A few months ago I decided to try and sort out the two dimensional spacetime case of the Wick rotation issues that have been bothering me for years now. It took me a while to understand what goes on in two dimensions, but when I did, this clarified a lot for me the underlying problems with Wick rotation that I’ve been confused about. I’ve just finished writing something up about this, which you can find here.
Instead of writing more about this tonight, I’ll leave that for tomorrow. Will either write some more in the posting, or in the comments.
Update: My thinking about the implications of this keeps changing and I’m making progress at better understanding what’s going on here. This will take a while and be significantly different than these notes, so I won’t write more about them for now.
Let me mention a couple of small things, I dont know how important they are.
1) Complexification of a complex vector space carries a natural quaternionic structure (and thus the same is true for complex vector bundles). Namely if V is a complex vector space, then V otimes_R C= V oplus V bar. And we can define a complex-antilinear endomorphism J that squares to -1: V \oplus V bar->V \oplus V bar, (v_1, v_2) mapsto (-v_2 bar, v_1 bar). It anticommutes with complex structure i by construction, and thus i, J and iJ satisfy quaternionic relations. Naturality refers to the fact that any complex linear map f:V->W induces a quaternion-linear map (f,f bar):V otimes_R C-> W otimes_R C.
2)Twistor bundle can be understood as a complex projectivisation of a spinc spinor bundle as well, Z^+=P(S^+_sigma), and therefore this works on any oriented smooth 4-manifold (one needs to choose at least a conformal structure to have Z^+ at all and a spinc structure sigma to fix this identification).
And the total space of Z^+ still carries a natural almost complex structure, but as in the usual case it is only integrable when X is anti-self dual (when X is self-dual same is true for Z^-). Which is a shame as when the total space is a genuine complex manifold we have access to Ward correspondence like in the ADHM construction, which connects things like instantons or monopoles on X to some special holomorphic vector bundles over Z^+.
I personally am not convinced that this is all there is to that, not even Kahler surfaces are all (anti-)self-dual as 4-manifolds. People have proposed different alternatives like constructing symplectic structures over the total space of Z^+ or other modified almost complex structures, but nothing too promising yet.
I have recently been thinking that because of the fact that SO(4)=Sp(1) times Sp(1)/ Z_2 closed oriented 4 manifolds should carry some kind of “quaternionic structure” after all, at least those with non vanishing Seiberg-Witten invariant. It is interesting that in certain cases we do have candidates, and of different nature. Like if X is spin, or (anti-) self-dual or symplectic. But nothing comes close to the beautiful theory we have for closed oriented Riemannian surfaces yet.
BagratM,
I should explain that I’m interested in a new understanding of local quantum field theory in four dimensions, so the global structure of the 4 manifold isn’t very relevant. If one can figure this out for S^4 that would be enough to do local physics. In particular, I’m not trying to get a topological field theory (although maybe they come as a by-product).
The new thing here is that you start with a Riemannian manifold (and its twistor space, or projective spin bundle), but you need to also have a new structure that gives you an appropriate notion of conjugation, in some sense interchanging positive and negative imaginary time and leaving invariant a subspace which captures the chiral degrees of freedom of a Lorentzian geometry.. If people have looked at this in 4d I’d be curious to hear about it, I’m not aware of such work, but haven’t looked hard.
Dear Peter,
thank for the interesting notes; here are some “first-impression considerations” that might be useful.
a.
The relevant structure here (in the case of a complex-space equipped with a conjugation J) is “split-quaternionic”. The split-quaternions being a (2,2)-signature real-subspace of the complex-space of the q-bit (while the quaternions are an Euclidean signature subspace).
This split-quaternionic structure is canonically induced on a Hilbert space by a choice of a “standard” real-subspace (as is well-known in first-quantized Tomita modular theory: see for example the work by J.Naudts and J.Zhang [ arXiv:2501.04010 ]); the J conjugation operator is a obtained by polar decomposition from the Tomita-conjugation real-structure associated to the standard real-subspace.
The “doubling” of degrees of freedom (of phase-space) due to a real-structure conjugation is a canonical feature of all complexifications of “ortho-symplectic spaces” (whenever the norm of the symplectic structure is less than 1 with respect to the norm induced by the metric) and more generally of Hitchin-Gualtieri “complex” generalized geometries; these things are described in detail in the slides of my talk at the recent workshop Quantum Geometry and Physics in Vercelli.
b.
Some previous studies on “reflection positivity” that might be of some relevance are in the works of P.Jorgensen, K.-H.Neeb and J.Olafsson [see the references in the book arXiv:1802.09037 ].
I will carefully read Your notes and I hope to be able to send further and more useful “second-impression comments” 😉
Best Regards.
Paolo