Explore the fascinating world of scattering amplitudes and their connection to the positive Grassmannian in this comprehensive lecture. Delve into the theoretical foundations of particle physics, starting with elementary particles and forces, and progressing through quantum field theory and scattering amplitudes. Discover the limitations of traditional Feynman diagrams and learn about innovative approaches like on-shell diagrams. Examine the mathematical underpinnings, including permutations, positive matrices, and the stratification of the positive Grassmannian. Investigate the surprising connections to cluster algebras and explore advanced concepts such as the Amplituhedron. Gain insights into cutting-edge research in theoretical physics and mathematics, with discussions on non-planar on-shell diagrams and resources for further study.
Scattering Amplitudes and Positive Grassmannian by Jaroslav Trnka
International Centre for Theoretical Sciences via YouTube
Overview
Syllabus
11:30 am | 10 DEC 2018
Scattering Amplitudes and the Positive Grassmannian
Scattering of elementary particles
Theorist's perspective What happens during scattering experiment?
New picture: on-shell diagrams Remarkable connection to mathematics
Motivation
Elementary particles and forces
Quantum field theory
Scattering amplitude
Perturbative expansion
Feynman diagrams
Search for new physics
Amplitudes of gluon's
Parke-Taylor formula
Amplitudes as a new field
Problem with Feynman diagrams
New objects
Kinematics of massless particles
Fundamental amplitudes
Definition of on-shell diagrams
On-shell diagrams
Amplitudes from on-shell diagrams
Simple examples
Simplicity revealed
Identity moves
Precise statement
Permutations
New look at permutations
Identity moves
Yang-Baxter moves
Reduced information
Positive Grassmannian
Positive matrices
Face variables
Perfect orientation
Boundary measurement
Entries of matrix
Positive matrix
Edge variables
Cell in the Positive Grassmannian
Example of configuration
Example of stratification
Relation to permutations
Boundary operator
Stratification of the positive Grassmannian
Summary of positive Grassmannian
Connection to cluster algebras
Seed and mutation
Square-move
Surprising connection
Connection
Cluster transformation
Momentum conservation
Geometry of delta function
Logarithmic form
Derivation: starting with 3pt
Amalgamation procedure
New procedure
Conclusion
Summary
Amplituhedron
Non-planar on-shell diagrams
Resources
Thank you!
Amplituhedron
Taught by
International Centre for Theoretical Sciences
