I’ve now finished with two things that I’ve been working on over the last year or so:

- The paper explaining my proposal for “Twistor Unification” is now done and uploaded to the arXiv, see here.
- I’ve finished lecturing for the course on quantum mechanics for mathematicians that I’ve been teaching this academic year. Because of the Covid-required online format for the lectures, they could easily be put on Youtube, where they’re available here. I’m hoping to never ever have to teach this way again, so don’t expect to ever again be producing Youtube lectures. The lectures pretty closely follow my book, and I had been hoping to work on improving and expanding the text. Unfortunately, partly due to laziness and partly due to the twistor stuff, while I found a lot in the book that needs improvement, I didn’t find the time to do the necessary rewriting and writing. I do however have a notebook full of notes on what needs to be done.

For the future, I’m hoping to go on some sort of vacation in a couple weeks, and soon get back to work on some of the major issues raised by the unification proposal (much of which is very sketchy, a lot to be done). I hope to do quite a bit of traveling the rest of this year, likely won’t be teaching in the fall, but probably will be teaching the quantum course again next year during the spring semester. At that point perhaps I’ll get finally get around to the project of rewriting and expanding the quantum book.

Another eclipse?

LMMI,

Unfortunately the only total solar eclipse in the next couple years is December 4 in Antarctica, which is not so easy to get to (could be done, but the cost is somewhat prohibitive). I’m hoping to make it to Europe late summer or fall, Covid permitting…

Thank you Prof. Woit for the book and the online course.

I am a self learner. I am still in Chapter 6. I am taking my time (doing it carefully and making notes in LaTeX) and enjoying the study. This may be the only good thing that has come out of COVID.

In the intro of the book it is stated

“Some of the main differences with standard physics presentations include:

• The role of Lie groups, Lie algebras, and their unitary representations is

systematically emphasized, including not just the standard use of these to

derive consequences for the theory of a “symmetry” generated by operators

commuting with the Hamiltonian.”

If the group is not that generated by an operator commuting with the Hamiltonian, then what does the group refer to? This is the conceptual problem of gauge field theory: we just posit a local non-abelian symmetry and plop there is the gauge field. But why demand this local non-Abelian symmetry?

From p. 34:

> An argument from beauty can be made …

You might get some backreaction from a comment like that

Martin van Staveren,

The reason for local symmetry (Abelian or non-Abelian) is that, while the theory of a free field is not invariant, the theory of a field coupled to a gauge field is, so this gives you an interacting theory. In the usual way of thinking about this, there’s not much representation theory going on in the case of gauge symmetry: you’re looking for states that are trivial representations of the gauge symmetry.

For the sort of thing I had in mind in what you quote, two examples are:

1. Lorentz boosts: there are a unitary symmetry of the state space in a relativistic theory, they don’t commute with the Hamiltonian.

2. For a system of oscillators, some quadratic operators in the P,Qs or in the a,a^dagger s that don’t commute with the Hamiltonian (e.g. different number of annihilation and creation operators) can be understood as the Lie algebra representation operators for a larger (symplectic) group than the usual unitary group of dimension n acting on n oscillators. This action of the larger symplectic group has physical significance, you can use it for instance to understand squeezed states in quantum optics (this is explained in detail in the book).

Martin,

I’m hoping so, that was meant to provocatively challenge a common point of view (one with a very eloquent expositor…).

“The reason for local symmetry … is that, while the theory of a free field is not invariant, the theory of a field coupled to a gauge field is, so this gives you an interacting theory”.

Yes, that what I meant, The need for a local symmetry follows from the need for an interacting theory. Usually the logic runs in the opposite direction.

Peter, I found your book outstandingly clear, benefiting from you teaching the subject and getting feedback from your students no doubt. But I feel the exercises should be more carefully integrated into the relevant sections rather than left at the back.

anon,

I agree completely. Probably the weakest aspect of the book pedagogically is that it needs more exercises, ideally I’d think at least twice as many, integrated with the chapters. This is among the things I’d hope to work on this past year while teaching the course, but never got to. Maybe next year….

I’m guessing this ranks pretty low on your list of priorities, but worth a try: Any chance of an expository piece, either here or elsewhere, meant to convey the main concepts of “Twistor Unification” to non-experts?

For instance, it appears you are identifying what I might have understood to be a “mathematical trick”, i.e. analytic continuation (a.k.a. Wick rotation in this instance, I think), with an actual physical process, i.e. electroweak symmetry breaking. I wonder to my naive self if this implies something about the nature of spacetime before and during the corresponding cosmological epoch. What does it imply in general to take Euclidean spacetime “as fundamental”? That Minkowski spacetime is not? And so forth. I probably can’t even formulate the right questions, which makes this attempt to understand all the more embarrassing. But I am interested, nonetheless, it what it all “means”.

LMMI,

I had been thinking that there are aspects of this that I should try and write about in a way that would be as widely accessible as possible, maybe as a blog post. An example would be the the basic picture of twistor theory and how it relates Euclidean and Minkowski space. It took me a while to get the right picture in mind, and what I wrote in the paper doesn’t easily convey that. So, may be will try that sometime soon.

On the analytic continuation business, there I did try and write things out in an expository manner in the paper. There’s a lot to say about this, but one simple fact is that if you say you are only going to think about Minkowski space-time and functions on Minkowski space-time, avoiding analytic continuation, that’s not actually an option. As you find in every textbook and as I work out in the paper, if you try and write down the simplest theory of free particles you end up with ill-defined formulas (this is true even for the propagator of the non-relativistic free particle). You have to do something to make sense of these formulas, the textbook tells you what to do as an “i\epsilon prescription”. When you do that, you’re defining your propagator as the boundary value (as \epsilon goes to zero from the positive direction) of an analytic continuation.

If you look at almost all the non-perturbative work on QFT, it is actually done in Euclidean space-time, with the idea that at the end of the calculation you analytically continue to Minkowski. The hard to believe claim I am making is that the way space-time symmetries behave under analytic continuation is much trickier than you might think (the behavior is straightforward for correlation functions, but for states you need to take into account the fact that you must break Euclidean invariance to define states and to know how to analytically continue to Minkowski).

I’ve been trying to think more about what this point of view implies for quantum gravity, and what strikes me is that how to even formulate the problem there is quite unclear. Beyond a recent paper of Kontsevich and Segal where they try and analytically continue in a space of metrics, I know of very little that seems to have been done about this (would love to hear from those who know more).

Peter,

”what this point of view implies for quantum gravity, and what strikes me is that how to even formulate the problem there is quite unclear.”

Canonical quantum field theory is formulated in curved spacetime (where analytic continuation makes sense only in special cases) by defining the propagators using the theory of linear hyperbolic PDEs rather than by an $i\epsilon$ prescription. See, e.g.,

Hollands, Stefan, and Robert M. Wald. “Quantum fields in curved spacetime.” Physics Reports 574 (2015): 1-35.

This works very well perturbatively. It also covers canonical gravity if one accepts that for higher and higher accuracy you need to use more and more coupling constants.

Arnold Neumaier,

Except, see page 68 of that paper

https://arxiv.org/abs/1401.2026

I don’t think that using microlocal analysis really gets you away from having to consider analytic continuation.

In general, I just don’t understand the point of view that one should avoid having analytic continuation a part of your fundamental theory. One gains nothing and loses a lot.

Peter,

Microlocal analysis allows and heavily uses analytic continuation on tangent and cotangent spaces (which are flat vector spaces), but not on the manifold!

‘page 68 of that paper’ is a distribution in flat space, hence has no bearing on curved space modeling.

‘I just don’t understand the point of view that one should avoid having analytic continuation a part of your fundamental theory. One gains nothing and loses a lot.’

Analytic continuation of curved space-time without symmetries is simply ill-defined. p.7/8 of the above paper by Hollands and Wald states:

”However, a general curved spacetime will not be a real section of a complex manifold that also contains a real section on which the metric is Riemannian. Thus, although it should be possible to define “Euclidean quantum field theory” on curved Riemannian spaces [65], there is no obvious way to connect such a theory with quantum field theory on Lorentzian spacetimes. Thus, if one’s goal is to define quantum field theory on general Lorentzian spacetimes, it does not appear fruitful to attempt to formulate the theory via a Euclidean approach.”

They don’t give a reference for their no-go statement, but I think this is a theorem of differential geometry. Thus one loses nothing but gains logical correctness.

Peter Woit: “In general, I just don’t understand the point of view that one should avoid having analytic continuation a part of your fundamental theory. One gains nothing and loses a lot.”

It’s just a sad fact of life that analytic continuation is a luxury that not everyone can afford (borrowing a social justice metaphor). For a non-flat Lorentzian manifold, the existence of a Wick rotation via analytic continuation is a highly restrictive condition (one of course has to first make a reasonable definition for what Wick rotation might even mean in general). Questions about it occasionally come up on MathOverflow. Here’s an example where there was a somewhat detailed discussion of the differential geometric obstructions.

Presumably, a satisfactory theory that would include both standard model and gravitational physics, would eventually have no problem describing particle creation/scattering in the vicinity of two merging black holes emitting gravitational waves into an expanding cosmological background. Such a theory does not yet exist, but if one were to also require the ability to Wick rotate as a fundamental property of such a theory, it immediately becomes highly suspect whether these two requirements are actually compatible.

Peter: are you submitting this manuscript for publication?

shantanu,

Will likely do so, need to look more into what an appropriate journal would be (e.g. non-problematic business model and copyright policy, able to referee an unusual manuscript).

Arnold Neumaier,

I haven’t followed through all the details in Hollands-Wald, but it looks to me like they still need to define distributions as boundary values of something holomorphic, so are assuming some analyticity of the manifold (so locally analytically continuing it) and the complex analytic story is not just on the tangent space.

Arnold Neumaier/Igor Khavkine,

I’m well aware of the usual problems with

1. Analytically continuing an arbitrary classical Lorentzian solution to a Euclidean one.

2. Making sense of Euclidean space-time gravitational path integrals.

The relation of Euclidean and Lorentz in quantized gravity theory has always seemed highly unclear to me (if anyone knows a good discussion, would like to hear about it). Perhaps the twistor picture gives a different perspective, as might the proposal here for unification with electroweak degrees of freedom.

I’m well aware of the usual problems with1. Analytically continuing an arbitrary classical Lorentzian solution to a Euclidean one.

2. Making sense of Euclidean space-time gravitational path integrals.

The relation of Euclidean and Lorentz in quantized gravity theory has always seemed highly unclear to me (if anyone knows a good discussion, would like to hear about it).—

I am pasting a response which might be relevant for this, from the comment section of an earlier blogpost of yours:

(Imaginary) Time Asymmetry(Aug 6, 2020)—————————————————————————————————————————-

The key idea behind this (first noted, to the best of my knowledge, by Hawking & Ellis (HE)) is based on the following mathematical result: given a Lorentzian metric g_L and a nowhere vanishing timelike direction field U, one can always construct a Euclidean metric g_E. As is well known, non-compact manifolds always admit such a vector field, as well as compact manifolds with Euler number zero.

Some references where this HE observation was discussed in the context of QFT are Candelas & Raine, PRD15, 1494 (1977) and Visser 1702.05572, while recent generalization and consequences for GR, euclidean QG, and euclidean action can be found in Kothawala, arXiv:1705.02504, arXiv:1802.07055. The latter references also discuss transition from euclidean to lorentzian, instead of just the euclidean phase.

Hope this helps.

Peter Woit wrote: “I haven’t followed through all the details in Hollands-Wald, but it looks to me like they still need to define distributions as boundary values of something holomorphic, so are assuming some analyticity of the manifold (so locally analytically continuing it) and the complex analytic story is not just on the tangent space.”

Just to clear out some potential miscommunications, here’s a bit of an explanation. The ideas about “wave front sets” and microlocal analysis referred to in the paper by Hollands & Wald do not require any kind of analyticlity of the spacetime. The basic idea of microlocal analysis is to look at the singularities of a distribution $D(x)$ by taking coordinate Fourier transforms $\hat{D}_\chi(p)$ of products $\chi(x)D(x)$, where $\chi(x)$ is a bump function with support small enough to fit into a single coordinate chart. As the Fourier transform of a compactly supported distribution, $\hat{D}_\chi(p)$ is of course analytic in the momentum variable $p$, but that is beside the point, as none of its $p$-local properties are invariant under changing the bump function $\chi(x)$ or changing the $x$-coordinate chart. What is invariant is the asymptotic behavior of $\hat{D}_\chi(p)$ for large $p$ (“high frequencies”). This information is collected geometrically into a subset of the cotangent bundle of the spacetime, the “wave front set”. The key “Hadamard property” identifying physically reasonable $n$-point functions in QFT on curved spacetimes, as referred to by Hollands & Wald, is formulated purely in terms of this wave front set. The Hadamard property and its relatives are considered to be appropriate generalizations of all the $\pm i\epsilon$ prescriptions floating around in flat space QFT.

When the spacetime is analytic, one can indeed analytically continue the spacetime coordinates into complex directions and define distributions as boundary values of holomorphic functions. In that case, there absolutely is a relationship between holomorphicity properties (the size of the domain of the holomorphic function whose limit is being taken) and the wave front set (of the distributional boundary value). This relationship is captured by the “analytic wave front set”.

A quite detailed exposition of the above ideas can be found in the lecture notes arXiv:1901.10175 (or a bit shorter and down-to-earth one in arXiv:1412.5945, risking a bit of self-promotion 😅).

DKepler/Igor Khavkine,

Many thanks for the clarifications and the references, those are very helpful.