Physics With Friends

This Session's Topic
Covariant quantization, the Polyakov string action, and the basics of conformal field theory

Overview
This is a collaborative study group dedicated to understanding the foundations, motivations, and current landscape of quantum gravity research. Our goal is to build a clear conceptual picture of why quantum gravity is challenging, what the major approaches attempt to do, and how they compare.

Developing a coherent understanding of quantum gravity requires engaging with the mathematics. As such, we will work through explicit equations and derivations to understand how gravitational dynamics emerge, how quantum consistency constrains theories, and why certain results are unavoidable rather than assumed.

Participants should have a basic familiarity with the principles of quantum mechanics, quantum field theory, and general relativity, and be comfortable with the Lagrangian formalism of physical theories.

What to expect

• We assign readings, lectures to watch, and curated exercises prior to each meeting, which will be announced below.
• During the meetings, one or more of the volunteers will walk through the assigned materials and moderate the discussion.
• Additional discussion between meetings will take place on our chat server, where we can exchange solutions, notes, interesting and relevant articles, etc. Notes and solutions may also be hosted in our Github repository.

What this group is NOT

• Not a popular science or sci-fi discussion group.
• Not a high-pressure, exercise-heavy bootcamp.
• Not a forum to strongly endorse or air grievances against any one approach to quantum gravity .

For casual physics chat or general discussion, please see our other Meetup groups.

Community & Continuity
We maintain a live chat server for continuing discussions between meetups, and a link can be provided upon request if you’d like to join. This even is part of the broader Physics With Friends community, which hosts multiple collaborative study tracks across physics and mathematics.

Last time, we explored how curvature acts like a force between neighboring geodesics. Through the Jacobi Equation, we saw how positive curvature produces relative acceleration toward one another, while negative curvature produces relative acceleration apart.

But our examples lived on two-dimensional surfaces. What happens when three-dimensional space itself is curved?

In Chapter 29, “Riemann’s Curvature,” of Tristan Needham’s Visual Differential Geometry and Forms, we move from curved surfaces to curved space. Needham develops the central ideas through visual examples in three dimensions, giving us something concrete to picture before the next chapter takes us into four-dimensional spacetime and Einstein’s theory of gravity.

The lecture that astonished Gauss

In 1854, Bernhard Riemann submitted three possible topics for a lecture required for his application to become an unsalaried faculty member at Göttingen. He expected Gauss to choose the first, based on his already completed doctoral research. Instead, Gauss chose his third proposal: “The Foundations of Geometry.”

That choice pushed Riemann into a new investigation of the very meaning of space and curvature. His resulting lecture, delivered to a largely nonmathematical audience, contained just one formula—and ideas that profoundly impressed Gauss. According to Dedekind’s later account, Gauss expressed unusual excitement afterward about the depth of Riemann’s ideas.

Riemann died of tuberculosis in 1866, aged just 39. His lecture was published after his death, yet its ideas eventually provided the mathematical foundation for Einstein’s theory of gravity.

One point, many curvatures

On a surface, a single number—the Gaussian curvature—describes the intrinsic curvature at a point. In three-dimensional space, different planes through the same point can have different curvatures. Space can focus neighboring geodesics in one direction while spreading them apart in another.

We’ll explore how sectional curvature captures this dependence on the plane we choose.

Carry a vector around a loop

Parallel transport gives us a way to carry a direction through curved space. When we bring a vector around a small closed loop, it can return pointing in a different direction.

This change, called holonomy, reveals the curvature. In three dimensions, the result depends on both the plane of the loop and the vector we carry around it. The returning vector can even tip out of the loop’s plane.

The Riemann curvature tensor organizes this information into a complete description of intrinsic curvature.

How curvature shapes motion

Returning to the Jacobi Equation, we’ll examine how curvature changes the relative motion of neighboring geodesics. Imagine a small circle of particles launched perpendicular to its plane. Their paths can focus, spread out, or stretch the circle in one direction while squeezing it in another.

This leads to the Ricci curvature tensor, which describes the initial acceleration of the area enclosed by this bundle of geodesics.

The three-dimensional picture prepares us for a crucial step: in four dimensions, a transverse circle becomes a sphere, and Ricci curvature governs the initial acceleration of its enclosed volume. This connection will become central to the next chapter’s treatment of the Einstein field equation.

Topics we’ll discuss

• Riemann’s remarkable lecture and Gauss’s response.
• Parallel transport in three dimensions, including Schild’s ladder.
• Covariant derivatives: comparing vectors at different points.
• The Riemann tensor as a measure of infinitesimal holonomy.
• Sectional curvature and its connection to geodesic deviation.
• Ricci curvature and the focusing of bundles of geodesics.

Throughout, we’ll follow Needham’s visual approach. Although the mathematics extends to higher dimensions, three-dimensional examples let us picture what curvature does.

Join us as we move from curved surfaces to curved space, building the geometric foundation for our next chapter on Einstein’s curved spacetime.

Background: Familiarity with basic calculus, geodesics, and the ideas of parallel transport and holonomy will be helpful. No prior knowledge of General Relativity is required.

Join us every other Tuesday for an open-ended casual discussion with our book club for Roger Penrose's The Road to Reality, the magnum opus of one of the greatest theoretical physicists of our time. In a single volume, Penrose takes the reader from basic mathematical and logical arguments through advanced math and then onto modern physics, relativity, quantum theory, and beyond.

Please have access to a copy of the book and have read at least one chapter prior to attending. We normally agree to read one particular chapter in advance and then spend our meeting time discussing that chapter and its problem set, sometimes branching off to other parts of the book. We support each other to go at whatever pace is comfortable and we discuss material that we find interesting or challenging. Those of us who want to work on the problem sets can help one another. Sometimes someone will want to make a mini-presentation on a topic of their liking. We also maintain a private chat room for those who want to stay in touch between meetings.