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Cat -> Quant (TWF 236, Morton's thesis)

  1. Jul 26, 2006 #1

    marcus

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    TWF 236 is out.

    http://math.ucr.edu/home/baez/week236.html

    It has a plug for Morton's thesis. We should keep an eye on what is happening with people using categorics to make sense of quantics.

    there seem to have been revelations: a veil falling now and then
    lightly to the ground

    the link to Morton's thesis is
    http://xxx.lanl.gov/abs/math.QA/0601458
    http://arxiv.org/abs/math.QA/0601458

    an exerpt from the abstract is:
    "... We will show how to construct a combinatorial model for the quantum harmonic oscillator in which the group of phases, U(1), plays a special role. We describe a general notion of "M-Stuff Types'' for any monoid M, and see that the case M=U(1) provides an interpretation of time evolution in the combinatorial setting, as well as other quantum mechanical features of the harmonic oscillator."

    the thesis is apparently slated for publication in a categorics journal, I will just quote TWF:

    "...However, none of this work dealt with the all-important phases in quantum mechanics! For that, we'd need a generalization of finite sets whose cardinality can be be complex. And that's what my student Jeffrey Morton introduces here:

    24) Jeffrey Morton, Categorified algebra and quantum mechanics, to appear in Theory and Application of Categories. Also available as math.QA/0601458.

    He starts from the beginning, explains how and why one would try to categorify the harmonic oscillator, introduces the "U(1)-sets" and "U(1)-stuff types" needed to do this, and shows how the usual theorem expressing time evolution of a perturbed oscillator as a sum over Feynman diagrams can be categorified. His paper is now the place to read about this subject. Take a look!"

    "TAC" is an online journal:
    http://www.tac.mta.ca/tac/

    We need Kea to provide some guidance here. At least someone who understands it better than I do.
     
    Last edited: Jul 26, 2006
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  3. Jul 26, 2006 #2

    marcus

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    BTW to a large extent I think the "philosophy" of something consists of finding different ways to look at it.
    philosophy of morals basically means finding different ways of looking at morals----ethics

    philosophy of science means different ways to look at science---what is science, what is not science etc.

    philosophy of physics, or of spacetime issues, will be asking basic questions like what is space? what is time? what is a physical theory?

    at times people have to get CREATIVE and find different ways to look at their things----or they cant get out of the box and solve their hard problems.

    Newton, it is said, was philosophically imaginative and asked fundamental questions about what is space and what is time, and arrived at some completely new (probably wrong) ideas which provided the philosophical basis for physics to advance for 300 years. At some point somebody has to get the completely new (probably wrong) ideas of what the basic things are, so that progress gets unstuck.
    Frequently these people strike others as excessively imaginative, but the proof of the pudding.

    Well now some of us are scandalized by categorics because it strikes us as excessively imaginative and what does all that have to do with physics:biggrin:

    But I think that what categorics is is this: it is simply how THE PHILOSOPHY OF MATHEMATICS appears to us at this historical period----that is as a bunch of people finding new ways to think about sets functions and numbers and hilberts and manifolds.

    And probably it is necessary to have this concerted effort to get a new take on the basic things-----it tends to lubricate people's minds and make them more nimble and inventive. (which they probably need to be at this stage in history).

    this is my personal nonexpert hunch.
    =======================

    so the illustration of what I'm saying is that you look at TWF 236 and see that some 80 percent seems to be about DIFFERENT WAYS OF LOOKING AT THE ORDINAL NUMBERS and trying to understand why a Greek would construct an ICOSAHEDRON
    in other words an attempt to capture the perspective of classical wisdom
    which we do by invoking the ancestors (personally I pray to Kepler:smile: )
    and then after about 95 percent has gone by, he says BY THE WAY THERE IS A DIFFERENT WAY TO LOOK AT A QUANTUM OSCILLATOR

    and by the way the creation operator and the annihilation operator have a COMBINATORIAL ANALOG involving putting balls in urns and taking them out again. to inspire is sometimes to infuriate.

    and then when 98 percent has gone by he says BTW check out Morton's thesis, you might get a new idea about quantum mechanics.
     
    Last edited: Jul 26, 2006
  4. Jul 26, 2006 #3
    ***
    Well now some of us are scandalized by categorics because it strikes us as excessively imaginative and what does all that have to do with physics:biggrin: **

    Hehe, it does not strike me as ``excessively imaginative''; look Marcus it is not hard at all to be imaginative/creative, the difficulty is to have a new idea that leads somewhere (having many ideas which were tested already is a proof of sanity) - and here history proves that good ideas had almost immediate impact. Do not take me wrong, no-one doubts that CT can offer some more abstract way of looking at things, and the formulae and diagrams do capture the eye. I liked the Bob Coecke paper (about quantum teleportation) John referred me to (only read the first 12 pages though) but I could hardly call it surprising from the physics point of view. I believe the task of physicists is to communicate pictures of things which can be understood by people `with a healthy peasant brain''. Sometimes a new language can shed new light upon a problem, but this is rarely so... and in any case new ideas are required. So, a good way to start would be to explain the new point of view behind the harmonic oscillator - 't Hooft offered such new perspective by, for example, considering a particle moving on a circle - or by considering a classical quantum system with a quantum constraint (Kleinert, Jizba and others have worked on that further). So, you dig into this and give the picture !

    Careful
     
    Last edited: Jul 26, 2006
  5. Jul 26, 2006 #4

    Kea

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    Sweet, Marcus, but I haven't actually looked at Morton's thesis.

    At the end of the day, I think there will be many ways to look at oscillators in an n-category context. I like to think of [itex]\mathbb{N}[/itex] as an object in the topos Set. The existence of this object is one of few axioms that characterises set theory completely. To do quantum physics we need to 'categorify' our toposes. Going to monoidal structures we see straight away how numbers are like dimensions of spaces.

    Imagine we are working in Cat. Remember that this is the analogue of Set one dimension up. This has a funny tensor product (Batanin used this term in his seminar yesterday, so it must be kosher). That is, if we compare [itex](a \otimes 1)(1 \otimes b)[/itex] to [itex](1 \otimes b)(a \otimes 1)[/itex] we don't get the same thing! This structure on Cat isn't the usual one, but it is more topological (meaning that homotopy theorists like it).

    Computer scientists like the funny tensor product too, because it helps understand program sequencing. That is, they use it to think about time in a computational setting. The broken interchange law is allowed for sesquicategories, which are like one-and-a-half dimensional pictures! For tensors of 2-dimensional categories (instead of 1-cats) we would end up with Gray tensor product, which we link to a physical notion of time via its role in mass generation. In both cases the breaking is associated to a quantum parameter: for 1D with [itex]\hbar[/itex] and for 2D with a deformation [itex]q[/itex], which is sometimes thought of as associated to exponentiation of [itex]\hbar[/itex], just like we would expect in going from one basic product to a higher level structure.

    Batanin was talking about trees with labels (really he was talking about something called coloured operads). Using [itex]\mathbb{N}[/itex] for labels along with the funny tensor product one gets an operad describing directed paths on a square lattice, and this can be related to some fancy Hochschild complexes and stuff, which is really very useful for calculating things - like Feynman graphs! That's right...take the simple physicist's way of doing things and get generalised (co)homology, which we can now justify on cosmological grounds.

    I think this is pretty cool.
     
    Last edited: Jul 26, 2006
  6. Jul 26, 2006 #5

    Kea

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    Yes, but what is a particle? What does the circle represent? Careful, we are thinking seriously about answering the questions raised by such so-called explanations.
     
  7. Jul 26, 2006 #6

    Kea

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    P.S. For the String theorists out there: Batanin's talk was really about some recent work of Tamarkin entitled What do DG-categories form? It turns out that one wants something that looks locally like an [itex]A_{\infty}[/itex] category, which is an [itex]E_{1}[/itex] structure in the sense that the square lattice paths mentioned above only need to have one bend in them. This is all about clever ways of understanding what weak n-categories should really be.
     
  8. Jul 27, 2006 #7

    marcus

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    Permit me to recommend it! After looking at it on the screen quite a bit yesterday, I decided to print it out.

    It is unusually well written. Largely self-contained (it has a pedagogical aspect). Explains things as it goes along without being tiresome.
    Aimed at communicating to both physicists and categoricians.

    ==============
    some more thoughts:
    combinatorics is about counting how many structures of some given type
    you can ENRICH combinatorics (make it more fun) if you contrive new structures to count!
    categorics inspires people to invent new structures and helps them be more inventive, so
    it will inevitably stimulate combinatorial ingenuity.

    ANOTHER WAY TO ENRICH combinatorics is to use the complex numbers (in place of the non-negative integers) for counting. To use better cardinals in other words. To include "phase" in the enumeration of things.
    One could almost define a mathematical field of research called "Quantum Combinatorics", by saying it is what you get if you do combinatorics but include phase in the enumeration.
    the idea "phazy cardinals" comes to mind.

    maybe this bunch of ideas is well known or familiar already, but not to me. I never heard of quantum combinatorics before. Now as I take another look at Morton's thesis I imagine that he is trying to invent a research area called quantum combinatorics. I see him devising what I would call "phazy cardinals" to count something he calls
    "U(1)-groupoids"
     
    Last edited: Jul 27, 2006
  9. Jul 27, 2006 #8
    :bugeye: So these are the questions you are worried about ??
    The circle could be the (imaginary) path traced out in the external environment (like the electron in the bubble chamber). Nobody knows what a particle is (and for someone who tinks QM you suddenly use ``to be'' quite often), the best one can do for now is to think about what a particle detection could be like ; the latter question being more than complicated enough (or why it is legitimate to think in terms of particles to start with). So, could you explain us what the new insight concering the harmonic oscillator is ? That tickles my curiosity.

    Careful
     
    Last edited: Jul 27, 2006
  10. Jul 28, 2006 #9

    Kea

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    So, it sounds like you now agree that understanding measurement is at least as fundamental as an a priori notion of particulate. We say that the mathematics of measurement involves the mathematics of logic (with category theory we get geometry and algebra as well). I fail to see how this can be controversial. A basic characterisation of particles in terms of simple propositional properties is an old idea, but getting it to work alongside geometry is tricky. It is no small fact that Gray's development of higher tensor products (for system combination) was based on a study of categorical analogues for basic set theoretic axioms, in particular the one pertaining to the existence of a set containing things with a given property (this is just the global version).

    As for particle creation etc, the monadic nature of the ideas mentioned above is a completely different way of thinking about oscillators to anything else of which I have heard.
     
  11. Jul 28, 2006 #10
    **So, it sounds like you now agree that understanding measurement is at least as fundamental as an a priori notion of particulate. **

    :frown: But Kea, I have said at least 25 times that understanding the interaction between microscopic and macroscopic objects is extremely important vis a vis the single event interpretation of QM (and the distinction between repeated single events and simultaneous multi particle events). However, this does NOT imply that I say that measurement is of some fundamentally different nature than the materialistic world around us : this excludes Copenhagen, MWI, Rovelli relational QM etc... . Really, ask John Baez what I am talking about, he should know some of this given his MIT past. What I am suggesting here (and others did for long time) is the possiblility of a very different physics behind the measurement in QM, not some fancy way of saying the same things differently.

    You are still mistakenly assuming that I am malicious towards CT, I have on the contrary always been open, which does not imply the latter needs to be mindless though. Consider that 95 percent of physicists would not even imagine asking about it.

    Careful
     
    Last edited: Jul 28, 2006
  12. Jul 29, 2006 #11

    john baez

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    thesis???

    No it doesn't! Jeffrey Morton has barely started writing his thesis, which will be about topological quantum field theory which include particles or strings as `topological defects'. This will make precise the 2-categorical idea I keep talking about:

    matter = objects
    space = morphisms
    time = 2-morphisms

    Morton's new paper is on another topic. He wanted to write about something else before getting into his thesis work, so he decided to tackle the interface between combinatorics, categories and Feynman diagrams.

    Jeff is slated to finish his thesis next June. It's time for him to get cracking.:grumpy:
     
    Last edited: Jul 30, 2006
  13. Jul 30, 2006 #12

    marcus

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    My error. Cant keep track of all your students and collaborators. Interesting group of people and ideas. Nice he got it accepted for publication in the Category Theory and Applications journal.
     
    Last edited: Jul 30, 2006
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