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 circumstances 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

Multicritical Matrix Model Miracles

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Well, that was my title for my seminar last Thursday at the KITP. My plan was to explain more the techniques behind some of the work I’ve been doing over the last few years, in particular the business of treating multicritical matrix models as building blocks for making more complicated theories of gravity.

chalkboard from KITP seminar

The seminar ended up being a bit scattered in places as I realised that I had to re-adjust my ambitions to match limitations of time, and so ended up improvising here and there to explain certain computational details more, partly in response to questions. This always happens of course, and I sort of knew it would at the outset (as was clear from my opening remarks of the talk). The point is that I work on a set of techniques that are very powerful at what they do, and most people of a certain generation don’t know those techniques as they fell out of vogue a long time ago. In the last few years I’ve resurrected them and developed them to a point where they can now do some marvellous things. But when I give talks about them it means I have a choice: I can quickly summarise and then get to the new results, in which case people think I’m performing magic tricks since they don’t know the methods, or I can try to unpack and review the methods, in which case I never get to the new results. Either way, you’re not likely to get people to dive in and help move the research program forward, which should be the main point of explaining your results. (The same problem occurs to some extent when I write papers on this stuff: short paper getting swiftly to the point, or long paper laying out all the methods first? The last time I did the latter, tons of new results got missed inside what people thought was largely just a review paper, so I’m not doing that any more.)

Anyway, so I ended up trying at least to explain what (basic) multicritical matrix models were, since it turns out that most people don’t know these days what the (often invoked) double scaling limit of a matrix model really is, in detail. This ended up taking most of the hour, so I at least managed to get that across, and whet the appetite of the younger people in the audience to learn more about how this stuff works and appreciate how very approachable these techniques are. I spent a good amount of time trying to show how to compute everything from scratch – part of the demystifying process.

I did mention (and worked out detailed notes on) briefly a different class of Click to continue reading this post →

Living in the Matrix – Recent Advances in Understanding Quantum Spacetime

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ribbon diagram that can be drawn on a torusIt has been extremely busy in the ten months or so since I last wrote something here. It’s perhaps the longest break I’ve taken from blogging for 20 years (gosh!) but I think it was a healthy thing to do. Many readers have been following some of my ocassional scribblings on social media (see sidebar), and will guess from those that the main news to report is that I’ve been getting on with the usual practices of my job (research, teaching and mentorship, science communication, etc) and life, including getting settled into a new city and a new working environment. The latter has all been rather fantastic, I’m happy to say! I hope to say a bit more about all these things at more-than-social-media length.

Let me end this little update with something juicy to dig into: Last week I gave a public lecture at the Kavli Institute for Theoretical Physics (KITP) here at UCSB. Its title was “Living in the Matrix – Recent Advances in Understanding Quantum Spacetime” and you can see a recording of it here. It is a little bit of an update on some aspects of research into understanding black holes at the quantum level using random matrix model methods to perform the gravitational path integral.

I put a lot of preparation into it, trying to motivate the research and give some ideas about how and why it proceeded the way it has. I was trying to do a bit more than just show a lot of pretty pictures and talk over them. I wanted to convey to the audience member a little bit of the sense of what it is like to think about some of the issues involved, and how a theoretical physicist tackles them. So you get to look over the shoulder of physicist as they write in their notebook.

I learned afterwards that people seemed to enjoy the talk, and so maybe you will too.

Enjoy.

–cvj

And so it begins…

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cvj sitting with mountain view
There’s not much in this post, but I wanted to mark a significant date. It is the first day of the rest of 2023, but in addition, it is the beginning of a new chapter for me. Yesterday was my last day as an employee of the University of Southern California, after 20 years (!) and today is my first day as an employee of the University of California, Santa Barbara. (I mentioned this was coming up in a previous post, and some of its significance is explained there.)

Of course, since I’m still on a research retreat at the Aspen Center for Physics, none of this seems entirely real just yet. Electronic signatures winging back and forth over the web don’t really substitute for the physical business of changing where you show up for work, who you meet there, and so forth, and so that will have to wait for little while. That having been said, lots of electronic welcomes and so forth did help a lot with that “new job smell”, if that’s a thing. But I’ll be showing up in person soon enough, and it’ll be fun to begin to construct a new routine.

Sadly I’ve not had a lot of time to properly sit and contemplate and let this all sink in, to be honest, so I’ll have to find time to do it later. (That photo, above, of me at the top of Aspen mountain is nice, but I was in that spot for about 5 minutes!) Right now, there’s travel, and then sorting and packing and more packing to do, along with family matters, and all that entails. I’ll have to leave the contemplation for a bit later.

Perhaps during a nice long run along the beach…!

-cvj

Rattle and Hum

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A lot of us have been waiting for a long time to hear this news! The NANOGrav collaboration has announced strong evidence of a background of low frequency gravitational waves emitted from supermassive black hole mergers. Their detection methods are pulsar timing arrays (still one of those fantastically simple, cool ideas I still wish I’d thought of). There’s a New York Times article by Katrina Miller here (“The Cosmos is Thrumming with Gravitational Waves…”), and here’s Yale’s Chiara Mingarelli (one of the team) describing some of what this means in simple terms:

-cvj

A Return (Again)

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About two years ago I wrote a post entitled “A Return”, upon moving to Princeton for a year (I was a Presidential Visiting Scholar at the Physics department). I reflected upon the fact that it was a return to a significant place from my past, where I’d been transformed in so many ways. Princeton was the first place I visited (not counting airports) in the USA, the location of my first postdoctoral appointment (at the Institute for Advanced Study (IAS)), and its was there that I did a deep enriching dive into the hubbub of Theoretical Physics, at one of the very top places in the world to do so.

Coastal view from UCSB campusAfter that, I moved West, to Santa Barbara, where my next postdoc position was at the Institute for Theoretical Physics at the University of California Santa Barbara (UCSB), now called the KITP. I was very lucky to be able to go from one top place to another, and (as I’ve recently talked about in a BBC interview here) additionally, my field was in a delicious turmoil of activity and discovery. I was able to be a part of the maelstrom (the “Second Superstring Revolution”, and all the gifts it gave us, including better understanding of the role in quantum gravity of extended objects beyond strings (such as D-branes), the physics of quantum black holes, the tools to unlock the holographic nature of quantum gravity more generally (through AdS/CFT), and so on. (I’ve blogged about many of these topics here, so use the search tool for more.)

I’ve been known to say that Princeton was the place where I found my physics voice (Edward Witten was a key guide at that time). Well, to continue the theme, Santa Barbara (with its wonderful research group made up of people from both the KITP and the wider Physics Department) was the place where I started to learn how to use that voice to sing (with the guidance of Joe Polchinski (who sadly passed away a few years ago)).

Well, as you may be guessing after that long introduction, I’m doing “A Return” again, but this time not with some boxes and suitcases of things for a year’s stay: I can now announce that I’ll be leaving the University of Southern California (USC) and (as of 1st July 2023) joining Click to continue reading this post →

The Life Scientific Interview

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After doing a night bottle feed of our youngest in the wee hours of the morning some nights earlier this week, in order to help me get back to sleep I decided to turn on BBC Sounds to find a programme to listen to… and lo and behold, look what had just aired live! The programme that I’d recorded at Broadcasting House a few weeks ago in London.

So it is out now. It is an episode of Jim Al-Khalili’s excellent BBC Radio 4 programme “The Life Scientific”. The show is very much in the spirit of what (as you know) I strive to do in my work in the public sphere (including this blog): discuss the science an individual does right alongside aspects of the broader life of that individual. I recommend listening Click to continue reading this post →

What a Week!

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Some Oxford scenesI’m sitting, for the second night in a row, in a rather pleasant restaurant in Oxford, somewhere on the walk between the physics department and my hotel. They pour a pretty good Malbec, and tonight I’ve had the wood-fired Guinea Fowl. I can hear snippets of conversation in the distance, telling me that many people who come here are regulars, and that correlates well with the fact that I liked the place immediately last night and decided I’d come back. The friendly staff remembered me and greeted me like a regular upon my return, which I liked. Gee’s is spacious with a high ceiling, and so I can sit away from everyone in a time where I’d still rather not be too cavalier with regards covid. On another occasion I might have sought out a famous pub with some good pub food and be elbow-to-elbow with students and tourists, but the phrase “too soon” came to mind when I walked by such establishments and glanced into the windows.

However, I am not here to do a restaurant review, although you might have thought that from the previous paragraph (the guinea fowl was excellent though, and the risotto last night was tasty, if a tiny bit over-salted for my tastes). Instead I find myself reflecting on Click to continue reading this post →

BBC Fun!

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BBC broadcasting house scenesAs I mentioned in the previous post, I had business at BBC Broadcasting House this week. I was recording an interview that I’ll fill you in on later on, closer to release of the finished programme. Recall that in the post I mentioned how amusing it would be for me (or W1A fans), as a Brompton rider, to arrive (as in the show W1A) at the building on a Brompton. In the end I did not rent one, although it was tempting, but I was pleased to see that upon my arrival there was someone folding a Brompton, just like Hugh Bonneville’s character in the show. It was a welcome sight. See the multi-component photo. After the interview, the producer (Lucy) kindly let me look around some of the parts of the building that were featured heavily in the show, and it was fun to see them in real life!

-cvj

W1A

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Brpmpton bicycle rental lockers.

Brompton bicycle rental lockers.


I’ll be visiting Broadcasting House during my time here in London this week, for reasons I’ll mention later. Needless to say (almost), as a Brompton rider, and fan of the wonderful show W1A, I feel a sense of regret that I don’t have my bike here so that I can ride up to the front of the building on it. you won’t know what I’m talking about if you don’t know the show. Well, last night I was a-wandering and saw the rental option shown in the photo. It is very tempting…

-cvj

Back East

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[I was originally going to use the title “Back Home”, but then somehow this choice had a resonance to it that I liked. (Also reminds me of a lovely Joshua Redman album…)]

An old favourite of mine, the BT tower, taken from near Great Portland Street, London.


So I am back in London, my home town. And since I’ve got 8 hour jet lag, I’m sitting up at 4:00am in a little hotel room, eating rich tea biscuits and drinking tea. It is as though the last 37 years since my undergraduate years never happened. (I was a student at Imperial College, living in a little room in a hall of residence not too much further away from the BT tower you see in the photograph, in a slightly different direction. Late nights with rich tea biscuits were my standard M.O.)

I’ve not been here since 2018, I’m shocked to realize, which is perhaps the longest period of time I’ve been away. Part of the reason is of course that two or more of those years had a Thing happen… you know, a global pandemic, and part of it was the recent arrival to the family, which served to delay any plans I had to do travel this far East away from Los Angeles.

I’m here for both business and personal matters – the best combination for any long trip (more later) – and hope to get some glimpses of the city (and a bit beyond) while I’m here, to see what’s changed and what’s the same, revisiting some old haunts along the way.

-cvj

Green for Go!

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Happy New Year!

I was greeted by great deal of green in Griffith Park today, and it was particularly lovely to look out over the whole park when I’d reached higher elevation, as the greens of various kinds stretched off in all directions, and even into the city. You’ve likely seen recent photos from me from a similar vantage point, where the dominant colours are brown and grey, so you can probably appreciate the contrast. The speed with which the green can come back in full strength never ceases to amaze me.

I’ve not done a run in this part of the park for a few weeks, for one reason or another, and so that made it all the more stark a contrast, I imagine, since there’s been a lot of rain here and there (and a lot over the last few days) and that has no doubt helped the transformation.

I’m going to take the green as an encouraging sign to press ahead (“green for go”) with this new year. There’s a great deal on my Click to continue reading this post →

A New Distribution

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[This is a relatively technical post.]
Probability distributionThe red curve in this figure is the probability distribution of the ground state energy E=s of the microstate spectra of quantum completions of JT gravity. Put differently (the way Wigner might have) if you ask what are all the discrete spectra that are compatible with the leading semi-classical result for quantizing JT gravity (the famous Schwarzian result for the density of states: \rho(E)=e^{S_0}(4\pi^2)^{-1}\sinh(2\pi\sqrt{E})), this curve gives the shape of the distribution of ground states. (The blue curve is simply the associated CDF.) I first uncovered this distribution in a paper last year, with further insights and generalizations in a paper earlier this year, along with the distributions for higher energy levels that follow from it. But the exciting new result of my paper from a few weeks ago is that I have now shown that it is a solution of an ordinary differential equation (or a family of them). This allows for some powerful universal things to be said analytically about the properties of the distribution!

This is fully analogous to what happened for the well-known Tracy-Widom distribution for the largest (or smallest) energy of Gaussian random hermitian matrices. While many workers (such as Forrester) had uncovered important aspects of the distribution, and while it was known that it can be expressed as a particular Fredholm determinant, Tracy and Widom broke new ground in 1994 by showing that the distribution was governed by a well known ODE – the Painleve II equation – and in particular can be given in terms of a special solution of it studied earlier by Hastings and McLeod. The result helped forge further connections between properties of random matrix theory and several interesting areas of mathematics and mathematical physics. Furthermore, Click to continue reading this post →

Arrival!

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Drawing of newborn babyA major part of the reason I’ve been very quiet here since the last post is because I was working on a different kind of project that has in fact taken up much of my bandwidth during the last many months.

This has been very welcome, as there is a new member of the family as a result! Attached is a drawing of our new daughter that I did a little bit after returning home from the delivery. (Click for larger view.)

Both mother and daughter are doing well, in case you’re wondering.

On past performance (see here) this means I should be getting busy and designing and drawing a 230 page non-fiction graphic narrative… but that seems a bit of a stretch to me right now!

(As I expect that there’ll be lots of sitting up on watch in the coming period, perhaps I’ll find some time to fill you in on several other things that have been on my mind, on the research front. There have been a couple of papers out that I’ve not spoken about yet. Stay tuned!)

-cvj