Since last summer Eric Weinstein has been running a podcast entitled The Portal, featuring a wide range of unusual and provocative discussions. A couple have had a physics theme, including one with Garrett Lisi back in December.

One that I found completely fascinating was a recent interview with Roger Penrose. Penrose of course is one of the great figures of theoretical physics, and someone whose work has not followed fashion but exhibited a striking degree of originality. He and his work have often been a topic of interest on this blog: for one example, see a review of his book *Fashion, Faith and Fantasy*.

Over the years I’ve spent a lot of time thinking about Penrose’s twistors, becoming more and more convinced that these provide just the radical new perspective on space-time geometry and quantization that is needed for further progress on fundamental theory. For a long time now, string theorists have been claiming that “space-time is doomed”, and the recent “it from qubit” bandwagon also is based on the idea that space-time needs to be replaced by something else, something deeply quantum mechanical. Twistors have played an important role in recent work on amplitudes, for more about this a good source is a 2011 Arkani-Hamed talk at Penrose’s 80th birthday conference.

One of my own motivations for the conviction that twistors are part of what is needed is the “this math is just too beautiful not to be true” kind of argument that these days many disapprove of. There are many places one can read about twistors and the mathematics that underlies them. One that I can especially recommend is the book Twistor Geometry and Field Theory, by Ward and Wells. A one sentence summary of the fundamental idea would be

A point in space time is a complex two-plane in complex four-dimensional (twistor) space, and this complex two-plane is the fiber of the spinor bundle at the point.

In more detail, the Grassmanian G(2,4) of complex two-planes in $\mathbf C^4$ is compactified and complexified Minkowski space, with the spinor bundle the tautological bundle. So, more fundamental than space-time is the twistor space T=$\mathbf C^4$. Choosing a Hermitian form $\Omega$ of signature (2,2) on this space, compactified Minkowski space is the set of two-planes in T on which the form is zero. The conformal group is then the group SU(2,2) of transformations of T preserving $\Omega$ and this setup is ideal for handling conformally-invariant theories. Instead of working directly with T, it is often convenient to mod out by the action of the complex scalars and work with $PT=\mathbf{CP}^3$. A point in complexified, compactified space-time is then a $\mathbf{CP}^1 \subset \mathbf{CP}^3$, with the real Minkowski (compactified) points corresponding to $\mathbf{CP}^1$s that lie in a five-dimensional hypersurface $PN \subset PT$ where $\Omega=0$.

On the podcast, Penrose describes the motivation behind his discovery of twistors, and the history of exactly how this discovery came about. He was a visitor in 1963 at the University of Texas in Austin, with an office next door to Engelbert Schucking, who among other things had explained to him the importance in quantum theory of the positive/negative energy decomposition of the space of solutions to field equations. After the Kennedy assassination, he and others made a plan to get together with colleagues from Dallas, taking a trip to San Antonio and the coast. Penrose was being driven back from San Antonio to Austin by Istvan Ozsvath (father of Peter Ozsvath, ex-colleague here at Columbia), and it turned out that Istvan was not at all talkative. This gave Penrose time alone to think, and it was during this trip he had the crucial idea. For details of this, listen to what Penrose has to say starting at about 47 minutes before the end of the podcast. For a written version of the same story, see Penrose’s article Some Remarks on Twistor Theory, which was a contribution to a volume of essays in honor of Schucking.

Very interesting post Peter.

I have read both “Fashion, Faith and Fantasy” and “The Road to Reality”, and I must say that I found Twistor Theory extremely complicated. It takes a hell of a lot of time to get to understand it just roughly.

There’s no doubt about Penrose’s originality and out of the box thinking, and how much Twistor Theory has influenced the amplituhedron and other Arkani-Hamed’s ideas on geometry of the Universe.

But back in 2003, he had a meeting with Witten in Princeton, and it seems that the latter managed to write a 70 page paper in which he basically showed that Twistor Theory was embedded into String Theory.

Has this been fully proven and that’s why T.T. has been basically abandoned as the possible theory of “where the Universe emerges/comes from”?

DB,

It’s not so much that twistor theory is complicated as that it involves some mathematical ideas that physicists are not familiar with (e.g. holomorphic vector bundles). The basic idea of twistors that I described in the post is actually simple, but radical. Not everything is about string theory or AdS/CFT, and trying to understand the twistor string or duality and N=4 super Yang-Mills is not a good way to try and understand twistor theory. One thing to say though is that Witten’s paper on the twistor string does contain a very lucid and explicit discussion of what’s called the Penrose transform (relating field equations on space-time and on twistor space).

New ideas are definitely needed to connect twistor theory to the Standard Model and to use it to quantize gravity. There’s a long history of people trying this, without success so far. I see various possibilities and am trying to write something up about those, we’ll see what comes of that project. I’m hopeful, but still confused about some crucial questions.

Thanks for your helpful answer Peter, and I’m looking forward to that future project!!

Best of lucks.

Dear Peter,

I share your fascination with twistor theory and spent some time during graduate school in Oxford learning it from Penrose and his students. Indeed, twistor theory turns out to be intimately connected to LQG and spin foam models; one way to explain why is to to show that both make use of the chiral structures that are revealed when you reduce general relativity to a topological field theory plus constraints. I have one paper about this: https://arxiv.org/abs/1311.0186, but it has been developed extensively by Wolfgang Wieland, Simone Speciale, Laurent Freidel and others. Some of the interesting papers are below.

Take care,

Lee

https://arxiv.org/abs/1207.6348

https://arxiv.org/abs/1901.08161

https://arxiv.org/abs/1107.5002

https://arxiv.org/abs/1602.01861

https://arxiv.org/abs/1807.11376

Admittedly and perhaps reasonably, this requires investing huge amounts of time for those of us with the math knowledge of a physics degree (grad studies didn’t add much in terms of fundamental math), simply for getting what it is about. Still, it might well turn out that reality needs math beyond this level.

So I think that the question that comes to mind is, what are the insights / results / meaning of twistors? Why are they worth pursuing? What are the consequences of complex two-planes as fibers of spinor bundles?

It is curious that the mathematics which makes twistors exist precisely in spacetime dimensions 3 and 4 and 6 and 10 (here) is the same that makes the Green-Schwarz superstring exist in exactly these dimensions (here).

Lee Smolin,

Thanks for the references. The way both twistor theory and LQG exploit chirality of 4d geometry in a fundamental way is fascinating.

Tulpoeid,

There’s a lot of sophisticated mathematics involved here, but the point about spinors is rather simple. All fundamental matter fields we know of involve a complex two-dimensional spinor degree of freedom. In the conventional approach to geometry, it is rather complicated to understand where these degrees of freedom come from. In the twistor approach, a point in spacetime is precisely given by a complex two-dimensional space (in 4d twistor space), and this is the spinor space. The spinor degree of freedom is not something added ad hoc later, it is built into the very definition of space-time (in mathematician’s lingo, the spinor bundle is “tautological”).

It’s a sort of line geometry for spacetime. Strange coincidence – Pluecker’s original paper on line geometry appeared in the same volume of “Philosophical Transactions of the Royal Society of London” as Maxwell’s “A Dynamical Theory of the Electromagnetic Field” – in 1865. That was the first complete presentation of Maxwell’s theory.

-drl

What I find fascinating about twistor theory is the way that Penrose uses the fundamental structure of the group SU(2,2), and the twistor space on which it acts, to model spacetime at a point, *before* doing the geometry to model spacetime on an extended scale. He uses this particular group because it is a double cover of SO(4,2), that he wants for some geometrical reason that I admit is opaque to me. Altogether there are about 8 different real forms of this group (depending exactly how you count them), coming from 5 different real Lie algebras so(6)=su(4), so(5,1)=sl(2,H), so(4,2)=su(2,2), so(3,3)=sl(4,R) and su(3,1). I am aware of significant work using su(4), and other work using so(5,1), both attempting to do similar things to what Penrose has done. But has anyone looked seriously at sl(4,R) or su(3,1)?

Peter, I just want to thank you for directing me to the Portal website, the discussion with Roger Penrose was fascinating , and there is much content of a similarly stellar nature there.

Robert Wilson,

SO(4,2) is the group of conformal transformations, the Poincare group is a subgroup. Relativistic massless particles transform according to irreducible infinite dimensional representations of this group (or its double cover). Understanding these reps tells you a great deal about relativistic massless particles. Note that the SU(2,2) representation theory story is a generalization of the SU(1,1)=SL(2,R) story (where the discrete series representations can be constructed using holomorphic methods on a hemisphere of the Riemann sphere). The “Penrose transform” is of great interest to those working on geometric constructions of irreps of real Lie groups like SU(2,2).

One of the wonderful aspects of the twistor story is that it relates what happens in different space-time signatures. SO(4,2) is the Minkowski space story, SO(5,1) is the conformal group of $S^4$, so gives a (compactified) Euclidean space story. SO(3,3) gives the signature (2,2) story, where your spinors are real, and you’re talking about real 2 planes in real 4 space. From what I’ve seen, people doing amplitudes calculations seem to exploit the ability to work in different signatures, with the real (2,2) signature case often useful. I don’t know of uses of the other signatures. SO(6)=SU(4) is a much simpler story (SU(4) acts transitively on G(2,4) and the irreps one gets are finite dimensional. I don’t know about use of the SU(3,1) real form.

“One of the wonderful aspects of the twistor story is that it relates what happens in different space-time signatures. SO(4,2) is the Minkowski space story, SO(5,1) is the conformal group of , so gives a (compactified) Euclidean space story. SO(3,3) gives the signature (2,2) story, where your spinors are real, and you’re talking about real 2 planes in real 4 space. ”

This is a really deep point. I have done of lot of living in SO(3,3) and often broke my head on the relation to SO(4,2) physically.

The books of Felix Klein are really good preparation for all this, even though they are old. They read like novels 🙂

-drl

Peter,

Thanks for that explanation. I understand that SO(4,2) is the only real form that contains the Poincare group, so that if you’re interested in gravity then this is the obvious choice. But it also contains a subgroup SO(1,1) that commutes with the Lorentz group and acts on the translation part of the Poincare group as scale-changes. This is fine if all you want to model is GR, but it seems to me to be a problem if you want a quantum gravity. So I’m not entirely surprised that particle physicists find the SO(3,3) signature useful. Do you know of a good reference to get me started on exploring what particle physicists do with SO(3,3)? To me, the interesting thing about this signature, apart from the fact that it is the only one with real spinors, is that there are subgroups of dimension 12, which do not occur in any other signature. Dimension 12 seems to be important for both bosons and fermions in the standard model.

D.R. Lunsford/Robert Wilson,

I know nothing about physical applications of the split signature case. Two good places to find information about the way different signatures are related in the twistor picture are the paper

Real methods in twistor theory, by Woodhouse

https://iopscience.iop.org/article/10.1088/0264-9381/2/3/006

and Tim Adamo’s lectures on twistor theory

https://arxiv.org/abs/1712.02196

In Adamo’s lectures you learn that

“In general, the idea in twistor theory is to work in the complexified setting, imposing reality conditions only at the end of a calculation. In the old days of the subject, these reality conditions were usually the Lorentzian ones, while early in the ‘twistor renaissance’ of 2004 the split signature reality conditions were preferred. Nowadays, Euclidean reality conditions seem to be the most useful when performing explicit calculations. So depending on what era of the literature you read, you can find any one of the three reality conditions given preference for a combination of physical and technical reasons.”

I’m not all that familiar with the 2004 era literature using the split signature that he’s referring to.

https://cerncourier.com/a/when-twistors-met-loops/

Dear Peter,

Eric Weinstein released the video of his 2013 Oxford talk on his approach to unified field theory, along with a preamble and some clarifying notes after the talk, see here:

https://www.youtube.com/watch?v=Z7rd04KzLcg&t=1s

To this physicist many parts are unclear, but perhaps you and others who follow this blog will be able to parse his constructions. I wonder in particular if this can be done in d=2 as a first toy model, before approaching d=4.