Roger Penrose, Reinhard Genzel, and Andrea Ghez share the Nobel Prize in Physics

I am absolutely delighted for Roger Penrose. He has contributed so much to mathematical and theoretical physics, it is a surprise to me that he hadn’t won the Nobel Prize sooner.

When I was first getting into mathematical physics twistor theory, originally proposed by Penrose in 1967, was a topic that I had been obsessed about, particularly as it then came to twistor string theory and the amplituhedron. It remains a notebook project for the future, but the great enthusiasm and interest I developed for the twistor programme I think aided my focus in mathematical physics before arriving at university. The geometry of twistors and the twistor description of massless fields, as well as his description of Grassmannian space and Grassmann algebra were things that left me inspired. In hindsight it is nice to think about how, as twistor space is chiral and treats the left and right handed parts of physical fields differently, my current thesis is building on what I think is a wider project involving the chiral string which is not so far removed.

In the world of modern mathematical and theoretical physics it seems one is never too far from a Penrose contribution. It was not too long ago when I was thinking about the Ward-Penrose transform, which has a wonderful relation to the geometry of strings. Then of course there are the beautiful Penrose Tilings. But I don’t think the discussion is complete without also recognising the importance of his views on different scientific and philosophical matters. Whether one agrees or disagrees, Penrose’s views on quantum mechanics and objective collapse, with his theory of dual fields, serve as more than sampling food for thought. Additionally, I’ve personally taken a lot from his views on mathematical Platonism in which he relates the Platonic mathematical to the physical and the physical to the mental, and so on. The topic of a separate essay, there is quite a lot here to unpack and develop in the context of the wider history of mathematical thought.

For the layperson or even the engaged student of physics, The Road to Reality is one popular book that I would not hesitant to recommend. It is a proper masterclass, which goes into many important ideas in modern physics and does not spare on the mathematical details. The stories of Am-tep and Amphos in the prologue followed by the discussion on the roots of science is perhaps one of my favourite introductions to a book.

Having said all that, it is not possible to write a post celebrating the work of Penrose without mention of the singularity theorems for which the Nobel Prize was awarded. He showed that, as a consequence of General Relativity, black holes are a deeply general phenomena predicted to emerge in cases of gravitational collapse irrespective of symmetry. A key concept here is that of a singular manifold and trapped surface as related to geodesic incompleteness.

The proof of the Penrose singularity theorem is quite nice. There isn’t enough space here to detail it in full, but a summary of what the proof looks like might be a nice way to conclude the present entry (should one like to work through the complete proof here is a good text as well as notes and review).

First some definitions. Let us consider some spacetime (M,g), which we may think of as being singular if it is not geodesically complete. Now, in a global analysis, let us define (M,g) to be a globally hyperbolic spacetime in which we denote S the Cauchy hypersurface and n some future pointing unit normal vector field. For a compact 2-dimensional submanifold \Sigma \in S with unit normal vector field v, the proposition is that v is trapped if the expansions \theta_{+} and \theta_{-} of the null geodesics are negative everywhere on \Sigma. The initial conditions for the null geodesics are assumed to be n + v and n - v, and they satisfy the null energy condition.

The proof then goes something like this. Let t : M \rightarrow \mathbb{R} be a global time function such that S = t^{-1}(0). The integral curves of grad \ t are timelike, and they only intersect the hypersurface S once and \partial I^{+}(\Sigma) once, where I^{+}(\Sigma) is an open set and defines the chronological future along each point of the compact surface. What we obtain is a continuous injective map with an open image \pi : \partial I^{+}(\Sigma) \rightarrow S. What is really cool is that if we have a point p that chronologically precedes the point q, with the future-directed chronological (timelike) curve from p to q, following a number of definitions and corollaries, it can be shown that should q = \pi(p) then in some neighbourhood of q are images of points in \partial I^{+}(\Sigma). In the full proof one can then show that there is a contradiction in the intersection of the integral curves of grad \ t, and since \Sigma is trapped there exists some \theta <  0 such that the null geodesics \theta^{+} and \theta_{-} orthogonal to \Sigma satisfy \theta^{+}, \theta^{-} \leq 0. One can complete the proof from this point, finding that indeed should (M,g) be a connected globally hyperbolic spacetime, where the Cauchy hypersurface S is non-compact and satisfies the null energy condition typically written T^{\mu \nu} n_{\mu} n_{\nu } \geq 0, then if S contains the trapped surface \Sigma the spacetime is singular. Like I said, it is quite nice!

Before this post grows too long. I would also like to leave a special note congratulating Reinhard Genzel and Andrea Ghez in sharing the Nobel Prize in Physics. I cannot profess to have read their papers, but a review of their history and extensive work shows that for decades they have made significant contributions to long-term scientific research focusing on Sagitarius A*! The astronomers in the blogosphere will be able to speak more to the methods both Genzel and Ghez have developed, as well as the technicalities of the brilliant techniques they have prioneered along the way to providing the most convincing evidence to date of the presence of a supermassive black hole at the centre of our very own galaxy! For myself, I look forward to reading a few of their respective papers. Meanwhile, here is a 2017 article discussing some of their research.

A poster from 2005 promoting a pure spinor workshop.

Pure Spinor Formalism

In recent days, pure spinors have become my life. And that is by no means a bad thing.

I don’t want to divulge too much at this time. The short of it is that I’ve been looking into the pure spinor formalism for a possible research project. Whether the project comes to fruition or not has yet to be determined. Regardless of the outcome, the time will have been well spent as I’ve immensely enjoyed learning the topic. What is intriguing is the power of the formalism when studying superstrings on different curved backgrounds. It is also useful when studying multiloop amplitudes. More personally, I have also found it nice to work through and think about because there is some connection with my interests in twistor theory, among other things.

As it is quite a rich area, there is a lot to comment on. Given time, I will type and upload my own notes as a sort of tour through the formalism. For now I’ve put together a select list of preprint papers that give an overview, organised by date. I haven’t listed everything, and the reader may find other works that adequately study pure spinors. For me, I found it useful to simultaneously read [2, 3, 6] as review, having then marched on from there.

[1] N. Berkovits, ‘Super-Poincare Covariant Quantization of the Superstring’, (2000) preprint in arXiv [arXiv:hep-th/0001035 [hep-th]].

[2] N. Berkovits, ‘ICTP Lectures on Covariant Quantization of the Superstring’ [lecture notes], (2002) preprint in arXiv [arXiv:hep-th/0209059 [hep-th]].

[3] N. Berkovits and D. Z. Marchioro, ‘Relating the Green-Schwarz and Pure Spinor Formalisms for the Superstring’, (2004) preprint in arXiv [arXiv:hep-th/0412198 [hep-th]].

[4] N.I. Farahat and H.A. Elegla, ‘Path Integral Quantization of Brink-Schwarz Superparticle’, EJTP 5, No. 19 (2008) 57–64.

[5] C.R. Mafra, ‘Superstring Scattering Amplitudes with the Pure Spinor Formalism’, (2008) preprint in arXiv [arXiv:0902.1552v3 [hep-th]].

[6] O. A. Bedoya and N. Berkovits, ‘GGI Lectures on the Pure Spinor Formalism of the Superstring’, (2009) preprint in arXiv [arXiv:0910.2254v1 [hep-th]].

[7] T. Adamo and E. Casali,’Scattering equations, supergravity integrands, and pure spinors’, (2015) preprint in arXiv [arXiv:1502.06826v2 [hep-th]].

[8] N. Berkovits, ‘Untwisting the Pure Spinor Formalism to the RNS and Twistor String in a Flat and AdS_5 \times S^5 Background’, (2015) preprint in arXiv [arXiv:1604.04617v2 [hep-th]].

[9] N. Berkovits, ‘Origin of the Pure Spinor and Green-Schwarz Formalisms’, (2015) preprint in arXiv [arXiv:1503.03080 [hep-th]]

*Image: A 2005 poster by the IHES promoting a pure spinor workshop.


The Amplituhedron

The amplituhedron – a newly discovered geometrical object that enlivens the imagination. As a student in the area of theoretical particle physics, it has energised my curiousity in a very unique and notable way.

I am absolutely fascinated with this object, so much so that it has been a week with some sleepless nights. Fueled mostly by museli, vegetables and countless cups of tea, each day has merged into the next, interspersed with very few breaks. I suppose in general such study patterns are the norm of my existence. But in this present moment of day to day existence, if not donating time to unavoidable school related demands and on preparing for exams, much of my day is completely and totally consumed by the study of the amplituhedron and the physics related to it.

It is not that I have just discovered this geometrical object or just learned of its discovery. In fact, I originally came across it several months ago. But I was so busy with other things, my interest was more gentle. Over that time I dabbled in and out, thinking mostly of the broad consequences, particularly in relation to the idea of spacetime potentially being emergent.

More recently, I have listened to almost every lecture by Nima Arkani-Hamed that I could find on Youtube and elsewhere on the web. Correlating, too, with my reading of Penrose (I am interested in Twistor Theory), I have begun to dig into numerous papers, such as Arkani-Hamed and Jaroslav Trnka’s original 2013 paper introducing the amplituhedron, as well as their 2017 paper (with Hugh Thomas) on unwinding the amplituhedron in binary.

Admittedly, my twistor maths skills and my understanding of Grassmannians and of projective space are not yet at the level they need to be for me to truly understand and theoretically encircle the object and the physics in relation to it. Likewise, my understanding of planar N = 4 supersymmetric Yang–Mills theory is currently weak, and I am only starting to scratch the surface of the basics. This includes just how, or why, the amplituhedron represents a solution for Super Yang-Mills.

Drawing of amplituhedron by Nima Arkani-Hamed

But I think my interest is such that I could see my masters being focused on the amplituhedron in some way. There is still time, and things could change. But the very broad ideas associated with Arkani-Hamed’s (et al.) work are just so exciting. Perhaps in the near future I’ll write more about what makes it so exciting. For me, it is much more than just the possibility of the idea of how we might be able to simplify our calculations of scattering amplitudes. Certainly, this is one of the popular advertisements for the theory and that is understandable. For anyone who has studied, likes to draw, or understands Fenyman diagrams as a common way to calculate scattering amplitudes in quantum field theory, it can be incredibly grueling and quite a grind. Things become so complicated, and the amount of diagrams increases so much with the increase in the amount of loops, that not only does it become difficult to make accurate calculations but, at least for me, it suggests something deeper is missing. Something is not quite right. But what?

It almost seems that with Fenyman diagrams, we’re picking up or obtaining a glimpse of something about nature – a fragment of a more total picture. Or, to borrow a line from David Skinner, it very much seems like we’re picking up pieces or looking at shards of a broken Ming vase.

Think of the scattering of gluons. Something so simple, such as two gluons colliding to produce four less energetic gluons – to calculate the amplitude in using the textbook approach by way of Feynman diagrams, this would involve 220 diagrams. We’re talking tens of thousands of terms – pages and pages!

In that the amplituhedron might simplify these calculations, this evokes in me a sense of curiosity that the suspicion of something being missing might be true. It is at least worth thinking about and pursuing, to whatever end. But it’s also the idea of the reformulation of the whole of QFT, and things like how one can arrive at the same equation for the loop amplitudes without spacetime, gauge symmetry, quantum mechanics and the use of the path integral. It is really just so very cool.

From the perspective of my current understanding, the geometry itself is rooted in energy-momentum space, with the amplitude being the volume of the amplituhedron. Again, quite amazing to think about. If I am right in my understanding, unitarity and locality are also not completely discarded, nor are they required; instead, they are seen as being emergent (along with spacetime and QM). Again, fascinating.

The ideas may be in their infancy, and there may be lots of speculative impulses, but the entire theory is incredibly intriguing nonetheless.

My plan of attack is first to continue working toward studying and more deeply understanding the mathematics of Twistor Theory, and then also Grassmanians and so on.

I’m sure more posts will be made along the way.