Twistors and Unification

Posted on March 10, 2026 by woit

The last few postings here have been about rather technical problems with the conventional understanding of how spacetime symmetries and Wick rotation work in the Standard Model. These were written partly because I think these problems deserve to be better known, partly because they motivate a different way of thinking about spacetime symmetries which doesn’t have these problems. This different point of view is that of Penrose’s twistor theory, supplemented by the consideration of the Wick rotation issue, something that I haven’t seen addressed elsewhere in the literature about twistors.

In this posting I’d like to outline the ideas about twistors and unification that I’ve been thinking about for quite a few years now. I’ve been making slow progress at better understanding the details of how to make this work, with still a lot to be done. As this has been going on, I’ve become more and more optimistic that this makes sense and is a very fruitful research direction. I also remain very much struck by the beauty of this framework and its deep connections to fundamental mathematical ideas. In this day and age, Dirac’s argument that

It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress.

has become very unpopular (due to people with an ugly and incoherent 11 dimensional theory hyping it as beautiful or “elegant”). In an environment with no hints coming in from experiment, it could be our only hope.

The twistor point of view fits perfectly with the “spacetime is right-handed” slogan. It is inherently chiral, something that Penrose saw as a problem (the “googly problem”) since he wanted parity-invariant gravity, but that seems to me a virtue. The fundamental spacetime structure is a complex three dimensional space $PT$, which breaks up into two pieces $PT^+$ and $PT^-$, with a common five real dimensional boundary, $N$. Minkowski space time is defined in terms of $N$ (all $\mathbf CP^1$s inside $N$). The fundamental fact behind Wick rotation (that time dependence breaks up into positive and negative energy pieces, corresponding to holomorphicity in the upper or lower half place) in twistor theory appears as the fact that dependence on Minkowski spacetime (in the form of $N$) breaks up into pieces holomorphic in $PT^+$ and holomorphic in $PT^-$ (with boundary data hyperfunctions on N).

As discussed in the last posting, one can think of $PT$ as the standard $\mathbf CP^3$, with an $SU(2,2)$ action. This gives a generalization of Minkowski spacetime with a larger symmetry group, the conformal group, replacing the Poincare group. Elementary particles correspond to representations of this group. To get a gravity theory, one needs to abandon this global symmetry, and think about a holomorphic theory of $\mathbf CP^1$s embedded in a more general complex three dimensional object. I won’t here go further into the issue of how to get quantum gravity this way, but I think this is a more promising starting point than any of the currently popular ones.

In QM, “Wick rotation” is what happens when you decide to study something holomorphic in the upper half complex time plane not by looking at its hyperfunction boundary values on the real time axis, but by restricting to the positive imaginary axis and studying the real analytic restriction there. Analogously, “Wick rotation” in twistor theory is what happens when one picks a fibration of $PT$ by $\mathbf CP^1$s in a way that gives an imaginary time direction (the direction normal to the threereal dimensional family of fibers of the fibration restricted to $N$).

Recently my point of view about all this has changed a bit. I started by thinking that what one wanted to take as fundamental was the Euclidean spacetime twistor picture (thus “Euclidean Twistor Unification”), where $PT$ is the space of complex structures on the tangent spaces of the four-dimensional real space-time. Then Minkowski space-time is a derived object you get once you pick an imaginary time direction and Wick rotate. More recently I’ve found it useful to think of the Minkowski twistor point of view outlined above as fundamental, with “Wick rotation” what happens when you decide you don’t want to deal directly with hyperfunction boundary values, but would rather pick an arbitrary direction normal to the boundary and look at real analytic values restricted to a coordinate in this direction.

From this second point of view, when one Wick rotates, one is making something like a choice of gauge. The twistor point of view treats the $SL(2,\mathbf C)_R$ and $SL(2,\mathbf C)_L$ local symmetries very differently. The $SL(2,\mathbf C)_R$ is what acts on the $\mathbf CP^1$s that describe points. The $SL(2,\mathbf C)_L$ locally acts on the tangent space to the space parametrizing the $\mathbf CP^1$s. When you Wick rotate, you break that symmetry. From the Wick rotated Euclidean point of view, only an $SU(2)_L\subset SL(2,\mathbf C)_L$ acts. From the Minkowski spacetime point of view this symmetry is an internal symmetry, spontaneously broken by your choice of how to do the Wick rotation.

To get the Standard Model using these ideas, one thing one needs to do is work out a consistent formalism based on the above, that looks like the standard electroweak theory with the Higgs when written down in terms of the usual spacetime description. I don’t at this point know how to do this, but it is not obvious that something like this can’t be done.

Part of the twistor picture, as explained in the last posting, has always been that one could understand the solutions to the Weyl equation in terms of sections of a holomorphic line bundle over $PT^+$ (the “Penrose transform”). To understand gauge fields satisfying the self-duality equation, one can use holomorphic vector bundles over $PT^+$ (the “Penrose-Ward correspondence”, although this is somewhat different). These include the tangent bundle, on which the $SL(2,\mathbf C)_R$ will act along the fibers, and the $SU(2)_L$ will act as described above (details to be sorted out…). Above $\mathbf CP^3$ there’s a canonical line bundle and a three complex dimensional quotient bundle, with a $U(1)\times U(3)$ local symmetry. That you’re on $\mathbf CP^3$ means you’ve modded out a $U(1)$, so have the Standard Model $U(1)\times SU(3)$.

Matter fields will be described in terms of holomorphic sections of these holomorphic bundles over $PT^+$. Their couplings to gauge fields will be described, at least in the self-dual case, by the holomorphic structure of the bundles. I don’t have details worked out, but it seems that you can perhaps write down all this using something like a Dolbeault operator on $PT^+$ for these various holomorphic bundles. Or, perhaps you should look at a Dirac operator instead of the Dolbeault operator? How do generations appear?

The above should give some idea of what I’ve been thinking about. There’s a huge amount of challenging work to be done in terms of reformulating conventional ideas about quantum field theory in a consistent holomorphic formalism that implements the above general picture. Personally I think all of this is vastly more promising and interesting than the currently popular but moribund research programs that try and improve on the Standard Model and quantize gravity. I’ll keep at the project of getting more details written down as I understand them, but this is slow going and it’s very unclear what parts to write up since, honestly, just about no one else seems interested in any of it.

I’m no longer teaching, so have more time to work on this, and more time to travel. The week after next I’ll be in Paris, after that will spend a couple more weeks traveling in Europe, details still up in the air. If you understand some of the above and would like to hear more about it, contact me and maybe my travel plans can include a visit. By mid-April, I’ll be back in New York, only definite travel plan after that to get to Madrid around August 12 and see the solar eclipse visible from near there.

Posted in Euclidean Twistor Unification | 2 Comments

Twistors and Wick Rotation

Posted on March 9, 2026 by woit

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.

Posted in Euclidean Twistor Unification | 3 Comments

Weyl Spinor Fields and Right-handed Spacetime

Posted on March 5, 2026 by woit

In this post I’ll discuss Weyl spinor fields and explain why Wick-rotating a single Weyl spinor field appears to be impossible. This motivates a proposal for a different way of thinking about the relation between spinors and vectors, described by the slogan “spacetime is right-handed”.

Written in energy-momentum space, the equation of motion of a right-handed Weyl spinor field is
$$(E-\boldsymbol \sigma\cdot\boldsymbol p)\widetilde \psi_R(E,\mathbf p)=0$$
Here $\psi$ is a two-component complex field. Since
$$(E+\boldsymbol \sigma\cdot\boldsymbol p)(E-\boldsymbol \sigma\cdot\boldsymbol p)=E^2-|\mathbf p|^2$$
solutions will also be solutions of the massless Klein-Gordon equation and satisfy $E=\pm|\mathbf p|$.

Defining helicity as the eigenvalue of the operator
$$\frac{1}{2}\frac{\boldsymbol \sigma\cdot\boldsymbol p}{|\boldsymbol p|}$$
solutions of the equation of motion have either

  • Positive energy $E$, so describe particles, and helicity $+\frac{1}{2}$.
  • Negative energy $E$, so describe anti-particles, and helicity $-\frac{1}{2}$.

The quantum theory is an infinite collection of complex harmonic oscillators (one for each value of $\mathbf p$), for each of which quantization proceeds as discussed here, with annihilation and creation operators satisfying anti-commutation relations.

The Standard Model description of matter particles is built out of copies of this quantum theory, with interactions determined by gauge symmetry (replacing derivatives by covariant derivatives), and mass terms coming from Yukawa couplings to the Higgs field.

The Wightman function is the two by two matrix
$$\widetilde W_2(E,\mathbf p)=(E-\boldsymbol \sigma\cdot\boldsymbol p)^{-1}=\frac{E+\boldsymbol \sigma\cdot\boldsymbol p}{E^2-|\mathbf p|^2}$$
The obvious way to do Wick rotation would be to analytically continue in the complex $E$ plane, getting a Schwinger function
$$\widetilde S_2(E,\mathbf p)=(iE-\boldsymbol \sigma\cdot\boldsymbol p)^{-1}$$

This runs into a fundamental inconsistency with the usual understanding of the transformation properties of vectors and spinors in Minkowski and Euclidean spacetime. For a detailed discussion of the usual story, see some notes here, but the bottom line is that the operator
$$(E-\boldsymbol \sigma\cdot\boldsymbol p)$$
identifies vectors $(E,\mathbf p)$ with two by two complex matrices, and these matrices are supposed to be maps from the space $S_R^*$ of dual right-handed spinors to $S_L$, the space of left-handed spinors.

In Minkowski spacetime $S_L$ and $S_R$ are complex conjugate representations. $S_L$ is just a name for the conjugate of $S_R$ and one only needs one kind of complex field. The problem is that in Euclidean spacetime $S_L$ has no relation to $S_R$, it’s a different representation. If you want a Euclidean field theory of spinors, with the usual relation of vectors and spinors, you need to add a left-handed spinor field. The OS construction of Euclidean fields does this, then doubles again the number of degrees of freedom in order to get a self-adjoint Schwinger function.

The proposal made here is that one should resolve this problem by only using one kind of Weyl spinor field ($S_R$) to describe spacetime vectors (as maps from $S_R^*$ to conjugates of $S_R$), both in Minkowski spacetime and in Euclidean spacetime. In Minkowski spacetime nothing changes (except not using the inappropriate notation $S_L$), but the Euclidean spacetime one Wick rotates to is different than the usual one. Only the $SU(2)_R$ factor of $Spin(4)$ acts non-trivially on vectors, with the $SU(2)_L$ acting trivially, available for use as a gauged internal symmetry.

Four dimensions is very special in that it’s the only dimension in which the rotation group breaks up into two independent pieces and the geometry can be thought of as decomposing into right-handed and left-handed parts. In the usual formalism, one exploits this in Euclidean spacetime (working with self-dual or anti-self-dual objects), but in Minkowski spacetime the two parts are complex conjugates and can’t be separated. I’m proposing a different point of view, in which both Minkowski and Euclidean spacetime just sees the right-handed part of the geometry, with the left-handed part appearing as internal degrees of freedom.

So far I’ve avoided writing about twistors, which provide a fundamentally chiral context for thinking about the implications of this “spacetime is right-handed” point of view. In twistor theory, a point in spacetime is tautologically the same thing as the space of spinors $S_R$ at that point. $S_L$ is something else. I’ll be away on a long weekend starting tomorrow, but will write about twistors in another posting soon.

Posted in Euclidean Twistor Unification | Leave a comment

Lorentz versus Euclidean Symmetry

Posted on March 4, 2026 by woit

Wick rotation changes the spacetime symmetries of a quantum field theory, changing between

  • The Lorentz group, by which I’ll mean either $SO(3,1)$ or $SL(2,\mathbf C)$, the double cover of the time orientation-preserving subgroup of $SO(3,1)$.
  • The four-dimensional Euclidean signature rotation group, by which I’ll mean either $SO(4)$ or its double cover $Spin(4)=SU(2)\times SU(2)$.

A very long time ago I got interested in the possibility that one of the two $SU(2)$ factors in the Euclidean symmetry could appear as an internal symmetry in Minkowski spacetime. For many years though I had given up on this idea, convinced that in any version of Wick rotation, this could not happen. If you look at Wightman and Schwinger functions, they are restrictions of a single holomorphic function. $SO(3,1)$ and $SO(4)$ show up as the symmetries that preserve the two different restrictions, related through analytic continuation between two different real forms of $SO(4,\mathbf C)$.

Back in 2020 at some point I realized that there is a significant difference between Minkowski and Euclidean spacetime theories. In Minkowski spacetime, the reconstruction of states and operators from the Wightman functions does not break the $SO(3,1)$ symmetry: there is no distinguished time direction. In Euclidean spacetime on the other hand, OS reconstruction of physical states and operators does break the $SO(4)$ symmetry, by the choice of an imaginary time direction, and thus an OS reflection operator. More physically, in Minkowski spacetime you don’t need a choice of time direction to do physics, while in Euclidean spacetime, you must choose a direction, the direction you plan to do Wick rotation in to recover real-time physics.

The realization that the Euclidean theory had to come with this extra piece of structure, which broke $SO(4)$ symmetry to $SO(3)$ symmetry, led me to the ideas in this quite speculative paper. At the time I was quite confused about the details of how Wick rotation worked in rigorous versions of quantum field theory, but by now I’m much less so. In an earlier post I started writing about this, here will explain what happens to spacetime symmetries under Wick rotation, at least for scalar field theories. Things get much more interesting when you look at spinors, which I’ll do in future posts.

What’s relatively easy to understand is what happens when you start with a Minkowski spacetime theory. The Wightman functions have $SO(3,1)$ symmetry and Wightman reconstruction gives a state space with a unitary representation of this group. The $SO(3,1)$ invariant version of the positive energy condition is to assume that the Wightman functions are supported in the positive light-cone (in energy-momentum space). If one then writes the inverse Fourier transform formula, but for complex spacetime coordinates $z=(z_0,\mathbf z)$
$$W(z)= \frac{1}{(2\pi)^2}\int_{\mathbf R^{3,1}} e^{-i(z_0E-\mathbf z\cdot\mathbf p)}\widetilde W(E,\mathbf p) dE d^3\mathbf p$$
one gets $W(x)$ for $z=x$ real, and an analytic continuation into a “tube” which is a subspace of $\mathbf C^4$ of the form $\mathbf R^{3,1}$ plus $i$ times the inside of a lightcone. This is a generalization of the Paley-Wiener theorem argument discussed in the earlier blog post.

Given this analytic continuation to some of $\mathbf C^4$, one acts holomorphically by the group $SO(4,\mathbf C)$ to get an analytic continuation to a larger region (the “extended tube”). One then uses symmetry under interchange of coordinates to get an analytic continuation to an even larger region (the “permuted extended tube”). The Bargmann-Hall-Wightman theorem says that one can do this with a single-valued holomorphic result. The region of holomorphicity now includes the Wick-rotated Euclidean subspace $\mathbf R^4$. One can easily see from the formula above that $SO(3,1)$ symmetry in $\mathbf R^{3,1}\subset \mathbf C^4$ becomes $SO(4)$ symmetry in the Wick rotated $\mathbf R^4$.

When one tries to go the other direction, starting with Schwinger functions with $SO(4)$ symmetry and doing OS reconstruction to get Wightman functions with $SO(3,1)$ symmetry, one runs into the trouble discussed earlier that there’s no good way to invert the formula for $W(z)$ above and get the Wightman function from the Schwinger function. What OS do is start by assuming some property of Schwinger functions and doing an elaborate analytic continuation argument (with results that are not very satisfactory). Even less satisfactory, but much simpler, is to just take as an axiom a property of Schwinger functions that ensures they come from some $\widetilde W(E,\mathbf p)$ as above. This is done for instance in section 5.6 of the book An Introduction to Non-Perturbative Foundations of Quantum Field Theory, by Franco Strocchi. When you do this, you can show that $SO(4)$ invariance of Schwinger functions implies $SO(3,1)$ invariance of Wightman functions.

The relation between the symmetry of Wightman functions and that of Schwinger functions is thus rather straightforward, but there’s still the question of the representation of the symmetry groups acting on states. On the Minkowski side, $SO(3,1)$ acts on test functions, and on the states constructed from these by Wightman reconstruction as a representation of the group. On the Euclidean side, $SO(4)$ acts on test functions on $\mathbf R^4$, but only the $SO(3)$ preserving a chosen imaginary time direction acts on OS reconstructed physical states.

How the $SO(3,1)$ representation on physical states is recovered is discussed by Seiler in section 8b here and by Klein and Landau here. Things are put in a more general representation-theoretical context by Frohlich, Osterwalder and Seiler here. The analytical continuation argument for how one gets an $SO(3,1)$ representation from the $SO(4)$ representation (+other data) is more involved than the argument above for Schwinger/Wightman functions.

My interest in this topic is in what happens for spinors, while these discussions are almost all about the real scalar theory. The original OS reconstruction paper does have a section about arbitrary spin fields, and the OS papers on fermion fields are designed to fit into this context.

In a post to come soon I’ll write about the issues with spinors and the possibility that the usual way to analytically continue them is not what one wants to do.

Posted in Euclidean Twistor Unification | Leave a comment

Hyperfunctions

Posted on March 2, 2026 by woit

I’ve been trying to understand (not entirely successfully yet…) Wick rotation using hyperfunctions, which are a sort of distributions not as well-known as they should be. Some notes about them are in a separate pdf.

What is well-known is that one often needs to generalize the notion of what a function is to include things like the Dirac delta function $\delta(x-a)$. The usual way to make this well-defined is the theory of distributions, but hyperfunctions provide an interesting and in many ways more useful alternative. See the notes for more detail, but an important example is that (up to a constant), the Dirac delta function is the function
$$\frac{1}{z-a}$$
of a complex variable $z=x+iy$, reinterpreted as a hyperfunction. This way of thinking about the delta function makes available the powerful methods of complex analysis.

There have been some previous attempts to do the sort of thing I’m thinking about. In particular see this paper. Such attempts to reformulate QFT, with Wightman functions taken to be hyperfunctions, have generally had as one motivation to resolve the problems with OS reconstruction discussed in an earlier blog posting. My own motivation is rather different (providing a formalism in which one can understand Wick rotation of spinor fields in a new way).

The idea of using hyperfunctions this way in QFT has attracted relatively little interest over the years. I’m guessing one reason for this is that hyperfunctions become much harder to work with when one is dealing with more than one complex variable. In the usual rigorous QFT framework one tries to understand Wick rotation by complexifying all spacetime variables, not just time, with just complexifying time violating usual ideas of the necessity of preserving Lorentz invariance.

In a later posting I’ll discuss how hyperfunctions show up in twistor theory, where the way complexification and hyperfunctions work is quite different.

Posted in Uncategorized | 8 Comments

Osterwalder-Schrader and Euclidean Spinor Fields

Posted on February 28, 2026 by woit

If one tries to Wick rotate a quantum field theory with spinor fields, it’s well-known that problems arise, something first recognized in Schwinger’s earliest papers on the subject. I’ll try and outline here the 1972 proposal by Osterwalder and Schrader (see here, here and here), which is the best known way to deal with the problem. Over the years there have been a large number of other efforts to address this issue, and I’ve put together a bibliography of those here. Many of these I’ve never been able to completely understand. I’ll concentrate on the Osterwalder-Schrader proposal since I do now understand it (I didn’t in 1984…), and it seems to correspond best to the conventional wisdom of the subject.

There’s a first indication that something funny is going on when you look at any QFT textbook discussing spinor field theory in the standard “Dirac spinor” formalism where spinors take values in $\mathbf C^4$. A good example is Pierre Ramond’s book, where chapter 5 deals with this, in both Minkowski and Euclidean signature. The Lagrangian in both cases can be written the same way, as
$$\overline \psi (i\gamma^\mu\partial_\mu -m)\psi$$
with
$$\psi=\begin{pmatrix}\psi_L\\ \psi_R\end{pmatrix}$$
If you read more closely you find out that the notation is hiding things:

  • In Euclidean signature, $\overline \psi =\psi^\dagger$, but in Minkowski signature $\overline \psi =\psi^\dagger \gamma_0$ (the “Dirac adjoint”).
  • In Euclidean signature the spin group is $Spin(4)=SU(2)_L\times SU(2)_R$, $S_L$ is the spin representation of $SU(2)_L$, and $S_R$ is the spin representation of $SU(2)_R$. In Minkowski signature the (time-orientation preserving) spin group is $Spin(3,1)=SL(2,\mathbf C)$, $S_L$ is the spin representation of $SL(2,\mathbf C)$, and $S_R$ is the complex conjugate of the spin representation.

Osterwalder and Schrader propose that Wick rotation of spinor fields involves a doubling of the number of degrees of freedom, giving up the Dirac adjoint relation, and taking $\psi$ and $\overline\psi$ to be independent fields (which they call $\psi_1$ and $\psi_2$). They show that one can then do the same kind of OS reconstruction argument as in their paper dealing with scalar fields. The OS reflection operator that in the scalar case both complex conjugated fields and reflected in imaginary time now also interchanges $\psi_1$ and $\psi_2$, as well as having a $\gamma_0$ factor that interchanges $S_L$ and $S_R$.

This proposal does what it is advertised to do, reconstructing the Wightman functions and state space of the usual Minkowksi spacetime theory, but the way it does this is somewhat unsettling. Wick rotation is not just a matter of putting some factors of i in the right place, but involves a significant change in the degrees of freedom of the theory when one passes from Minkowski to Euclidean. For scalars, OS showed that the Wick rotation of complex conjugation surprisingly also now involved a reflection in spacetime. For spinors this becomes an even more intricate piece of structure one must add to the Euclidean theory to do reconstruction.

Osterwalder-Schrader and most later authors ignore something even more problematic about Wick rotating spinors, something pointed out by Ramond in his book: it doesn’t work for a Weyl spinor field. The basic building blocks of matter fields in the Standard Model are two-component spinor fields, with the simplest building block the theory of a chiral (say right-handed) massless Weyl fermion. This theory is simple to write down, and at first glance has a simple Wick rotation, just by taking time to be complex and proceeding as for scalars.

But this runs into a fundamental issue with how the transformation properties under spacetime rotation change as one goes from Minkowski to Euclidean. It appears that if one wants to describe a massless neutrino of one chirality in Euclidean QFT, one must quadruple the number of degrees of freedom (first double the degrees of freedom to get four-component Dirac spinors, then double again according to Osterwalder-Schrader).

I’ll leave for another time discussion of the details of how spacetime rotations change under Wick rotation in the usual formalism. I’ve outlined a proposal for a very different way of understanding this issue in my Spacetime is Righthanded paper.

Posted in Uncategorized | 7 Comments

Harmonic Oscillators

Posted on February 27, 2026 by woit

This is related to the Osterwalder-Schrader posting, but is much, much more elementary. I’ll write up some basic facts about the quantum harmonic oscillator and explain what bothers me about the relation to Osterwalder-Schrader.

Every quantum mechanics course covers the quantum harmonic oscillator, generally in the Schrödinger picture, with states functions of space and time. The Hamiltonian is a second order pde, one finds its eigenfunctions and eigenvalues. For a version of this that I wrote, see chapter 22 here.

Free quantum field theories are just infinite collections of such harmonic oscillators, but in QFT one wants to use the Heisenberg picture (the Schrödinger picture would be very awkward). For a single quantum harmonic oscillator in the Heisenberg picture, one has two operators $Q(t),P(t)$, with Hamiltonian
$$H=\frac{1}{2}\left(P^2 +\omega^2 Q^2\right)$$
(here I’m rescaling so that $m=\hbar=1$). The Heisenberg equations of motion are
$$
\frac{d}{dt}Q=i[H,Q]=P,\ \ \frac{d}{dt}P=i[H,P]=-\omega^2Q
$$
Subsituting the first in the second, one gets the second-order equation of motion
$$\left(\frac{d^2}{dt^2}+\omega^2\right)Q=0$$

Such equations can most easily be solved by complexifying (allowing not just real, but complex linear combinations of solutions). Using complex linear combinations of operators, one can write
$$a=\sqrt{\frac{\omega}{2}}Q+i\sqrt {\frac{1}{2\omega}}P,\ \ a^\dagger=\sqrt{\frac{\omega}{2}}Q-i\sqrt {\frac{1}{2\omega}}P$$
which turns the first order equations into
$$\frac{d}{dt}a=-i\omega a,\ \ \frac{d}{dt}a^\dagger=i\omega a^\dagger$$
with solutions
$$a(t)=a(0)e^{-i\omega t},\ \ a^\dagger(t)=a^\dagger(0)e^{i\omega t}$$
The Heisenberg commutation relations are the time-independent
$$[a,a^\dagger]=1$$
and the Hamiltonian is
$$H=\frac{\omega}{2}(aa^\dagger +a^\dagger a)$$
One can then easily show that the state space has a basis
$$\ket{0},\ket{1},\ket{2},\ldots$$
with
$$H\ket{n}=\omega (n+\frac{1}{2})\ket{n}$$

This in some sense is the simplest possible quantum system and easily extends to a quantum field theory describing arbitrary numbers of non-relativistic particles of mass $m$. Just put together an infinite collection of such oscillators, with operators $a_{\mathbf p},a^\dagger_{\mathbf p}$, parametrized by the possible momenta $\mathbf p$, with
$$\omega=\omega_{\mathbf p}=\frac{|\mathbf p|^2}{2m}$$
If one wants to describe fermions, just change commutation relations to anti-commutation relations. This system is exactly the starting point of many-body physics methods for dealing with condensed matter systems.

The usual field operators are the Fourier transforms of these operators parametrized by momenta to operators parametrized by space:
$$\widehat{\psi}(t,\mathbf x)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbf R^3} e^{i\mathbf p \cdot \mathbf x}a_{\mathbf p}(t) d^3\mathbf x$$

This is a wonderfully simple story, but it bothers me that it doesn’t seem to fit at all the Euclidean QFT philosophy of starting with an imaginary time theory, then using OS reconstruction to get the physical theory.

The simplest case of the Osterwalder-Schrader theory would describe a harmonic oscillator in a more complicated way, using not the first-order equations of motion but the second order equation. Still complexifying, $a$ satisfies the second-order equation
$$\left(\frac{d^2}{dt^2}+\omega^2\right)a=\left(\frac{d}{dt}+i\omega\right)\left(\frac{d}{dt}-i\omega\right)a=0$$
This has twice as many solutions as our earlier version, with the new solutions complex conjugates of the old ones. Physically the problem with them is that they have negative energy.

One can deal with the new solutions by defining a separate state space and separate operators $b,b^\dagger$, solving the negative energy problem by interchanging the role of annihilation and creation operators. Now, besides states of quanta, one also has “anti-quanta”, which one can metaphorically describe as “quanta traveling backwards in time.”

This is a theory of a quantum complex harmonic oscillator, with two adjoint operators
$$a(0)e^{-i\omega t} +b^\dagger (0) e^{i\omega t}\ \ \text{and}\ \ a^\dagger(0)e^{i\omega t} +b(0) e^{-i\omega t}$$
To get back to the usual state space with just quanta, one can identify quanta and anti-quanta, i.e. $a=b, a^\dagger=b^\dagger$. Then there is just one kind of operator, the self-adjoint
$$a(0)e^{-i\omega t} +a^\dagger (0) e^{i\omega t}$$

This last theory is a relativistic real scalar field in 0+1 dimensions. It has a sensible imaginary time version and the OS reconstruction theorem applies. For more about the details of how this works, see for examples section VII.4 of this paper.

A simple question that’s bothering me is that I haven’t run across a discussion of OS reconstruction that applies to the case of the complex harmonic oscillator. If someone is aware of such a thing, please let me know about it.

For the case of the simplest possible description of the harmonic oscillator as given in the beginning of this posting, I’ve always been bothered not just by the fact that something like Osterwalder-Schrader doesn’t seem to apply, but even more by the fact that it’s hard to come up with a consistent path integral formalism that would describe it, even in imaginary time.

During one period in my life I spent a great deal of time thinking about this. There’s a whole subject of “coherent state path integrals” (although they’re not really integrals), with a large literature. A good discussion of the subject is chapter 6 (“path integrals and holomorphic formalism”) of Jean Zinn-Justin’s Path Integrals in Quantum Mechanics (for a more public domain version see here).

Besides the harmonic oscillator case (quantization of $\mathbf C$) case, even simpler should be the spin degree of freedom (quantization of the Riemann-sphere). I ended up convinced that the only way to make sense of such a path integral would be with a supersymmetric path integral, of the sort been related to the index theorem. For an early write up of some of this, see here.

My current point of view is that what one wants is not a purely Euclidean path integral, but a formalism holomorphic in the time variable, so in the realm of complex analysis rather than real analysis. Still stuck on some of the details of this, hope to soon have the energy to get back to that and get something written up.

In case it’s not clear, the ultimate motivation of this is to come up with a better way of understanding some of the things that are confusing about the Standard Model, in particular the treatment of chiral spinor fields. I’ll try to write soon the promised blog entry about the other Osterwalder-Schrader paper, the one dealing with Euclidean Fermi fields.

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Looks Like it is Happening…

Posted on February 24, 2026 by woit

Update: Sorry, but a commenter points out that this may just be an artifact of counting based on when most recently modified, not on original submission date.

Numbers using original, not most recent, submission dates

For 12/1 to 12/31 the numbers were
2022: 800
2023: 811
2024: 815
2025: 855

For 1/1 to 2/1
2022:510
2023:490
2024:501
2025:544
2026:617

For 2/1 to 2/15
2022:255
2023:221
2024:280
2025:276
2026:311

These do show significant increases year to year for the last couple months, but not the near doubling indicated by the other numbers. The hep-th arxiv apocalypse is not here yet.

For a while now I’ve been speculating about what would happen when AI agents started being able to write papers indistinguishable in quality from those that have been typical of the sad state of hep-th for quite a while. Sabine Hossenfelder today has AI Is Bringing “The End of Theory”, in which she gives her cynical take that the past system of grant-holding PIs using grad students/postdocs to produce lots of mediocre papers with the PI’s name on them is about to change dramatically. Once AI agents can produce mediocre papers much more quickly than the grad students/postdocs, then anyone can play and we’ll get flooded by such papers from not just those PIs, but everyone else.

I decided to take a look at the arXiv hep-th submissions, and quickly generated the following numbers, by simple searches using
https://arxiv.org/search/advanced
to find all hep-th submissions in various date ranges.

For 12/1 to 12/31 the numbers were
2022: 634
2023: 684
2024: 780
2025: 1192

For 1/1 to 2/1
2022:583
2023:531
2024:626
2025:659
2026:1137

For 2/1 to 2/15
2022:299
2023:266
2024:271
2025:333
2026:581

From this very limited data it looks like submission numbers in the last couple months have nearly doubled with respect to the stable numbers of previous years.

I thought about spending more time I don’t have lookng into this, then realized “this is a job for AI!”. Surely an AI agent could do a lot better job than me in gathering such data, figuring out things like whether you can recognize the AI agent papers or not, and writing up a detailed analysis. I’m still resisting learning how to use AI agents, so someone else will have to do this.

One of my main problems with the comments here has been that it’s increasingly hard to tell the difference between human and AI generated ones. In this case, maybe the AI generated ones would be better than those from meatspace. So, unless you have something really substantive (like an explanation for why these numbers don’t mean what it looks like they mean, or know what the arXiv is doing about this) please resist commenting. I’ll moderate comments for things like irrelevance and hallucinations, but won’t delete comments just because they are non-human.

Posted in Uncategorized | 18 Comments

Osterwalder-Schrader

Posted on February 19, 2026 by woit

I’ve been trying to write up some new ideas about Wick rotation for a long time now, keep getting stuck as it becomes clear at various points that I haven’t gotten to the bottom of what is going on. To take a little break from that I thought it might be useful to write some more informal things here on the blog, about parts of this story that I do understand.

One thing I want to write about are two important papers by Konrad Osterwalder and Robert Schrader. The first is their Axioms for Euclidean Green’s Functions, published in Communications in Mathematical Physics in 1973. I’ll refer to this as the OS reconstruction paper. The second is Euclidean Fermi fields and a Feynman-Kac formula for Boson-Fermion models, published in Helvetica Physica Acta, also in 1973, which I’ll refer to as the Euclidean Fermi fields paper.

Konrad Osterwalder was the instructor in my Math 55 class at Harvard, my first semester there in the fall of 1975. I just found out while looking for some information about him that he passed away quite recently (December 19 last year). Back in 1975 Osterwalder was an assistant professor in mathematical physics at Harvard, and the Math 55 class he taught followed quite closely chapters 0-4 of Loomis and Sternberg’s Advanced Calculus, which at the time was the standard textbook for the course.

During my last term at Harvard (spring 1979) I took an upper level graduate class from Arthur Jaffe on the foundations of QFT. As a requirement of the course, I had to pick a relevant paper and write about it. The paper I picked was the Osterwalder-Schrader Euclidean Fermi fields paper. I was pretty much mystified by it, and remained so for many, many years. I’m planning to write something about this paper in a later blog post, here will concentrate on the OS reconstruction paper.

For some amusing commentary on the story of the OS reconstruction paper, see Slava Rychkov’s talk “CFT Osterwalder Schrader Theorem” at this meeting in 2019, where he says:

these papers appeared in Communications in Mathematical Physics. If you start reading these papers you immediately get a headache. The first ten pages are just notation. You have to go through then another theorem, lemma, lemma theorem, Hille-Yosida theorem, things like that.

Very few people have read these papers and very few people know what has actually been done there. It’s almost irresistible, people love to cite these papers because it’s like a feeling of ancient magic books, the scriptures. Many normally very careful people misquote these papers and miscite them by attributing to them results which are not there.

The background for all of this is the story of Euclidean or imaginary time quantum field theory, which starts with Julian Schwinger’s 1958 paper On the Euclidean Structure of Relativistic Field Theory. For some relevant history, see here. One way of looking at quantum field theory is that it’s all about the vacuum expectation values of field operators, the Wightman functions. What Schwinger was suggesting was that one could define quantum field theories in terms of the analytic continuation of Wightman functions, evaluated at imaginary time (these are now known as Schwinger functions). This fits well with the modern point of view that QFTs should be defined by path integrals, since it is only imaginary time path integrals for which one can hope to have something one can make rigorous, not a purely formal object.

Schwinger was well aware that if you tried to define a QFT as a set of Schwinger functions, something missing was a way of recognizing when these corresponded to a physical theory in real time. In the discussion session of his presentation at the 1958 ICHEP, he said

The question of to what extent you can go backwards, remains unanswered, i.e. if one begins with an arbitrary Euclidean theory and one asks: when do you get a sensible Lorentz theory? This I do not know. The development has been in one direction only: the possibility of future progress comes from the examination of the reverse direction, and this is completely open.

The OS reconstruction paper was based on a crucial new idea for how to recognize a Schwinger function corresponding to a physical real-time theory, the condition of “reflection positivity”. Jaffe recounts here how this came about.

A crucial property for a quantum theory is that it has a Hermitian inner product on states, with states having positive norm in this inner product. The Hermitian nature of the inner product of two state vectors involves complex conjugation on one of them. On functions of time, this is just complex conjugation of the value of the function. When you work with complex time $z=t+i\tau$ instead of real time, the complex conjugation takes $z=t+i\tau$ to $\overline z=t-i\tau$. This is a reflection $\tau \rightarrow -\tau$ in the imaginary time axis, sometimes called the Osterwalder-Schrader reflection.

By the way, this seems to me a first indication of the possibility I’ve been trying to understand of spacetime transformations in Euclidean spacetime turning into internal symmetries in Minkowski spacetime (here reflection in time is turning into pointwise complex conjugation).

What Osterwalder and Schrader did in the OS reconstruction paper was provide a theorem stating when Schwinger functions came from Wightman functions. As Rychkov notes, this paper is very hard going. After spending a lot of time with it, I realized one reason why the whole thing is difficult, which I’ll try and explain here. This is something that has held up what I’ve been trying to do with Wick rotation. Quite possibly I’m missing something and maybe someone will explain to me what it is.

Analytically continuing from real time to imaginary time is relatively easy, because it’s an example of what mathematicians know as the Paley-Wiener theorem. If you have a function $f(t)$ with Fourier transform $\widetilde f(E)$ that is only supported at positive energy, you can do inverse Fourier transformation to complex values of time by
$$F(z)=\frac{1}{\sqrt{2\pi}}\int_0^\infty e^{-izE}\widetilde f(E)dE$$
Because
$$e^{-izE}=e^{-itE}e^{\tau E}$$
this integral will give a result holomorphic in $z$ for $\tau<0$, with boundary value at $\tau=0$ the original function $f(t)$. The “Wick rotation” to a function of $\tau$ is given by $F(-i\tau)$. If you change the conventions I’m using for $2\pi$ factors and the sign of the exponent, this is just the Laplace transform of $\widetilde f(E)$
$$\int_0^\infty e^{-\tau E}\widetilde f(E)dE$$

The problem is that going in the other direction is much trickier. Given a function $f_S(\tau)$ (S for “Schwinger”), if you try to analytically continue to get $f(t)$ by first inverting the Laplace transform to get $\widetilde f(E)$ (then inverse Fourier to get $f(t)$), there’s a problem. When you look up the formula for inverse Laplace transform it basically says “first analytically continue to $f(t)$, then Fourier transform to get $\widetilde f(E)$.”

The argument in the OS reconstruction paper is tricky, partly because they can’t directly do this inverse Laplace transform. Instead, given $f_S(\tau)$, they define a function of $E$ by Laplace transform, but this function is not $\widetilde f(E)$ (does anyone know of a nice relation between them?), although it has properties they can use to prove their reconstruction theorem.

It turned out that the proof in the OS reconstruction paper was flawed. Their lemma 8.8 claimed to show that the way they were dealing with this problem for a single variable would continue to work for multiple variables, but this was wrong, with a counterexample soon found. They later wrote a second paper, which fixes the problem, but at the cost of a very difficult argument, and assuming a particular property of the Schwinger functions. The Rychkov talk linked to above explains that when he tried to understand the exact relationship between Euclidean and Minkowski in conformal field theory, he was shocked to realize that the OS reconstruction theorem did not apply, because there was no viable way of knowing if the Euclidean Schwinger functions had the necessary property. At this point, the best way to try and understand the OS reconstruction paper is not by reading it, but by looking at explanations from Rychkov (videos here and here, or section 9 of a paper with collaborators).

The OS reconstruction argument is an impressive and important piece of mathematical physics, but its impenetrability has had the unfortunate effect of convincing most people (myself included for many years…) that the relation between Minkowski and Euclidean quantum field theories is something straightforward and well-understood. This matches up with an equally unfortunate conviction that the problems of defining QFTs by path integrals are not serious, with Minkowski vs. Euclidean nothing but a different sprinkling of factors of i in an integral.

This story is just one aspect of fundamental problems about understanding QFTs which go much deeper than that of not being able to provide rigorous proofs. Already by the time I was a student it was clear there was a mismatch between the scalar QFTs studied by mathematical physicists using Euclidean methods and the ones relevant to the real world. In addition, the bottom line about such scalar QFTs turns out to be that they exist and are non-trivial only in two and three spacetime dimensions, must be trivial in four or more spacetime dimensions.

The Standard Model QFT is mainly built on spinor Fermi fields and Yang-Mills gauge fields. I’m sure that’s why back in 1979 I was interested in the Osterwalder-Schrader Fermi fields paper (much more about this in another blog posting). Attempts to fully understand Yang-Mills gauge fields soon moved to the discretized lattice gauge theory. During my graduate student years I was seduced by the simplicity of Euclidean spacetime pure Yang-Mills lattice gauge theory, which is basically a geometrically beautiful statistical mechanics system that can be studied with statistical mechanics methods, including straightforward Monte-Carlo calculations. That experience, coupled with not understanding the subtlety of the OS reconstruction theorem, left me convinced that the way to understand QFT was using path integrals in a Euclidean spacetime theory, with the question of the relation to physics just one of how to do the analytic continuation to real time after the theory was solved.

More and more I’ve become convinced that this was a misguided point of view. A better starting point may be the following. A fundamental aspect of quantum theory is the existence of the Hamiltonian H and a unitary operator $U(t)=e^{-itH}$ which represents translation in time and provides the dynamics of the theory. The significance of Wick rotation is that it is telling you that if you think of time as complex variable, positivity of the energy implies that $U(t)$ is only part of the story, a boundary value of a holomorphic representation $U(z)=e^{-izH}$ of a holomorphic semigroup (complex time translations with imaginary time one sign only). The fundamental quantum field theory of the real world likely should not be thought of as a statistical system, but as having a holomorphic aspect, involving much deeper mathematics.

Update: Glad to see a relevant comment from Yoh Tanimoto. In order to make clear exactly what is bothering me about the OS reconstruction argument and explain the comments about Laplace and inverse Laplace in the posting, here are some more details (I’m simplifying by ignoring spatial variables).

OS are getting Wightman distributions W from Schwinger functions S in 4.1. There in 4.12 they define the Fourier transform $\widetilde W$ as the unique thing whose Laplace transform is S. They do this by invoking the lemma 8.8 that gets them in trouble. If they had a formula for the inverse Laplace transform, explicitly giving $\widetilde W$ in terms of S, they wouldn’t have trouble.

What’s really bothering me is what they do in section 4.3 (which, besides being a complicated argument, is atrociously written, hard to decode). There they are trying to show that the following constructions of the physical state space are the same:

  • Euclidean construction $\mathcal K$ as test functions on the positive imaginary time line modulo those null in the inner product given by $S$ with the OS reflection.
  • Usual Wightman real time reconstruction of the state space $\mathcal H$ as test functions of real time modulo those null in the inner product given by W.

Here the problematic Laplace transform is that of equation 4.20. They Laplace transform (NOT inverse Laplace transform) the first kind of test function to get the Fourier transform of the second kind of test function. I guess they are able to get an identification of the two Hilbert spaces this way, but I’m wondering why you don’t instead use the inverse Laplace transform and be matching analytic continuations of the two kinds of test functions. Presumably because without a formula for the inverse Laplace transform you can’t match the norm using S with the norm using W.

To say a bit more about my motivation, it’s that I’d like to know why there’s not a more straightforward version of the Wick rotation between two different definitions of the physical state space. I’d like a holomorphic construction that specializes to the two different cases (have been trying to do this using hyperfunctions).

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Weil Anima

Posted on February 10, 2026 by woit

Dustin Clausen is giving a course at the IHES starting today on “Weil Anima” (or maybe “Weil-Moore Anima”. They’ve already put up video of the talk here (also available on Youtube). For an earlier talk by Clausen on this material from last year, see “Refining Weil groups” at the Manin memorial conference.

I just watched the IHES Clausen talk, which was a mind-blowing experience, highly recommended if you’ve been following this subject in recent years. Unfortunately I can’t do any sort of justice here to the sweep of fascinating new ideas about math at its deepest level that the talk covers, mainly because of lack of competence, but also because of lack of time.

Trying to find time to think about the talk clarified for me the current state of my intellectual life. Thinking about the larger world (or even the local world outside my office window…) has become thoroughly depressing. I really should stop following the news in any form to preserve my mental health. This argues for concentrating on thinking about the problems in math, physics and their overlap that I’d most like to understand better.

As far as deep ideas about physics goes, paying attention to what’s been going on in the subject long ago became depressing. The situation in mathematics is completely different, with Scholze, Clausen and many others making dramatic progress towards unearthing a wonderful world of new structure at the deepest level in mathematics. A big problem is that this is intellectually extremely challenging material to follow, and I don’t right now have the necessary time. Besides the Clausen talks, this month there’s a program in Marseilles on Langlands-related stuff, with videos starting to appear that would doubtless be helpful if I had the time to watch them.

I’m trying to get the details of a new way of thinking about Wick rotation written down, and to make progress on that is a full-time effort. It’s rewarding in its own way, as I’m learning new things, but it’s extremely slow, spending most of the time stuck on trying to understand things like where signs come from. Learning beautiful new ideas that someone much more industrious spent years slogging through trying to get straight is a lot more appealing.

Finally, as a long-term goal, there’s the tantalizing fact that the twistor line plays a central role in the new ideas Clausen is talking about, and at the same time it plays a central role (as a point of space-time) in what I’ve been trying to do. That deep ideas about physics and about mathematics are closely linked is something I firmly believe, so I don’t think these two very different contexts for the twistor line have nothing to do with each other. I’d love to be thinking more about this, but need to stop blogging and get back to getting the Wick rotation paper finished…

Posted in Langlands | 3 Comments