Week 02.03.2026 – 08.03.2026

Energy conditions were originally formulated as pointwise bounds on contractions of the stress–energy tensor and have played a central role as assumptions in many foundational results of classical general relativity, most notably the singularity theorems. However, these conditions are generically violated by quantum fields, which admit states with locally negative energy density. Such violations are nevertheless constrained: quantum energy inequalities impose bounds on the magnitude and duration of negative energy.

In this course, I will first introduce the classical energy conditions and review their physical motivation and known violations. Then I will provide a brief introduction to quantum field theory on curved spacetimes and demonstrate how quantum energy inequalities can be derived. Finally, I will discuss in detail the average null energy condition and the limitations it imposes to causality violating spacetimes.

Course plan:
Lecture 1: Classical energy conditions and their violations
Lecture 2: Quantum field theory on curved spacetimes
Lecture 3: A derivation of a quantum energy inequality
Lecture 4: The average null energy condition​

I will discuss scattering on the Coulomb branch of planar N=4 SYM at finite ’t Hooft coupling. This describes a family of classical open-string S-matrices that smoothly interpolates between perturbative parton scattering at weak coupling and flat-space string scattering at strong coupling. I will focus on the four-point amplitude and discuss its remarkably rich structure: nonlinear Regge trajectories, dual conformal invariance, an intricate spectrum of bound states with an accumulation point, and a two-particle cut. Using dispersion relations and S-matrix bootstrap techniques, these properties can be incorporated to constrain the amplitude at finite ’t Hooft coupling, and I will discuss bounds on Wilson coefficients, couplings to bound states, and the overall shape of the amplitude.

This is based on https://arxiv.org/abs/2510.19909.