Schedule

Google Calendar Link (UCR login required)

Welcome to grad school. 

• F: We will not be meeting today. Recommended alternative activities: discover where the botanical garden is, find the shelves in Orbach library that carry books in your subdiscipline of physics, watch the freshmen scramble to find their first lectures of college.

Assignments

See Week 1-2 assignments.

Suggested Reading

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

• 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.

• Simon Collison's "Maturity and the Weight of Learning" is a brilliant article in The Manual, a craft journal for web designers. It was published in the early 2010s. If you replace 'web design' with physics and references to tools with gestures to artificial intelligence, then the piece is directly applicable to our profession. 

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How (and why) do physicists use index notation for everything? How do we solve the ever-present eigenvalue problem in linear algebra?

• Mon 9/28: Logistics. How is math ifferent from physics? The Green's functoin problem. 

• Mon,9/28, discussion section: Dimensional analysis review.

• Wed 9/30: Index notation. Tensors. Funny vector spaces. 

• Fri 10/2: Inner products and metric spaces, hermiticity.  How tensors transform under a transformation matrix T: one factor of T for each upper index, one factor of T-inverse for each lower index. Two special classes of matrices: rotations (isometries) and Hermitian matrices. 

Assignments

• See Week 1-2 Assignments on the HW page for homework and submission links.

• This week you have to submit a brief survey, a video introduction, and your first short homework (all due Wed) 

Suggested Reading

See the linear algebra / tensor analysis / moment of inertia sections of our references page.

• Some background on the UCR Burros, "The Wild History of UCR’s Infamous Burros," by David Danelski for UCR Magazine.

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What is the infinite-dimensional limit of ordinary linear algebra? And why should we think about the "vector space" of functions as such a space?

"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 10/5: Histogram space, inner products, boundary conditions, differential operators. Lecture notes: from histograms to function space. 

• Mon 10/5 (Dis): Fourier Basis

• Wed 10/7: The continuum limit: function space. Lecture notes.

• Fri 10/9: The ubiquitous harmonic oscillator. What could go wrong? Topics: reminder of Fourier transform conventions, what is a Green's function, the Green's function equation, solving for the harmonic oscillator Green's function ... and getting stuck when we have a singular integral. Lecture notes.

Assignments

See assignments page. Due this Friday: the first long homework. 

Suggested Reading

For more on the continuum limit of "histogram space," see the continuum limit section of our references.

More formalism

• 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. 

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How is ℂ different from ℝ2? In what ways is our description of nature "intrinsically" complex? How is calculus on the complex plane different from multivariable calculus on two-dimensional real space?

• Mon 10/12: The complex plane, complex functions as maps. Riemann sheets. Notes.

• Mon 10/12: student presentation of long homework problems, any discussion points. See assignments page for presenters.

• Wed 10/16: "The intergral of an analytic function over a closed curve is zero." We show this three ways. The first way draws on the Riemann sum definition and reminds us that Δz = dz changes along a curve. The second way is similar to the surface integral of the curl of a function and shows that the integral along some boundary can be turned into integrals around little plaquettes on the interior. The final way integrates around a small circle around a point. Note how each method uses the analyticity (differentiability, Taylor expansion) of a function. Notes.

• Fri 10/17: Integration of analytic and meromorphic functions: the residue theorem. Notes. Some important examples:

  • 1/(x^2+1): simple example. You extend the integral over the real line into a closed integral along the upper or lower half plane. We showed that you get the same result either way. The integral along the arc vanishes as the radius goes to infinity. (We commented on the Riemann sphere, see the HW2 extra credit)

  • x^(1/3)/(x^2+1): example of a branch cut. The lesson: you have to parameterize the entire curve and you cannot cross a branch cut. This means that if you make a 2π loop around the origin, then your path has picked up an exp(2πi) factor. Do not be too quick to call this factor 1 when there's a function that required branch cuts: z^(1/3) of  exp(2πi)  is not 1.

  • We commented on the HW2.C extra credit problem from classical mechanics. In an idealized 1/r gravitational potential, what is the time it takes for a test mass to reach the origin? This is an integral of the form sqrt(x/(x-a)). It has an unusual branch cut structure: the line segement from 0 to a. We explained curves away from this segment do not have any issues with the integrand being multivalued. Recall that the motivation for the branch cut is that f(z) and f(e^{2πi} z) might map to different values even though z and e^{2πi} z are supposed to be the same number. The branch cut defines Riemann sheets that specify how z and e^{2πi} z are actually not the same numbers when e^{2πi}  came from a path that winds around the complex plane.

  • 2025 Notes + more 2025 Notes

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 through a conformal mapping problem. I find the act of working through it to be especially insightful. 

    • Lectures and Problems : A Gift to Young Mathematicians  by V. I. Arnold, Dmitry Fuchs, Mark Saul. A collection of essays; see specifically the chapter "Geometry of Complex Numbers, Quaternions, and Spins"

    • "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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How may we use the power of complex analysis (the residue theorem) to calculate real integrals that show up in physics? 

• Mon 10/19: Two more examples of the residue theorem for real integrals. The first example reminds us that exponential factors converge (or diverge) faster than any power law, so they typically determine the convergence of an integrant over the 'large arc.' This is important because our Fourier transform contains an exponential factor. The second example introduces the idea of principal values of real integrals and connects to pole-pushing. Notes.

• Wed 10/21: Kramers-Kronig relations and causality. 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. Comparison to the damped harmonic oscillator.

  • Notes

  • [2025 Notes]

• Fri 10/23: Making sense of the choice of sign for iε. Comparison to two other methods for solving differential equations: (i) completeness (resolution of the identity in terms of eigenfunctions), (ii) patching two homogeneous solutiosn together. Connection of our Fourier approach to boundary conditions in the time domain and in the frequency domain.  Notes.

  • 2025 Notes, + other 2025 Notes with a derivation of some distribution identities. These mainly follow Appel's ch. 15.2.

  • Kramers-Kronig relations, principal value, causality, an example. Notes.

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Suggested Reading

• See the Complex Analysis references on our references page.

• 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.

• For those in particle physics, Scott Willenbrock discusses the role of the complex pole in the unstable particle propagator in "A Brief History of Mass"(arXiv:2503.07866).

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Treatment of the wave equation as a "harmonic oscillator" in spacetime.

• Mon 10/26: The heat equation and the Schrodinger equation. Guessing the Green's function for the wave equation.  Notes.

  • [2025 Notes]

• Wed 10/28: waves in electromagnetism. Notes.

  • Part of our discussion (the apparent paradox of long-range EM wves) draws from the excellent textbook, Fly By Night Physics by Tony Zee. 

  • Our discussion of the Lorenz gauge (and the explanation that it is a valid gauge choice) comes from a note by Alexsandro Mota, arXiv:2509.00187

  • John Baez (UCR's very own proto-blogger) has a nice 'Mathstodon' thread on the Lorenz versus "Lorentz" gauge. He cites this history of physics paper: "Ludwig Valentin Lorenz is the discoverer of the 'Lorenz gauge'!" arXiv:0803.0047

  • "Why Gauge?," Rovelli  arXiv:1308.5599

• Fri 10/30: Hints for solving the Green's function in 3D. Investigation of dimensional reduction to 2D.

  • Extended notes from 2025.

  • Solving the wave equation in 2 and 3 dimensions. 2026: Notes. (We didn't end up taking about Huygen's principle.)

  • Flatland by Edwin Abbott Abbott on Project Gutenberg. (wikipedia) You may also want to check out the annotated edition by Ian Stewart.

  • Extra dimensions via PhD Comics and physicsts at UC Irvine.

Assignments

• See Homework page

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)

    • "An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism," George Green arXiv:0807.0088, a reproduction of Green's original article on what we now know as the Green's function of an operator. It's nice that the Green's function is a counter-example to Stigler's law of eponymy. I found this essay in Ch. 8 of Robert Herman's ODE notes (UNCW Math 463 2025), which opens with a quote from Julian Schwinger: "The young theoretical physicists of a generation or two earlier subscribed to the belief that: If you haven’t done something important by age 30, you never will. Obviously, they were unfamiliar with the history of George Green, the miller of Nottingham."

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Why are all of our favorite quantities fields that obey wave-like equations?

• Mon 11/2: Waves in (2+1) dimensions, accelerated charges, why the sky is blue. Notes. (The notes include a discussion of Huygen's principle which we did not get to in class.)

  • 2025 notes: A bird's eye view of field theory..

• Wed 11/3: The microphysics of fields. Dispersion, group and phase velocities. The Kelvin wake. Notes.

• Fri 11/6: What does it mean to write down a Lagrangian?

Assignments

• See Homework page

Additional References

• Kelvin wakes

  • Kelvin wakes from Dietrich Zawischa (original)

  • "Ship wakes: Kelvin or Mach angle?" Marc Rabaud, Frédéric Moisy arXiv:1304.2653

  • "Ship waves in the presence of uniform vorticity," Simen Ellingsen arXiv:1401.1682

  • "Lord Kelvin Wipes Out on Speed Boat Wakes? Classic analysis of wedge-like plumes not the whole story, two physicists argue" Adrian Cho in Science (2013)

  • "Elementary derivation of the wake pattern of a boat," Frank S. Crawford, Am. J. Phys. 52, 782–785 (1984)

  •  See also the treatment in Stone & Goldbart and the back-of-the-envelope discussion in Zee's Fly By Night Physics.

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Why are all of our favorite quantities fields that obey wave-like equations?

• Mon 11/9:  Gaussian integrals, field theory, a hint of perturbation theory.

• Wed 11/11:  No class today, federal holiday: Veteran's Day.

• Fri 11/14: 

Assignments

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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Your graduate coursework will make you an expert at taking a theory, say Maxwell's electromagnetism, and calculating the probability of some outcome: P( experiment | theory ). How, then, do we begin to answer the question that we actually care about: what is the probability of a given theory (model) given some experimental observations?

• Mon 11/16: [notes] Basics of probability. Richard Gott's experiment. Bayes' theorem.

  •  [notes] Entropy,  [notes] Entropy,  Monty Hall from Bayes' theorem.

• Wed 11/18: [notes] Likelihood. 

• Fri 11/21: [notes] Gaussian = free theory, Feynman diagrams, towards interacting theories.

Assignments

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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Perturbations about the Gaussian: what do we do when the universe is not quadratic?

• Mon 11/23: Interacting theories. [notes] 

• Wed 11/25: Introduction: mass renormalization.

• Fri: 11/28: no class, Thanksgiving

Assignments

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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• Mon 11/30: Spherical Harmonics and the CMB

• Wed 12/2: Correlation functions of the CMB

• Fri: 12/4: TBD. Possibilities include writing and speaking as a physicist, more on divergences in field theory

Assignments

Suggested Reading

• • • The Los Alamos Primer on Particle Physics (1984) is contains some of the most fantastic magazine-length public science writing of its era. (Not to be confused with what is known as the Los Alamos Primer.) If you have followed week 0 (dimensional analysis) and week 10 (introduction to renormalization) then I expect you will really appreciate Geoff West's contribution, "Scale & Dimension: from animals to quarks." It is an excellent thing to read when you're traveling somewhere over winter break.

    • There are many well written explanations of renormalization from a modern perspective. Here's one by João Melo from not too long ago. arxiv:1909.11099

    • For an intuitive picture of renormalization from 1969, see Charles Kaufman's "An Illustration from Classical Physics of Renormalization Mathematics" Am. J. Phys. 37, 560–561 (1969) https://doi.org/10.1119/1.1975679

    • Unlike other foundational ideas in theoretical physics, our understanding of renormalization is remarkably recent. Here's a brief history by Kerson Huang in memory of Ken Wilson. arxiv:1310.5533. 

    • "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. 

    • "Everything's a Mass on a Spring," ThatBlackChemist, Instagrasm, Dec 3, 2025. (courtesy of Chanda Prescod-Weinstein) One thing I'd like you to take away from our class: if (almost) everything is a mass on a spring, then (almost) all of the calculations we do are Green's functions.

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