • Speaking of "tensors transform as tensors," you may enjoy following this Reddit thread to spot which comments are accurate and which are not.

• Wed 10/1: Notes (handwritten). Bases, Fibonacci sequences as a vector space, eigenvectors and eigenvalues, eigenvector expansions as a solution to the Green's function problem. 

  • Brandon asked about the moment of inertia tensor. There is a partial explanation in Chapter 7.6 of the notes. A useful way to see what's going on is to consider a change of units where ω is measured in degrees per second, not radians per second.  This is a rescaling of ω to Aω. Because the kinetic energy does not care what angular unit you use, it must transform as (1/A)^2, which is not what you would expect from the transformation of a (1,1)-tensor under rescaling. (You would expect such a tensor to not transform under a rescaling.)

  • We talked about the homework being an opportunity to practice our science. Here's are pop culture references to Allen Iverson and Ted Lasso from our discussion.

• Fri 10/3: We reviewed "multiplication by 1" as a change of basis using the resolution of the identity, 1=|e><e|. This let us identify the rotation matrix between vectors. (See p2. of Wed notes) We began our discussion of function space from a discretized "histogram space." Notes (we did the first 2 pages).

  • Question in class: how do we think of a matrix M acting on a row vector <w|? Think about M as a (1,1)-tensor: it's a bilinear function that takes in a vector and a row vector to give a numer. This means that if you feed it a row vector, it will output a row vector: M(__, w) is a function that takes in a vector and returns a vector, which is the definition of a (0,1)-tensor, or a row vector.

  • Note: I use the word matrix to mean (1,1)-tensor, which is not a universal notation.

  • The free action as an inner product: S = <q, Aq>, where A is Hermitian. One can derive the Euler-Lagrange equation of motion straightforwardly. 

  • "Local" matrices are those that have components close to the diagonal. We gave some examples of non-local matrices: finite transformations (which are the exponetiation of local matrices) and discrete transformations like time-reversal, partiy. (There is also charge conjugation for 'internal' spaces.)

  • Ongoing puzzle: why are most of the "matrices" in physics local? That is, why are the operators that we care about written as derivatives? (And why two derivatives?)

Assignments

• Video Introduction: please record a short (~5min) video to introduce yourself to the class. Due Wednesday 10/4.

  • Submission link

  • Recording tips. 

  • Link to Flip Tanedo's video introduction

• Pre-Class Survey, due Wednesday 10/1

• Short Homework 1: due Wednesday 10/1

  • Please note that while you may edit your responses in the form, the pdf you upload cannot be replaced. Please double check your file before you upload for this and all of your uploaded pdfs.

Upcoming Assignments: Please look over the long homework and ask questions in class this week. 

• Long Homework 1: due Friday 10/10. [Submission link]

• Explainer Video 1: due Friday 10/17. More instructions below.

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.

  • We talked about how the N to infinity limit of histogram space is not well defined. We can see this in the challenges in defining the inner product. The formal reason is that the normalizable states do not contain the limit points of all possible sequences of normalizable states.

    •  I refer to Appel's Mathematics for Physics & Physicists, Chapter 9 for a thorough discussion. In quantum mechanics, we assume a Hilbert space, which is a space where the N to infinity limit makes sense.

    • See also Karen Smith's notes on infinite dimensional vector spaces from her Math 513 course in U. Mich. These are very well written for a physics audience.

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"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 10/6: We talked about how to write components of the adjoint matrix in terms of the original matrix (LHW1, Problem 1). There was a key trick: canceling tensors on both sides of an equation when the equation is true for any values of those tensors. We also talked about the physical meaning of the eigen-motions of the spring problem on LHW1. We discussed the meaning of the determinant.

• Wed 10/8:  Histogram space continued. Inner products, Hermiticity. Series solutions. Note: SHW2 has been updated (redundant problem removed, determinant problem added). [Notes]. 

• Fri 10/10:  We talked about complex functions, introduced the Cauchy-Riemann equations.

Assignments

• Long Homework 1: due Friday 10/10 [Submission link]

Upcoming:  Starting week 3 we begin our full cadence of assignments. Something is due every Wed and Fri.

• Short Homework 2 (Updated 10/7): due Wednesday 10/15. [Submission link]

• Explainer Video 1:  due Friday 10/17 

  • More instructions below.

  • [Submission link]

  • Assignments are on our internal sheet

• Peer Review 1: due  Wednesday 10/22 [submission link]

  • Assignments are on our internal sheet

• Long Homework 2: due Friday 10/24.  [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. 

    • 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. 

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Complex spaces.

• Mon 10/13: Analytic functions are differentiable. Complex functions as maps of the complex plane to itself. Notes.

• Wed 10/15: Integration of analytic and meromorphic functions: the residue theorem. Notes.

• Fri 10/17: Using the residue theorem to solve real integrals.  Notes.

Assignments

Please go over the long homework this week, even though it is due next week. Let us use class time to discuss the homework.

• Short Homework 2 (updated 10/7): due Wednesday 10/15.   [Submission link]

• Explainer Video 1:  due Friday 10/17 

  • [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

Due next week

• Peer Review 1: due  Wednesday 10/22 [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

• Long Homework 2: due Friday 10/24. [Submission link]

Suggested Reading

• • • Most of the material this week is a review of topics you may have seen as an undergrad, but framed with an eye on how we intend to use them. Please refer to your favorite mathematical physics textbook for additional resources.

    • We mentioned conformal mapping and its applications to the early aerospace industry. A simple example of a conformal map that takes the circle to an approximate airplane wing cross section is the Joukowski transform. You can find a bit about it on this old NASA page. 

    • He's an example of someone working throguh a conformal mapping problem. I find the act of working through it to be especially insightful. 

    • "Physics of the Analytic S-Matrix" by Sebastian Mizera (arxiv:2306.05395) is an excellent set of lecture notes on the analytic properties of scattering. The first section covers some of the key ideas of our course.

    • "A brief Introduction to Dispersion Relations and Analyticity" by Zwicky (arxiv:1610.06090) is an older set of lecture notes on the analytic properties of quantum field theory.

    • If you want to see why I have been spending time thinking about the analytic properties of quantum mechanics, see "Undecay" by Megias, Perez-Victoria, and Quiros (arxiv:2310.16593). It is a study of the propagator (Green's function) of a certain class of theories called unparticles (alternatively, non-compact extra dimensional theories or near-conformal strongly coupled theories). 

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Residue theorem gymnastics. And finally, the Green's function. 

• Mon 10/20: Using the residue theorem for real integrals. Notes. In class we discussed real integrals with branch cuts and re-did the example from Friday with a different contour. During our discussion we also brought up the generalized Stokes theorem.

• Wed 10/22: We worked out the Green's function for the harmonic oscillator. The choice of contour depends on the sign of the exponential, which itself came from the eigenbasis of the harmonic oscillator differential equation. [Notes]

• Fri 10/24: Making sense of the choice of sign for iε. The damped harmonic oscillator. [Notes]

Assignments

• Long Homework 2 (link udpated 10/20, thanks Michael P. ): due Friday 10/24. [Submission link]

• Peer Review 1: due  Wednesday 10/22.  [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

Upcoming

• Short Homework 3: due Wednesday 10/29. [submission link]

• Explainer Video 2: due Friday 10/31. [submission link]

  • Assignments are on our 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

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

• Mon 10/27: Discussion: Kramers-Kronig relations, principal value, 

• Wed 10/29:

• Fri 10/31:

Assignments

• Short Homework 3: due Wednesday 10/29. [submission link]

• Explainer Video 2:  due Friday 10/31 [submission link]

  • Assignments are on our internal sheet

Upcoming: please review the long homework this week so we may discuss it in class.

• Peer Review 2: due  Wednesday 11/5 [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

• Long Homework 3: due Friday 11/7.  [submission link]

Suggested Reading

• • • See Ch. 15.2 of Appel's Mathematics for Physics & Physicists for a discussion of using the principal value to determine the Green's function of the "ε=0" harmonic oscillator. Appel points out as a distribution, this latter case is not part of a convolution algebra. 

    • 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.

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

• Mon 11/3:

• Mon 11/3 (discussion hour):

• Wed 11/5:

• Fri 11/7: no class

Assignments

• Peer Review 2: due Wednesday 11/5. [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

• Long Homework 3: due Friday 11/7. [submission link]

Upcoming:

• Explainer Video 3:  due Friday 11/14 [submission link]

  • Assignments are on our internal sheet

• Short Homework 4: due  Wednesday 11/12 [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)

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

• Mon 11/10:

• Wed 11/12: 

• Fri 11/14:

Assignments

• Short Homework 4: due Wed 11/12 [submission link]

• Explainer Video 3: due Friday 11/14  [submission link]

  • Assignments are on our internal sheet

Upcoming:

• Peer Review 3: due Wed, 11/19 [submission link]

  • Assignments are on our internal sheet

• Long Homework 4: due Friday 11/21 [submission link]

Suggested Reading

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

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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.

• Mon 11/17:

• Wed 11/19:

• Fri 11/21:

Assignments

• Peer Review 3: due Wed 11/19 [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

• Long Homework 4: due Friday 11/21. [submission link]

Upcoming

• Explainer Video 4: due Friday 11/28 [Mon Dec 1 is ok]   [submission link]

• Last 2 weeks' assignments are to be determined.

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

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Perturbations about the Gaussian: what to do when the universe is not quadratic.

• Mon 11/24:

• Mon 11/24 (discussion hour):

• Wed 11/28:  no class

• Fri: 11/28: no class, Thanksgiving

Assignments

• Explainer Video 4: due Friday 11/28 [Mon Dec 1 is ok]   [submission link]

  • Assignments are on our internal sheet

• Last 2 weeks' assignments are TBD

  • [SHW5 Submission link]

  • [LHW5 Submission link]

• Peer Review 4: due Wed 12/3 [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

Suggested Reading

• • • "Feynman Diagrams," Weinzierl arXiv:2501.08354

    • 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.

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A further investigation of analyticity in physics.

Mon 10/1:

Wed 10/3:

Fri 10/5:

Assignments

• Last 2 weeks' assignments are TBD

  • [LHW5 Submission link]

• Peer Review 4: due Wed 12/3 [submission link]

  • Please fill out the same form once for each of the three peers you are reviewing

  • Assignments are on our internal sheet

Suggested Reading

• • • "Physics of the Analytic S-Matrix" (arXiv: 2306.05395) by Sebastian Mizera. This is an excellent summary of topics in this course and a peek at how it applies to quantum mechanics and quantum field theory. 

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