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Category theory and beyond the standard model

  1. Dec 24, 2006 #1

    CarlB

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    Kea tells me that a good first book to read to learn category theory is

    Sets for Mathematics $45
    F. William Lawvere and Robert Rosebrugh
    Advanced undergraduate or beginning graduate students need a unified foundation for their study of mathematics. For the first time in a text, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specifcation of the nature of categories of sets.

    Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion. The distinctive features of Cantorian abstract sets, as contrasted with the variable and cohesive sets of geomtry and analysis, are made explicit and taken as special axioms. Functor categories are introduced to model the variable sets used in geometry and to illustrate the failure of the axiom of choice. An appendix provides an explicit introduction to necessary concepts from logic, and an extensive glossary provides a window to the mathematical landscape.
    http://www.amazon.com/Sets-Mathematics-F-William-Lawvere/dp/0521010608/

    It is also available in hardback, but I have the paperback, linked above.

    When I opened the book and read at random, I was briefly convinced that it wasn't telling me anything I didn't already know. All the usual undergraduate analysis is here. It just doesn't seem very difficult. On the other hand, Kea tells me that category theory has great possibilities for moving beyond the standard model.

    This combination, ease of material, and power of use, make the book very attractive to me. You can be sure I will be reading it, and making comments here. Hopefully Kea will drop by and contribute to how this is related to braids and all that.

    I can hardly wait to see what sort of connection there is between the axiom of choice and physics, though I always had my doubts about how physicists treated "infinity".
     
    Last edited: Dec 24, 2006
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  3. Dec 24, 2006 #2

    john baez

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    It's not supposed to be. As the the title suggests, it's an introduction to the manipulations with sets that mathematicians use in daily life - which are best explained using category theory.

    Nothing about braids, tangles, topological quantum field theory, or spin foams here! For that, you'll have to go elsewhere... but, if you read this book, you'll absorb some of how category theorists think about math. Products, coproducts, equalizers, coequalizers, pushouts, pullbacks, and exponentials - all that kind of important basic stuff.
     
  4. Dec 24, 2006 #3

    CarlB

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    It appears that the majority of these things are stuff that I already knew, but with the following changes, so far:

    (1) Old terminology has become obsolete. What I called a "range" in 1976 is now called a "codomain". And there's no entry in the index for "open", but I know they discuss topology.

    (2) There is a very strong concentration on mappings.

    (3) A lot of things I was taught in separate classes are now united under one topic. For example, De Morgan's law and differentiation.

    (4) It seems that some of the concepts of computer science have leaked into the definitions.

    I guess I can see how some of this might be useful in physics. I recall that Kea told me that what got her interested in this was in the improvement in the traditional sloppy physics use of infinite sets. The stuff I'm doing right now involve discrete degrees of freedom and these problems don't occur, but my next book has to deal with this.

    The concentration on mappings, rather than sets and functions (which reminds me of digital design), is a unification in the sense that the number of types of objects has been reduced to just one (a logic block that has inputs and outputs). This is something I wholeheartedly agree with. It is the reason I geometrize the density operator formalism rather than following Hestenes and geometrize state vectors. Hestenes has to geometrize two types of objects; I only have one, it is trivial, and unlike Hestenes' spinor geometrization, the geometrization of the operators is unique. To see the confusion / lack of uniqueness / that comes from trying to geometrize spinors, see yet another great article by Baylis:
    http://www.arxiv.org/abs/quant-ph/0202060

    So far, the changes in terminology seem like an improvement, but where I found (graduate) abstract algebra the most difficult was in the huge number of definitions that proliferated in it. I could work homework problems by writing down enough definitions, but I could / would never memorize enough to survive a test. So I'm a little worried about how many definitions they are going to throw at me. I can't imagine being 50 is going to help.

    Hey, if there's someone else wants to learn this, post a message and I'll quit reading till you've got your copy. At $45 it's an inexpensive textbook.

    At this point, I'm quite convinced that the way you guys are going about this is difficult, divorced from physical principles, and will leave you forever unable to make much in the way of useful calculations. The problem is the same one that has plagued QM for decades. Symmetry principles give too many arbitrary constants. Abstract mathematics isn't going to solve that problem. You will always be drowning in experimentally determined constants.

    You need to use geometry, not symmetry, at the foundations of physics; and try to describe what is, rather than what its symmetries are. You need to postulate simple equations of motion, rather than simple symmetries (that then have to be broken, LOL). All this means giving up too many things that you know are true. Like the fist-trapped raccoon, all you have is a shiny bauble in your paw.
     
  5. Dec 25, 2006 #4

    Kea

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    A jolly hello from a well fed Kea!

    I'm sorry Carl, if you end up finding the book a waste of time because it's too simple. I had the impression that you wanted to build up a collection of books starting from the bottom up, and one can't do better than Lawvere et al, who are masters. I've seen the book in a library, but I don't actually have a copy.

    Naturally, I agree with Carl about symmetry (:mad:), but until we have the full derivation of lepton masses worked out one can't really expect people to drop such ingrained prejudices. I was just looking through an old QFT textbook from the 1970s. It is striking how convinced the authors were about massless neutrinos given the helicity argument in the context of parity violation.

    When I get a computer again, I will try and draw some diagrams here. Unfortunately, I don't know when that will be. Happy New Year.

    :smile:
     
  6. Dec 25, 2006 #5

    john baez

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    Merry Christmas to you too!

    I don't just study symmetry, by the way - I love geometry and think about it a lot. This hasn't succeeded in letting me figure out the Theory of Everything, but I'm probably just being dumb.
     
  7. Dec 25, 2006 #6

    CarlB

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    Oh no! The book is exactly what I asked for and exactly what I needed. I'm quite stupid and have to learn things by starting at the very very bottom. Learning QM was very very difficult for me because they start out at such an abstract level.

    From reading Lawvere, I realized that my problem is not with symmetry in physics per se, but instead excessive abstraction in general. As one gets away from the physics, one ends up having difficulty distinguishing between reality and imitations of reality. In this, I guess I am becoming more and more of an admirer of Newton.

    And as far as breaking down ingrained prejudices, you'll be happy to hear that my New Years resolution is to get published in peer reviewed journals. I will break the derivation out into about 12 pieces and publish separately. It will be great entertainment.

    They had very good reason for believing in zero neutrino masses. A great (though not yet released) source on this is

    The Standard Model, A Primer
    Cliff Burgess and Guy Moore 2007
    In this chapter we encounter the first (and, as of this writing, only) known case where there is good evidence that the standard model does not provide a good description; the phenomenon of neutrino oscillation. (p 395)
    ...
    Requiring [tex]k_{mn}\nu[/tex] to be this small requries the eigenvalues of the matrix [tex]k_{mn}[/tex] must be [tex]< 10^{-11}[/tex]. Indeed, the cosmological bound constrains the sum of eigenvalues, and so requires [tex]\tr k_{mn} < 4 \times 10^{-12}[/tex]. This is to be contrasted with the other Yukawa coupling matrices, for which [tex]\tr f_{mn} ~ 10^{-2},[/tex] [tex]tr h_{mn} ~ 2.5 \times 10^{-2},[/tex] and [tex]gr_{mn} ~1.[/tex] Of course we do not understand why any of the fermion masses are what they are :rofl: , but one must always pause when arbitrarily setting dimensionless couplings to be zero to more than ten decimal places.

    These same models similarly require extremely small choices to be made for the dimensionful Majorana mass parameters, [tex]M_n.[/tex] For instance, for the Dirac-neutrino to be correct the masses, [tex]M_n,[/tex] must satisfy the extraordinary bound [tex]M_n < 10^{-20}\mu,[/tex] where [tex]\mu[/tex] is the mass parameter appearing in the Higgs potential, see Eq. (2.26), and indeed is the only other mass parameter at all in the standard model. By contrast, the normal situation in the standard model is that all couplings which can be present on the grounds of symmetry and renormalizability are actually measured to be nonzero. The sole exception is the [tex]\Theta_3[/tex] term in Eq. (2.13), which satisfies [tex]|\Theta_3| < 10^{-9}.[/tex] (The puzzle as to why this is so is discussed in Section 11.4) A Dirac-neutrino scenario must explain why the same should be true (with extraordinary accuracy) for the Majorana neutrino masses.
    ...
    In the standard model without right-handed neutrinos, the hypercharges of all particles are determined uniquely (up to overall normalization) by gauge-anomaly cancellation, as we saw in Subsection 2.5.3. This guarantees that the quark hypercharge assignments ensure that the neutron electric charge is precisely zero. [note, compare with Sweetser's gravity and electromagnetic unification theory, which denies that the neutron's electric charge is not quite exactly zero] Adding an N field to each generation adds a new contribution to Eq. (2.131), which turns out to make it hold automatically. [which is what you want, no?] In the theory with an N particle added, the anomaly-cancellation conditions remain satisfied even if we shift each particle's hypercharge y by d times its B-L charge, for any d. Such a shift would change the electric charge of the neutron by d. Since the neutron's electric charge is measured to differ from zero by no more than [tex]10^{-21},[/tex] a theory with an N field and exact B-L conservation requires an extraordinary fine tuning of particle hypercharges. (On the other hand, when a Majorana-mass term like [tex]M\bar{N}N[/tex] is present, gauge invariance of that term demands d=0, and so again ensures the exact neutrality of the neutron.)
    (pp 423-4)
    http://www.amazon.com/Standard-Model-Primer-Cliff-Burgess/dp/0521860369

    Uh, the comments in [ ] and the laughing face are my annotations to the original text.

    In short, the expectation that the neutrino mass was zero is the same defect which accompanies other attempts to enforce symmetry (beauty) onto non symmetrical situations. The circle is symmetrical, the earth's orbit is circular (to first order), therefore we assumed that the orbits of planets are exactly circular. More observations, and you get epicycles. And later, if nothing else had been found, we'd have been using epicycles on epicycles. As soon as you admit "symmetry breaking" in your toolbox, you can do any old thing with it.

    I have an early copy of the book because I bought a (defective) copy from the publisher at the DPF meeting in Hawaii.
     
    Last edited: Dec 25, 2006
  8. Dec 25, 2006 #7

    CarlB

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    Sorry if I confused you with the John Baez who asked what the fool's golden ratio was:
    https://www.physicsforums.com/showthread.php?t=143745

    whose question I answered precisely in post #6 of the above, but who then managed to write a "This Week’s Finds in Mathematical Physics" that didn't mention my contribution here:

    http://math.ucr.edu/home/baez/week241.html

    By the way, I added a page with photos of pyrite crystals showing the two types of pyritohedron, with faces labeled, here:
    http://www.brannenworks.com/xtl/pyrite/

    I sent a link to the above to sci.physics.research, but the moderators threw it away. This sort of treatment of amateurs is quite common in the physics community, and believe me, if they expect better treatment the other way around, they are living in a fantasy land. Humans aren't like that.

    As a good start in understanding geometry, you would be well advised to study how geometry provided the first evidence for the existence of atoms, namely, the fact that the angles of crystal faces can be defined by rational ratios, the Miller Indices. Anyone who has to have an amateur explain crystal faces to them certainly has quite a lot left to learn about the application of geometry to physics.

    The most elegant way of defining crystal faces in geometry was discovered fairly recently, by David Hestenes, and relies on his geometric (Clifford) algebra:
    http://modelingnts.la.asu.edu/html/SymmetryGroups.html
     
    Last edited: Dec 25, 2006
  9. Dec 26, 2006 #8

    CarlB

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    One can take pretty much any mathematics and call it "geometry" by putting the right spin on it. When I say "geometry", what I mean is stuff that has an immediate and obvious meaning in terms of things that would be recognized as "geometry" by the Egyptians who built the pyramids.

    With regard to the geometry of a wave function, I am saying that you have to tie the theory back to the set of stresses you can place onto the manifold you choose for spacetime. For example, with the geometric algebra, a scalar corresponds to a localized compression (if positive) or a rarefaction (otherwise). A vector corresponds to a local stress in the given direction. A bivector implies a localized region rotated in a manner defined by an oriented plane. A trivector implies a localized region which is hard to describe, but is the only remaining localized deformation.

    All these stresses are easy to demonstrate with rubber bands and balls in 3 dimensions and are quite intuitive. You can show the equivalent in 2 dimensions by twisting a paper napkin in various directions. That their generalization to 5 dimensions is sufficient to derive the elementary particles means a hell of a lot.

    The "trouble with physics" is that it is an accreted science and has become divorced from physical reality and physical intuition. Graduate students who want to understand reality are told to "shut up and calculate."

    Many years ago, one of my students told me that I should read Herman Hesse's book "The Glass Bead Game". I was a young man with a young man's love and respect of science and mathematics, and I thought her advice was a bit insulting. Perhaps for that reason, I didn't read the book immediately, even though I loved pretty much all of Hesse's other books. It was only after a decade or two that I got around to reading "The Glass Bead Game".

    The "Glass Bead Game" is set in the distant future. It is set in a sort of a giant university which is devoted to the life of the mind. The intellectuals who live there play the Glass Bead game. The rules of the game are never specified, but they are known to be so sophisticated that they are difficult to imagine. Wikipedia calls it an "abstract synthesis of all arts and scholarship". It proceeds by players making deep connections between similarly unrelated topics." Sound familiar???

    Part of the theme of the Game is that none of the practitioners have any desire to make the game easier to learn. It reminds me of the distate that Schwinger supposedly had for Feynman's "bringing field theory to the masses". Of course I am also a great worshiper of Schinger and the stuff I am doing is based on his "Measurement Algebra" that also treats the vacuum as a convenient mathematical fiction. And now that I think of it, I am a great worshipper of Einstein and Feynman too.

    The problem arises from assuming that the course of physics to the present day cannot possibly have had any errors, for example that the principles of symmetry have been proven beyond all doubt.

    To the extent that QM and GR do not fit cleanly with each other and can be united, each theory impeaches the other. Going to higher levels of abstraction can hide the incompatibility in the ontology of the two theories, but they cannot possibly repair it.

    The inherent arrogance of people in academia is based in a certain part on the fact that the competition for academic jobs is fierce, therefore the survivors must be the very best, LOL. Of course this is nonsense of the sort that can be proposed only by someone who can ignore basic economic theory. If you want to see examples where only the very best are employed, you have to look in occupations where the pay is high, for example, pro sports.

    In short, what I'm saying here is that mathematical sophistication is a great thing for solving problems. But it makes a lousy foundation, no foundation at all, really. Every other science, for example, biology or chemistry, is based on a foundation of real objects that exist in three real dimensions and have real existence, with an ontology that makes complete sense intuitively. Physics does not. That is the "trouble with physics".

    Carl
     
    Last edited: Dec 26, 2006
  10. Dec 27, 2006 #9

    Demystifier

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    Not with the whole physics. Only with quantum physics, I would say.
     
  11. Dec 27, 2006 #10

    Hurkyl

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    It's rather arrogant to think that the biases one picked up when they were a little kid happen to give a completely accurate picture of physical situations one never experienced, don't you think?

    "Shut up and calculate" is only the first step -- it's not until you forget your prejudices about how the universe should operate, that you stand a chance to develop an intuition about how it actually does operate.


    (The same happens in mathematics too -- I've even seen a functional analysis textbook explicitly state in the preface that the reader should relearn linear algebra in the infinite-dimensional case -- he should forget all of the intuition he developed for finite-dimensional linear algebra because it's more likely to lead him astray than to be helpful)
     
    Last edited: Dec 27, 2006
  12. Dec 27, 2006 #11

    john baez

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    That's the same John Baez - me. I'm sorry I didn't thank you for answering that question. I've now corrected week241 to thank you for that help.
     
  13. Dec 27, 2006 #12
    Even though it is not directed to me, I shall respond to it just because I am fed up with such nonsense. First of all, CarlB didn't say anything about preconceptions from his childhood :rolleyes: so it is hardly fair to say so. Second, and this is what bugs me, every researcher has some set of prejudices about how the universe operates. No-one who is not sufficiently convinced about his/her (educated) picture of the world, is going to sacrifice a significant part of his/her life to this question and/or do good work with relevance for physics. The textbook example you give is irrelevant since people actually learned new tools were necessary to understand infinite dimensional algebra, so that they can now tell this to you. Concerning our quest for a deeper understanding of the universe, there is by far no such luxury yet (and indeed the very formulation of the problem allows for some liberties), so there is a set of ``valid prejudices'', but prejudices are needed nevertheless. If you want to do research in LQG -say- you need to be convinced that background independence is an indispensable ingredient to the puzzle.

    Careful
     
  14. Dec 27, 2006 #13

    CarlB

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    When I was a graduate student I agreed with you completely. But particle physics was still making real progress at that time, in terms of new predictions matching new experiments. Plus, I was naive like most youth, and, in the absence of the internet, I had been carefully exposed only to the arguments that were in favor of the accepted conclusions.

    Now, with the internet, it's a lot easier to hear the other side of things. For example, here's Julian Schwinger talking about a version of QM where the creation and annihilation operators are not physical, but instead are mathematical fictions convenient for calculation:
    http://www.pnas.org/cgi/reprint/46/2/257.pdf

    At this time, with physics a little stuck, I think it is time to look around and consider the possibility that things are not as they seem. As your post accurately describes, this is something we have all been through already. Are we ever anything other than naive little children? What I'm saying is that if we want to rewrite the foundations, we may have to do it again, and the natural place to start is with the assumption that our naive view of the world has something to do with reality.

    I think that the excessive abstraction of physics is best illustrated by the quote of Einstein from a few months before his death, something to the effect that the passage of time is an illusion, and that all time exists in the same object. What I'm proposing is that this is abstract nonsense, and that there is a "now", and that Einstein is dead, and that we are alive but soon will not be, and that all these are fundamental facts about the universe that are more significant than simply noting relative coordinates of two events in a particular choice of coordinates, t_a < t_b.

    There is a lot of discussion about the absence of an arrow of time in the physics equations. I think it is clear where the absence comes from, the equations were not written with one, so it can only appear statistically. One can then make a "formal argument" along the line that time is an illusion, etc.

    If these sorts of arguments had solved the various physics riddles that have survived this past quarter century, or even allowed a unification of QM and GR, then I think one could make a strong argument that formal nonsense is sensible. However, this is not the case.

    Carl
     
    Last edited: Dec 27, 2006
  15. Dec 27, 2006 #14

    CarlB

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    Dr. Baez,

    I hope you will accept my apologies for being snippy. It's difficult for amateurs and cranks to not walk around with a bit of a chip on their shoulders.

    Carl
     
  16. Dec 27, 2006 #15

    Hurkyl

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    Surely there must exist at least one objective researcher!

    One can be convinced for reasons other than preconceived bias. But I don't see why one must be preconvinced of something in order to do successful research anyways.

    Might not someone do research in LQG simply because they find it interesting? Or because previous work suggests it may be a fruitful approach? Why do you think a mind closed to the alternatives is a prerequisite for researching LQG?
     
    Last edited: Dec 27, 2006
  17. Dec 27, 2006 #16

    Hurkyl

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    You mean the one where heavy objects fall faster than light objects, the Sun orbits the Earth, and electrons race through power lines at high speed?

    That paragraph wasn't meant to be flippant. I'm trying to point out that this isn't a natural place to start: your choice is presumably a by-product of your experience and education.


    But this is getting far afield from the point I wanted to make, which was formal, not physical -- specifically, elevating one's intuition above one's reason, and the implication that just because you find something unintuitive, so must everyone else.


    Just to make it clear, I'm not saying you've chosen a bad place to start: I'm saying that you're making unwarranted statements in your rationalization of that choice.


    I don't see how one is more abstract than the other.


    Rereading your post again, I noticed this:

    This is one of the very things that the process of abstraction is meant to do -- you start with some notion, which may be very complicated, or it may be a poorly expressed intuitive notion. You study the notion to distill its essential features -- i.e. create an abstraction. Now that you know the essential features, you can focus your study on those. Also, someone who doesn't understand the complicated notion, or hasn't been able to learn your intuitive notion through osmosis, is able to start working with the abstraction, making it easier to build up their intuition.
     
    Last edited: Dec 27, 2006
  18. Dec 27, 2006 #17
    Does not exist. I am quite astonished I should hear this from someone who adopts the MWI interpretation of QM :biggrin:


    It depends upon what you call succesful and on the research you are doing.

    I did not say such thing, I said that you strongly need to believe in background independence in order to do LQG. Why ? Because I cannot come up with any other sensible reason.
     
    Last edited: Dec 27, 2006
  19. Dec 27, 2006 #18

    john baez

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    Sure! It's very noble of you to offer them. And, I'd be unhappy being on bad terms with the guy who told me the Miller indices of the pyritohedron!

    Yes, I know a bunch of smart people interested in math and physics who are not in academia, who have to put up with a lot of condescension from academics. But, as George Herbert said, living well is the best revenge.
     
  20. Dec 29, 2006 #19

    Kea

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    Carl, we're just old folks. Teenage undergraduates take courses in category theory now (in a few places, anyway). It it no more obscure to them than Clifford algebras or crystallography.

    Have you been into your friend's liquor cabinet? This is highly uncharacteristic. Happy New Year anyway!
     
  21. Dec 29, 2006 #20

    Hurkyl

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    Aww, I wish I could have taken a category theory course when I was a teenage undergraduate. :cry: (Or even have heard of it!)
     
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