Towards a Grand Unified Theory of Mathematics and Physics

A draft of an essay I’ve written, with plans to submit it to the FQXI essay contest, is available here. Constructive comments welcome…

People who have a take on the subject that has nothing to do with what I’m writing about are encouraged to submit their own essays to FQXI, but not to post them here.

: Thanks to all commenters for often helpful comments. I’ve revised the essay a bit, mostly by adding some material at the end, material that to some extent addresses important issues raised by some commenters.

: The essay has been submitted and is posted here.

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The Singular Universe and the Reality of Time

Lee Smolin has a new book out last month, co-written with philosopher Roberto Unger, entitled The Singular Universe and the Reality of Time. To get some idea of what he’s up to, there’s a review by Bryan Appleyard at The Sunday Times (non paywalled version here), another Bryan Appleyard piece here, and interviews with John Horgan and at Scientia Salon. In other news about Smolin, he’s one of the winners of this year’s first Buchalter Prize in cosmology.

The book is written in a rather unusual style, with the first two thirds or so by Unger, the rest a shorter contribution from Smolin, together with a section discussing where they disagree. It’s neither a popular science book, nor a technical work of philosophy, but something somewhere in between, best perhaps compared to something one rarely now sees, a work of “Natural Philosophy”. I found the long section by Unger rather hard going and not very rewarding, and realized that I have a fundamental problem with this sort of writing. Arguments about physics and mathematics made in natural language leave me often unable to figure out exactly what is being claimed. Sometimes this is because I’m not familiar enough with a philosphical tradition being invoked and its associated use of terms, sometimes I suspect it’s because natural language is just too imprecise and ambiguous.

The Smolin section is shorter and written with more precision, making it easier to get an idea of what he’s trying to claim. To seriously address all his arguments would be a large project I’m not able to undertake, but here is a list of “hypotheses” or “principles” that he arrives at:

  • The uniqueness of the universe.
  • The reality of time.
  • Mathematics as the study of evoked relationships, inspired by observations of nature.

For the first of these I don’t really disagree. Smolin takes this as an hypothesis of no “multiverse”, an hypothesis that science may be able to confirm or disconfirm. Our current best understanding of science shows no evidence for a multiverse, so anyone who wants to posit one needs to come up with some significant evidence for one, experimental or theoretical, and I haven’t seen that happening. It’s entirely possible that a compelling theory may emerge that naturally implies a multiverse, but that’s not currently the case. Unlike Smolin, I wouldn’t take this as an hypothesis, more just would say that the question of multiple universes is well worth ignoring until someone comes up with a good reason to pay attention.

For the second, one problem is that I’m not exactly sure what it means. I guess that when I hear the word “real” I’m always rather suspicious that a meaningless distinction is being invoked (i.e. is the wave-function “real”?), and start trying to remember what it was I once understood about ontological commitments from reading Quine long ago. Part of what Smolin is referring to I think I’m sympathetic with: the nature of time remains mysterious in a way that space isn’t. While relativity treats them on an equal footing, in quantum theory this is not so clear. My suspicions about this mystery though tend to focus on the analytic continuation between Minkowski and Euclidean signature, which I’d guess is quite different than Smolin’s concerns (see hypothesis three…)

What Smolin seems to have in mind here is the hypothesis that physical laws are not “timeless”, but can evolve in time, with an example the ideas about “Cosmological Natural Selection” he has worked on. One problem with this is that the question then becomes “what law describes the evolution of physical laws?”, with an answer re-introducing “timeless” laws. Smolin refer to this as the “meta-law dilemma” and devotes a chapter to it, but I don’t think he has a convincing solution.

On the third hypothesis, about the nature of mathematics and its relationship to physics, I just fundamentally and radically disagree. For a shorter version of Smolin’s argument, see this essay, which he has recently submitted to the FQXI essay contest. I’ve been writing something about how I see the topic, will blog about it here very soon. What I’m writing isn’t a response to Smolin’s arguments, but a positive argument for the unity of math and physics at the deepest level.

My problems with Smolin’s point of view aren’t especially about his arguments concerning Platonism and whether mathematical objects are “real” (see earlier comments about what’s “real”), they’re about arguments like this one, where he argues that the explanation for the “unreasonable effectiveness of mathematics” in physics is not some deep unity, but just

mathematics is a powerful tool for modelling data and discovering approximate and ultimately temporary regularities which emerge from large amalgamations of elementary unique events.

The argument essentially is that mathematics is nothing more than a calculational tool that just happens to be useful sometimes in physics. This is a common opinion among physicists, and a big problem for me is that here Smolin is not taking a provocative minority point of view, but just reinforcing the strong recent intellectual trend amongst the majority of physicists that the “trouble with physics” is too much mathematics. As I’ve often pointed out, the failures of recent theoretical physics are failures of a wrong physical idea, rather than due to too much mathematics, with the multiverse just an endpoint of where you end up if you throw away all non-trivial mathematical structure in pursuit of a bad idea.

In his essay, Smolin gives a discussion of mathematics itself which I think few mathematicians would recognize, defining it as “the study of systems of evoked relationships inspired by observations of nature”, and consisting in bulk just of elaborations of the concepts of number, geometry, algebra and logic. I started my career in physics departments, and I’m well aware of how mathematics looks from that perspective (even if you have a lot of interest in math, like I did). My experience of moving to work in math departments made clear to me that the typical ideas of physicists about what mathematics is and what mathematicians do are highly naive, with Smolin’s a good example.

I’ll end with just one example of what I see is wrong about the conventional physics view that Smolin represents. A big application of mathematics to physics is the use of the rotation group SO(3). In that case it’s true that many of the applications can be thought of as concerning approximate aspects of complicated physical systems, and derived from working out precisely the implications of our experience dealing with the 3d physical world. But, besides the chapter on angular momentum operators (and thus SO(3) representation theory) in every quantum mechanics textbook, there’s an earlier chapter where the Heisenberg commutation relations are given as fundamental postulates of the theory. A concise way of stating this postulate is that quantization is based on a specific unitary representation of a Lie algebra (the Heisenberg Lie algebra). This is not approximate, but the fundamental definition of what we mean by quantum theory. The structure here is very deep mathematics (appearing for instance in number theory, the theory of theta functions and of Abelian varieties), and is far removed from the kinds of mathematics that one runs into as typical approximate calculational tools when studying physical problems. This is just an example, but there are many others. I don’t think that if you look at them you can sustain the argument that deep mathematics and deep physics are not close cousins with a unity we only partially understand.

Anyway, more detail to come about this…

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Short Items, and a Quick Book Review

  • Peter Orland has a new blog, Ensnared in Vacuum, where he’s writing about some non-perturbative QFT questions.
  • Physics Today this month has book reviews of two books about theology and the multiverse (one of which I wrote about here). There was a time when I would have thought that discussions of theology wouldn’t be what Physics Today covers, but evidently that’s no longer the case.
  • On a related topic, Kate Becker at The Nature of Reality has an article entitled Does Science Need Falsifiability? It’s about the campaign by physicists like Sean Carroll and Lenny Susskind against the Popperazi who keep pointing out that giving up on falsifiability puts physics in danger of becoming, well, theology. Frank Wilczek has a very sensible take on the subject:

    “I think falsifiability is not a perfect criterion, but it’s much less pernicious than what’s being served up by the ‘post-empirical’ faction,” says Frank Wilczek, a physicist at MIT. “Falsifiability is too impatient, in some sense,” putting immediate demands on theories that are not yet mature enough to meet them. “It’s an important discipline, but if it is applied too rigorously and too early, it can be stifling.”

    On Twitter, the usually mild-mannered Wilczek makes clear his feeling about this

    Not often I refer to a “pernicious” “faction”, but appropriate here.

    Mysteriously, he has a new website, for a company “Wolfcub Vision, Inc”.

  • Frank Close has a new book out, Half-Life, which is essentially a biography of the physicist Bruno Pontecorvo. It’s also a gripping spy story, investigating the question of exactly why Pontecorvo fled with his family to the Soviet Union in 1950. There’s no smoking gun found, but all the evidence Close lays out makes the case that it is quite likely that Pontecorvo had been spying for the Soviets, fleeing when warned that he was in danger of being exposed.

    Freeman Dyson has a much better review of the book in the New York Review of Books than I could ever write. He argues that Pontecorvo made a mistake by fleeing to enforced isolation in Russia, that in the worst case if caught he would have spent a few years in jail, then could have resumed his career. That things would go this way would not however have been clear to Pontecorvo: the Rosenbergs were arrested just before he fled, and things didn’t work out so well for them.

    Besides the fascinating spy story, there’s also a lot of history of nuclear physics during the 30s, 40s and 50s, much of which I wasn’t aware of, as well as quite a bit about Pontecorvo’s later work on neutrinos. If you’re interested in the history of 20th century physics, this is something you’ll find well worth reading.

: For another new book, Steven Weinberg’s To Explain the World, I fear that I don’t have the time to read it and write a review. However, here are two interesting reviews, pro and con.

: For two hints about “Wolfcub Vision”, a commenter points out that Wolf cub=Wilczek in Polish, and a correspondent points me here.

Update: David Mumford has a posting Is it Art? at his blog, motivated by my friend Dan Rockmore’s equations project. An article about the recent panel discussion of this at Yale is here.

Posted in Book Reviews, Multiverse Mania | 16 Comments

IPMU Conversation with Edward Witten

I recently heard from Hirosi Ooguri that a transcript of a long conversation with Witten held at the time of his Kyoto Prize award has just appeared in the Kavli IPMU Newsletter. It’s a truly fascinating document, giving some great insights into Witten’s work at the boundary of math and physics and how he sees the state of ideas in this area. It’s wonderful that he was induced to give such a thoughtful and extensive explanation of both the history and significance of these various topics.

Just to pick out a couple examples, the discussion of geometric Langlands describes a lot of detailed history that I was unaware of. I had noticed that in their first massive (still unpublished) paper that started the subject, Beilinson and Drinfeld credit Witten with “the main idea” (I wrote about this in detail here). But there was nothing in what Witten has written that corresponds to what they did, and experts I talked to didn’t see how this came from Witten. Witten tells the true story this way:

Actually, the very little bit of what Beilinson and Drinfeld were saying that I could understand made me wonder if the work of Nigel Hitchin would be relevant to them, so I pointed out to them Hitchin’s paper in which he had constructed commuting differential operators on the moduli space of bundles on a curve. Differently put, Hitchin had in a certain sense quantized the classical integrable system that he had constructed a few years before. Although I understood scarcely anything of what Beilinson and Drinfeld were saying, I did put them in touch with Hitchin’s work, and actually, in their very long, unpublished foundational paper on geometric Langlands that you can find on the web, Beilinson and Drinfeld acknowledged me very generously, far overestimating how much I had understood. All that had really happened was that based on a guess, I told them about Hitchin’s work, and then I think that made all kinds of things obvious to them. Maybe they felt I knew some of those things, but I didn’t. But anyway, there were ample reasons in those years to think that geometric Langlands had something to do with physics, but as you can see I still couldn t make any sense out of it.

He also describes how he came to the idea of interpreting geometric Langlands as a form of mirror symmetry, inspired by things he learned from David Ben-Zvi at lectures about the Langlands program bringing together mathematicians and physicists at the IAS.

He contrasts his work in recent years relating Khovanov homology and gauge theory with the geometric Langlands work, saying that he thinks the Khovanov homology ideas are in a form such that mathematicians are more likely to be able to appreciate their roots in gauge theory:

I think it s actually very difficult to see what advance in the near term could make the gauge theory interpretation of geometric Langlands accessible for mathematicians. That’s actually one reason why I m excited about Khovanov homology. My approaches to Khovanov homology and to geometric Langlands use many of the same ingredients, but in the case of Khovanov homology, I think it is quite feasible that mathematicians could understand this approach in the near future if they get excited about it. I believe it will be more accessible. If I had to bet, I think I have a decent chance to live to see gauge theory and Khovanov homology recognized and appreciated by mathematicians, and I think I’d have to be lucky to see that in the case of gauge theory and the geometric Langlands correspondence – just a personal guess

About the geometric Langlands story, he thinks there is still much to be understood, including its connection to conformal field theory:

In fact, part of the original work of Beilinson and Drinfeld on geometric Langlands has still not been understood to my satisfaction. Here I have in mind the use of conformal field theory at what they call the critical level (level -h, where h is the dual Coxeter number) to construct the A-model dual of certain B-branes (the ones that are associated to opers, in the language of Beilinson and Drinfeld). Davide Gaiotto and I obtained a few years ago a reasonable understanding of what electric-magnetic duality does to the variety of opers, but I still do not really feel I understand its relation to conformal field theory. However, in the last few years physicists working on supersymmetric gauge theories in four dimensions and their cousins in six dimensions have made several discoveries involving the role of conformal field theory at the critical level, so the time may well be right to resolve this point.

Among the many other highly interesting comments, one was Witten’s take on the possible connection of quantum field theory to number theory. He has a long history with this, going back to conversations with Atiyah in 1977 in which Atiyah suggested some connection between Langlands and Montonen-Olive. Witten writes

I was skeptical about Montonen-Olive duality, I didn’t seriously try to relate it to Langlands duality and I didn’t try to learn what Langlands duality was. I did not learn anything more about these matters until the late 1980s. Then I learned just superficially about the Langlands correspondence. If one knows even a little bit about the Langlands correspondence and a little bit about conformal field theory on a Riemann surface, one can see an analogy between them. I wrote a paper that was motivated by that but then I realized that my understanding was too superficial to lead to anything deep, so I abandoned the matter for a number of years.

Later of course, he followed work on geometric Langlands and ultimately found the connection to gauge theory he worked out with Kapustin. As far as current prospects for connections to number theory, he has quite a few comments, but thinks the subject is still a dream that is not ripe:

For me personally−it’s a dream that eventually number theory would make contact with physics some time, but I doubt it will be soon. There are all kinds of areas where specific number theory formulas appear in physics, and these may be clues that the dream will come true one day. But to really get me excited, somehow the number theory would have to enter the physics in a more structural way. I m not that interested in a specific formula that comes out of a physics calculation in a more or less ad hoc fashion. Number theory would have to be more integrated with the physics to get me excited, and I don t see that happening soon. In my work, I concentrated on the geometric form of the Langlands correspondence because I could see that there was hope to really understand it in the context of the physics-based tools that were at hand. There might be something like that one day for the Langlands correspondence of number theory, but probably a lot is missing and we do not know what has to happen first.

This just gives a taste of the conversation, there is lots, lots more there, on a wide range of topics. Highly recommended reading for anyone with an interest in this area, I’ve never seen anything like it.

Posted in Langlands | 23 Comments

Planck Data Out

Long awaited data from the Planck satellite was released today, papers available here. The accompanying press release leads with results about the timing of the first stars, 500 million years or so after the big bang, with little mention of the very early universe. This is also the main topic of BBC News coverage.

This paper reports a bound on r of .08-.09, exactly what Shaun Hotchkiss was predicting earlier this week here. This appears to be pretty much the end of the line for hopes that Planck would see primordial gravitational waves, with the paper seemingly pointing to other experiments being necessary to get below r=.05 (see page 35).

The BBC News story also characterizes these bounds as ruling out the simplest inflationary models, requiring they be supplemented by “exotic physics”.

What is clear from the Planck investigation is that the simplest models for how that super-rapid expansion worked are probably no longer tenable, suggesting some exotic physics will eventually be needed to explain it.

“We’re now being pushed into a parameter space we didn’t expect to be in,” said collaboration scientist Dr Andrew Jaffe from Imperial College, UK. “That’s OK. We like interesting physics; that’s why we’re physicists, so there’s no problem with that. It’s just we had this naïve expectation that the simplest answer would be right, and sometimes it just isn’t.”

For about as long as I can remember, string theorists and multiverse fans have been pointing to Planck data as the test of their ideas. For cosmic strings, the last Planck data release had a paper ruling them out. I don’t see a paper on this topic out or projected for the new data, it seems that this is now something not even worth looking for.

We’ve also been hearing for years that Planck will test supposed evidence of bubble collisions indicating other universes, see for instance this article about this paper, where the article states that

Data from the Planck telescope should resolve the question once and for all.

I don’t see anything in the new data even looking for this. Has it already been ruled out, without any publicity, or did the Planck people think it was something not worth even looking for?

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A Letter to the AMS

Leonid Reyzin at Boston University has drafted a letter in response to the recent article published in the Notices by Michael Wertheimer of the NSA (discussed here). He’s collecting signatures, and if you’re a member of the AMS I urge you to consider contacting him and adding yours. If you know others who might be interested in signing, please forward the link to them.

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Short Items

  • Science magazine this week has an article and a podcast about the NSA and the AMS. AMS president David Vogan is portrayed as outraged at the NSA’s misuse of mathematics, but without much support for doing anything about it:

    But after all was said and done, no action was taken. Vogan describes a meeting about the matter last year with an AMS governing committee as “terrible,” revealing little interest among the rest of the society’s leadership in making a public statement about NSA’s ethics, let alone cutting ties. Ordinary AMS members, by and large, feel the same way, adds Vogan, who this week is handing over the presidency to Robert Bryant, a mathematician at Duke University in Durham, North Carolina. For now, U.S. mathematicians aren’t willing to disown their shadowy but steadfast benefactor.

  • Two odd things from the piece and the podcast:

    1. The NSA budget is highly classified. Essentially nothing is known about it, with estimates of its total ranging from $8 billion to $25 billion/year (by the way, can anyone tell me why that number is a secret?). Precisely one line item in their budget is publicly reported: the $4 million to the AMS-administered grant program. The AMS seems to be the only organization in the world that the NSA has a publicly disclosed relationship with.
    2. The reporter said he tried but was unable to get in contact with Richard George, the ex-NSA person who published a piece in the Notices claiming the NSA backdoor was “just innuendo”.
  • On a much more positive note, this week the New Yorker has a really wonderful piece about Yitang Zhang.
  • There’s an interview at the Huffington Post with Lenny Susskind about The Future of Physics. It looks like his point of view is that there is no known alternative, no matter what happens at the LHC, that fine-tuning is evidence for the multiverse is evidence for string theory. The only alternative now: hope for an unforseen surprise.

    [Oops, I should have noticed this was a republication of a 2006 interview. The “expect the unexpected” thing didn’t work out…]

Update: Paul Frampton tells me that he has published a book telling the story of what happened to him. It’s now available on Amazon.

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It Really Is Just Dust

The Planck collaboration has an inimitable way of releasing important new results, they like to do it in French (see here for instance). Tonight a French Planck website contains the long-awaited news of the results from the BICEP2/Keck/Planck collaboration to reanalyze the BICEP2 data on polarized B-modes, in a way that allows proper estimation of the contribution of dust. The bottom line is that the BICEP2 claims of seeing a primordial r=.16-.20 that got a huge amount of media attention last year have been shot down. The new analysis says that r is less than .13. I don’t see a paper yet, rumor is that the paper will be on the arXiv Monday night.

My understanding is that full polarization results from Planck’s own data are still not quite ready, as has been the case for quite a while now.

Hat tip for this news to Steve (retired LLNL physicist). Thanks Steve!

Update: The Planck collaboration now has information available about this (in English!). An NSF version is here, NBC News here. It still looks like we’ll have to wait until next week for the paper, which has been submitted to PRL.

Update: The paper is now available here.

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Mathematics Without Apologies

If you’d asked me ten years ago to describe a book I’d love to read that could be characterized as part of an “incredibly unlikely trend in books about math for the general public”, I might have chosen “brilliant meditations on the practice of mathematics and on mathematics at the deepest level, from first-rate mathematicians, focusing on the Langlands program, with expert-level discussion of the subject.” And yet, here we are, not much more than a year after Edward Frenkel’s Love and Math, with the publication last week of another very different but equally fascinating example of exactly this trend: Michael Harris’s Mathematics Without Apologies. If you are interested at all in what mathematics really is and what the best mathematicians really do (and you’re up for an intellectual challenge) I highly recommend that you get a copy and set some time aside for delving into this unusual book.

While Harris shares many of Frenkel’s themes and concerns, his style is very different, favoring density, indirectness, the post or post-post-modern, and deep engagement with history, philosophy and sociology. Only one of these two authors assumes a familiarity with Max Weber. Where Frenkel is ever guileless and straight-forward, Harris has a whole chapter on the “trickster”, taking some pride in being known for “Harris’s tensor product trick.” While reading, more than once one wonders whether one is really supposed to take something seriously (for instance, there’s quite a long bit about Thomas Pynchon’s novels and conic sections…).

Normally when I’m reading a book I want to later write about, my practice is to fold down the corners of pages that contain something new, unexpected, especially insightful, or something I’d really like to argue with. Then I can start writing by reviewing those pages. My problem with this book is that I ended up folding down the corners of a large fraction of the pages, so when I sat down to write, my usual method would force me to reread pretty much the entire book. Not a bad idea, since I’m convinced I missed a lot the first time through, but other tasks beckon and it’s not a quick read.

I’m not sure I can do much better here than randomly list a few of the themes of the book: the pleasures of doing mathematics, the role of pure mathematicians in society (Wall Street!) and many forms of art and culture, how best to explain number theory to an insightful actress, the philosophy of mathematics and philosophy of Mathematics (two different things), Indian Metaphysics, n-categories, the yoga of motives, Voevodsky’s univalent foundations, the life and thought of Alexander Grothendieck and Robert Langlands, etc., etc. There’s also serious doses of sex (including an extensive discussion of Frenkel’s film), drugs (from Erdos to Andreas Floer to late nights at Oberwolfach) and rock and roll (from the “Math Rock” genre which I’d never heard of before to the IAS house band “Do Not Erase”).

Harris manages to move back and forth between the deepest ideas about mathematics at the frontiers of the subject, insightful takes on the sociology of mathematical research, and a variety of topics pursued in a sometimes gonzo version of post-modern academic style. You will surely sometimes be baffled, but definitely will come away knowing about many things you’d never heard of before, and with a lot of new ideas to think about.

For some more about the book, including some early versions of some chapters, see Harris’s website here.

Update: Princeton University Press now has a Q and A with Harris about the book up here.

: The book now has a blog.

Posted in Book Reviews | 8 Comments

Snowpocalypse 2015

For the last few days the media in New York have been filled with continuous frantic warnings of the deadly storm of the century bearing down on the city. Grocery stores have been emptied, with long lines of desperate people trying to stock up on supplies.

Midday yesterday Columbia announced that classes were canceled starting at 3pm, Barnard went one hour better, canceling classes starting at 2pm. The city announced that it would be illegal to be in the parks after 6pm (a snow-covered branch might fall on you), the transit system would start shutting down at 7pm and by 11pm there would be no public transit, and all roadways in the entire tri-state area would be closed to non-emergency traffic. The mayor’s office warned people not to try and order takeout delivery since it would be illegal for the delivery people to travel on the streets to deliver it.

By late afternoon the university was deserted, and stores on Broadway had signs announcing early closing due to the impending disaster. Weather reports the day before had said the storm would start at 1pm Monday, but by early evening there hadn’t been much more than snow flurries, with maybe an inch or two total accumulation. When I went to sleep around midnight, the city was completely locked down, with the TV news channels filled with blaring warnings of the two to three feet of snow about to arrive, interspersed with press conferences from public officials telling people to barricade themselves in their homes and not go outside.

The strange thing about this was that if you actually looked at the weather report, they were now forecasting 3-5 inches of snow overnight. Waking up in the morning and looking out the window, all that was visible were more flurries, and a total accumulation of 2-3 inches, with the streets clear. Turning on the TV news, the huge “Blizzard of 2015″ logos were still up, and camera crews seem to have been sent out to search the region (mostly unsuccessfully) for a snow drift to put a reporter in front of. The contrast between looking out the window and watching TV was pretty dramatic.

Anyway, my class today is canceled, so students will have to wait until Thursday to hear more about the mathematics of quantization of the harmonic oscillator (complex structures, squeezed states, coherent states). Lecture notes still being worked on, but this is chapter 21 of the current notes.

Columbia never used to shut down at all, New York City never used to shut down the transit system, and the states never used to shut down all roadways. Until the past decade or so people tried to go about their business here in the winter, taking action to shut things down only once snow had arrived and was causing a problem. The US has now become a nation of hysterics, with media-driven hype frightening everyone about everything, and public officials desperately taking action to protect the citizenry from imaginary threats.

Luckily for us all, people cowering in their homes do have the internet and can still learn quantum mechanics. MIT has just announced that edX will have an online version of their quantum course, Mastering Quantum Mechanics, which looks quite good and will start February 10. The instructor will be Barton Zwiebach, and I’m glad to see that one of the topics covered will be squeezed and coherent states of the harmonic oscillator.

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