## Tuesday, December 10, 2013

### The first Amplituhedron paper is out

We've been using the word "Amplituhedron" since September 2013 but only now, the first preprint with this word in the title was released:
The Amplituhedron
The authors, Nima Arkani-Hamed and Jaroslav Trnka ["Yuh-raw-sluff Turn-kuh" if you allow me to bastardize a Czech name), are preparing two more papers, "Into the Amplituhedron" and "Scattering Amplitudes from Positive Geometry", as well as a third paper along with Andrew Hodges, "Three Views of the Amplituhedron".

The today's paper has 36 pages of JHEP $$\rm\LaTeX$$.

These pages are divided to 14 short sections and it seems that one should be able to read the whole paper. So I hope that some of the TRF readers will try to look, too.

1. Dear Lubos this was one of the most enjoyable posts I have ever read on TRF.
And not only because the Middle Earth is damn real and important but also because Tolkien succeeded to give such a clear account of it even if he considered that the climate variability was neither defining nor interesting.
For many reasons the Lord of the Rings belongs to the 3 books that have most influenced my life.
That's why considering my expertise, I would strongly object to your qualification of Sauron as denier.
First of all he was a supreme ruler and could pretty much impose whatever he wanted on (almost) everybody so that there was nothing he could deny.
Second the Nazguls and the armies of Morgoth had hardly any valid opposition
The only microscopic detail between him and the absolute power for eternity was a ring held by a hobbit who for some reasons wanted to deny the above reality
.
So I humbly submit that the only few deniers in the Middle Earth were Frodo Baggins, Gimli and more generally most Dwarfs because as their racial anthem goes :
"... Some folks we never forget
Some kind we never forgive
Haven't seen the back of us yet
We'll fight as long as we live ...."
.

2. Andrew Hodges, aged 64, is an inspiration to us all for showing that steady work through out our lives can pay dividends later on at the cutting edge of research.

3. Yay, finally! Will read that soon.

4. In addition, this "insight" shows how the physical laws are important for the IPCC people. How can they use their climate models to model the Middle Earth climate, if different physical laws are valid there? No Middle Earth magic is allowed by our physical laws.

This article just shows that IPCC is completely ignorant about physics and they propagate only their wishes.

5. "hidden infinite dimensional symmetries" Congratulations for breaking the 10^500 acceptable vacua barrier! Think of the shelf length savings when Phys. Rev. D distributes on Blu-Ray quadruple BD-XL DVDs.

6. I wouldn't normally reply but because this particular idiocy of yours might be more widespread, let me reply.

There has never been any "10^500 barrier". For centuries, some sets in maths and physics were small, some were large, and some were infinite.

In this particular case, the talk isn't about the number of solutions - which may be small, large, or infinite as well - but about the dimensions of a Lie group. Infinite-dimensional Lie groups have been known for a century or so, too.

The fact that hardcore cranks like you have a trouble to imagine large sets, large groups, or work with infinite sets doesn't pose any problem in maths or physics whatsoever.

7. So is this a revolutionary development or not? And if it is, is it one in physicist's understanding or in their ability to calculate? Is there a difference? It seems clear that Lubos's friend Nima thinks the answers to these questions is yes. But what does Lubos think? Inquiring Lilliputians want to know.

8. Luboš, I empirically demonstrated your axions as dark matter are gravitationally short by a factor of 8.8×10^18. You have not corrected, recanted, or even admitted error - my arithmetic and arXiv:1306.5534 observation. If I am a hardcore crank for suggesting spacetime geometry be tested with test mass geometry, you are no less a hardcore crank for demanding physics as mathematical derivation is immune to empirical falsification. Have some greater respect for Galileo and less for Aristotle.

arXiv:1109.1963 for periodic lattices damn continuous geometry GR, then
http://elib.mi.sanu.ac.rs/files/journals/publ/69/7.pdf
Section 2: eleven pairs of enantiomorphic space groups.
http://www.math.ru.nl/~souvi/papers/acta03.pdf
Section 3ff.

Math is self-consistent, science is that plus empirical.

9. I haven't responded to your *axion* rant because that case was clearly just another demonstration of your psychopathology whose nonsensical nature is comprehensible to everyone - and I am not a psychiatrist who could help you with your problem.

If a theory assumes more axions, they will have a greater gravitational impact. A theory with them can't be "gravitationally short".

10. What a waste of time and effort! RWhat a waste of time and effort!

Ramanujan may or may not have had a clear idea of what a proof was, but you certainly have no idea at all what “demonstrate” means, because you have “demonstrated” nothing at all.

Some of the things that you say about “Ramanujan and Feynman” are no doubt true, but they would be equally true about “Ramanujan and Newton”, “Ramanujan and Fermat”, “Ramanujan and Galois”, “Ramanujan and Abel” and so on, an so on. Nobody is denying Ramanujan’s genius and that one can find various things in common between different ones. It’s just that the choice of “Ramanujan and Feynman” is completely arbitrary.

Other things that you say are just rubbish, which shows that you picked up all your knwledge of this topic quickly, probably from a source like the Wikipedia. The stupidest thing of all is the claim that Ramanujan’s genius was somehow amplified by his lack of formal education. In fact, Hardy, who knew him better that anyone else, once said something similar: “if Ramanujan had been better educated he would have been less of Ramanujan”. However, when he considered what he had said, Hardy recanted this statement as possibly the silliest one he had made in his life. Clearly, a better educated Ramanujan would have been a far more wonderful one.

Ramanujan was one of the very greatest natural geniuses mathematics has ever know, along with Archimedes, Newton, Galois, Abel, Gauss and Riemann but his contribution was very narrow. Over 30 years I have published about 50 papers in various areas of topology, geometry, algebra, and probability theory and have never had a chance to mention his name, unlike all the others about (except Archimedes).

You mention that Ramanujan was (quite deservedly!) made a Fellow of the Royal Society. As you must know, it was due to Hardy that this happened. Therefore, the following assessment by Hardy of Ramanujan’s importance carries more weight than that of someone like you (don’t you agree?):

http://uzweb.uz.ac.zw/science/maths/zimaths/ramanhdy.htm

11. Nima makes many fascinating and
provocative statements in his talks and papers.
Examples are

1) There are no fields in nature just
particles. Humans define fields as a convenience for representing local
interactions.

2) There are no local observables. As
a consequence space time is "doomed".

3) There is no such thing as gauge
symmetry, only gauge redundancy.

4) The union of quantum mechanics and
gravity is not an issue, what we don't understand is gravity at short distances.

5) There must be a reformulation of
QFT in which locality (space-time) and unitarity are emergent phenomena.

As I understand it the Amplituhedron allows
one to calculate in a systematic manner the integrand for any N=4 SYM scattering
amplitude for any loop order and any number of particles. This seems to be of
interest for practical and fundamental reasons.
On the fundamental side we have an example of a QFT than can be represented
with no Langrangian, no Hilbert space, no space-time or gauge symmetry
(redundancy). On the practical side we seem to have a calculational tool.

I am left asking many questions though

1.
Can
the integrands be integrated in a systematic manner?

2.
Can
this be extended to QCD and used as tool for estimating LHC backgrounds?

3.
How
do you do something like Dim-Reg in this framework in order to get RG flows?

4.
Does
a similar formulation exist when fermions are part of the scattering process?

5.
What
about correlation functions, can they also be evaluated outside the std. QFT
framework?

JR

12. This discussion is somewhat interesting to me.

I wouldn't compare myself to Feynman or any of the great geniuses of physics (unless it is to raid their papers/memories for hints as to their styles of thinking.) I've noticed before that I myself think in terms of visualizations and "wordless algorithms" (akin to the process thinking you refer to perhaps?) I have difficulty with long chains of logic when I can't "see" what each step is "doing". On the other hand, I get the impression that some others tend to think of things far more in terms of symbolic manipulation. (I'm not dinging them - I wish I were better at that too so I could understand what the heck they are talking about. Without a guiding picture or process, I'm flying blind. With a guiding picture, I almost don't care about the particular symbols/nomenclature involved in expressing what I am doing. However if I could simply churn through long chains of expressions like a machine, disposing of relationships between complicated mathematical objects as easily as simple scalars like some of my math professors seem to be able to do, I could probably arrive at conclusions much faster, with the proper rigour, and not get lost 20 levels down some deep logic tree.)

I am not a physicist by formal education, but I do enjoy understanding the world, and so I've been studying quantum physics a bit in my spare time. As far as I can tell from where I am in my understanding so far - quantum field theory is a generalization of the original 1920s quantum mechanics to include the creation and destruction of particles. The Feynman diagram approach, in terms of the traditional wavefunction picture with which I am currently most comfortable, appears to me to be a sort of Green's function approach to the propagation of the wavefunction state. (Am I incorrect in thinking this?) It also seems to vastly simplify the generalization of these dynamics to creation and anhilation of particles. (I suppose without this you would need to have a ton of "potential particle" dimensions floating around in your model of the Hilbert space that the creation operators would populate.)

One thing that has been bugging me is that if quantum mechanics ever needed to be nonlinearized, (extended to handle something like space-time curvature depending on amplitude and feeding back into propagation of amplitude to handle gravity), then that would break a great many of the tools used to analyze quantum physics. Green's functions don't apply (in general) to nonlinear equations (and so Feynman diagrams couldn't be directly applied?). Neither do nice linear-algebra of operators, arbitrary bases, etc. We'd have to go back to the wavefunction PDEs and start over, with all the nice mathematical tools broken.