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September 19, 2007

Deep Beauty: Understanding the Quantum World

Posted by John Baez

There were many amazing mathematicians in the 20th century — people with world-transforming powers, like Gödel, Mac Lane and Grothendieck. But surely, no matter how short your list of greats, John von Neumann would have to be on it. From topics so abstract as the foundations of set theory and quantum mechanics, to topics so practical as game theory, the Manhattan project and the first computers, he seemed to be everywhere… right at the cutting edge.

Soon there will be a symposium honoring the 75th anniversary of von Neumann’s book The Mathematical Foundations of Quantum Mechanics:

Hans Halvorson is a philosopher at Princeton. I really like his idea that new mathematics may be needed to make quantum mechanics more intelligible, and that philosophers should get involved. I think it’s true.

Unfortunately attendance is by invitation only. Participants include Časlav Brukner, Jeffrey Bub, Bob Coecke, Andreas Doering, Lucian Hardy, Chris Isham, Simon Kochen, Klaas Landsman, Miklós Rédei, and Stephen Summers. Café regulars will recall that Coecke, Doering and Isham have all visited here. A number of graduate students are also attending the symposium, including my student Mike Stay and Isham’s student Jamie Vicary, both café regulars themselves.

I’ll try to report more on this symposium later. Right now you can see a very preliminary draft of a paper Spans in Quantum Theory, which is what I’ll be talking about.

Among other things, I want to start by making the point that categories of states and processes in quantum theory tend to be symmetric monoidal categories with duals, also known as dagger-compact categories. This point has already been made by Bob Coecke — and since he’ll also be at the symposium, he and I should figure out how to join forces in explaining this to everyone. But then, I want to explain how such categories naturally arise as categories of spans, and what spans have to do with Feynman path integrals. And then, I want to talk about spans of groupoids, and the whole groupoidification program. The fun part will be seeing how much I can cover — intelligibly! — in an hour.

By the way: speaking of the “intelligibility” of quantum mechanics, I can’t help but remember a line by von Neumann that always bugged me. He’s supposed to have said, in response to annoying questions from some student:

In mathematics you don’t understand things. You just get used to them.

I don’t agree with that, and it’s hard for me to imagine a good mathematician saying that — unless he had something very specific in mind. Maybe he was just trying to get rid of the student?

Posted at September 19, 2007 9:21 PM UTC

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Re: Deep Beauty: Understanding the Quantum World


I don’t agree with that, and it’s hard for me to imagine a good mathematician saying that — unless he had something very specific in mind.

It’s clearly an outrageous exaggeration at best, probably for the humor value. However, there’s a kernel of truth to it. There’s a lot of mathematical machinery that can’t be understood via a single ah-ha moment: mastering it requires seeing lots of different examples, recognizing new examples one wasn’t expecting, seeing how the machinery really tells one what one wants to know, etc. This is intrinsically a slow process, and there’s no sense sitting around waiting to magically “get it”.

Posted by: Anonymous on September 20, 2007 5:16 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I think of, sympathize with, and use that quote all the time. It’s hard for me to imagine a good mathematician not thinking it. Perhaps we should take a poll!

Posted by: Allen Knutson on September 20, 2007 3:10 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I didn’t see Anon’s response when I posted mine, and I disagree with that one too! So I guess I should say what truth I find in the von Neumann quote.

There are a number of results I’ve seen the proofs of – even short ones – that I’ve initially found unsatisfying and unintuitive. Is that a failure of understanding? Perhaps. But I’ve often taken it as more of a sign that I should replace my axiom system, and redefine such things as “intuitive”.

This takes effort – basically, for a while I have to work at remembering when faced with a new problem to check if this not-yet-intuitive theorem may help. Once it does so enough times, poof, it feels intuitive.

For example, I am in the middle of this process with Frobenius splitting. The proofs using it are short and incontrovertible, and I’m getting a sense of when it’s useful. If you “understand” Frobenius splitting then more power to you, sir.

Posted by: Allen Knutson on September 20, 2007 3:18 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I agree that there are lots of things in mathematics that are hard to understand, that we must temporarily accept in order to get on with business. I also don’t think we reach a definitive understanding of anything in a finite amount of time. But, I’d never say that in math we “don’t understand things, just get used to them”. I spend a lot of time trying to understand math, and I often make progress.

Most of all I like to tackle facts that are easy to prove but hard to understand. I’ve spent over a decade trying to understand why there are 4 normed division algebras over the reals, with dimensions 1, 2, 4, and 8. I know plenty of proofs: proofs that use Clifford algebras, proofs that use string diagrams… but I still feel dissatisfied. I’m not sure why. Maybe it’s because all the proofs seem a bit magical. Maybe it’s because I don’t know whether all these proofs are fundamentally the same.

So, I wouldn’t say I understand this fact. But, I’m certainly not just “getting used to it”, which suggests some sort of passive resignation. I’m gonna keep fighting to understand this stuff until I die!

Posted by: John Baez on September 20, 2007 5:15 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

In contrast to a quote from Thom:
very easy to see, very hard to prove

jim

Posted by: jim stasheff on September 20, 2007 5:23 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Several months ago I tackled the proof of a proposition generalizing a crucial result by someone else whose proof I thought was particularly opaque and unsatisfying. I thought that my particular approach and more advanced machinery would certainly make a more transparent and intuitive proof. After sweating over it for a week, I finally had it nailed down, but in the end my proof was no less transparent than the original, and I still have a hard time believing that it actually works. What does this mean when this happens? Does this necessarily imply that we just are not understanding something from the correct viewpoint yet, or are there some things that will always remain mysteriously opaque?

Posted by: Richard on September 22, 2007 3:30 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Maybe he’s not talking about the understanding of the proofs, he’s talking about why one need to use/introduce certain concepts/methods in mathematics at all. We understand them only through their results.

Posted by: tytung on September 20, 2007 5:49 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I think naively that the all point is in the meaning of the word “understand”, and in the word “things”. After all, what does it mean to understand, in general, and in particular, understanding maths ? If one means to be able, say, to reproduce a proof, justifying how every logical step follows from the precedent one, then I think you can say you can understand “things”, in this case the proof. But it occurred a lot of time to me that even if i did this, i wasn’t feeling somehow comfortable with the results, and then one day I woke up and thought:”That’s how it was!”. Probably it was induced from “getting used” to the results, or, in some irrational way, from convincing oneself of the results, beyond the mathematical proof, or maybe from “understanding”…
Anyway, I do agree with John when he says we can never learn everything about a subject or a concept in a finite amount of time, but that we try to add bits from time to time…

Posted by: Alessandro on September 21, 2007 5:48 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

What does it mean to “understand” something? To me, you understand something if you can calculate with it and have a mental picture of what is going on. What more can there be to understanding?

E.g., I think I understand the real numbers - I can calculate with them, and my mental picture is a bunch of points on a line. Now, there are constructions of the real numbers from set theory, but I never understood those constructions and now I have forgotten everything about them. Does this limit my understanding of real numbers?

Another example is the Dirac delta function. Mathematicians often claim that nobody understood the delta function before Schwartz developed distribution theory, but I think that is nonsense. Dirac had a mental picture (an infinitely high, infinitely thin peak) and he certainly could calculate, so in the sense above he had an understanding. That mathematicians didn’t understand, or didn’t like, what Dirac was doing is another thing.

Ironically, von Neumann seems to have been very hostile to Dirac’s techniques. In his mathematical foundations of QM, he argues that his Hilbert space methods are vastly superior to Dirac, because he does not have to pretend the existence of something know not to exist (i.e. the delta function).

Posted by: Thomas Larsson on September 22, 2007 1:27 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

What does it mean to “understand” something? To me, you understand something if you can calculate with it and have a mental picture of what is going on. What more can there be to understanding?

Maybe asking “why” you calculate in the way you do and not in an another..
After all, you have first to define rules for those calculations, and then show (within the standards of mathematics) that there is a model consistent with them, otherwise your calculations will make only sense in a formal way.

Posted by: Alessandro on September 22, 2007 5:21 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

As I recall, Dirac never did anything with the delta `function’ that required it to be a true function, i.e. he would integrate against test functions.

I think some of the disciples were less creful in transmitting the message.

Posted by: jim stasheff on September 22, 2007 9:45 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Dirac talks about generalized functions, and this seems to have irritated von Neumann and other mathematicians. But of course he used the delta function in ways that make sense, which essentially means that he followed the rules of distribution theory.

I have recently been led to use delta functions in an unconventional and potentially dangerous way. A decade ago, I asked you about commutation relations in jet space. More precisely, the question was prompted by your secret paper and formulated for the anti-bracket, but this is not essential. If

[φ(x), π(y)] = i δ(x-y),

what is the bracket between the Taylor coefficients φm and πn, where m and n are multi-indices? The answer is obviously

m, πn] = i(-1)nm+nδ(0),

where ∂m denotes the m:th order derivative. That this must be true follows immediately if you pretend that δ is a smooth function (e.g. a narrow Gaussian), but it took me almost a decade to write down this formula because I had a mental block against δ(0).

Note that the formula that you might naively expect,

m, πn] = i δnm,

only works if π has an upper multi-index.

Posted by: Thomas Larsson on September 23, 2007 1:24 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Hello Thomas, it has been over three years since I last chatted with you and Urs Schreiber back in the old sci.physics.strings.

For another “unconventional” use of the Dirac delta function see this paper [1] which uses the function within a Bayesian probability context for use in artificial intelligence. By the way, I am now working with a company doing research into intersections between probability, AI and mathematical finance which is why I happen to know about this paper. (One of these days, when I get some extra time, I will try to get some grip on what this “progic” is about.)

Regarding my previous question in this thread, I have found a paper [2] entitled “A Bundle Representation for Continuous Geometries” which I will have to read when I go to the MIT Science Library in a few days. (I also received a potentially useful tip from Chris Isham via email.)

I would now like to take this moment to personally thank John Baez for introducing me to the concepts of buildings, categorified bundles/gerbes and n-categories (as well as Cayley and Jordan algebras). Wow, does John know a lot about mathematics !


[1] http://www.springerlink.com/content/d3640651w5532735/

[2] John Harding and Melvin Janowitz, Advances in Applied Mathematics, v19 (2), pp.282-293 (1997)

Posted by: Charlie Stromeyer Jr on September 24, 2007 2:41 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

You’re welcome, Charlie! Nothing makes me happier than turning someone on to a cool new concept.

Posted by: John Baez on September 24, 2007 3:56 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Another reminder NOT to raise and lower indices casually

Posted by: jim stasheff on September 24, 2007 5:04 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I think one can detect a student’s (lack of) understanding by the way she looks. So, probably future software could detect that too. Perhaps another method to estimate the degree of understanding is the degree at which readers are troubled by typing-errors in formulas, e.g. because “understanding” enables to correct them very fast. Else, my impression is that “understanding” is the build-up of a conscious (visual) mental model + a subconscious model of the speakers/writers way to think, probably provided by mirror neurons.

Conc. Neumann: I heard that he intended to become a banker until his father insist him to study math. Did that lack of intrinsic initial motivation show up later in some way?

Posted by: anonymous on September 22, 2007 5:52 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I think some mathematicians felt von Neumann was too well-adjusted to be one of them — he dressed in suits, liked parties, etc. Maybe he didn’t suffer from the low-level Asperger syndrome so common in mathematicians. But on the other hand:

John was intellectually precocious and had a photographic memory. As a young child he learned to speak German, French, and classical Greek, and amassed an encyclopaedic knowledge of historical events. A favorite party trick of his was to memorize a page of the telephone book. He would ask a visitor to the Neumann household to select a page of the book, which he would then read through a few times. He would then hand the book back to the visitor and ask them to quiz him on the page, say by giving him a name and asking for the phone number, or getting him to recite a sequence of names, addresses and numbers in order. He was rarely wrong.

When John entered the local school in 1911, aged 8, the mathematics teacher recognized his genius at once, and arranged for special tuition from Gabor Szego, a well known mathematician at the University of Budapest. This was surely made easier by the fact that at the same school, just one year ahead of John, was another future mathematical giant, Eugene Wigner. By the time John completed his high school education in 1921, his prowess in mathematics was clear to all, and he was accepted to study mathematics at the University of Budapest.

Posted by: John Baez on September 24, 2007 3:11 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Thanks! Here an online Asperger test. How did von Neumann (or poeple who knew him) describe his way of thinking?

Posted by: anonymous on September 24, 2007 8:26 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

John von Neumann was famous for the speed and accuracy of his mental calculations.

Most mathematicians know the joke where one of von Neumann’s friends challenged him to quickly solve the famous bee and train puzzle.

The puzzle goes like this: two trains are 20 miles apart on the same track, each heading towards each other at 10 miles per hour. A bee takes off from the nose of one train at 20 miles per hour, towards the other train. As soon as the bee reaches the other train, it turns around and heads off at 20 miles per hour back towards the first train. It continues to do this, going back and forth over and over, infinitely many times, until the trains collide, killing the bee. How far did it fly?

This puzzle is also called ‘the mathematician’s trap’, because it tempts you to set up an infinite geometric series and figure out how to sum it. That takes a while. But, you can avoid the trap by simply figuring out how long it takes for the trains to collide and multiply this time by the bee’s speed!

Anyway, when a friend of von Neumann posed this puzzle to him, he instantly replied: “Twenty miles.”

His friend said “Ha! Good! That was quick — most mathematicians make the mistake of trying to sum the geometric series.”

Von Neumann looked puzzled and replied: “But I did sum the series.”

Anyway, I can’t find any evidence that von Neumann planned to be a banker until his father convinced him to do math. On the contrary, his father Max was a banker, and according to Keith Devlin:

Unfortunately, John’s love for mathematics did not meet with his father’s approval when it came to deciding what to study at university. Max Neumann wanted his son to prepare for a career in business. When John made it clear that business studies were not to his liking, father and son settled on chemistry as a compromise subject. And so John entered the University of Berlin to major in chemistry, to please his father, while at the same time enrolling at the University of Budapest to study mathematics, arranging to be tutored – at a distance – by a young assistant there. Being unable to attend any of the mathematics lectures in Budapest clearly did not hamper John’s progress; the following year he wrote his first mathematics paper, jointly with his tutor. In 1926, he was awarded a diploma in chemical engineering from the University of Zurich – where he transferred from Berlin in 1923 – and a doctorate in mathematics from Budapest. His thesis topic in mathematics was set theory, the foundational subject still in its infancy at that time.

By his mid-twenties, von Neumann – clearly now a mathematician rather than a chemist – was a mathematical celebrity, famous throughout the international mathematical community. After obtaining his Ph.D. he spent a year with the great David Hilbert in Goettingen and lectured in Berlin and Hamburg. In addition to his work in set theory, he did groundbreaking work in measure theory, the theory of real variables, and game theory. Not content with that, he also turned his attention to quantum theory, where in the two year period 1927-29 he more or less single-handedly developed the entire mathematical foundation of the subject, pioneering work that led to an invitation to go to Princeton. His research output during this early European part of his professional career was about one paper a month. (Tenure committees at present day universities generally view one or two papers a year as indicative of a good mathematical researcher!)

Posted by: John Baez on September 24, 2007 5:38 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Hello everyone, I have an inquiry about generalizing von Neumann’s concept of continuous geometry. Please consider this question:

Is it possible to have a “space” X such that an infinite X is the limit of a sequence of increasingly large finite instances of X?

If the answer to this question is “yes” then such a construction may have important implications for mathematics. For instance, the celebrated topologist Michael Freedman has suggested it might help
with unsolved problems in computational complexity and geometry [1]. Also, the theoretical physicist and string theorist Raphael Bousso once posed this question:

“Is there a sequence of theories with finite-dimensional Hilbert spaces such that string theory emerges in the infinite-dimensional limit?” [2]

I am further reminded of Edward Witten’s conjecture about U(∞)-valued Chan–Paton factors [3], and of various string theory papers about infinite towers of states. (By the way, even though I am not a physicist I am supposed to have some expertise in the math of high energy physics as you can see, for example, in the Acknowledgements sections of two papers [4, 5]. My own ideas in this area involved finding new connections within M-theory via Jordan algebras, but I don’t wish to discuss this now because it would be too off-topic.)

The great John von Neumann introduced the concept of continuous geometry which can be viewed as the limit of increasingly large finite instances of projective geometry [6]. Is there a way to sufficiently generalize von Neumann’s concept? For example, should I consider using bundles?

(I did find a paper about a tower of n-gerbes with the {infinity}-gerbe [7]. Would it make any sense to try to define a notion of convergence for such a tower?)

Thank you,
Charlie Stromeyer Jr.

[1] http://forum.wolframscience.com/showthread.php?threadid=288

[2] http://www.physics.ucsb.edu/~giddings/Mquest.html

[3] http://xxx.lanl.gov/abs/hep-th/0007175

[4] http://xxx.lanl.gov/abs/gr-qc/0212096

[5] http://xxx.lanl.gov/abs/hep-th/0407122

[6] http://planetmath.org/encyclopedia/VonNeumannLattice.html

[7] http://xxx.lanl.gov/abs/math/0301271

Posted by: Charlie Stromeyer Jr on September 23, 2007 12:59 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Your question seems very open-ended, and I can’t really answer it, but I urge you to dig deeper into von Neumann algebras, particularly the hyperfinite type II 1II_1 factor, since:

  • It’s approximable by finite-dimensional matrix algebras — that’s what ‘hyperfinite’ means.
  • It’s self-similar — in fact it’s the noncommutative geometry analogue of the Cantor set!
  • It’s deeply connected to Dynkin diagrams, topological quantum field theory, and many other things, thanks in large part to the work of Vaughan Jones.

So, it may not be what you’re looking for, but it’s a wonderful thing. Here’s a decent place to start, after you’ve read the basics in the link above:

  • F. M. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter graphs and towers of algebras, MSRI Series, Berkeley, Springer, New-York, 1989.

You mention Michael Freedman: he’s now working on topological quantum computers, which are based on precisely this sort of math.

To add to the fun, the radical logician Jean-Yves Girard is now trying to apply the hyperfinite type II 1II_1 factor to logic.

Posted by: John Baez on September 24, 2007 4:41 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Thanks for the suggestion, John, and I will dig deeper into von Neumann algebras if only because thinking about infinity is my favorite use of mathematics.

I first read about these algebras years ago in TWF 175 and so I should credit you with introducing me to this concept as well. I then read further about von Neumann algebras in the book “Jordan, real and Lie structures in operator algebras” by S. Ayupov et al.

You taught me that the reals (R), the complexes (C) and the quaternions (H) can be obtained from the octonions (O), and then I learned, for example, that the concept of Hilbert Space can be defined over O, and also that there are Jordan operads.

Posted by: Charlie Stromeyer Jr on September 25, 2007 9:53 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

The operad for Jordan algebras! I’d sort of forgotten about that.

I remember talking with you a lot on sci.physics.research. Then you sort of disappeared — from my viewpoint, that is.

Posted by: John Baez on September 26, 2007 3:47 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Well, I migrated from sci.physics.research over to sci.physics.strings, but then (due to the demands of work) I did not have time to think about quantum theory, e.g., the last time I thought about string theory was well over three years ago - not since my last post on s.p.s.

The first paper to succeed in defining Jordan operads is the paper “Jordan Triples and Operads” by A.V. Gnedbaye and M. Wambst in Journal of Algebra 231, 744-757 (2000). From page 745:

“Jordan algebras are not quadratic in the standard sense of operads. Nevertheless, as pointed out by Loday, the triple Jordan product which can be associated with any Jordan algebra satisfies a quadratic relation. It is then natural to consider ternary algebras, the Jordan triple systems, which are quadratic in the sense of algebras. In this article, we describe the quadratic operad of Jordan triple systems and prove that its dual is the quadratic operad of partially associative and partially antisymmetric ternary algebras.”

I rate this paper an A- overall only because I wish the authors had said something about potential applications of their work.

As you know, the role of von Neumann and Jordan algebras for quantum theory is an unfinished story. I first learned something about quantum mechanics when I was 15 years old, and now (a full 20 years later) I can say with all confidence that I still do not really understand quantum theory.

(A further reason why I prefer mathematics over physics might have something to do with the fact that three of my ancestors worked for the NSA. Perhaps not surprisingly, it appears that only a tiny amount of what they worked on has been declassified, but what little has been declassified is historically interesting if you want to see it, e.g., in this NSA tribute webpage:

http://www.nsa.gov/honor/honor00029.cfm

Posted by: Charlie Stromeyer Jr on September 26, 2007 9:02 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

So I read the paper “A Bundle Representation for Continuous Geometries” and it is related to the von Neumann algebra you mention above.

J. von Neumann introduced continuous geometry as a point-free generalization of projective geometry. He discovered that a continuous geometry is a complete, irreducible, complemented and modular lattice which is both join and meet continuous. Years later, F. Maeda showed that a continuous geometry is a Z-lattice.

The above paper shows that any continuous geometry can be represented as the continuous sections of a bundle whose stalks are all irreducible continuous geometries. The topology of this bundle is such that the subspace topology on each stalk is the usual metric space topology of an irreducible continuous geometry.

This bundle representation is related to the Pierce sheaf of the continuous geometry. A Pierce sheaf is the name for a sheaf naturally associated with each bounded lattice.

Further, the dimension functions on the stalks form a continuous map from the bundle into the reals. As the authors note, this is similar to the classical representations of von Neumann algebras as rings of continuous functions.

What the authors do not mention is that the type II_1 factor gives rise to a lattice with no atoms (i.e. no minimal non-zero elements) and its dimension function takes all possible values in the interval [0, 1]. The lattice is modular but not orthomodular.

(I will also mention that I found an example of convergence to an infinite set in the book “Fuzzy Topology” by N Palaniappan. The set of fuzzy points to which a fuzzy net converges is, in general, infinite - yet putting a restriction on the supports obtains uniqueness of convergence related to the iterated limits theorem.)

Posted by: Charlie Stromeyer Jr on September 27, 2007 9:45 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

So I followed your advice to dig deeper into von Neumann algebras and I have found various papers in category theory based upon these algebras which I will have to start reading when I return to the MIT Science Library (as well as some papers on Pierce sheaves).

Here is a question you may be able to answer:

Chris Isham told me via email that I might consider the theory of measures on infinite dimensional vector spaces. Such a measure can be thought of as an inverse limit of measures defined on finite dimensional vector spaces. Since I am not an expert on the theory of measures would you please explain to me if/how this concept might translate over to the setting of groupoids and vector spaces? Thanks.

Posted by: Charlie Stromeyer Jr on September 28, 2007 5:22 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Charlie wrote:

Chris Isham told me via email that I might consider the theory of measures on infinite dimensional vector spaces. Such a measure can be thought of as an inverse limit of measures defined on finite dimensional vector spaces. Since I am not an expert on the theory of measures would you please explain to me if/how this concept might translate over to the setting of groupoids and vector spaces?

I’m not sure what you’re asking here, so I’ll just blabber random words.

The concept Chris Isham mentioned is often called a ‘cylinder measure’. Cylinder measures are very important in quantum field theory, where one wants to integrate over infinite-dimensional vector spaces. I believe there’s a good introduction to cylinder measures in the first volume of Choquet-Bruhat et al’s Analysis, Manifolds and Physics.

Along with Irving Segal and Zhengfang Zhou, I helped write a book which includes a very detailed treatment of cylinder measures. However, Segal demanded that we call them ‘distributions’ instead of cylinder measures. You can get this book free online — but unless you know a fair amount of analysis, you’ll pay a high price in blood, sweat and tears when you try to actually read it, since I helped write it before I learned the importance of a gentle expository style, and Segal didn’t believe in babying the reader.

In a feeble attempt to make sense of your actual question, I’ll say that Jeff’s paper thoroughly groupoidifies the quantum harmonic oscillator with one degree of freedom — which amounts to groupoidifying the theory of integrals with respect to a Gaussian measure on \mathbb{R}. Generalizing to finitely many degrees of freedom is trivial, and it amounts to groupoidifying the theory of integrals with respect to a Gaussian measure on n\mathbb{R}^n. Everything still works when n=n = \infty, but then the Gaussian measure is really a cylinder measure.

Posted by: John Baez on September 28, 2007 11:31 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Thanks, John, because you did answer my question and because you taught me a new concept. As you could tell, I had not heard of ‘cylinder measure’ before, and nor have I read Jeff’s paper that you referred me to. I may have some more question(s) on this topic after I look at some relevant papers at the MIT Science Library.

In the meantime, here is something for Urs Schreiber: I found a recent paper about von Neumann bimodules by Michael Skeide.

Posted by: Charlie Stromeyer Jr on September 29, 2007 5:35 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

John, since you are a top expert on cylinder measures and quantum gravity you may want to look at this paper called “Quantization by non-Abelian promeasures” by C J S Clarke in J. Phys. A: Math. Gen. 23 (1990) pp. 4463-4470. The abstract is:

“A new method is proposed for the non-perturbative quantization of certain nonlinear field theories (group-bundle theories), based on a generalization of the idea of a promeasure from vector spaces to infinite-dimensional Lie groups. The quantum theory is not automatically finite, but there is a natural way of imposing a momentum cut-off, leading to the possibility of renormalization. The method relies on the geometrical structure of the classical theory and so may provide clues for the quantization of gravity.”

I am just now looking at this paper for the first time and will try to think if it has any meaning in a gerbed context (and I would welcome any advice on this matter).

Also, for Urs Schreiber: I have found an example of Hopf-bimodules based upon Hopf-von Neumann algebras within the context of quantum groupoids in the work of Michel Enock, e.g., in his paper “On Lesieur’s Measured Quantum Groupoids” which is in the arxiv as 0706.1472

Later, I will say something about how von Neumann algebras are related to the quadratic operad of Jordan Triple Systems which I mentioned earlier in this thread.

Posted by: Charlie Stromeyer Jr on September 30, 2007 5:13 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Charlie wrote:

C J S Clarke wrote:

A new method is proposed for the non-perturbative quantization of certain nonlinear field theories (group-bundle theories), based on a generalization of the idea of a promeasure from vector spaces to infinite-dimensional Lie groups.

Interesting. My work on loop quantum gravity also used a generalization of promeasures (= cylinder measures) from vector spaces to certain infinite-dimensional Lie groups. If you’re interested, try this:

I wonder which infinite-dimensional Lie groups Clarke was studying…

Posted by: John Baez on October 1, 2007 5:37 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Clarke uses his own “kink theories” but I have not yet looked at his earlier paper he refers to which rigorously describes these theories that he says “are ones where the set of field values at a point forms a Lie group, with the group operation being independent of any choice of gauge. The collection of all these field values at all points forms, in geometrical language, a fibre bundle whose fibres are groups.” In his approach, what is needed is the gauge-invariance of the promeasures.

Posted by: Charlie Stromeyer Jr on October 1, 2007 8:40 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

John, while you are at your conference, I learned some things that I want to share with the Cafe. The first two involve measures and an answer to your question about which Lie groups C J S Clarke was studying. The third is an affirmative answer to my question earlier in this thread from the mathematician Steve Vickers.

1) The limit arising within von Neumann’s continuous geometry is called “profinite” and you can read more about profinite topology and groups in this wikipedia entry on profinite groups.

In category theory terms, this is a special case of a (co)filtered limit construction.

Berkeley mathematician Mark Haiman generalized von Neumann’s concept in his paper “On realization of Bjorner’s ‘continuous partition lattice’ by measurable partitions” in Trans. Amer. Math. Soc. v343(2) (1994) 695-711. Abstract:

Bjorner showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Bjorner’s construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as a lattice of measurable partitions. This new lattice contains the Bjorner lattice and shares its key properties. Furthermore its automorphism group is the full automorphism group (mod 0) of the unit interval with Lebesgue measure, whereas, as we show, the Bjorner lattice possesses only a proper subgroup of these automorphisms.

2) C J S Clarke studies ‘kink theories’ which “are ones where the set of field values at a point forms a Lie group, with the group operation being independent of any choice of gauge. The collection of all these field values at all points forms, in geometrical language, a fibre bundle whose fibres are groups.”

As an example, in his “Quantization by non-Abelian promeasures” paper he uses the (3+1)-dimensional sine-Gordon equation over a fixed (curved) spacetime as a bundle with fiber SO(n) or SO(p,q).

Then, “If we take the Lie algebra of the group formed by the field values at a point, and do this at every point, we get a related field theory whose bundle is a vector bundle. Linear equations of motion in this Lie-algebra theory correspond to nonlinear equations in the Lie-group-bundle theory, in a natural way.”

At the end of the paper, Clarke has to generalize a Gaussian distribution to a nonabelian Lie group, and does so with a heat equation kernel.

This allows him to define L^2(G,u_q’) for the quantum Hilbert space. Since I am not a physicist, I will ask the question:

Does this space L^2(G,u_q’) have any physical significance?

3) Earlier in this thread, I asked whether there could be an infinite X as the limit of increasingly large finite instances of X.

Steve Vickers says this is possible, and here I take the liberty of pasting his reply so that you know to whom it is attributed:


Dear Charlie,

Jon Awbrey already hinted at an important part of the background in his reply to you [over three years ago in [1]]:

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Topological Information Spaces

Charlie,

Will look up some references later –
or you can probably search-engineer
your own well enough – but there’s
a lot of topological thinking in the
Dana Scott, Barwise, Seligman terrain
of thought, going way back to the 60’s,
partly due to the topological character
of “information domains” that constitute
complete partial orders, if memory serves,
which it often doesn’t.

Jon Awbrey

Dana Scott used his (information) domains to interpret computable functions as continuous maps. The approach is now well established in computer science, with applications (“denotational semantics”) to giving mathematical meanings to computer programs.

The domains are non-Hausdorff and can appear strange to a traditional topologist. They make essential use of the specialization order, x specializes y if y in the closure of {x} - in other words, every open containing y also contains x. (This is just equality in a T1 space.) High in the specialization means high in information content (more open neighbourhoods), and the low points can often be understood as approximations to the high ones. This leads to your “infinite X is the limit of a sequence of increasingly large finite instances of X”.

Steve Vickers.

Posted by: Charlie Stromeyer Jr on October 5, 2007 12:26 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Thanks to what Steve Vickers told me, I am now reading about specialization order and Scott topology in the book “Continuous lattices and domains” by G. Gierz et al.

Speaking of lattices, the above paper by Mark Haiman refers to a paper by V.A. Rohlin called “On the fundamental ideas of measure theory” in Amer. Math. Soc. Transl. 1(10) (1962) 1-54 which includes, e.g., a fully developed theory of measurable partitions of the real unit interval.

Also, speaking of lattices again, Vaughan Jones told me via email that Kaplansky proved that a complete orthocomplemented modular lattice is a continuous geometry. For anyone who might be interested, this lattice is related to quantum probability theory in the work of Zdenka Riecanova, including a recent generalization of the concept.

Posted by: Charlie Stromeyer Jr on October 6, 2007 7:20 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

In reply to what Steve Vickers said above, Professor Dana Scott told me via email about his initial example in the history of (profinite) interval analysis - a subject which is new to me but which Bas Spitters is an expert on.

More specifically, Prof. Scott referred me to the work of Prof. Abbas Edalat which is fascinating because it contains a variety of examples of profinite maths. For instance, Prof. Edalat defines a notion of derivative of a real-valued function on a Banach space called the L-derivative which is constructed by a generalization of Lipschitz constant of a map. For functions on finite Euclidean spaces, any continuous function and its L-derivative are profinite.

This is cool because I am trying to understand whatever I can about profinite maths. With the sole exception of von Neumann’s continuous geometry, the subject of profinite maths is new to me, and I like this wikipedia entry on profinite topology and groups because both the links within and the references below are clickable.

Posted by: Charlie Stromeyer Jr on October 10, 2007 5:19 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

There’s now an official webpage for this conference, which contains a subliminal advertisement for the nn-Category Café.

Posted by: John Baez on September 26, 2007 3:43 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Here’s a nice photo of some of the conference participants. This photo was taken by Jamie Vicary, who therefore alas remained invisible himself. Thanks for the self-sacrifice, Jamie!

From left to right: Simon Kochen, Jeffrey Bub, Bob Coecke, Peter Woit, John Baez, Mike Stay, Andreas Döring, Camm Maguire and Chris Heunen.

Posted by: John Baez on October 9, 2007 3:51 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

No wonder Dr. Bub didn’t respond to my request for some face time while I was up near College Park this weekend. Maybe in December…

Posted by: John Armstrong on October 9, 2007 4:13 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

A second version of Stephen Summer’s contribution – Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State – is available.

An idle thought: might we be able to think of the vacuum state in a galoisian way? Or, in Lautmannian terms, is there a perfectedness to the vacuum state?

Posted by: David Corfield on February 23, 2009 12:14 PM | Permalink | Reply to this

modular objects determined by the vacuum state; Re: Deep Beauty: Understanding the Quantum World

“… modular objects determined by the vacuum state and algebras of observables localized in certain regions of Minkowski space encode a remarkable range of physical information, from the dynamics and scattering behavior of the theory to the external symmetries and even the space–time itself….”

Wow!

Posted by: Jonathan Vos Post on February 23, 2009 6:22 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Does anybody Know Where could I find the book “Deep beauty” edited by Halvorson, to download online???

Posted by: Jacobo Cabrera Cifuentes on March 12, 2011 7:32 PM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

I don’t know if you should be asking this here.

Posted by: David Roberts on March 14, 2011 4:33 AM | Permalink | Reply to this

Re: Deep Beauty: Understanding the Quantum World

Of course he shouldn’t. It’s illegal! And I’m one of the authors of this volume! Do you post comments on musician’s websites asking how you can illegally download their albums?

On the other hand, some chapters from this book are freely and legally available online. For example:

The best place to look for others is the arXiv.

Posted by: John Baez on March 14, 2011 5:34 AM | Permalink | Reply to this

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