I’ve finally managed to write up something short about an idea I’ve been working on for the last few months, so now have a preliminary draft version of a paper tentatively entitled Spacetime is Right-handed. One motivation for this is the problem of how to Wick rotate spinor fields, given that Minkowski and Euclidean spacetime spinors are quite different. In particular, it has always been a mystery why a Weyl spinor field has a simple description in Minkowski spacetime, but no such description in Euclidean spacetime, where the Euclidean version of Lorentz symmetry seems to require introducing fields of opposite chirality. The argument of this paper is that the relation between Euclidean and Minkowski is not the usual chirally-symmetric analytic continuation but something where both sides use just one chirality (“right-handed”). It’s quite remarkable that the dynamics of gauge fields and of GR also has a chiral-asymmetric formulation.

In the ideas about unification using QFT formulated in Euclidean twistor space that I’ve been working on the past few years, it was always unclear why, when you analytically continued back to Minkowski signature, the left-handed Euclidean spin symmetry would not go to the Lorentz boost symmetry, but to an internal symmetry. One goal of this paper is to answer that question.

This past weekend I recorded a podcast with Curt Jaimungal, which presumably will at some point appear on his Theories of Everything site. It includes some discussion of the ideas behind the new paper.

Interesting — you write: “One goal of this paper is to answer that question.”

Is there an answer in the paper? Might you be able to summarize, if so?

Thank you!

PaulS,

What I’m showing in the first section is that if you identify the Lorentz group as the right $SL(2,C)_R$ factor in the orthogonal group (actually spin double cover) of complex spacetime, and identify complex spacetime as the tensor product of the spinor rep for this group and its conjugate (so, spacetime is righthanded in the sense of built out of righthanded spinors), then Euclidean spacetime will have a distinguished (imaginary time) direction and only the $SU(2)_R \subset SL(2,C)_R$ will act on it (not the $SU(2)$ from the left-handed factor, this acts trivially on spacetime).

In this paper I haven’t discussed at all how these left-handed $SU(2)$s behave in the Euclidean theory, this is what I’m trying to do with twistors. The argument about them is just that they are not going to show up as Minkowski space-time symmetries as they do in the usual analytic continuation (analytically continuing to Lorentz boosts).

So, this is part of your theory of everything?

Thank you for the detailed anwer! Is there an experiment that can demonstrate that “Spacetime is Right-handed”? How would right-handed space differ from left-handed space, or ambidextrous space? Thank you!

Hayun,

This is related to the twistor unification stuff, as explained above. But there’s also a more general point, about the not well known fact that the elements of our best fundamental theory (matter fields, gauge fields, GR) can be formulated in chiral variables of one handedness. This suggest to me, independent of the motivation from twistors, that one should be thinking about the SM/GR in this kind of spinorial language, not the usual chirally symmetric vectors.

Paul S.,

If you’re just looking at Minkowski space-time, you won’t see this kind of chiral asymmetry, it just appears when you try to Wick rotate. Questions about what it implies for observed physics then depend on how the theory works in Euclidean spacetime signature, which isn’t a topic of this paper.

I’m not especially expert on various approaches to quantum gravity, possibly the asymmetry I’m pointing to may be relevant in some of them.

What breaks the symmetry? Minkowski space is left-right symmetric. The Wick rotation is a formal process. So one can expect that the mathematics has a sign convention or something that causes the asymmetry. But is there a physical reason for it?

Peter, if space is right-handed, doesn’t this imply there’s also an arrow for time in your paper; from the CPT theorem?

flippiefanus/Interested amateur,

Some comments:

The proposed asymmetry I’m writing about is not a meaningful distinction if you just look at Minkowski space-time. As far as that is concerned, there are not two SL(2,C)s, one right, one left, but just a single SL(2,C), which is the usual Lorentz symmetry, no distinguished time direction. Since there’s only one, it’s meaningless to say it’s the right one, the left one or the diagonal one.

It does become meaningful once you start thinking about Wick rotation, or even just introducing the “i epsilon” terms that are essential to make sense of propagators (I don’t share the view that this is “unphysical”). Once you do that, you are writing formulas with time complex. Then you can ask what is the symmetry group of complexified space-time, and the usual story is that there is a left SL(2,C) and a right SL(2,C), with the physical Lorentz group the (conjugate) diagonal. What I’m suggesting is that this is the wrong way to complexify, that you should take the physical Lorentz group to be the right SL(2,C).

This is different than the usual complexification, which has been the one used in rigorous QFT as in Streater-Wightman. I haven’t checked, but quite possibly doing this you lose the CPT theorem. But, I’m really only interested in the Standard Model, and that respects CPT by construction. What I’m pointing out is that it may be possible to set up the SM and GR in a way that behaves quite differently when you Wick rotate, with, in particular, the left-handed part of rotations in Euclidean signature not acting at all on Minkowski spacetime, instead acting like an internal symmetry.

A comment about the “arrow of time”. Once you introduce those “i epsilon”s and start thinking about imaginary time, you find that the theory is very much asymmetric in imaginary time. By Fourier transform, positivity of the energy picks out one sign of imaginary time in which you can define things (the imaginary time propagator only goes in one direction).

Of course this may all be silly, but to quote one of the greats on the bigger picture:

— Michael Atiyah [Farmelo (2009)]

Are there any specific experimental or observational implications that might arise from your proposed “right-handed” chirality approach to the relation between Euclidean and Minkowski spacetime?

Michael Sykutera,

If I wanted to emulate string theorists, I’d say this approach predicts SU(2) gauge fields with chiral couplings, so is experimentally vindicated by the weak interactions. But, more seriously, this is just part of a larger project of reformulating the standard model in a way that exploits four dimensional spinor and twistor geometry, hopefully explaining more than the standard model does, but that is still a long ways off.

I my efforts to get my pea brain around the concept of Wick rotation, I’ve encountered many an internet forum wherein perplexed grad students learning QFT try to figure out what it “means” to implement such a thing. The stock answer seems to be it’s just to get sensible answers out of path integrals, and it’s all very rigorously dealt with, Osterwalder-Shrader, no hocus-pocus here, etc. It’s kind of a trick. Maybe they bring up connections to statistical mechanics if they want to entertain a little mysticism, but otherwise the attitude of the experts is all very blasé. It’s a cog in the machinery of QFT, more-or-less. Is that an accurate assessment of the typical attitude about the subject?

I ask because I’m trying to get a sense of why you’ve had a difficult time getting anyone interested in this idea. There are myriad possible answers to such a question, of course, but to narrow it down to the more “professional” sorts of objections you might encounter, one might be that your approach assigns a great deal more significance to the act of analytically continuing from Euclidean to Minkowskian spacetime than most folks would ever imagine. They might object that you’re taking a basic fact about mathematics, i.e. that not everything is integrable, and elevating it to some deep hint about the nature of reality. And that’s just not justified.

Am I being at all accurate here? I wish to make abundantly clear I am not at all attempting to debate the merits of your idea (which I am demonstrably incapable of doing), just trying to both understand it and also imagine how those endowed with the ability to debate it might react.

Thanks!

LMMI,

Yes, many people may think “just a trick”, but there are many arguments against this point of view. To pick one relevant to my experience: our non-perturbative info about QCD all comes from Euclidean spacetime lattice calculations. To get a workable definition of QCD that you can do numerical calculations with, you have to formulate the theory in Euclidean spacetime. The path integral for QCD simply is meaningless in Lorentz spacetime, well-defined and calculationally tractable in Euclidean spacetime.

When you look at chiral spinors and try to Wick rotate you realize something strange is going on. This paper tries to explain that, arguing that the correct Wick rotation gives you something very unexpected, a different Euclidean spacetime, with what you thought were SU(2)_L spacetime rotations actually an internal symmetry. This gives a very different way of thinking about unification which seems to me very much worth pursuing.

Abit off-topic to the spinors but, why is the “i epsilon” considered a trick? When I first learned it, I thought about it as a trick, but now I think about it in terms of the Schwinger Keldysh path integral (closed time contour), where it appears naturally from causality. In a similar way it makes sense to me that the imaginary time correlators have an asymmetry, when one starts implementing them with a Konstantinov-Perel’-contour with KMS boundary conditions. This is also pretty much my justification for Wick rotations.

Micha,

You’ll have to ask those who consider it a “trick”, not me. By the way, Schwinger was the one who first advocated Euclidean QFT as a consistent formalism and a fundamental idea.

Perhaps a naive question, but if I’m understanding correctly your proposal is essentially to identify one of the SU(2) factors in the (Euclidean) Lorentz group SO(4)=SU(2) x SU(2) as the SU(2) of the weak interactions. Does this not run afoul of the Coleman Mandula theorem, which proves that no non-trivial S-matrix can involve a mixing of spacetime and internal symmetries?

Gavin,

The main point of this paper is that the left Euclidean SU(2) acts trivially on spacetime, only the right handed SU(2) acts on spacetime

So the left SU(2) is an internal symmetry commuting with spacetime symmetries. Just like standard model and no conflict with Coleman Mandula.