By Thomas L. heath
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Additional info for Diophantus of Alexandria. A Study in the History of Greek Algebra
By successively attaching BCFW-bridges to a small set of ‘simple’ diagrams, a very complex collection of diagrams can be produced (both planar and nonplanar 20 Introduction to on-shell functions and diagrams alike). Indeed, as we will soon understand, it turns out that all (physically relevant) on-shell diagrams can be constructed in this way. 6 On-shell recursion for all-loop amplitudes While on-shell diagrams are interesting in their own right, for planar N = 4, we will see that they are of much more than purely formal interest.
66]). We can specify a k-plane in n dimensions by giving k vectors cα ∈ Cn , whose span defines the plane. We can assemble these vectors into a (k×n) matrix C, whose components are cαa for α =1, . . , k and a=1, . . , n. Under GL(k)-transformations, C → · C—with ∈ GL(k)—the row vectors will change, but the plane spanned by them is obviously unchanged. Thus, the Grassmannian G(k, n) can be thought of as the space of (k×n) matrices modulo this GL(k) “gauge” redundancy. From this, we see that the dimension of G(k, n) is k×n − k2 = k(n − k).
Indeed, as we will soon understand, it turns out that all (physically relevant) on-shell diagrams can be constructed in this way. 6 On-shell recursion for all-loop amplitudes While on-shell diagrams are interesting in their own right, for planar N = 4, we will see that they are of much more than purely formal interest. Scattering amplitudes at all loop orders can be directly represented and computed as on-shell scattering processes. This is quite remarkable, considering the ubiquity of “off-shell” data in the more familiar Feynman expansion.
Diophantus of Alexandria. A Study in the History of Greek Algebra by Thomas L. heath