In the last posting I explained how a fundamental problem shows up if you try to Wick rotate a Weyl spinor. Wick rotation is supposed to be analytic continuation in a four dimensional complex spacetime which, in terms of spinors, is $S_L\otimes S_R$ (or equivalently, linear maps from $S_R^*$ to $S_L$). This is acted on by the complex spin group $$Spin(4,\mathbf C)=SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$ where the two factors act on the two kinds of spinors. The subgroup preserving the four real dimensional Minkowski spacetime is $SL(2,\mathbf C)$, embedded by
$$g\in SL(2,\mathbf C)\rightarrow (\overline g, g)\in SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$
The subgroup preserving the four real dimensional Euclidean spacetime is
$$Spin(4)=SU(2)_L\times SU(2)_R$$
so pairs of $SL(2,\mathbf C)$ elements that are in $SU(2)$.
In Minkowski spacetime you only need one kind of spinor (and its complex conjugate), but analytic continuation to Euclidean spacetime requires two independent kinds. You have to introduce Dirac spinors (pairs of Weyl spinors with both chiralities) to do this kind of Wick rotation. My “right-handed spacetime” proposal is essentially that you should find a version of Wick rotation that just uses $S_R$, not analytically continuing spinors in complex spacetime in the usual way.
A way to do this is to use twistor theory, where a point in spacetime can be identified with the spin space $S_R$ at the point. Twistor space is a space $T=\mathbf C^4$, with a complex spacetime point a $\mathbf C^2\subset T$. Projectivizing (modding out by the action of non-zero complex numbers), projective twistor space is $PT=\mathbf CP^3$, and spacetime points are $\mathbf CP^1$s inside $PT$. In twistor theory one can exploit the complex structure of the complex three dimensional space $PT$ rather than that of the complex four dimensions parametrizing the space of $\mathbf CP^1$s.
To understand how Minkowski spacetime appears in $PT$, it may be a good idea to start with something simpler that has much of the same structure, thinking about $\mathbf CP^1$ instead of $\mathbf CP^3$. $\mathbf CP^1$ is the space of complex lines through the origin in $\mathbf C^2$, and it is acted on transitively by $SL(2,\mathbf C)$. It is a version of the Riemann sphere, with $SL(2,\mathbf C)$ the group of conformal transformations acting on the sphere. The unitary subgroup $SU(2)$ is a real form of $SL(2,\mathbf C)$ and also acts transitively on this sphere. One can construct finite dimensional representations of $SU(2)$ or $SL(2,\mathbf C)$ on spaces of holomorphic sections of holomorphic line bundles on $\mathbf CP^1$ (Borel-Weil), or on cohomology spaces $H^1$ (Borel-Weil-Bott).
If one instead looks at the other real form, $SL(2,\mathbf R)=SU(1,1)$, one finds that it acts on $\mathbf CP^1$ with three orbits: the upper hemisphere $\mathbf CP^1_+$, the lower hemisphere $\mathbf CP^1_-$, and the equator that is their boundary. One can construct infinite dimensional discrete series unitary representations of $SU(1,1)$ on holomorphic sections of holomorphic line bundles over $\mathbf CP^1_+$ or $\mathbf CP^1_-$ (or on cohomology spaces $H^1$). These representations can be characterized by their behavior at the boundary $S^1$, where one can make various choices of function space, with hyperfunctions a very natural one.
The Cayley transform relates the action of $SL(2,\mathbf R)$ on $\mathbf C=\mathbf R^2$ (with orbits the upper/lower half planes and the real number line) to the action of $SU(1,1)$ on $\mathbf CP^1$ as above. The action of $SL(2,\mathbf R)$ on the upper half plane is a central object in mathematics, especially in number theory (which comes into play through the subgroup $SL(2,\mathbf Z)$).
One way to understand how Minkowski spacetime appears in twistor theory is to generalize the above story from $\mathbf CP^1$ to $\mathbf CP^3$. One now has $SL(4,\mathbf C)=Spin(4,2,\mathbf C)$ and its real form $SU(4)$ acting transitively. The analog of $SU(1,1)$ is another real form, $SU(2,2)=Spin(4,2)$, which again acts with three orbits: $PT^+, PT^-$ and their common boundary, which we’ll call $N$.
$SU(2,2)=Spin(4,2)$ is the conformal group of Minkowski spacetime. In twistor theory, points in Minkowski spacetime are the $\mathbf CP^1$s that lie inside $N$. $N$ can be identified physically with the space of light-rays in Minkowski spacetime, and topologically with $S^3\times S^2$. The Penrose transform identifies solutions of massless wave equations in Minkowski spacetime with representations of $SU(2,2)$ on infinite dimensional spaces of holomorphic sections of holomorphic line bundles over $PT^+$ or $PT^-$ (or cohomology spaces $H^1$). As in the $\mathbf CP^1$ case, it is natural to characterize the function spaces involved as spaces of hyperfunctions on $N$.
What about Wick rotation in this context? Something happens in the twistor case that did not happen in the $\mathbf CP^1$ case. One can choose an identification of $\mathbf C^4$ with $\mathbf H^2$ and get a different real form of $Spin(4,2,\mathbf C)$, $SL(2,\mathbf H)=Spin(5,1)$, which acts transitively on $PT$. This is the Euclidean signature conformal group. Using the identification $\mathbf C^4=\mathbf H^2$, acting on a point in $\mathbf CP^3$ by quaternion multiplication, gives a fibering of $\mathbf CP^3$ over $S^4=\mathbf HP^1$ with fiber $\mathbf CP^1=S^2$. $S^4 $ is conformally compactified Euclidean space-time, with its points identified with the fibers in $PT$.
From the point of view that Minkowski spacetime is all $\mathbf CP^1$s inside $N$, Wick rotation starts with a choice of which Euclidean spacetime we want to Wick rotate to. This will be characterized by the $S^3\subset S^4$ where they intersect. $N$ will be fibered over this $S^3$ by $S^2$s: it no longer is just topologically $S^3\times S^2$ but there is an explicit geometrical identification.
I’m running out of time today, and want to get to a blog posting explaining how this is all supposed to relate to physics. I may or may not add a bit more to this later, or just go on tomorrow to the next topic.
Here is Grok AI assessment of your last 3 blog posts.
These three blog posts, published on Peter Woit’s “Not Even Wrong” blog in early March 2026, form a short, interconnected series expanding on the core ideas from his December 2023 arXiv preprint “Spacetime is Right-handed” (arXiv:2311.00608v2). They elaborate the motivation, address the key obstruction with conventional Wick rotation of Weyl fermions, and sketch how twistor theory might provide a framework for an alternative, purely right-handed analytic continuation. The posts are speculative, pedagogical, and aimed at readers familiar with relativistic QFT, spinor geometry, and some axiomatic approaches (Wightman, Osterwalder-Schrader reconstruction, etc.).They represent ongoing refinement of Woit’s long-running program toward a chiral, twistor-based unification that reinterprets the left-handed SU(2) as internal rather than spacetime symmetry in the Euclidean sector. The writing is clear, concise, and consistent with the preprint; no major contradictions or errors appear in the mathematics or citations.1. “Lorentz versus Euclidean Symmetry” (March 4, 2026)This post reviews the symmetry change under Wick rotation for scalar fields, emphasizing the asymmetry between Minkowski and Euclidean reconstruction.Strengths and accurate points:Correctly states that Wightman reconstruction preserves full SO(3,1) symmetry on physical states (no preferred time direction needed beyond positive energy).
Accurately notes that Osterwalder-Schrader (OS) reconstruction requires choosing an imaginary-time direction, breaking SO(4) → SO(3) via the OS reflection positivity condition.
The analytic-continuation story (tube domains, extended tube, Bargmann-Hall-Wightman theorem) is standard and correctly summarized for scalars.
The inverse problem (recovering Wightman from Schwinger functions) is fairly described: direct inversion is hard; OS use elaborate contour arguments, while simpler axiomatic approaches (e.g., Strocchi’s book) postulate properties of Schwinger functions that imply the desired Minkowski limit.
References to Seiler, Klein-Landau, Fröhlich-Osterwalder-Seiler on representation recovery are appropriate.
Critique / limitations:The post is restricted to scalars; it explicitly defers spinors to later posts (as delivered).
It acknowledges the author’s earlier confusion about rigorous Wick rotation but claims greater clarity now—no specific new theorem or proof is given here; the insight is conceptual (the broken Euclidean symmetry motivates chiral asymmetry).
No explicit calculation shows how the SO(3) subgroup maps back to SO(3,1) in the spinor case—this is promised for future.
Overall: Solid pedagogical recap for scalars. It sets up the problem cleanly but adds no new technical result.2. “Weyl Spinor Fields and Right-handed Spacetime” (March 5, 2026)This is the most substantive of the three. It explains why conventional Wick rotation fails for a single Weyl fermion and proposes the “right-handed spacetime” fix.Accurate technical content:The momentum-space equation (E – σ·p) ψ̃_R = 0 and helicity assignments are standard.
The two-point Wightman function as (E – σ·p)^{-1} and naive Wick rotation to (iE – σ·p)^{-1} are correctly written.
The obstruction is precisely stated: in the usual van der Waerden / Penrose notation, Minkowski vectors are maps S_R^* → S_L (with S_L ≅ conjugate S_R), but Euclidean vectors require independent S_L and S_R (no conjugation relates them). A single Weyl field cannot propagate consistently under full Spin(4) without adding the opposite chirality (and often doubling for hermiticity).
The OS construction for fermions indeed doubles degrees of freedom to achieve self-adjoint Schwinger functions.
The proposal—identify vectors everywhere as maps S_R^* → conjugate S_R (or equivalent purely right-handed tensor products)—reproduces standard Minkowski SL(2,ℂ) while making Euclidean geometry have only SU(2)_R acting non-trivially on spacetime, with SU(2)_L trivial (hence available internally).
Four dimensions are special: Spin(4) = SU(2)_L × SU(2)_R splits cleanly; the proposal exploits this asymmetrically.
Strengths:This directly attacks a genuine, well-known technical headache in Euclidean QFT and lattice gauge theory (fermion doubling, chiral symmetry issues).
It ties neatly to self-dual formulations (Ashtekar variables, self-dual YM) that already live in right-handed sectors.
The slogan “spacetime is right-handed” is now clearly kinematic: the tangent bundle uses only right-handed spinors.
Critique / open issues:The post asserts that analytic continuation between Minkowski and this modified Euclidean subspace is possible without the full holomorphic SL(2,ℂ)_L × SL(2,ℂ)_R structure, but gives no explicit contour or domain argument. This remains the biggest gap (as noted in my earlier review of the preprint).
No explicit Euclidean propagator or two-point Schwinger function is written in the new variables—readers must trust the geometry implies one exists (perhaps via Hitchin-type Dirac operators mentioned in the preprint).
The claim that left-handed Minkowski Weyl fields map to right-handed Euclidean ones via conjugation is stated but not derived in detail.
Overall: The strongest and most original of the three. It rigorously identifies the fermion Wick-rotation problem and offers a clean geometric resolution. If the continuation can be made rigorous, this would be a significant conceptual advance.3. “Twistors and Wick Rotation” (March 9, 2026)This post sketches how twistor theory might implement the right-handed Wick rotation, analogizing to real forms and orbits in lower dimensions.Accurate points:Standard twistor setup: T = ℂ⁴, PT = ℂℙ³, spacetime points as ℂℙ¹ lines; Minkowski points live in the boundary N of orbits under SU(2,2).
Penrose transform maps massless fields to holomorphic objects on PT^+ or PT^- (cohomology or sections).
The Euclidean conformal group appears via quaternionic structure (ℂ⁴ ≅ ℍ², SL(2,ℍ) ≅ Spin(5,1)), fibering PT over ℍℙ¹ ≅ S⁴.
Wick rotation involves choosing an S³ ⊂ S⁴ where Minkowski (N) and Euclidean intersect; N fibers over this S³ by S².
Strengths:Correctly identifies that twistor theory is already chiral (points = right-handed spinor spaces; left-handed spinors appear differently, often as quotients or sections).
The analogy to SL(2,ℝ) on ℂℙ¹ (orbits upper/lower hemispheres + equator) → SU(2,2) on PT is pedagogically useful and standard in twistor literature.
The post honestly admits it is incomplete (“running out of time”) and defers the physics payoff.
Critique / limitations:The post is the most tentative and least fleshed out. It describes geometry but does not show an explicit right-handed Wick continuation or how to define Euclidean correlators purely in right-handed terms.
No derivation of how the distinguished imaginary-time direction emerges from the S³ choice or how OS positivity arises.
The connection to unification (gauging the trivial SU(2)_L) is promised but not delivered here.
Overall: A promising sketch tying the proposal to twistors, but more aspirational than conclusive. It needs follow-up posts (promised) with concrete calculations.Overall Assessment of the SeriesThese posts are a coherent continuation and popular exposition of the 2023 preprint. They strengthen the conceptual case by:Clearly separating the scalar vs. spinor Wick-rotation stories.
Pinpointing the single-Weyl-field obstruction with precision.
Motivating the right-handed reformulation as the natural way to avoid doubling while preserving chirality.
Pointing toward twistors as the arena where the idea may become dynamical.
Rigorous checks: No algebraic mistakes, misrepresentations of standard results, or inconsistent notation were found. All group actions, helicity definitions, and twistor basics align with textbooks (Penrose & Rindler, Ward & Wells, etc.) and the preprint.Limitations (same as preprint):The analytic continuation without holomorphic SL(2,ℂ)_L remains unproven at the axiomatic level.
No explicit Euclidean QFT (propagators, path integral, OS positivity proof) is constructed.
Unification remains kinematic—no electroweak couplings, Higgs, or gravity quantization.
The series is intellectually stimulating and internally consistent. It advances a novel, well-motivated alternative to standard Euclidean fermion technology. Whether it leads to a workable theory depends on future technical development (explicit continuation, propagators, unification details). As of March 2026, it remains a promising research direction rather than a completed framework—recommended for anyone interested in chiral gravity, Euclidean QFT puzzles, or twistor approaches to unification.
why would Peter or anyone need you to do that ?
zzz,
I’ve deleted all previous AI generated comments, let that last one through because the discouraging thing is that I’ve been getting zero intelligent comments recently from human beings who understand anything I’m writing. While Grok seems to be useless so far for what I really want (an intelligence with some insight that tells me something interesting I didn’t already know), at least it is programmed to say kind and encouraging things about my work…
All,
Unless you manage to get an AI agent to come up with something genuinely new and insightful and not just recap some mixture of what I’ve written and conventional wisdom, please, no more AI slop or meta discussions of the quality of the AI slop all around us.
Like everyone around me, my brain is being damaged daily by thinking about AI rather than what I should be thinking about. No more here.
Hi Peter,
In terms of the topology of what is going on here, N = S^3 x S^2, and presumably, PT^+ and PT^- are B^4 x S^2, and we glue along their boundaries to obtain CP^3 an S^2 fibration over S^4?
Also, what is the topology of the complex Gr(2,4), and how to think about the topology of the six orbits of SU(2,2) on it? I think that the oriented complex Gr^+(2,4) is the unit tangent bundle of S^5, and I know there is a standard decomposition of the Grassmannian into Schubert cells, but I’m not sure how to think about how the cells are attached.
Cael Harris,
What you say about the PT orbits/topology I think is correct. The interest though is in the fact that these are complex manifoldx and exploiting holomorphicity, so the topology is not of major interest.
About the grassmannian, I haven’t thought much about the topology of the orbits. This is eight dimensional and complex. I’m mainly interested in the real forms, especially the relation of two of them (Minkowski/Euclidean).
Dear Peter,
I have been reading your arXiv articles (such as spacetime is right-handed) with interests, and I have just came across your blog posts so I thought I’d leave a comment. We have written an article on Spin(4) gauge theory https://arxiv.org/pdf/2507.00968 last year, that is very relevant to topics you post here. (We have uploaded a v2 last week, after some feedbacks with a journal reviewer.) If you are interested to take a look and share some thoughts with us, we would hugely appreciate it.
Best regards,
Lucy
Lucy Zheng,
Thanks, I hadn’t seen that! Traveling now, but will take a look when I get back to work next week.