I’ve completely re-organized and largely rewritten my paper from earlier this year on Euclidean Spinors and Twistor Unification. Soon I’ll upload this as a revision to the arXiv, for now it’s available here. This new version starts from a very basic point of view about 4d geometry, leaving the technicalities about Euclidean QFT for spinors and the expository material about twistors to appendices.

Most ideas I’ve worked on over the years that seemed initially promising ultimately became more and more problematic the more I looked at them. This set of ideas keeps looking more and more solid. There are several (to me at least…) attractive aspects:

- Spinors are tautological objects (a point in space-time is a space of Weyl spinors), rather than complicated objects that must be separately introduced in the usual geometrical formalism.
- Analytic continuation between Minkowski and Euclidean space-time can be naturally performed, since twistor geometry provides their joint complexification.
- Exactly the internal symmetries of the Standard Model occur.
- The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
- One gets a new chiral formulation of gravity, unified with the Standard Model.
- Conformal symmetry is built into the picture in a fundamental way.

There’s more in this version about how quantum gravity fits into this, when formulated in terms of chiral variables (i.e. Ashtekar variables). This gives a new context for old questions about quantizing in these variables (this is in Eucldean signature, the other chirality is not space-time geometry but internal Yang-Mills geometry, and the imaginary time component of the vierbein is distinguished and given the dynamics of a Higgs field). I haven’t spent much time on this yet, but suspect this new context may help overcome problems that people trying to pursue quantum gravity in this chiral connection framework have run into in the past.

One common reaction I’ve gotten to these ideas is the one I myself had in the past: analytic continuation relates expectation values of field operators in Euclidean and Minkowski signature, so my left-handed SU(2) after analytic continuation gives part of Lorentz symmetry, not an internal symmetry. What took me a long time to realize is just how different Euclidean and Minkowski signature QFT is. Yes, Schwinger functions and Wightman functions can be related by analytic continuation (in a rather subtle way, the Wightman functions aren’t functions, but boundary values of holomorphic functions). But at the level of states and operators things are very different. It’s just not true that there is some holomorphic formulation of QFT states and operators, with Euclidean and Minkowski space restrictions related by analytic continuation. There’s a lot of explanation about this in the paper.

One objection I’ve run into is that by distinguishing a direction in Euclidean space I’m breaking Lorentz symmetry. What’s true is quite the opposite: having such a distinguished direction is needed to get Lorentz symmetry after analytic continuation. If you want to start in Euclidean space and get Lorentz symmetry, you have to do something like distinguish a direction and get an Osterwalder-Schrader reflection in that direction, which you need to get from SO(4) to SL(2,C). From the other direction, if you start in Minkowski space-time and analytically continue, you have a choice of lots of possible Euclidean slices to analytically continue to. You need to pick one, and that will distinguish an imaginary time direction. This is most easily seen in the twistor formalism, where the Minkowski space-time geometry is determined by a quadratic form that picks out a 5-dimensional hypersurface in PT. This will project down to an imaginary time = 0 subspace of Euclidean space-time, which picks out the imaginary time direction.

This looks like real progress, Here’s hoping your theory is well-received. 2 questions:

Does point 3 imply that you get the 3 generations of particles of the SM ?

Does Euclidean Twistor Unification make any specific prediction at this stage ?

Thomas,

No, this doesn’t explain why multiple generations, in particular why 3 generations.

What I find compelling about this is that it provides a remarkable geometric framework that has exactly the local symmetries of GR and the standard model. But it involves formulating the theory on twistor space rather than spacetime, which I don’t understand quite how to do correctly. Perhaps if that can be done, one would have a well-defined theory more constrained than the standard model, so able to make new predictions.

Good readability, one typo, last sentence before Section 6, ‘recently’ =recent

Would love to know if Penrose is aware of this!

Excellent! Thanks! This is very good!

-drl

Towards the bottom of p4 the expression for the determinant should be:

x_0^2 – x_1^2 + x_2^2 – x_3^2

Very interesting. From my very incomplete understanding, what you are saying is that if you set up the integration measure for some partition function not on 4d real space (Euclidean with a Minkowski analytical continuation) but in complex space, reparametrized using twistors, GR general covariance and gauge invariance under SU(3)XSU(2)XU(1) could follow from some version of invariance of the functional integral under field redefinitions.

Is this it or is this a misunderstanding?

Bertie/SteveB,

Thanks, will fix the typos.

Haven’t heard from Penrose, but he’s been rather busy, I hear being a Nobelist takes a lot of time.

lun,

That’s not really it.

One role for twistors is that they tautologically describe spinors on complexified space time. So they give an arena where you can do analytic continuation of spinor fields between Euclidean and Minkowski 4d real slices.

Unlike most attempts to use twistor theory in physics, I’m looking not at the Minkowski slice but at the Euclidean slice. There you have usual 4d Euclidean space time, and usual Euclidean QFT as a sort of stat mech, but also

1. A distinguished imaginary time direction (this determines how you are going to recover Lorentz symmetry and the Minkowski theory).

2. Euclidean twistor space, i.e. twistor theory, restricted to the Euclidean slice. This is a bundle over Euclidean space time, fiber a sphere, which you can interpret in various ways (projective spinors at the point, orthogonal complex structures at the point).

The claim is that these elements determine local SU(3)xU(1)xSU(2)xSU(2) gauge symmetry on twistor space. The first three factors can be thought of as internal symmetries, with the SU(2) spontaneously broken by the distinguished imaginary time direction. Gauge theory with the last SU(2) gives a chiral spin connection that is used in one formulation of GR (general covariant if you use the usual Palatini action in a chiral form). It may be that this setup gives some different possibilities for the action than just the usual term.

Peter,

”if you start in Minkowski space-time and analytically continue, you have a choice of lots of possible Euclidean slices to analytically continue to. You need to pick one, and that will distinguish an imaginary time direction. ”

This means that there are many conjugate Euclidean formulations of the same Minkowski theory, breaking explicit Poincare covariance in each Euclidean formulation and restoring it only after the Osterwalder-Schrader reconstruction.

You also didn’t mention that analytic continuation is not possible in most curved spacetimes – only in the quite restrictive class of spacetimes compatible with a complex structure.

In my view, both observations send the same message: The Euclidean formulation is only a tool, rather than a basis for QFT.

Arnold Neumaier,

I agree that explicit SO(4) invariance is broken by having a Euclidean QFT + analytic continuation to Minkowski space-time. Note that it is not Poincare that is broken, it is just Euclidean rotations. My argument is that this is not a bug but a feature: it corresponds to the spontaneous symmetry breaking of the electroweak theory.

For one common objection about a non-existence problem of the analytic continuation, see the 2002 survey by Gibbons, where section 20.3.1 is titled “A non-problem”, and starts with

“Let us comment on an often repeated objection. This is the mathematically correct but physically totally irrelevant statement that ‘not ever Lorentzian spacetime M_L admits an analytic continuation M_C containing a Riemannian section M_R.’ This is also true for example of Yang-Mills connections on Minkowski spacetime, they don’t always analytically continue to give real connections on Euclidean space. The answer is the same in both cases. All that is needed is the behavior of quantities like Z(\Sigma,h) as a function of boundary values. Physically the individual interior spacetimes, Riemannian or Lorentzian, that go into the functional sum have no particular significance except in the classical limit, when one is preferred, or, more interestingly in the special case when both Lorentzian and Riemannian real sections exist simultaneously in the same complex spacetime M_C.”

The special case of S^4 I discuss in the paper is an example of the last phenomenon Gibbons mentions.

Hi Peter, could you give the title of Gibbons survey ? He seems to have a lot of preprints in 2002, and it is not clear to me which one is relevant here.

Thanks !

martibal,

The full reference is reference 18 in the latest version of the paper.

Peter,

“My argument is that this is not a bug but a feature: it corresponds to the spontaneous symmetry breaking of the electroweak theory.”

But the latter has physical consequences whereas the former is a nonphysical artifact of the Euclidean view. No experimental result hints at a broken Poincare symmetry!

Arnold Neumaier,

You keep talking about broken SL(2,C) invariance, but what I’m pointing out is that SO(4) invariance is not a symmetry of the states or operators of the theory. To get SL(2,C) invariance on states you have to have something like an Osterwalder-Schrader reflection, which breaks SO(4). This breaking of SO(4) does not break SL(2,C), it’s required to get SL(2,C).

Electroweak symmetry breaking is a breaking of gauge degrees of freedom, so the SO(4) breaking will be of that nature.

To get SL(2,C) invariance on states you have to have something like an Osterwalder-Schrader reflection – yes, but my point is that what you get is a representation of the Poincare group with a distinguished time coordinate, which is unphysical!

Starting with Euclidean theory you just can’t get a Poincare group representation at all, unless you have a distinguished imaginary time direction in Euclidean space (to do OS reflection). For a general context see for example

https://www.jstor.org/stable/2006979

This particular construction is in coordinates and has a distinguished real-time coordinate, but since it’s a Lorentz group representation, it’s unitarily equivalent to any choice of real-time coordinate.

Whether you can map fields on Lorentzian-signature manifolds and Euclidean-signature manifolds is irrelevant (there is such a bijective mapping, as far as I know, between Lorentzian manifolds and Euclidean manifolds with a vector field everywhere. But it does not matter).

The real issue is whether {\em amplitudes}, not fields on manifolds, can be continued. That’s what OS positivity is for.

Sorry, what I wrote needs an edit:

Whether you can map fields BETWEEN Lorentzian-signature manifolds and Euclidean-signature manifolds is irrelevant (there is such a bijective mapping, as far as I know, between Lorentzian manifolds and Euclidean manifolds with a FIDUCIAL vector field everywhere. But it does not matter).

The real issue is whether {\em amplitudes}, not fields on manifolds, can be continued. That’s what OS positivity is for.

My previous comment about “predictions” didn’t satisfy some, lots of comments demanding “predictions” deleted. If one wanted to play the game played by string theorists over the years, one could claim for this set of ideas “predictions” of four dimensions, chiral spinor fields, U(1) x SU(2) x SU(3) gauge symmetries and matter fields with the right quantum numbers, as well as gravitation.

But that’s a waste of time, the problem with string theory (or any speculative theory) has always been much more complex than “does it make “predictions”?”, and I’ll continue to delete comments that want to go on about this. The interesting question is whether there are new ideas here that will be fruitful and explain things about the Standard Model + GR we’ve previously had no explanation for. I think so, but if you don’t, do something else. There’s a lot I still don’t understand about the implications of these ideas, maybe further effort will show they can’t work, or maybe they’ll lead to progress. Time will tell.

In any case, we’re still a long ways from any danger of decades of fundamental research in theoretical physics going nowhere because of being dominated by these ideas…

Peter, Peter, and Arnold: Concerning the issue of distinguished time coordinate emerging when going from Euclidean to Lorentzian, this (inevitable fact) is most clearly evident when working with the covariant formalism for Wick rotation (using a timelike direction field) which I believe is what Peter Orland was referring to; I have left references about this earlier on this blog. The key point is that the partition function (for gravity+matter) should now presumably also include the path integral over this timelike field. (Gibbons, on the other hand, does *not* talk about this additional path integral.) I do not see any reason why Lorentz invariance will necessarily be broken in all this. In fact, at the quantum level, having such a direction field might be a bonus since we know that description of quantum vacuum is observer dependent.

Peter: you should give an ILQG seminar at PSU or the quantum gravity seminar series at Perimeter or other places which run a seminar series.

DK, Yes, I mentioned that vector field in my first version of my post, but edited it away, after deciding I was veering off topic.

Shantanu,

I’m speaking at Brown next week. If other people are interested in hearing about this, I’ll be happy to talk elsewhere.

This level of math is way beyond me. But I am curious were the justification for the statement

“since we know that description of quantum vacuum is observer dependent.”

comes from.

Dr. Woit,

Would you be able to share the slides from your Brown talk in this blog?

Kingsuk Maitra,

Just finishing the slides this evening, will post here after the talk.

does Euclidean Twistor Unification combine space-time and internal symmetries and does Coleman–Mandula theorem apply ?

Anonymous,

No, I’m not embedding both space-time and internal symmetries in a larger symmetry group and having that act on the states. That’s what Coleman-Mandula says you can’t do (because you’ll end up with a trivial theory).

For me a lot of the appeal of this picture is exactly that it gives a sort of unification that does not involve embedding everything in a bigger grand unification group, that you then have to break somehow to get back to the SM and usual space-time symmetries.

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