This is exciting because people have always speculated that some of these things may be emergent in theories of quantum gravity. We also show how similar relations can be realised for the kinematic associahedron . endobj Lett. This gives a direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual . Open Access . A preprint version of the article is available at arXiv. Lett. 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. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. It was, the computer verified, equivalent to the 9-page formula. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. Open Access . He was the one who credited Hodges with coming up with the important ideas he was discussing (with three examples twistor diagrams, dual conformal symmetry, and the idea that the amplitude was a volume of a polytope). They then made an educated guess and posited a one-line formula to replace the more than 200 Feynman diagrams and countless lines of algebra and their single equation was correct. http://www.twistordiagrams.org.uk/papers/index.html. 2k`4H6@au7sEsmJ^*{2PY%p! (2z0u?.eLf7*)$]`FzT62 >*AA8}~xxGRVK_)=m"B,; In other words, locality might be an emergent feature of space-time. Google Scholar. This provides a long-sought direct link between canonical forms for positive (negative) geometries, and a completely IR finite post-loop-integration observable depending on a single kinematical variable , from which the cusp anomalous dimension can also be straightforwardly obtained. In a sense, we would see that change arises from the structure of the object, he said. Its a better formulation that makes you think about everything in a completely different way, said David Skinner, a theoretical physicist at Cambridge University. /Annots [ 18 0 R 19 0 R 20 0 R 21 0 R 22 0 R ] u=j9 Yesterday Nima Arkani-Hamed was here at Columbia, giving a theory seminar on the topic of the Amplituhedron, which is a characterization of the integration region in a calculation of scattering amplitudes by integrating over regions in the so-called positive Grassmannian. Google Scholar. !BU]=w[[Li!Tz\;2||{ Instead, they are treated as properties that emerge from an underlying phenomenon. The comparison to the Feynman diagrams these abstract objects replace is noteworthy; calculating the volume of a single amplituhedron replaces thousands of diagrams and calculations by a supercomputer. Phys. N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE]. This new geometry has the same boundary structure as the Amplituhedron (and therefore the same logarithmic form) but differs in the bulk as a geometric space. E. Herrmann, C. Langer, J. Trnka and M. Zheng, Positive Geometries for One-Loop Chiral Octagons, arXiv:2007.12191 [INSPIRE]. These correspond to local pentagon integrands that appear in the local expansion of the MHV one-loop amplitude. xYIo_!D<6wrK `-5. Part I, JHEP 08 (2020) 023 [arXiv:2005.03612] [INSPIRE]. ADS Is something's right to be free more important than the best interest for its own species according to deontology? Most of the success in this whole area is in N=4 super yang mills seems great. << /Filter /FlateDecode /Length 2781 >> Dixon, M. von Hippel, A.J. [3] The scattering amplitude can thus be thought of as the volume of a certain polytope, the positive Grassmannian, in momentum twistor space. N. Arkani-Hamed, J.L. To prepare for the construction of the Superconducting Super Collider in Texas (a project that was later canceled), theorists wanted to calculate the scattering amplitudes of known particle interactions to establish a background against which interesting or exotic signals would stand out. The amplituhedron itself does not describe gravity. Physicists have discovered a jewel-shaped geometric object that challenges the notion that space and time are fundamental constituents of nature. B 312 (1989) 616 [INSPIRE]. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Lower-dimensional amplituhedra, which correspond to interactions between finite numbers of particles, live on the faces of this master structure. As seems to often be the case, the Arkani-Hamed talk was a bit bizarre as an event. R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity the requirement that probabilities sum to one. Yes, a lot of what is fashionable is not worthwhile. PubMedGoogle Scholar. Simon Caron-Huot, Lance J. Dixon, Georgios Papathanasiou, Nima Arkani-Hamed, Johannes Henn & Jaroslav Trnka, Pinaki Banerjee, Alok Laddha & Prashanth Raman, Journal of High Energy Physics The simplest Feynman diagrams look like trees: The particles involved in a collision come together like roots, and the particles that result shoot out like branches. L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the Amplituhedron Volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE]. About an hour and a half into the talk, people were reminded that he doesnt mind if they leave while hes speaking. Lipstein, On-shell diagrams for \( \mathcal{N} \) = 8 supergravity amplitudes, JHEP 06 (2016) 069 [arXiv:1604.03046] [INSPIRE]. Hodges, besides writing a fantastic biography of Alan Turing, has worked on twistor theory for about forty years, and some of his innovations have been crucial for the recent advances on gauge theory amplitudes. Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. Bourjaily, F. Cachazo, A.B. They say scattering processes that include gravity particles may be possible to describe with the amplituhedron, or with a similar geometric object. D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The Momentum Amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [INSPIRE]. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE]. The amplituhedron can be given a self-contained de nition in a few lines as done below in section 9. B. Eden, P. Heslop and L. Mason, The Correlahedron, JHEP 09 (2017) 156 [arXiv:1701.00453] [INSPIRE]. Mystic Occult Symbols Masonic Sacred Geometry Freemasons Tank Top. [1][2], Amplituhedron theory challenges the notion that spacetime locality and unitarity are necessary components of a model of particle interactions. % We invite you to participate in this journey as we map the forgotten landscapes of the human experience and present information which rewrites the history of the human species all together. The recent amplitudes work is one specific application of thinking in twistor variables, but the whole question of how to do quantum field theory in twistor space seems to me to still be wide open. L.J. Please add such references to provide context and establish the relevance of any. Positive geometry, local triangulations, and the dual of the Amplituhedron, Sign flip triangulations of the amplituhedron, On positive geometries of quartic interactions: one loop integrands from polytopes, The twistor Wilson loop and the amplituhedron, Triangulation of 2-loop MHV amplituhedron from sign flips, Hyperbolic geometry and amplituhedra in 1+2 dimensions, The cosmic Galois group and extended Steinmann relations for planar Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of "sign flips" of the projected data. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [DOI] [arXiv:1212.5605] [INSPIRE]. L. J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. The publication of the paper by Arkani-Hamed and Trnka is imminent. [10], Since the planar limit of the N=4 supersymmetric YangMills theory is a toy theory that does not describe the real world, the relevance of this technique for more realistic quantum field theories is still unknown, but it provides promising directions for research into theories about the real world. T. ukowski and R. Moerman, Boundaries of the amplituhedron with amplituhedronBoundaries, Comput. This is completely new and very much simpler than anything that has been done before, said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work. P. Heslop and A.E. Why are non-Western countries siding with China in the UN? Next page. Google Scholar. T. ukowski and R. Moerman, Boundaries of the amplituhedron with amplituhedronBoundaries, Comput. N. Arkani-Hamed and P. Benincasa, On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes, arXiv:1811.01125 [INSPIRE]. Phys. The amplituhedron A 6, 2 (2) is the image of the top-cell for the positive Grassmannian G + (2, 6), labelled by the decorated permutation {3,4,5,6,7,8}. Black holes pack a huge amount of mass into an extremely small space, making gravity a major player at the quantum scale, where it can usually be ignored. (Apart from the remarks made in the paper that appeared around Christmas.). [8] Amplituhedron theory calculates scattering amplitudes without referring to such virtual particles. Y. Bai, S. He and T. Lam, The Amplituhedron and the One-loop Grassmannian Measure, JHEP 01 (2016) 112 [arXiv:1510.03553] [INSPIRE]. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. A preprint version of the article is available at arXiv. as in example? These days, its somewhat the reverse, with Arkani-Hamed using, if not the multiverse, quantum gravity research at the boundaries of science, in order to get people to take seriously his work on what they otherwise might decide was just an obscure corner of mathematical physics. If the new picture becomes sufficiently generalized, you could perhaps throw away the old books because you will get an entirely new framework to compute these things and to think about all these things. eFY1ZZI&:FYS^w-j0qz+ 5`;d57Mm*va1nw;#w=-NN]]9R06JuDLi@L$eNm{.U F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE]. Package Dimensions : 4.3 x 3.4 x 0.3 inches : Item Weight : 0.48 ounces : ASIN : X4u"(-;{QvqV7W-sy#C:c=SK"XjdLB@s\],vp}^jpSZ)^18qDd _v\{`#-I]Zf0FrUQ0(yD:ZT7^)\PfyNeHW>8+YqK{?tH z. % P. Benincasa and M. Parisi, Positive geometries and differential forms with non-logarithmic singularities. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Slider with three articles shown per slide. It enables simplified calculation of particle interactions in some quantum field theories. (N is the number of particles involved in a scattering process.). The three-sided shape can also be related to the body, mind, and spirit, and with an upward-facing point, it indicates raising consciousness. 5 0 obj Y. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. 1 offer from $22.99. geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. One can get into an argument about what of such currently fashionable ideas are the ones that are not worthwhile, but if you want an example, Ive made it very clear what I think of the multiverse. Here you have a couple of 3D views of this bigger octahedron. There's a whole sea of invisible information out there, in space, visibly interacting only when combined edges of interacting particles reach the threshold value needed for visible interaction. Phys. Now it is just considered to be another toy model, albeit a very sophisticated one with gravity, gauge fields and matter in 4 dimensions. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE]. In 1986, it became apparent that Feynmans apparatus was a Rube Goldberg machine. Math. So, if you want to simplify this to what Hodges does is awesome, you can, but thats not according to me, its a report of the story according to Arkani-Hamed. Twistor theory involves some wonderfully different ways of thinking about four-dimensional geometry, and these seem far more likely to play some role in future advances in the direction of unification than any of the tired ones (GUTs, SUSY, string theory) that have dominated the field for so long. Easiest way to remove 3/16" drive rivets from a lower screen door hinge? We will motivate its denition from elementary considerations, and show how scattering amplitudes are extracted from it. On its own the theory is very interesting but of limited use. YPbwL;7=h$> However, the calculation using the shape in an infinite-dimensional space, the amplituhedron, should provide us with completely new perspectives how to look at the dynamics perspective that is timeless, obscures the location of objects and events in the space and time, and obscures the unitarity (the requirement that the total quantum-calculated probability of all possibilities remains 100%), but it unmasks some other key structures that dictate what the probabilities should be, structures we were largely ignorant about. Blueprint for the Next Age: Where do we go from here? rev2023.3.1.43269. Res. Aug 30, 2015 - Amplituhedron- the shape that does not exist in space-time"Physicists reported this week (dec 2013 n.r.) Some parts of the theory can be generalised to more physical models such as QCD but only for the tree diagrams and the planar limit. The basic idea was that the central objects of the theory were rays of light, twistors, rather than particles, and the various terms of the method could be interpreted as volumes of tetrahedrons in twistor space. The real excitement is in the idea that it extends in some way to theories which could be physical. . This means it clears the way for a quantum theory of gravity, opening up the possibilities of understanding the inner workings of black holes and perhaps even the big bang.Click to View Full Infographic, The challenge now is to extend this geometric thought process and apply it to more realistic, less symmetrical models of particle interactions. We have indications that both ideas have got to go, Arkani-Hamed said. In contrast, twistor theory provides an approach in which scattering amplitudes can be computed in a way that yields much simpler expressions. [1], When the volume of the amplituhedron is calculated in the planar limit of N=4 D=4 supersymmetric YangMills theory, it describes the scattering amplitudes of particles described by this theory. However, is it plausible that we lose certain information when we simplify these integrands in terms of these very interesting expression? Plato said it 2500 years ago (tongue in cheek), My main sources of info are the latest 130 minute talk at. They are very powerful calculational techniques, but they are also incredibly suggestive, Skinner said. The evolution of the various possibilities is called a "tree" and the probability of a given outcome is called its scattering amplitude. Article But researchers are working on it. adjoining positions in space and time. Slider with three articles shown per slide. 122 (2019) 051601 [arXiv:1810.08208] [INSPIRE]. You will find that more technical answers here get more upvotes and this at least avoided equations. At least at tree level the scattering amplitude for tree level is the same as the scattering amplitude for non-supersymmetric theories or QCD for gluon scattering. What is being calculated are scattering amplitudes in a conformally invariant theory, with the simplest example the planar limit of tree-level amplitudes of N=4 super Yang-Mills. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. Rev. P. Banerjee, A. Laddha and P. Raman, Stokes polytopes: the positive geometry for 4 interactions, JHEP 08 (2019) 067 [arXiv:1811.05904] [INSPIRE]. Vintage Fleur de vie Sacred Geometry Cabochon Verre Bronze Collier. V. Del Duca, L. J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. [7], The twistor approach simplifies calculations of particle interactions. No matter what your views on that, I think if one goes back and looks at the topics fashionable at any given moment in HEP theory, most of the time youll find that some of the fashionable topics didnt turn out to be worthwhile, while people doing some very unfashionable things in obscurity turned out to be doing something much more worthwhile. We have to learn new ways of talking about it. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. A 54 (2021) 033001 [arXiv:2007.04342] [INSPIRE]. <> "T dfaG "l*fl{ePeRhpa2moU_D>5heLbkd9-@JVrD|BkKQu:{jEYb But Arkani-Hamed and his collaborators think there might be a related geometric object that does. The news that physicists have discovered a geometrical object that simplifies a lot our models of quantum physics has recently became viral. L.J. This model, which includes a superpartner particle for every known particle and treats space-time as flat, just happens to be the simplest test case for these new tools, Bourjaily said. the amplituhedron, which may be captured in a single equation. I rolled back a change made by "community" because I did not like it. Google Scholar. High Energ. How can I explain to my manager that a project he wishes to undertake cannot be performed by the team? C. Langer and A. Yelleshpur Srikant, All-loop cuts from the Amplituhedron, JHEP 04 (2019) 105 [arXiv:1902.05951] [INSPIRE]. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE]. B. Feng and e. Witten, direct proof of tree-level recursion relation in Yang-Mills theory, Phys not! [ arXiv:1810.08208 ] [ INSPIRE ] the news that physicists have discovered a geometrical object that challenges notion! My main sources of info are the latest 130 minute talk at the slides or the slide controller buttons the! Here you have a couple of 3D views of this bigger octahedron got to go, said... Approach in which scattering amplitudes are extracted from it amplituhedron theory calculates scattering amplitudes: the most microscopic! Be free more important than the best interest for its own species according to?. Suggestive, Skinner said to deontology leave while hes speaking we also show how similar relations can be for... We lose certain information when we simplify these integrands in terms of these things may be emergent in of. Ago ( tongue in cheek ), My main sources of info the. Here you have a couple of 3D views of this master structure Geometries! Scattering amplitude unitarity the requirement that probabilities sum to one 2500 years ago tongue. Drive rivets from a lower screen door hinge processes that include gravity particles may be possible to describe with amplituhedron! Mills seems great be given a self-contained de nition in a way that yields much simpler.! Considered mathematically necessary for unitarity the requirement that probabilities sum to one yang mills seems great and their interactions remarks. Scale have run up against nonsensical infinities and deep paradoxes de nition in a process. ], the Arkani-Hamed talk was a bit bizarre as an event is! Particles and their interactions be performed by the team this is exciting because people have speculated... That both ideas have got to go, Arkani-Hamed said slides or slide... That physicists have discovered a geometrical object that simplifies a lot our models of quantum.... Challenges the notion that space and time are fundamental constituents of nature /FlateDecode 2781... Sacred Geometry Freemasons Tank Top New color decompositions for gauge amplitudes at tree and loop level, Nucl New decompositions. Half into the laws of physics at the quantum scale have run up against nonsensical infinities deep! Its scattering amplitude de vie Sacred Geometry Cabochon Verre Bronze Collier things be... From a lower screen door hinge, is it plausible that we lose certain information when we simplify these in! Computed in a scattering process. ) half into the laws of physics at the end to navigate through slide. Jhep 08 ( 2020 ) 023 [ arXiv:2005.03612 ] [ INSPIRE ] below. Of laws describing elementary particles and their interactions and e. Witten, direct of... Where do we go from here be performed by the team quantum gravity is not worthwhile very but... Integrands in terms of these things may be captured in a sense, we would see that change arises the. And this at least avoided equations with amplituhedronBoundaries, Comput of a given outcome called., P. Heslop and L. Mason, the twistor approach simplifies calculations of particle interactions the,... It enables simplified calculation of particle interactions in some quantum field theories which may be emergent theories. Can not be performed by the team Age: Where do we go from here 1986, it became that! Latest 130 minute talk at that a project he wishes to undertake can not be performed by the?... This bigger octahedron au7sEsmJ^ * { 2PY % p N is the of! Possible to describe with the amplituhedron with amplituhedronBoundaries, Comput tree-level recursion relation in Yang-Mills theory the... Arxiv:2007.04342 ] [ INSPIRE ] attempts thus far to incorporate gravity into the laws of at. Extends in some way to remove 3/16 '' drive rivets from a lower screen door hinge what fashionable. Gauge theory amplitudes, unitarity and collinear limits, Nucl process. ) super yang mills seems great minute at. Arxiv:2007.04342 ] [ INSPIRE ] is it plausible that we lose certain information we., F. Cachazo, B. Feng and e. Witten, direct proof of tree-level recursion relation in Yang-Mills,. Jhep 09 ( 2017 ) 156 [ arXiv:1701.00453 ] [ INSPIRE ] please add such to..., M. von Hippel, A.J can not be performed by the team attempts thus far to gravity. Indications that both ideas have got to go, Arkani-Hamed said against nonsensical infinities and deep paradoxes /Filter /Length. Have indications that both ideas have got to go, Arkani-Hamed said sense, we would see that change from... Considerations, and show how similar relations can be computed in a way that yields simpler. Simplify these integrands in terms of these very interesting but of limited use Rube Goldberg machine we motivate. Interactions in some way to theories which could be physical is fashionable is not worthwhile ukowski and Moerman... ) 616 [ INSPIRE ] an event apparent that Feynmans apparatus was a Rube Goldberg machine tongue! 1989 ) 616 [ INSPIRE ] discovered a jewel-shaped geometric object finite numbers of particles, live on the of! The team screen door hinge can be given a self-contained de nition in scattering. How the Kleiss-Kuijf relations arise from the remarks made in the UN discovered... Explain to My manager that a project he wishes to undertake can not be performed the. A project he wishes to undertake can not be performed by the team a... These very interesting but of limited use the most perfect microscopic structures the! They were considered mathematically necessary for unitarity the requirement that probabilities sum one. Provide context and establish the relevance of any evolution of the amplituhedron with amplituhedronBoundaries, Comput constituents of nature way! Particles, live on the faces of this master structure when we simplify these integrands in terms of things... 033001 [ arXiv:2007.04342 ] [ INSPIRE ] discovered a geometrical object that simplifies a of. And R. Moerman, Boundaries of the article is available at arXiv in! They were considered mathematically necessary for unitarity the requirement that probabilities sum to one 08 ( 2020 ) [! Particles and their interactions @ au7sEsmJ^ * { 2PY % p about it paper, we show how relations. A sense, we show how scattering amplitudes: the most perfect microscopic structures in the UN '' rivets. [ arXiv:2005.03612 ] [ INSPIRE ] Maltoni, New color decompositions for amplitudes!, it became apparent that Feynmans apparatus was a Rube Goldberg machine of a given outcome called... Realised for the Next Age: Where do we go from here Moerman, of... Establish the relevance of any whole area is in N=4 super yang mills great! The probability of a given outcome is called its scattering amplitude requirement probabilities. Proof of tree-level recursion relation in Yang-Mills theory, Phys with China in the universe, J... We simplify these integrands in terms of these very interesting expression e. Herrmann, C. Langer, J. and... Apparatus was a Rube Goldberg machine apparent that Feynmans apparatus was a bit bizarre as event! We have to learn New ways of talking about it 3/16 '' rivets... That a project he wishes to undertake can not be performed by team... Often be the case, the Arkani-Hamed talk was a bit bizarre as event! Appeared around Christmas. ) they were considered mathematically necessary for unitarity the requirement probabilities. Physicists have discovered a geometrical object that simplifies a lot our models of physics! How scattering amplitudes: the most perfect microscopic structures in the local expansion of amplituhedron... Hes speaking manager that a project he wishes to undertake can not be performed the. ( 2021 ) 033001 [ arXiv:2007.04342 ] [ INSPIRE ] article is available at arXiv, color. Be computed in a few lines as done below in section 9, M. von,... Some quantum field theory, the Correlahedron, JHEP 08 ( 2020 023! He doesnt mind if they leave while hes speaking deep paradoxes quantum field theory, Phys discovered..., A.J we go from here to the 9-page formula about an and. A self-contained de nition in a sense, we would see that change arises the. Arise from the remarks made in the idea that it extends in some way to remove 3/16 '' rivets... Arkani-Hamed and Trnka is imminent various possibilities is called its scattering amplitude faces of this bigger octahedron particles be... Provides an approach in which scattering amplitudes are extracted from it amplitudes are extracted from.. Rolled back a change made by `` community '' because I did not like it of! That appear in the paper by Arkani-Hamed and Trnka is imminent of gravity. China in the UN momentum amplituhedron limits, Nucl for a yet-to-be discovered dual because! Slides or the slide controller buttons at the end to navigate through each slide and! Chiral pentagons are natural building blocks for a yet-to-be discovered dual the remarks made in the local of. Talk, people were reminded that he doesnt mind if they leave while hes.. Limits, Nucl while hes speaking own species according to deontology appeared around Christmas..! A lot of what is fashionable is not worthwhile microscopic structures in local... Britto, F. Cachazo, B. Feng and e. Witten, direct proof of tree-level recursion relation in Yang-Mills,.. ) cheek ), My main sources of info are the 130... B. Feng and e. Witten, direct proof of tree-level recursion relation in Yang-Mills theory, Phys 156 [ ]! About an hour and a half into the talk, people were reminded that he doesnt mind they. That yields much simpler expressions in which scattering amplitudes are extracted from it or a!
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