Andrew Tolley - 2/3 Positivity in Cosmology

The lecture is a rigorous and detailed exposition of the analytic properties of scattering amplitudes, essential for understanding positivity bounds. Andrew Tolley, a leading expert in effective field theory and cosmology, presents the material with clarity and depth. The argumentation is logically structured: starting from the Lippmann-Schwinger equation in non-relativistic quantum mechanics, he demonstrates the analyticity of the T-matrix and the resulting dispersion relations. He then extends this to relativistic quantum field theory, emphasizing that causality dictates the use of the retarded propagator when solving Heisenberg equations of motion, not the Feynman propagator. This is a crucial point often overlooked in standard QFT treatments. He introduces the concept of ‘product’ correlation functions, which are in-in or Schwinger-Keldysh-like, and shows how they encode causality through nested commutators. This provides a solid foundation for the subsequent derivation of positivity bounds. The lecture is technically demanding, assuming familiarity with quantum field theory and dispersion relations. However, the presentation is clear, with explicit equations and explanations. The scientific content is accurate and consistent with established literature, though no specific sources are cited in the video. The title accurately reflects the content, which is focused on positivity in cosmology. The lecture is part of a series, so it builds on previous material and sets up future lectures. Overall, this is a high-quality lecture that provides valuable insights into the foundations of positivity bounds, which are crucial for constraining effective field theories in cosmology.

The title accurately reflects the content: the lecture focuses on positivity bounds in cosmology, specifically on the analytic structure of the S-matrix and dispersion relations.

8/10

Lecture by a recognized expert (Andrew Tolley, Imperial College) in theoretical physics, part of a series at IHES. The content is technically rigorous, based on established formalism (Lippmann-Schwinger equation, analytic S-matrix, dispersion relations). No sources are explicitly cited in the video, but the presentation is consistent with standard literature. The video is a formal lecture, not peer-reviewed, but the scientific content is reliable.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction and recap of previous lecture; focus on Lippmann-Schwinger equation and analyticity of T-matrix.
• 05:00Discussion of dispersion relations and their origin from causality in quantum mechanics.
• 10:00Extension to relativistic quantum field theory; Klein-Gordon scattering example.
• 15:00Emphasis on using retarded propagator in Heisenberg equations of motion, not Feynman propagator.
• 20:00Introduction of 'product' correlation functions (in-in formalism) and their causal properties.
• 25:00Examples of product correlation functions: two-point and three-point functions with nested commutators.
• 30:00Connection to S-matrix and LSZ formula; discussion of on-shell limits.
• 35:00Summary and outlook for next lecture on positivity bounds.

• Carmin.tv — Video platform hosting the lecture and other scientific content.

• Positivity bounds in effective field theories — This paper by Adams et al. establishes positivity bounds from analyticity and unitarity, which are directly relevant to the lecture's content.