Grassmannian Origin of Scattering Amplitudes

Abstract

Quantum field theory (QFT) is our central theoretical

framework to describe the microscopic world, arising from the union

of quantum mechanics and special relativity. Since QFTs play such a

central role in our understanding of Nature, a deeper study of their

physical properties is one of the most exciting directions of

research in theoretical physics. This has led to the discovery of

many important theoretical concepts, such as supersymmetry and

string theory. One of the most prominent physical observable in any

QFT is the scattering amplitude, which describes scattering

processes of elementary particles. Theoretical progress in

understanding and computing scattering amplitudes has accelerated in last few years with the discovery of amazing new mathematical structures in a close cousin of QCD, known as N=4 Super-Yang-Mills theory (SYM).

In the first chapter we study integrands of loop amplitudes in

planar N=4 SYM and show their astonishing simplicity when

written in terms of special set of chiral integrals. In chapter two

we show how to reconstruct the multi-loop integrand recursively

starting from tree-level amplitudes. This approach makes the

long-hidden Yangian symmetry of the theory completely manifest and provides a Lagrangian-independent approach for determining the integrand at any loop order. In chapter three we demonstrate that

the problem of calculating of scattering amplitudes in planar N=4 SYM can be completely reformulated in a new framework in terms

of on-shell diagrams and integrals over the positive Grassmannian

G(k,n). Remarkably, the building blocks for amplitudes play a

fundamental role in an active area of research in mathematics

spanning algebraic geometry to combinatorics. In chapter four we

sketch the argument that the amplitude itself is represented by a

single geometrical object defined purely using a new striking

property -- positivity -- and all physical concepts like

unitarity and locality emerge as derived concepts, each having a

sharp geometric interpretation.