A New Equation?

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Some years ago I speculated that it would nice if a certain mathematical object existed, and even nicer if it were to satisfy an ordinary differential equation of a special sort. I was motivated by a particular physical question, and it seemed very natural to me to imagine such an object… So natural that I was sure that it must already have been studied, the equation for it known. As a result, every so often I’d go down a rabbit hole of a literature dig, but not with much success because it isn’t entirely clear where best to look. Then I’d get involved with other projects and forget all about the matter.

Last year I began to think about it again because it might be useful in a method I was developing for a paper, went through the cycle of wondering, and looking for a while, then forgot all about it in thinking about other things.

Then, a little over a month ago at the end of March, while starting on a long flight across the continent, I started thinking about it again, and given that I did not have a connection to the internet to hand, took another approach: I got out a pencil and began mess around in my notebook and just derive what I thought the equation for this object should be, given certain properties it should have. One property is that it should in some circumstances reduce to a known powerful equation (often associated with the legendary 1975 work of Gel’fand and Dikii*) satisfied by the diagonal resolvent {\widehat R}(E,x) {=}\langle x|({\cal H}-E)^{-1}|x\rangle of a Schrodinger Hamiltonian {\cal H}=-\hbar^2\partial^2_x+u(x). It is:

4(u(x)-E){\widehat R}^2-2\hbar^2 {\widehat R}{\widehat R}^{\prime\prime}+\hbar^2({\widehat R}^\prime)^2 = 1\ .

Here, E is an energy of the Hamiltonian, in potential u(x), and x is a coordinate on the real line.

The object itself would be a generalisation of the diagonal resolvent {\widehat R}(E,x), although non-diagonal in the energy, not the Click to continue reading this post →

Valuable Instants

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This week’s lectures on instantons in my gauge theory class (a very important kind of theory for understanding many phenomenon in nature – light is an example of a phenomenon that is described by gauge theory) were a lot of fun to do, and mark the culmination of a month-long theme on topological objects and non-perturbative effects. I always enjoy teaching this stuff, including the history!

–cvj

93 minutes

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Thanks to everyone who made all those kind remarks in various places last month after my mother died. I’ve not responded individually (I did not have the strength) but I did read them all and they were deeply appreciated.

Yesterday would’ve been mum‘s 93rd birthday. A little side-note occurred to me the other day: Since she left us a month ago, she was just short of having seen two perfect square years. (This year and 1936.) Anyway, still on the theme of playing with numbers, my siblings and I agreed that as a tribute to her on the day, we would all do some kind of outdoor activity for 93 minutes. Over in London, my brother and sister did a joint (probably chilly) walk together in Regents Park and surrounds.

I decided to take out a piece of the afternoon at low tide and run along the beach. It went pretty well, Click to continue reading this post →

A Long Goodbye

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I’ve been very quiet here over the last couple of weeks. My mother, Delia Maria Johnson, already in hospital since 5th November or so, took a turn for the worse and began a rapid decline. She died peacefully after some days, and to be honest I’ve really not been myself since then.

My mother Delia at a wedding in 2012

There’s an extra element to the sense of loss when (as it approaches) you are powerless to do anything because of being thousands of miles away. On the plus side, because of the ease of using video calls, and with the help of my sister being there, I was able to be somewhat present during what turned out to be the last moments when she was aware of people around her, and therefore was able to tell her I loved her one last time.

Rather than charging across the world on planes, trains, and in automobiles, probably being out of reach during any significant changes in the situation (the doctors said I would likely not make it in time) I did a number of things locally that I am glad I got to do.

It began with visiting (and sending a photo from) the Santa Barbara mission, a place she dearly loved and was unable to visit again after 2019, along with the pier. These are both places we walked together so much back when I first lived here in what feels like another life.

Then, two nights before mum passed away, but well after she’d seemed already beyond reach of anyone, although perhaps (I’d like to think) still able to hear things, my sister contacted me from her bedside asking if I’d like to read mum a psalm, perhaps one of her favourites, 23 or 91. At first I thought she was already planning the funeral, and expressed my surprise at this since mum was still alive and right next to her. But I’d misunderstood, and she’d in fact had a rather great idea. This suggestion turned into several hours of, having sent on recordings of the two psalms, my digging into the poetry shelf in the study and discovering long neglected collections through which I searched (sometimes accompanied by my wife and son) for additional things to read. I recorded some and sent them along, as well as one from my son, I’m delighted to say. Later, the whole thing turned into me singing various songs while playing my guitar and sending recordings of those along too.

Incidentally, the guitar-playing was an interesting turn of events since not many months ago I decided after a long lapse to start playing guitar again, and try to move the standard of my playing (for vocal accompaniment) to a higher level than I’d previously done, by playing and practicing for a little bit on a regular basis. I distinctly recall thinking at one point during one practice that it would be nice to play for mum, although I did not imagine that playing to her while she was on her actual death-bed would be the circumstance under which I’d eventually play for her, having (to my memory) never directly done so back when I used to play guitar in my youth. (Her overhearing me picking out bits of Queen songs behind my room door when I was a teenager doesn’t count as direct playing for her.)

Due to family circumstances I’ll perhaps go into another time, Click to continue reading this post →

Magic Ingredients Exist!

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I’m a baker, as you probably know. I’ve regularly made bread, cakes, pies, and all sorts of things for friends and family. About a year ago, someone in the family was diagnosed with a severe allergy to gluten, and within days we removed all gluten products from the kitchen, began to be very selective about restaurants we ate at, and generally had to rethink a number of aspects of our lives as a family.

This had a big impact on my baking, to put it mildly. With the aid of some excellent gluten-free general purpose flours (mostly the ones made by Bob’s Red Mill) certain kinds of things could be readily made the way I used to make them (more or less – you quickly notice you have to increase moisture content a bit because such flours are more absorptive), such as scones, biscuits (American), and basic pastry, but other things really needed to be seriously re-thought, or abandoned altogether rather than make a terrible facsimile of it (particularly yeast breads, especially light fluffy loaves or buns, light cakes, anything that needs the structure gluten provides to rise and form a crumb, etc…)

For almost a year I just removed a lot of the baking I do from regular rotation, and resigned myself to not making certain kinds of things any more. It was a very painful goodbye (I’ve been baking breads and cakes for many decades), but I was fine with it, given the life-threatening health issues I’d seen gluten cause, up close.

At the same time, I began to be increasingly stunned by the situation concerning gluten-free bread and bread-like products you can find on sale. While some good breads can be found (with persistence), so very much of it is entirely, in the eating, devoid of joy, much of it is sometimes like eating solidified ash. But still they charge you huge amounts of money for it. I’ve seen all kinds of mediocre loaves of bread up or near (sometimes beyond!) the $20 price point, and people buy it without (it seems) batting an eyelid. Why? Because it is hard to find, and (I thought!) hard to make.

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Hope

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The delicious chaos that (almost always) eventually tames into a tasty flaky pastry crust… it’s always a worrying mess to start out, but you trust to your experience, and you carry on, with hope. #thanksgiving

Rolling out gluten-free flaky pastry dough…

Decoding the Universe!

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I realised just now that I entirely forgot (it seems) to post about an episode of PBS’ show Nova called “Decoding the Universe: Cosmos” which aired back in the Spring. I thought they did a good job of talking about some of the advances in our understanding that have happened over the last 50 years (the idea is that it is the 50th anniversary of the show) in areas of astrophysics and cosmology. I was a contributor, filmed at the top of Mount Wilson at the Observatory where Hubble made his famous discoveries about the size of the universe, and its expansion. I talk about some of those discoveries and other ideas in the show. Here’s a link to the “Decoding the Universe” site. (You can also find it on YouTube.)

If you follow the link you’ll notice another episode up there: “Decoding the Universe: Quantum”. That’s a companion they made, and it focuses on understanding in quantum physics, connecting it to things in the everyday world. and also back to black holes and things astrophysical and cosmological. It also does a good job of shining a light on many concepts.

I was also a contributor to this episode, and it was a real delight to work with them in a special role: I got to unpack many of the foundational quantum mechanical concepts (transitions in atoms, stimulated emission, tunnelling, etc) to camera by doing line drawings while I explained – and kudos Click to continue reading this post →

Bluesky!

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For those of you who keep up with my social media posts, you’ve probably been expecting that I’d eventually announce that I’m transitioning from Twitter to something else… and it is Bluesky. I’ll stay on Twitter for a bit longer while I settle in (and while I wait for people to see the change, etc.), but consider following @asymptotia.bsky.social asap. (I’ll continue posting at the Facebook and Instagram accounts for now.)

This change fits nicely with the fact that I have plans to somewhat increase my post frequency here on the blog – as time allows – and so I’ll post links to them on the new social media platform, and maybe welcome some new communities too. Truth be told, for the longest while I’ve been very tied up with too many projects (some of which I can’t tell you about yet!) to be as frequent a poster as I’d like to be, but I’ll do what I can.

-cvj

Westminster Wonders

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Never toured the inside of the Houses of Parliament before, seeing all the red and green colour coded areas (lords and commons – look at the benches next time you see debates in either place) and busts and statues of some of the shapers, for better or worse, of much of the fabric of UK democracy. (Thanks to my sister for this wonderful opportunity on Friday!) BTW, most of the interesting stuff I saw was off limits to photos, sorry!

–cvj

Running London

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During the pandemic shutdown I regularly ran these london streets and bridges -virtually- on a treadmill watching a YouTube video of such a run.

This morning (actually 8 days ago since I see now I forgot to hit “publish”) was the first time I did it for real! I wonder if any of the many other runners I saw were on the video?

(Do have a look, if you wish, on my social media accounts for other posts from this London visit.)

–cvj

Tumble Science Podcast Episode

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tumble_episode_logoFor some weekend listening, there’s a fun and informative podcast for youngsters called Tumble Science Podcast. I learned of it recently because they asked to interview me for an episode, and it is now available! It is all about time travel, and I hope you (and/or yours) have fun listening to it. The link is here.

There’s an accompanying blog post here.

More generally, listen to more episode on Tumble’s website, or at the pod link here.

Enjoy!

–cvj

When Worlds Collide…

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This morning I had a really fantastic meeting with some filmmakers about scientific aspects of the visuals (and other content) for a film to appear on your screens one day, and also discussed finding time to chat with one of the leads in order to help them get familiar with aspects of the world (and perhaps mindset) of a theoretical physicist. (It was part of a long series of very productive meetings about which I can really say nothing more at the current time, but I’m quite sure you’ll hear about this film in the fullness of time.)

Then a bit later I had a chat with my wife about logistical aspects of the day so that she can make time to go down to Los Angeles and do an audition for a role in something. So far, so routine, and I carried on with some computations I was doing (some lovely clarity had arrived earlier and various piece of a puzzle fell together marvellously)…

But then, a bit later in the morning while doing a search, I stumbled upon some mention of the recent Breakthrough Prize ceremony, and found the video below (one of several). It’s as though I fell asleep at my desk and was having one of those strange dreams where two parts of your life that have little to do with each other get intertwined: Robert Downey Jr (RDJ) and Da’Vine Joy Randolph (DJR) doing a “bit” about statistical physics, quantum field theory, and symmetries, and then John Cardy and Alexander Zamolodchikov come on stage…

-cvj

Catching Up

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KITP UCSB by cvj
Since you asked, I should indeed say a few words about how things have been going since I left my previous position and moved to being faculty at the Santa Barbara Department of Physics.

It’s Simply Wonderful!

(Well, that’s really four I suppose, depending upon whether you count the contraction as one or two.)

Really though, I’ve been having a great time. It is such a wonderful department with welcoming colleagues doing fantastic work in so many areas of physics. There’s overall a real feeling of community, and of looking out for the best for each other, and there’s a sense that the department is highly valued (and listened to) across the wider campus. From the moment I arrived I’ve had any number of excellent students, postdocs, and faculty knocking on my door, interested in finding out what I’m working on, looking for projects, someone to bounce an idea off, to collaborate, and more.

We’ve restarted the habit of regular (several times a week) lunch gatherings within the group, chatting about physics ideas we’re working on, things we’ve heard about, papers we’re reading, classes we’re teaching and so forth. This has been a true delight, since that connectivity with colleagues has been absent in my physics life for very many years now and I’ve sorely missed it. Moreover, there’s a nostalgic aspect to it as well: This is the very routine (often with the same places and some of the same people) that I had as a postdoc back in the mid 1990s, and it really helped shape the physicist I was to become, so it is a delight to continue the tradition.

And I have not even got to mentioning the Kavli Institute for Theoretical Physics (KITP) Click to continue reading this post →

Recurrence Relations

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(A more technical post follows.)

By the way, in both sets of talks that I mentioned in the previous post, early on I started talking about orthogonal polynomials P_n(\lambda)=\lambda^n+\mbox{lower powers}, and how they generically satisfy a three-term recurrence relation (or recursion relation):

\lambda P_n(\lambda) = P_{n+1}(\lambda) +S_n P_n(\lambda) +R_n P_{n-1}(\lambda) \ .

Someone raised their hand and ask why it truncates to three terms and on the spot I said that I’d forgotten the detailed proof (it has been many years since I thought about it) but recall that it follows straightforwardly from orthogonality. Lack of time meant that I did not want to try to reconstruct the proof on the board in real time, but it is a standard thing that we all use a lot because it is true for all the commonly used families of polynomials in lots of physics, whether it be Hermite, Legendre, Laguerre, Chebyshev, etc. Anyway, I finally got around to reminding myself of it and thought I’d record it here. Now all I have to do in future is point people here as a handy place to look it up. ([Although you can find equivalent discussions in several sources, for example this nice YouTube lecture here, which is part of a lovely series of lectures on polynomial approximation of functions, which is a fascinating topic in its own right.]

Ok, I should quickly remind what the setup is. The polynomials are normalised so that the nth one is P_n(\lambda)=\lambda^n+\mbox{lower powers} (they’re “monic”) and they are orthogonal with respect to the measure w(\lambda)d\lambda where w(\lambda) is called the “weight function” (it has some suitable properties we won’t worry about here). In the case of random matrix models we have w(\lambda) = \exp\{-N V(\lambda)\} for some potential V(\lambda) (here N is the size of the matrix; in this problem it is just a normalisation choice – you can just as well absorb it into the potential).

So we have the inner product:

\langle P_n, P_m\rangle\equiv \int w(\lambda) P_m(\lambda) P_n(\lambda) d\lambda = h_n\delta_{mn}\ ,

defining the orthogonality, where the h_n are some positive non-vanishing normalisation constants. Ok now we are ready for the proof.

Imagine there are terms in the recursion beyond the three terms. Let’s write these “remainder” terms as a linear combination of all lower polynomials up to degree n-2, so the recursion is tentatively:

\lambda P_n = P_{n+1} +S_n P_n +R_n P_{n-1} + \sum_{k=0}^{n-2} T_kP_k.

Taking the inner product \langle P_m, \lambda P_n\rangle for m=n-1, n or n+1 just tells you the definition of the recursion coefficients S_n and R_n in terms of ratios of inner products for those m, and for m any higher you get zero since the polynomial is then of too high order to give anything non-zero.

So S_n = \frac{\langle \lambda P_n,P_n\rangle}{\langle P_n,P_n\rangle} and R_n = \frac{\langle \lambda P_n,P_{n-1}\rangle}{\langle P_{n-1},P_{n-1}\rangle} .

Then you take the inner product \langle P_m, \lambda P_n\rangle for the cases m < n-2.

But this is also (by definition; I can let the lambda act in the opposite direction inside the integral) \langle\lambda P_m, P_n\rangle, which vanishes since the degree of the first entry, m+1, is less than n, and so it can only contain polynomials of degree less than n which are orthogonal to P_n. Therefore the inner product says T_m \langle P_m,P_m\rangle=0 in all those cases, which means that T_k=0 for k=0, 1,...n-2.

That’s it. All done. Except for the remark that given the expression for S_n above, when the weight function is even, the S_n vanish. (This is the case for even potentials in the case of random matrix models.)

Ok, one more useful thing: It is clear from the definition of the inner product integral that h_{n+1}=\langle P_{n+1},\lambda P_n\rangle. But you can also write this as h_{n+1}=\langle \lambda P_{n+1}, P_n\rangle and use the recursion relation \lambda P_{n+1} = P_{n+2}+S_nP_{n+1}+R_{n+1}P_n, and all these terms vanish in the integral except the last, and so we get h_{n+1}= R_{n+1}\langle P_n,P_n\rangle = R_{n+1}h_n.

Hence we’re derived an important relation: R_n=\frac{h_n}{h_{n-1}}\ .

(We essentially got this already in the earlier equation for R_n; just rearrange the action of \lambda up there again.)

–cvj