The amplituhedron looks like an intricate, multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated, “scattering amplitudes,” which represent the likelihood that a certain set of particles will turn into certain other particles upon colliding

The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.

“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at Harvard University and one of the researchers who developed the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.”

*Illustration by Andy Gilmore. Artist’s rendering of the amplituhedron, a newly discovered mathematical object resembling a multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated — the probabilities of outcomes of particle interactions.*

Arxiv - Scattering Amplitudes and the Positive Grassmannian

Locality is the notion that particles can interact only from adjoining positions in space and time. And unitarity holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one. The concepts are the central pillars of quantum field theory in its original form, but in certain situations involving gravity, both break down, suggesting neither is a fundamental aspect of nature.

In keeping with this idea, the new geometric approach to particle interactions removes locality and unitarity from its starting assumptions. The amplituhedron is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.

“It’s a better formulation that makes you think about everything in a completely different way,” said David Skinner, a theoretical physicist at Cambridge University.

The amplituhedron itself does not describe gravity. But Arkani-Hamed and his collaborators think there might be a related geometric object that does. Its properties would make it clear why particles appear to exist, and why they appear to move in three dimensions of space and to change over time.

Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery of the amplituhedron could cause an even more profound shift, Arkani-Hamed said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of the universe arose out of pure geometry.

“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless.”

**Abstract**

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams

as objects of fundamental importance to scattering amplitudes. W e show that the all-loop integrand in N =4 super Yang-Mills (SYM) is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful strati cation of the Grassmannian G(k; n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations|in particular, BCFW deformations correspond to simple adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with \positive" coordinates i and an invariant measure of the form Qid log i which determines the on-shell function associated with the diagram.

This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. The Yangian invariance of scattering amplitudes is transparently represented by di eomorphisms of G(k; n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop integrand for scattering amplitudes|presenting all integrands in a novel \d log" form which is a direct reflection of the underlying positive structure.

**Outlook**

We have explored much of the remarkable physics and mathematics of scattering amplitudes in planar N = 4 SYM, as seen through the lens of on-shell diagrams as the primary objects of study. Let us conclude by making some brief comments on further avenues of research.

One immediate extension of our work is the continued study of theories with N < 4 SUSY, whose most basic features we sketched out in section 14. For N 1, all-loop BCFW recursion holds just as for N = 4, together with its realization in terms of on-shell diagrams. For N = 0 SUSY, the forward limit of tree amplitudes are singular, and thus don't directly give us the single-cuts of the loop-integrand. More thought is needed to establish a connection between on-shell diagrams and the full amplitude, though it is likely that fully understanding the on-shell diagrams will continue to play an important role in determining N =0 amplitudes as well. The general connection between on-shell diagrams and the Grassmannian has nothing to do with any particular theory, only with the general picture of amalgamating basic three-particle amplitudes, and the connection to the positive Grassmannian in particular holds for any planar theory. Only the form on the Grassmannian changes from theory to theory. As briefly discussed in section 14, the essential physical novelty of gauge theories with N 2 supersymmetry is the presence of UV-divergences. The most physical, Wilsonian, way to think about UV-divergences makes critical use of o -shell ideas, and so a major challenge is fi nding the correct way of thinking about such physics in a directly on-shell language. It is fascinating to see that the UV and IR singularities, together with UV/IR decoupling, is reflected directly in on-shell diagrams through simple structures in the Grassmannian. A clear goal would be to understand the physics of the renormalization group along these lines. Another obvious extension is to push beyond the planar limit, starting already with N = 4 SYM; in this case, there is no longer an obvious notion of \the loop integrand", and thus we must learn how to establish a connection between on-shell diagrams and the full scattering amplitude along the lines of the BCFW construction in the planar limit. It is also very likely that on-shell ideas can be used to determine other observables in gauge theories beyond scattering amplitudes, including all correlation functions. These objects also have discontinuities and cuts, and the on-shell diagrams for leading singularities of form-factors and correlation functions are exactly the same as the (in general non-planar) on-shell diagrams we have been considering. The structure of cuts has already proved to powerful in determining form-factors. For scattering amplitudes, we have seen that o -shell notions like virtual loop integration variables can be fully understood in on-shell terms. It is tempting to try and compute completely o -shell objects like correlation functions in the same way . SOURCE - simonsfoundation, arxiv

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