Spacetime is Right-handed v. 2.0 and Some Notes on Spinors and Twistors

I’ve just replaced the old version of my draft “spacetime is right-handed” paper (discussed here) with a new, hopefully improved version. If it is improved, thanks are due to a couple people who sent helpful comments on the older version, sometimes making clear that I wasn’t getting across at all the main idea. To further clarify what I’m claiming, here I’ll try and write out an informal explanation of what I see as the relevant fundamental issues about four-dimensional geometry, which appear even for $\mathbf R^4$, before one starts thinking about manifolds.

Spinors, twistors and complex spacetime

In complex spacetime $\mathbf C^4$ the story of spinors and twistors is quite simple and straightforward. Spinors are more fundamental than vectors: one can write the space $\mathbf C^4$ of vectors as the tensor product of two $\mathbf C^2$ spaces of spinors. Very special to four dimensions is that the (double cover of) the complex rotation group $Spin(4,\mathbf C)$ breaks up as the product
$$Spin(4,\mathbf C)=SL(2,\mathbf C)\times SL(2,\mathbf C)$$
where these two factors act on the spinor spaces.

While spinors are the irreducible objects for understanding complex four-dimensional rotations, twistors are the irreducible objects for understanding complex four-dimensional conformal transformations. Twistor space $T$ is a $\mathbf C^4$, with complex conformal transformations acting by the defining $SL(4,\mathbf C)$ action. A complex spacetime point is a $\mathbf C^2\subset T$ and conformally compactified complex spacetime is the Grassmannian of all such $\mathbf C^2\subset T=\mathbf C^4$. One of the spinor spaces at each point of complex spacetime is tautologically defined: it’s the point $\mathbf C^2$ itself (the other is of a different nature, with one definition the quotient space $T/\mathbf C^2$).

Real forms

While the twistor/spinor story for complex spacetime is quite simple, the story of real spacetime is much more complicated. When several different real spaces complexify to the same complex space, these are called “real forms” of the space. A real form can be characterized by a conjugation map $\sigma$ (an antilinear map on the complex space satisfying $\sigma^2=1$), with the real space the conjugation-invariant points. Using the obvious conjugation on $\mathbf C^4$, we get an easy to understand real form: the $\mathbf R^4$ with real coordinates, rotation group $SL(2,\mathbf R)\times SL(2,\mathbf R)$ and conformal group $SL(4,\mathbf R)$. Unfortunately, this real form seems to have nothing to do with physics, its invariant inner product is indefinite of signature $(2,2)$.

The real spacetime with Euclidean signature inner product has an unusual conjugation that is best understood using quaternions. If one picks an identification of the twistor space $T$ as $T=\mathbf C^4=\mathbf H^2$, then the conjugation is multiplication by the quaternion $\mathbf j$. The Euclidean conformal group is the group $SL(2,\mathbf H)$. The spinor spaces $\mathbf C^2$ are identified with two copies of the quaternions $\mathbf H$, with the rotation group now the group $Sp(1)\times Sp(1)$ of pairs of unit quaternions.

In this case the conjugation acts in a subtle manner. Since $\mathbf j^2$ is $-1$ rather than $1$, it’s not a conjugation on $T$, but is one on the projective space $PT=\mathbf CP^3$. It has no fixed points, so the twistor space has no real points. What is fixed are the quaternionic lines $\mathbf H\subset \mathbf H^2$, each of which corresponds to a point in the (conformally compacified, so $S^4=\mathbf HP^1$) real Euclidean signature spacetime. Using the decomposition as a tensor product of spinors, the action by $\mathbf j$ squares to $-1$ on each factor, but $1$ on the tensor product, where it gives a conjugation with fixed points the Euclidean spacetime.

The real spacetime with Minkowski signature is another real form of a subtle sort, with very different subtleties than in the Euclidean case. The conjugation $\sigma$ in this case doesn’t take the twistor space $T$ to itself, but takes $T$ to its dual space $T^*$. It takes spinors of one kind to spinors of the opposite kind (at the same time conjugating spinor coordinates to get anti-linearity). The Minkowski signature conformal group is the group $SU(2,2)$ and the rotation group is the Lorentz group $SL(2,\mathbf C)$ (acting diagonally on the two spinor spaces, with a conjugation on one side).

Some philosophy

The usual way in which the above real forms get used is that mathematicians ignore the Minkowski story and use the Euclidean signature real form to do four-dimensional Riemannian geometry, with the $Sp(1)\times Sp(1)$ decomposition at the Lie algebra level corresponding to the decomposition of two-forms into self-dual and anti-self-dual. Physicists on the other hand (especially Penrose and his school, but also those trying to do quantum gravity using Ashtekar variables) ignore the Euclidean story and use the Minkowski signature real form. In various places Penrose is quoted as explicitly skeptical of any relevance of the Euclidean story to physics. Working just with the Minkowski real form, one struggles with the fact that the Lorentz group is simple, but that one can get a very useful self/anti-self dual decomposition if one makes one’s variables complex.

The point of view I’m taking is that Wick rotation tells one that one should look simultaneously at both Euclidean and Minkowski real forms, understanding how to get back and forth between them. This is standard in usual geometry where one just looks at vectors, but looking at spinors and twistors shows that something much more subtle is going on. The argument of this new paper is that when one does this, one finds that the spacetime degrees of freedom can be expressed purely in terms of one kind of spinor (right-handed by convention), the one that twistor theory tautologically associates to each point in spacetime. The other (left-handed) half of the spinor geometry involves a purely internal symmetry from the point of view of Minkowski spacetime. This should correspond to the electroweak gauge theory, exactly how that works is still under investigation…

Update: Now posted on the arXiv here. Only reaction on social media I’ve seen so far is from Strinking42069, which seems to be a parody account trying to make fun of string theorists.

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18 Responses to Spacetime is Right-handed v. 2.0 and Some Notes on Spinors and Twistors

  1. RD says:

    `Off-topic comments better be interesting…`

    Apologies for instantly derailing the comment thread, but Ed Witten was the guest on ‘The Life Scientific’ this morning on BBC Radio 4. It’s a regular, weekly show in which Jim Al-Khalili interviews various members of the scientific community , lightly touching on the technical aspects of their work without alienating the distinctly non-technical Radio 4 audience.

    Jim is a physicist himself. You made need to create a free account to listen to it:

  2. Peter Woit says:

    Thanks. That is a nice summary of Witten’s career, including explaining how much the quark confinement problem has motivated him as well as explanations of his relation to mathematics. I do find his excellent advice about the danger of getting hung up on preconceptions in one’s research darkly humorous given his continued faith in and promotion of M-theory unification.

    This is an old story, with nothing really new here, unlikely anyone has anything new to say about it. I’d rather have discussion of some new ideas….

  3. Jim Eadon says:

    Dear Professor Woit,
    You mention people modeling quantum gravity Does gravity (in the General Relativity sense) crop up in your set-up (as far as I can tell it doesn’t). It seems to me (I’m no expert) you are modeling Special Relativity plus Euclidean space via twistors and so on, and potentially obtaining Electroweak fields for free, so to speak.
    Perhaps a snappy tagline on what you are claiming could come in handy (more understandable than the abstract right-handed concept)

  4. Peter Woit says:

    Jim Eadon,
    I’m making an argument at the level of fundamental symmetries, but yes, this includes an argument (explained a bit in a subsection) for a particular approach to quantum gravity (Euclidean quantum gravity using chiral variables). I even write down a Lagrangian.

    For much much more about exactly the chiral variables approach, I just noticed two new papers on the arXiv:
    These are in Lorentzian signature, and struggle with exactly the problem I mention in the paper. This problem goes away in Euclidean signature, but the issue of relating Euclidean quantum gravity to physical quantum gravity is obscure. I hope those expert in the subject of Ashtekar variables and chiral formulations will take a look at this.

  5. Peter P says:

    “The other (left-handed) half of the spinor geometry involves a purely internal symmetry from the point of view of Minkowski spacetime. This should correspond to the electroweak gauge theory…”

    Terms leading to neutral currents and coupling to Higgs require both chiralities (uL and uR). Arguably, iso-singlet contributions into neutral currents are smaller than iso-triplet ones. Currently, they are still required to match experiments. The overall idea is intriguing: matching right-handed spacetime with left-handed inner DoFs (matter?).

  6. Interested amateur says:

    On the subject of twistors, a lecture was given by Nima Arkani-Hamed a few months ago as part of the IAS-DIAS organised conference, The Amplituhedron at 10: Hidden Mathematical Structures of the Amplituhedron: Why Is the Universe Big?

    In particular at 1:27:38, Nima is asked how his physics program adds anything beyond Heisenberg’s S-matrix theory. Worth a post by itself sometime in the future, maybe?

  7. Low Math, Meekly Interacting says:

    Hopefully not too OT, but I’m not convinced of the parody hypothesis. The truth of internet personae can be so much stranger than any imaginable fiction one must keep an open mind pending definitive evidence.

  8. Peter Woit says:

    Interested amateur,
    I’ve often written here about the amplitudes program and Arkani-Hamed’s vision for it. Earliest post I can find is from nearly fifteen years ago, see
    At the time I was quite interested to see that people were working on formulating QFT in twistor space, that’s something that has continued.

    The amplitudes program has had a huge sociological success, it’s now a very large enterprise. In his talk and at the point you mention, I think Arkani-Hamed makes clear what his vision is for it: find some new geometrical principle which will give S-matrix amplitudes, replacing local QFT and replacing spacetime. He’s a huge optimist about whatever idea he’s working on at the moment, we’ll see what happens.

    I’m all for new ideas about reworking QFT in terms of different geometrical fundamentals, and twistors are among the most promising such ideas. But I don’t think it’s so healthy that the sociology of the field is that one very speculative and not obviously successful idea about this attracts a large fraction of the people and other resources of the field. We’d be better off if people interested in new geometrical ideas about fundamental physics were investigating a much wider range of possibilities. For instance, the geometry of the Wick rotation between spinors in Euclidean and Minkowski spacetimes…

  9. Peter Woit says:

    Hard to guess what the true story of Stringking42069 is, maybe it is an actual string theorist whose brain has been rotted by too much testosterone and 420. But it’s very hard to take at face value the combo of informed and accurate information about string theory research with a huge amount of material that makes the author appear to be a complete idiot.

  10. suomynona says:

    It appears fairly obvious that Stringking42069 is not any kind of serious account. If not parody, Stringking42069 is at the very least an intentionally highly exaggerated personality. Even for an anonymous internet commenter, they’re far too unhinged for them to be serious and posting in good faith while still having enough faculties to accurately understand the HET community. A pretty dead giveaway are the numbers ‘420’ and ’69’ in their Twitter/X handle. These are numbers which are, quite obviously, associated with an intentional joke.

  11. Marvin says:

    Hi Peter,

    When you say in your article:”Spacetime (both Minkowski and Euclidean)
    can be said to be “right-handed”, and we have seen that this goes beyond the spin 1/2 matter degrees of freedom, with Yang-Mills and gravitational dynamics also described using right handed spinors.”

    Are you expecting some measurable astronomical effect following the fact that spacetime had this preferred chirality. For what I know, exept for small anomalies (like the “Axis of evil”), there is no such significant effect visible. Or may be you have your own idea on this.

  12. Peter Woit says:

    What I’m claiming is that you can describe all those degrees of freedom in a way that is completely chirally asymmetric when you look at complex spacetime (or Wick rotate to Euclidean spacetime). Just looking at Minkowski spacetime nothing is different, and you just have the usual theories, which are thought of as chirally symmetric. The main point is that if you Wick rotate one of these theories normally thought of as chirally symmetric, you get something very definitely chirally asymmetric.


  13. tulpoeid says:

    Btw I think Stringking is as much a parody of string theorists as Motl is. Parody, yes, maybe; intentional, not really.
    (Also, he’s apparently the most interesting strings have to offer nowadays.)

  14. Marvin says:

    Thanks Peter: very clear explanation!

  15. Hayun says:

    Unlike other TOE, your argument is that gravity and gauge theories are completely decoupled?”

  16. Peter Woit says:

    What I’m pointing out here is just that the usual YM and gravity theories can be expressed in terms of purely chiral variables. They are coupled in just the usual manner (the YM action involves the Hodge star, which depends on the metric, or vierbein). Chiral fermions are coupled to YM and gravity by the usual covariant derivatives.

  17. Plebian Science Beggar says:

    Are there any interesting new predictions of Twistor Theory? I get that in theory it recovers the familiar models, but on the margins has it yielded new observable stuff that the old theories don’t predict?

    BTW, long-time fan and lurker and first-time poster. It would mean a whole lot to get a response from you. Tear me to pieces, please!

  18. Peter Woit says:

    Twistor theory isn’t really a “theory”, although I suppose you could try to follow the string theorists and go on about how it’s a “framework”.

    What twistor theory does is point out that there’s mathematically a different way of describing Minkowski spacetime, one that has some very powerful and appealing characteristics: it makes conformal symmetry manifest and allows the use of complex variable methods. You can try and write down the full standard model and GR in this description of spacetime, but it’s very awkward, definitely doesn’t give what you would like (a theory that reproduces the successful parts of the standard theory and also gives something different and testable).

    What I’m trying to do is to investigate a new set of ideas based on twistor theory (looking simultaneously at the Minkowski and Euclidean versions) which look to me to have promising properties. Still a ways though away from showing exactly how to reproduce the standard theory this way (and, ideally, also something new and different), but what I do understand looks promising.

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