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I’m encountering a sporadic bug over the past few months with the way WordPress renders or displays its LaTeX images on this blog (and occasionally on other WordPress blogs).  On most computers, it seems to work fine, but on some computers, the sizes of images are occasionally way off, leading to extremely distorted and fairly unreadable versions of the images appearing in blog posts and comments.  A sample screenshot (with accompanying HTML source), supplied to me by a reader, can be found here (in which an image whose dimensions should be 321 x 59 are instead being displayed as 552 x 20).  Is anyone else encountering this issue?  The problem sometimes can be resolved by refreshing the page, but not always, so it is a bit unclear where the problem is coming from and how one might mitigate it.  (If nothing else, I can add it to the bug collection post, once it can be reliably replicated.)

It’s time to (somewhat belatedly) roll over the previous thread on writing the first paper from the Polymath8 project, as this thread is overflowing with comments.  We are getting near the end of writing this large (173 pages!) paper, establishing a bound of 4,680 on the gap between primes, with only a few sections left to thoroughly proofread (and the last section should probably be removed, with appropriate changes elsewhere, in view of the more recent progress by Maynard).  As before, one can access the working copy of the paper at this subdirectory, as well as the rest of the directory, and the plan is to submit the paper to Algebra and Number theory (and the arXiv) once there is consensus to do so.  Even before this paper was submitted, it already has had some impact; Andrew Granville’s exposition of the bounded gaps between primes story for the Bulletin of the AMS follows several of the Polymath8 arguments in deriving the result.

After this paper is done, there is interest in continuing onwards with other Polymath8 – related topics, and perhaps it is time to start planning for them.  First of all, we have an invitation from  the Newsletter of the European Mathematical Society to discuss our experiences and impressions with the project.  I think it would be interesting to collect some impressions or thoughts (both positive and negative)  from people who were highly active in the research and/or writing aspects of the project, as well as from more casual participants who were following the progress more quietly.  This project seemed to attract a bit more attention than most other polymath projects (with the possible exception of the very first project, Polymath1).  I think there are several reasons for this; the project builds upon a recent breakthrough (Zhang’s paper) that attracted an impressive amount of attention and publicity; the objective is quite easy to describe, when compared against other mathematical research objectives; and one could summarise the current state of progress by a single natural number H, which implied by infinite descent that the project was guaranteed to terminate at some point, but also made it possible to set up a “scoreboard” that could be quickly and easily updated.  From the research side, another appealing feature of the project was that – in the early stages of the project, at least – it was quite easy to grab a new world record by means of making a small observation, which made it fit very well with the polymath spirit (in which the emphasis is on lots of small contributions by many people, rather than a few big contributions by a small number of people).  Indeed, when the project first arose spontaneously as a blog post of Scott Morrrison over at the Secret Blogging Seminar, I was initially hesitant to get involved, but soon found the “game” of shaving a few thousands or so off of H to be rather fun and addictive, and with a much greater sense of instant gratification than traditional research projects, which often take months before a satisfactory conclusion is reached.  Anyway, I would welcome other thoughts or impressions on the projects in the comments below (I think that the pace of comments regarding proofreading of the paper has slowed down enough that this post can accommodate both types of comments comfortably.)

Then of course there is the “Polymath 8b” project in which we build upon the recent breakthroughs of James Maynard, which have simplified the route to bounded gaps between primes considerably, bypassing the need for any Elliott-Halberstam type distribution results beyond the Bombieri-Vinogradov theorem.  James has kindly shown me an advance copy of the preprint, which should be available on the arXiv in a matter of days; it looks like he has made a modest improvement to the previously announced results, improving k_0 a bit to 105 (which then improves H to the nice round number of 600).  He also has a companion result on bounding gaps p_{n+m}-p_n between non-consecutive primes for any m (not just m=1), with a bound of the shape H_m := \lim \inf_{n \to \infty} p_{n+m}-p_n \ll m^3 e^{4m}, which is in fact the first time that the finiteness of this limit inferior has been demonstrated.  I plan to discuss these results (from a slightly different perspective than Maynard) in a subsequent blog post kicking off the Polymath8b project, once Maynard’s paper has been uploaded.  It should be possible to shave the value of H = H_1 down further (or to get better bounds for H_m for larger m), both unconditionally and under assumptions such as the Elliott-Halberstam conjecture, either by performing more numerical or theoretical optimisation on the variational problem Maynard is faced with, and also by using the improved distributional estimates provided by our existing paper; again, I plan to discuss these issues in a subsequent post. ( James, by the way, has expressed interest in participating in this project, which should be very helpful.)

Once again it is time to roll over the previous discussion thread, which has become rather full with comments.  The paper is nearly finished (see also the working copy at this subdirectory, as well as the rest of the directory), but several people are carefully proofreading various sections of the paper.  Once all the people doing so have signed off on it, I think we will be ready to submit (there appears to be no objection to the plan to submit to Algebra and Number Theory).

Another thing to discuss is an invitation to Polymath8 to write a feature article (up to 8000 words or 15 pages) for the Newsletter of the European Mathematical Society on our experiences with this project.  It is perhaps premature to actually start writing this article before the main research paper is finalised, but we can at least plan how to write such an article.  One suggestion, proposed by Emmanuel, is to have individual participants each contribute a brief account of their interaction with the project, which we would compile together with some additional text summarising the project as a whole (and maybe some speculation for any lessons we can apply here for future polymath projects).   Certainly I plan to have a separate blog post collecting feedback on this project once the main writing is done.

The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments.  We are still writing the paper, but it appears to have stabilised in a near-final form (source files available here); the main remaining tasks are proofreading, checking the mathematics, and polishing the exposition.  We also have a tentative consensus to submit the paper to Algebra and Number Theory when the proofreading is all complete.

The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not  need any of the advanced exponential sum estimates).  The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.

The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments.    As with the previous thread, the main focus on the comments to this thread are concerned with writing up the results of the Polymath8 “bounded gaps between primes” project; the latest files on this writeup may be found at this directory, with the most recently compiled PDF file (clocking in at about 90 pages so far, with a few sections still to be written!) being found here.  There is also still some active discussion on improving the numerical results, with a particular focus on improving the sieving step that converts distribution estimates such as MPZ^{(i)}[\varpi,\delta] into weak prime tuples results DHL[k_0,2].  (For a discussion of the terminology, and for a general overview of the proof strategy, see this previous progress report on the Polymath8 project.)  This post can also contain any other discussion pertinent to any aspect of the polymath8 project, of course.

There are a few sections that still need to be written for the draft, mostly concerned with the Type I, Type II, and Type III estimates.  However, the proofs of these estimates exist already on this blog, so I hope to transcribe them to the paper fairly shortly (say by the end of this week).  Barring any unexpected surprises, or major reorganisation of the paper, it seems that the main remaining task in the writing process would be the proofreading and polishing, and turning from the technical mathematical details to expository issues.  As always, feedback from casual participants, as well as those who have been closely involved with the project, would be very valuable in this regard.  (One small comment, by the way, regarding corrections: as the draft keeps changing with time, referring to a specific line of the paper using page numbers and line numbers can become inaccurate, so if one could try to use section numbers, theorem numbers, or equation numbers as reference instead (e.g. “the third line after (5.35)” instead of “the twelfth line of page 54″) that would make it easier to track down specific portions of the paper.)

Also, we have set up a wiki page for listing the participants of the polymath8 project, their contact information, and grant information (if applicable).  We have two lists of participants; one for those who have been making significant contributions to the project (comparable to that of a co-author of a traditional mathematical research paper), and another list for those who have made auxiliary contributions (e.g. typos, stylistic suggestions, or supplying references) that would typically merit inclusion in the Acknowledgments section of a traditional paper.  It’s difficult to exactly draw the line between the two types of contributions, but we have relied in the past on self-reporting, which has worked pretty well so far.  (By the time this project concludes, I may go through the comments to previous posts and see if any further names should be added to these lists that have not already been self-reported.)


The main objectives of the polymath8 project, initiated back in June, were to understand the recent breakthrough paper of Zhang establishing an infinite number of prime gaps bounded by a fixed constant {H}, and then to lower that value of {H} as much as possible. After a large number of refinements, optimisations, and other modifications to Zhang’s method, we have now lowered the value of {H} from the initial value of {70,000,000} down to (provisionally) {4,680}, as well as to the slightly worse value of {14,994} if one wishes to avoid any reliance on the deep theorems of Deligne on the Weil conjectures.

As has often been the case with other polymath projects, the pace has settled down subtantially after the initial frenzy of activity; in particular, the values of {H} (and other key parameters, such as {k_0}, {\varpi}, and {\delta}) have stabilised over the last few weeks. While there may still be a few small improvements in these parameters that can be wrung out of our methods, I think it is safe to say that we have cleared out most of the “low-hanging fruit” (and even some of the “medium-hanging fruit”), which means that it is time to transition to the next phase of the polymath project, namely the writing phase.

After some discussion at the previous post, we have tentatively decided on writing a single research paper, which contains (in a reasonably self-contained fashion) the details of the strongest result we have (i.e. bounded gaps with {H = 4,680}), together with some variants, such as the bound {H=14,994} that one can obtain without invoking Deligne’s theorems. We can of course also include some discussion as to where further improvements could conceivably arise from these methods, although even if one assumes the most optimistic estimates regarding distribution of the primes, we still do not have any way to get past the barrier of {H=16} identified as the limit of this method by Goldston, Pintz, and Yildirim. This research paper does not necessarily represent the only output of the polymath8 project; for instance, as part of the polymath8 project the admissible tuples page was created, which is a repository of narrow prime tuples which can automatically accept (and verify) new submissions. (At an early stage of the project, it was suggested that we set up a computing challenge for mathematically inclined programmers to try to find the narrowest prime tuples of a given width; it might be worth revisiting this idea now that our value of {k_0} has stabilised and the prime tuples page is up and running.) Other potential outputs include additional expository articles, lecture notes, or perhaps the details of a “minimal proof” of bounded gaps between primes that gives a lousy value of {H} but with as short and conceptual a proof as possible. But it seems to me that these projects do not need to proceed via the traditional research paper route (perhaps ending up on the blog, on the wiki, or on the admissible tuples page instead). Also, these projects might also benefit from the passage of time to lend a bit of perspective and depth, especially given that there are likely to be further advances in this field from outside of the polymath project.

I have taken the liberty of setting up a Dropbox folder containing a skeletal outline of a possible research paper, and anyone who is interested in making significant contributions to the writeup of the paper can contact me to be given write access to that folder. However, I am not firmly wedded to the organisational structure of that paper, and at this stage it is quite easy to move sections around if this would lead to a more readable or more logically organised paper.

I have tried to structure the paper so that the deepest arguments – the ones which rely on Deligne’s theorems – are placed at the end of the paper, so that a reader who wishes to read and understand a proof of bounded gaps that does not rely on Deligne’s theorems can stop reading about halfway through the paper. I have also moved the top-level structure of the argument (deducing bounded gaps from a Dickson-Hardy-Littlewood claim {DHL[k_0,2]}, which in turn is established from a Motohashi-Pintz-Zhang distribution estimate {MPZ^{(i)}[\varpi,\delta]}, which is in turn deduced from Type I, Type II, and Type III estimates) to the front of the paper.

Of course, any feedback on the draft paper is encouraged, even from (or especially from!) readers who have been following this project on a casual basis, as this would be valuable in making sure that the paper is written in as accessible as fashion as possible. (Sometimes it is possible to be so close to a project that one loses some sense of perspective, and does not realise that what one is writing might not necessarily be as clear to other mathematicians as it is to the author.)

[This guest post is authored by Ingrid Daubechies, who is the current president of the International Mathematical Union, and (as she describes below) is heavily involved in planning for a next-generation digital mathematical library that can go beyond the current network of preprint servers (such as the arXiv), journal web pages, article databases (such as MathSciNet), individual author web pages, and general web search engines to create a more integrated and useful mathematical resource. I have lightly edited the post for this blog, mostly by adding additional hyperlinks. - T.]

This guest blog entry concerns the many roles a World Digital Mathematical Library (WDML) could play for the mathematical community worldwide. We seek input to help sketch how a WDML could be so much more than just a huge collection of digitally available mathematical documents. If this is of interest to you, please read on!

The “we” seeking input are the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU), and a special committee of the US National Research Council (NRC), charged by the Sloan Foundation to look into this matter. In the US, mathematicians may know the Sloan Foundation best for the prestigious early-career fellowships it awards annually, but the foundation plays a prominent role in other disciplines as well. For instance, the Sloan Digital Sky Survey (SDSS) has had a profound impact on astronomy, serving researchers in many more ways than even its ambitious original setup foresaw. The report being commissioned by the Sloan Foundation from the NRC study group could possibly be the basis for an equally ambitious program funded by the Sloan Foundation for a WDML with the potential to change the practice of mathematical research as profoundly as the SDSS did in astronomy. But to get there, we must formulate a vision that, like the original SDSS proposal, imagines at least some of those impacts. The members of the NRC committee are extremely knowledgeable, and have been picked judiciously so as to span collectively a wide range of expertise and connections. As president of the IMU, I was asked to co-chair this committee, together with Clifford Lynch, of the Coalition for Networked InformationPeter Olver, chair of the IMU’s CEIC, is also a member of the committee. But each of us is at least a quarter century older than the originators of MathOverflow or the ArXiv when they started. We need you, internet-savvy, imaginative, social-networking, young mathematicians to help us formulate the vision that may inspire the creation of a truly revolutionary WDML!

Some history first.  Several years ago, an international initiative was started to create a World Digital Mathematical Library. The website for this library, hosted by the IMU, is now mostly a “ghost” website — nothing has been posted there for the last seven years. [It does provide useful links, however, to many sites that continue to be updated, such as the European Mathematical Information Service, which in turn links to many interesting journals, books and other websites featuring electronically available mathematical publications. So it is still worth exploring ...] Many of the efforts towards building (parts of) the WDML as originally envisaged have had to grapple with business interests, copyright agreements, search obstructions, metadata secrecy, … and many an enterprising, idealistic effort has been slowly ground down by this. We are still dealing with these frustrations — as witnessed by, e.g., the CostofKnowledge initiative. They are real, important issues, and will need to be addressed.

The charge of the NRC committee, however, is to NOT focus on issues of copyright or open-access or who bears the cost of publishing, but instead on what could/can be done with documents that are (or once they are) freely electronically accessible, apart from simply finding and downloading them. Earlier this year, I posted a question about one possible use on MathOverflow and then on MathForge, about the possibility to “enrich” a paper by annotations from readers, which other readers could wish to consult (or not). These posts elicited some very useful comments. But this was but one way in which a WDML could be more than just an opportunity to find and download papers. Surely there are many more, that you, bloggers and blog-readers, can imagine, suggest, sketch. This is an opportunity: can we — no, YOU! — formulate an ambitious setup that would capture the imagination of sufficiently many of us, that would be workable and that would really make a difference?

Things are pretty quiet here during the holiday season, but one small thing I have been working on recently is a set of notes on special relativity that I will be working through in a few weeks with some bright high school students here at our local math circle.  I have only two hours to spend with this group, and it is unlikely that we will reach the end of the notes (in which I derive the famous mass-energy equivalence relation E=mc^2, largely following Einstein’s original derivation as discussed in this previous blog post); instead we will probably spend a fair chunk of time on related topics which do not actually require special relativity per se, such as spacetime diagrams, the Doppler shift effect, and an analysis of my airport puzzle.  This will be my first time doing something of this sort (in which I will be spending as much time interacting directly with the students as I would lecturing);  I’m not sure exactly how it will play out, being a little outside of my usual comfort zone of undergraduate and graduate teaching, but am looking forward to finding out how it goes.   (In particular, it may end up that the discussion deviates somewhat from my prepared notes.)

The material covered in my notes is certainly not new, but I ultimately decided that it was worth putting up here in case some readers here had any corrections or other feedback to contribute (which, as always, would be greatly appreciated).

[Dec 24 and then Jan 21: notes updated, in response to comments.]

Lars Hörmander, who made fundamental contributions to all areas of partial differential equations, but particularly in developing the analysis of variable-coefficient linear PDE, died last Sunday, aged 81.

I unfortunately never met Hörmander personally, but of course I encountered his work all the time while working in PDE. One of his major contributions to the subject was to systematically develop the calculus of Fourier integral operators (FIOs), which are a substantial generalisation of pseudodifferential operators and which can be used to (approximately) solve linear partial differential equations, or to transform such equations into a more convenient form. Roughly speaking, Fourier integral operators are to linear PDE as canonical transformations are to Hamiltonian mechanics (and one can in fact view FIOs as a quantisation of a canonical transformation). They are a large class of transformations, for instance the Fourier transform, pseudodifferential operators, and smooth changes of the spatial variable are all examples of FIOs, and (as long as certain singular situations are avoided) the composition of two FIOs is again an FIO.

The full theory of FIOs is quite extensive, occupying the entire final volume of Hormander’s famous four-volume series “The Analysis of Linear Partial Differential Operators”. I am certainly not going to try to attempt to summarise it here, but I thought I would try to motivate how these operators arise when trying to transform functions. For simplicity we will work with functions {f \in L^2({\bf R}^n)} on a Euclidean domain {{\bf R}^n} (although FIOs can certainly be defined on more general smooth manifolds, and there is an extension of the theory that also works on manifolds with boundary). As this will be a heuristic discussion, we will ignore all the (technical, but important) issues of smoothness or convergence with regards to the functions, integrals and limits that appear below, and be rather vague with terms such as “decaying” or “concentrated”.

A function {f \in L^2({\bf R}^n)} can be viewed from many different perspectives (reflecting the variety of bases, or approximate bases, that the Hilbert space {L^2({\bf R}^n)} offers). Most directly, we have the physical space perspective, viewing {f} as a function {x \mapsto f(x)} of the physical variable {x \in {\bf R}^n}. In many cases, this function will be concentrated in some subregion {\Omega} of physical space. For instance, a gaussian wave packet

\displaystyle  f(x) = A e^{-(x-x_0)^2/\hbar} e^{i \xi_0 \cdot x/\hbar}, \ \ \ \ \ (1)

where {\hbar > 0}, {A \in {\bf C}} and {x_0, \xi_0 \in {\bf R}^n} are parameters, would be physically concentrated in the ball {B(x_0,\sqrt{\hbar})}. Then we have the frequency space (or momentum space) perspective, viewing {f} now as a function {\xi \mapsto \hat f(\xi)} of the frequency variable {\xi \in {\bf R}^n}. For this discussion, it will be convenient to normalise the Fourier transform using a small constant {\hbar > 0} (which has the physical interpretation of Planck’s constant if one is doing quantum mechanics), thus

\displaystyle  \hat f(\xi) := \frac{1}{(2\pi \hbar)^{n/2}} \int_{\bf R} e^{-i\xi \cdot x/\hbar} f(x)\ dx.

For instance, for the gaussian wave packet (1), one has

\displaystyle  \hat f(\xi) = A e^{i\xi_0 \cdot x_0/\hbar} e^{-(\xi-\xi_0)^2/\hbar} e^{-i \xi \cdot x_0/\hbar},

and so we see that {f} is concentrated in frequency space in the ball {B(\xi_0,\sqrt{\hbar})}.

However, there is a third (but less rigorous) way to view a function {f} in {L^2({\bf R}^n)}, which is the phase space perspective in which one tries to view {f} as distributed simultaneously in physical space and in frequency space, thus being something like a measure on the phase space {T^* {\bf R}^n := \{ (x,\xi): x, \xi \in {\bf R}^n\}}. Thus, for instance, the function (1) should heuristically be concentrated on the region {B(x_0,\sqrt{\hbar}) \times B(\xi_0,\sqrt{\hbar})} in phase space. Unfortunately, due to the uncertainty principle, there is no completely satisfactory way to canonically and rigorously define what the “phase space portrait” of a function {f} should be. (For instance, the Wigner transform of {f} can be viewed as an attempt to describe the distribution of the {L^2} energy of {f} in phase space, except that this transform can take negative or even complex values; see Folland’s book for further discussion.) Still, it is a very useful heuristic to think of functions has having a phase space portrait, which is something like a non-negative measure on phase space that captures the distribution of functions in both space and frequency, albeit with some “quantum fuzziness” that shows up whenever one tries to inspect this measure at scales of physical space and frequency space that together violate the uncertainty principle. (The score of a piece of music is a good everyday example of a phase space portrait of a function, in this case a sound wave; here, the physical space is the time axis (the horizontal dimension of the score) and the frequency space is the vertical dimension. Here, the time and frequency scales involved are well above the uncertainty principle limit (a typical note lasts many hundreds of cycles, whereas the uncertainty principle kicks in at {O(1)} cycles) and so there is no obstruction here to musical notation being unambiguous.) Furthermore, if one takes certain asymptotic limits, one can recover a precise notion of a phase space portrait; for instance if one takes the semiclassical limit {\hbar \rightarrow 0} then, under certain circumstances, the phase space portrait converges to a well-defined classical probability measure on phase space; closely related to this is the high frequency limit of a fixed function, which among other things defines the wave front set of that function, which can be viewed as another asymptotic realisation of the phase space portrait concept.

If functions in {L^2({\bf R}^n)} can be viewed as a sort of distribution in phase space, then linear operators {T: L^2({\bf R}^n) \rightarrow L^2({\bf R}^n)} should be viewed as various transformations on such distributions on phase space. For instance, a pseudodifferential operator {a(X,D)} should correspond (as a zeroth approximation) to multiplying a phase space distribution by the symbol {a(x,\xi)} of that operator, as discussed in this previous blog post. Note that such operators only change the amplitude of the phase space distribution, but not the support of that distribution.

Now we turn to operators that alter the support of a phase space distribution, rather than the amplitude; we will focus on unitary operators to emphasise the amplitude preservation aspect. These will eventually be key examples of Fourier integral operators. A physical translation {Tf(x) := f(x-x_0)} should correspond to pushing forward the distribution by the transformation {(x,\xi) \mapsto (x+x_0,\xi)}, as can be seen by comparing the physical and frequency space supports of {Tf} with that of {f}. Similarly, a frequency modulation {Tf(x) := e^{i \xi_0 \cdot x/\hbar} f(x)} should correspond to the transformation {(x,\xi) \mapsto (x,\xi+\xi_0)}; a linear change of variables {Tf(x) := |\hbox{det} L|^{-1/2} f(L^{-1} x)}, where {L: {\bf R}^n \rightarrow {\bf R}^n} is an invertible linear transformation, should correspond to {(x,\xi) \mapsto (Lx, (L^*)^{-1} \xi)}; and finally, the Fourier transform {Tf(x) := \hat f(x)} should correspond to the transformation {(x,\xi) \mapsto (\xi,-x)}.

Based on these examples, one may hope that given any diffeomorphism {\Phi: T^* {\bf R}^n \rightarrow T^* {\bf R}^n} of phase space, one could associate some sort of unitary (or approximately unitary) operator {T_\Phi: L^2({\bf R}^n) \rightarrow L^2({\bf R}^n)}, which (heuristically, at least) pushes the phase space portrait of a function forward by {\Phi}. However, there is an obstruction to doing so, which can be explained as follows. If {T_\Phi} pushes phase space portraits by {\Phi}, and pseudodifferential operators {a(X,D)} multiply phase space portraits by {a}, then this suggests the intertwining relationship

\displaystyle  a(X,D) T_\Phi \approx T_\Phi (a \circ \Phi)(X,D),

and thus {(a \circ \Phi)(X,D)} is approximately conjugate to {a(X,D)}:

\displaystyle  (a \circ \Phi)(X,D) \approx T_\Phi^{-1} a(X,D) T_\Phi. \ \ \ \ \ (2)

The formalisation of this fact in the theory of Fourier integral operators is known as Egorov’s theorem, due to Yu Egorov (and not to be confused with the more widely known theorem of Dmitri Egorov in measure theory).

Applying commutators, we conclude the approximate conjugacy relationship

\displaystyle  \frac{1}{i\hbar} [(a \circ \Phi)(X,D), (b \circ \Phi)(X,D)] \approx T_\Phi^{-1} \frac{1}{i\hbar} [a(X,D), b(X,D)] T_\Phi.

Now, the pseudodifferential calculus (as discussed in this previous post) tells us (heuristically, at least) that

\displaystyle  \frac{1}{i\hbar} [a(X,D), b(X,D)] \approx \{ a, b \}(X,D)


\displaystyle  \frac{1}{i\hbar} [(a \circ \Phi)(X,D), (b \circ \Phi)(X,D)] \approx \{ a \circ \Phi, b \circ \Phi \}(X,D)

where {\{,\}} is the Poisson bracket. Comparing this with (2), we are then led to the compatibility condition

\displaystyle  \{ a \circ \Phi, b \circ \Phi \} \approx \{ a, b \} \circ \Phi,

thus {\Phi} needs to preserve (approximately, at least) the Poisson bracket, or equivalently {\Phi} needs to be a symplectomorphism (again, approximately at least).

Now suppose that {\Phi: T^* {\bf R}^n \rightarrow T^* {\bf R}^n} is a symplectomorphism. This is morally equivalent to the graph {\Sigma := \{ (z, \Phi(z)): z \in T^* {\bf R}^n \}} being a Lagrangian submanifold of {T^* {\bf R}^n \times T^* {\bf R}^n} (where we give the second copy of phase space the negative {-\omega} of the usual symplectic form {\omega}, thus yielding {\omega \oplus -\omega} as the full symplectic form on {T^* {\bf R}^n \times T^* {\bf R}^n}; this is another instantiation of the closed graph theorem, as mentioned in this previous post. This graph is known as the canonical relation for the (putative) FIO that is associated to {\Phi}. To understand what it means for this graph to be Lagrangian, we coordinatise {T^* {\bf R}^n \times T^* {\bf R}^n} as {(x,\xi,y,\eta)} suppose temporarily that this graph was (locally, at least) a smooth graph in the {x} and {y} variables, thus

\displaystyle  \Sigma = \{ (x, F(x,y), y, G(x,y)): x, y \in {\bf R}^n \}

for some smooth functions {F, G: {\bf R}^n \rightarrow {\bf R}^n}. A brief computation shows that the Lagrangian property of {\Sigma} is then equivalent to the compatibility conditions

\displaystyle  \frac{\partial F_i}{\partial x_j} = \frac{\partial F_j}{\partial x_i}

\displaystyle  \frac{\partial G_i}{\partial y_j} = \frac{\partial G_j}{\partial y_i}

\displaystyle  \frac{\partial F_i}{\partial y_j} = - \frac{\partial G_j}{\partial x_i}

for {i,j=1,\ldots,n}, where {F_1,\ldots,F_n, G_1,\ldots,G_n} denote the components of {F,G}. Some Fourier analysis (or Hodge theory) lets us solve these equations as

\displaystyle  F_i = -\frac{\partial \phi}{\partial x_i}; \quad G_j = \frac{\partial \phi}{\partial y_j}

for some smooth potential function {\phi: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}}. Thus, we have parameterised our graph {\Sigma} as

\displaystyle  \Sigma = \{ (x, -\nabla_x \phi(x,y), y, \nabla_y \phi(x,y)): x,y \in {\bf R}^n \} \ \ \ \ \ (3)

so that {\Phi} maps {(x, -\nabla_x \phi(x,y))} to {(y, \nabla_y \phi(x,y))}.

A reasonable candidate for an operator associated to {\Phi} and {\Sigma} in this fashion is the oscillatory integral operator

\displaystyle  Tf(y) := \frac{1}{(2\pi \hbar)^{n/2}} \int_{{\bf R}^n} e^{i \phi(x,y)/\hbar} a(x,y) f(x)\ dx \ \ \ \ \ (4)

for some smooth amplitude function {a} (note that the Fourier transform is the special case when {a=1} and {\phi(x,y)=xy}, which helps explain the genesis of the term “Fourier integral operator”). Indeed, if one computes an inner product {\int_{{\bf R}^n} Tf(y) \overline{g(y)}\ dy} for gaussian wave packets {f, g} of the form (1) and localised in phase space near {(x_0,\xi_0), (y_0,\eta_0)} respectively, then a Taylor expansion of {\phi} around {(x_0,y_0)}, followed by a stationary phase computation, shows (again heuristically, and assuming {\phi} is suitably non-degenerate) that {T} has (3) as its canonical relation. (Furthermore, a refinement of this stationary phase calculation suggests that if {a} is normalised to be the half-density {|\det \nabla_x \nabla_y \phi|^{1/2}}, then {T} should be approximately unitary.) As such, we view (4) as an example of a Fourier integral operator (assuming various smoothness and non-degeneracy hypotheses on the phase {\phi} and amplitude {a} which we do not detail here).

Of course, it may be the case that {\Sigma} is not a graph in the {x,y} coordinates (for instance, the key examples of translation, modulation, and dilation are not of this form), but then it is often a graph in some other pair of coordinates, such as {\xi,y}. In that case one can compose the oscillatory integral construction given above with a Fourier transform, giving another class of FIOs of the form

\displaystyle  Tf(y) := \frac{1}{(2\pi \hbar)^{n/2}} \int_{{\bf R}^n} e^{i \phi(\xi,y)/\hbar} a(\xi,y) \hat f(\xi)\ d\xi. \ \ \ \ \ (5)

This class of FIOs covers many important cases; for instance, the translation, modulation, and dilation operators considered earlier can be written in this form after some Fourier analysis. Another typical example is the half-wave propagator {T := e^{it \sqrt{-\Delta}}} for some time {t \in {\bf R}}, which can be written in the form

\displaystyle  Tf(y) = \frac{1}{(2\pi \hbar)^{n/2}} \int_{{\bf R}^n} e^{i (\xi \cdot y + t |\xi|)/\hbar} a(\xi,y) \hat f(\xi)\ d\xi.

This corresponds to the phase space transformation {(x,\xi) \mapsto (x+t|\xi|, \xi)}, which can be viewed as the classical propagator associated to the “quantum” propagator {e^{it\sqrt{-\Delta}}}. More generally, propagators for linear Hamiltonian partial differential equations can often be expressed (at least approximately) by Fourier integral operators corresponding to the propagator of the associated classical Hamiltonian flow associated to the symbol of the Hamiltonian operator {H}; this leads to an important mathematical formalisation of the correspondence principle between quantum mechanics and classical mechanics, that is one of the foundations of microlocal analysis and which was extensively developed in Hörmander’s work. (More recently, numerically stable versions of this theory have been developed to allow for rapid and accurate numerical solutions to various linear PDE, for instance through Emmanuel Candés’ theory of curvelets, so the theory that Hörmander built now has some quite significant practical applications in areas such as geology.)

In some cases, the canonical relation {\Sigma} may have some singularities (such as fold singularities) which prevent it from being written as graphs in the previous senses, but the theory for defining FIOs even in these cases, and in developing their calculus, is now well established, in large part due to the foundational work of Hörmander.

I recently finished the first draft of the last of my books based on my 2011 blog posts (and also my Google buzzes and Google+ posts from that year), entitled “Spending symmetry“.    The PDF of this draft is available here.  This is again a rather  assorted (and lightly edited) collection of posts (and buzzes, and Google+ posts), though concentrating in the areas of analysis (both standard and nonstandard), logic, and geometry.   As always, comments and corrections are welcome.


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