This Week's Finds in Mathematical Physics (Week 257)

  1. Also available as http://math.ucr.edu/home/baez/week257.html

    October 14, 2007
    This Week's Finds in Mathematical Physics (Week 257)
    John Baez

    Time flies! This week I'll finally finish saying what I did on
    my summer vacation. After my trip to Oslo I stayed in London,
    or more precisely Greenwich. While there, I talked with some good
    mathematicians and physicists: in particular, Minhyong Kim, Ray
    Streater, Andreas Doering and Chris Isham. I also went to a
    topology conference in Sheffield... and Eugenia Cheng explained
    some cool stuff on the train ride there. I want to tell you about
    all this before I forget.

    Also, the Tale of Groupoidification has taken a shocking new
    turn: it's now becoming available as a series of *videos*.

    But first, some miscellaneous fun stuff on math and astronomy.

    Math: if you haven't seen a sphere turn inside out, you've got
    to watch this classic movie, now available for free online:

    1) The Geometry Center, Outside in,
    http://video.google.com/videoplay?docid=-6626464599825291409

    Astronomy: did you ever wonder where dust comes from? I'm
    not talking about dust bunnies under your bed - I'm talking
    about the dust cluttering our galaxy, which eventually clumps
    together to form planets and... you and me!

    These days most dust comes from aging stars called "asymptotic giant
    branch" stars. The sun will eventually become one of these. The
    story goes like this: first it'll keep burning until the hydrogen in
    its core is exhausted. Then it'll cool and become a red giant.
    Eventually helium at the core will ignite, and the Sun will shrink
    and heat up again... but its core will then become cluttered with even
    heavier elements, so it'll cool and expand once more, moving onto the
    "asymptotic giant branch". At this point it'll have a layered
    structure: heavier elements near the bottom, then a layer of helium,
    then hydrogen on the top.

    (A similar fate awaits any star between 0.6 and 10 solar masses,
    though the details depend on the mass. For the more dramatic
    fate of heavier stars, see "week204".)

    This layered structure is unstable, so asymptotic giant branch
    stars pulse every 10 to 100 thousand years or so. And, they
    puff out dust! Stellar wind then blows this dust out into space.

    A great example is the Red Rectangle:

    2) Rungs of the Red Rectangle, Astronomy picture of the day,
    May 13, 2004, http://apod.nasa.gov/apod/ap040513.html

    Here two stars 2300 light years from us are spinning around
    each other while pumping out a huge torus of icy dust grains and
    hydrocarbon molecules. It's not really shaped like a rectangle
    or X - it just looks that way. The scene is about 1/3 of a light
    year across.

    Ciska Markwick-Kemper is an expert on dust. She's an astrophysicist
    at the University of Manchester. Together with some coauthors, she
    wrote a paper about the Red Rectangle:

    3) F. Markwick-Kemper, J. D. Green, E. Peeters, Spitzer
    detections of new dust components in the outflow of the Red
    Rectangle, Astrophys. J. 628 (2005) L119-L122. Also available
    as arXiv:astro-ph/0506473.

    They used the Spitzer Space Telescope - an infrared telescope on
    a satellite in earth orbit - to find evidence of magnesium and
    iron oxides in this dust cloud.

    But, what made dust in the early Universe? It took about a
    billion years after the Big Bang for asymptotic giant branch stars
    to form. But we know there was a lot of dust even before then!
    We can see it in distant galaxies lit up by enormous black holes
    called "quasars", which pump out vast amounts of radiation as
    stuff falls into them.

    Markwick-Kemper and coauthors have also tackled that question:

    4) F. Markwick-Kemper, S. C. Gallagher, D. C. Hines and J. Bouwman,
    Dust in the wind: crystalline silicates, corundum and periclase in
    PG 2112+059, Astrophys. J. 668 (2007), L107-L110. Also available
    as arXiv:0710.2225.

    They used spectroscopy to identify various kinds of dust in
    a distant galaxy: a magnesium silicate that geologists call
    "forsterite", a magnesium oxide called "periclase", and aluminum
    oxide, otherwise known as "corundum" - you may have seen it on
    sandpaper.

    And, they hypothesize that these dust grains were formed in the
    hot wind emanating from the quasar at this galaxy's core!

    So, besides being made of star dust, as in the Joni Mitchell
    song, you also may contain a bit of black hole dust.

    Okay - now that we've got that settled, on to London!

    Minhyong Kim is a friend I met back in 1986 when he was a grad
    student at Yale. After dabbling in conformal field theory, he
    became a student of Serge Lang and went into number theory. He
    recently moved to England and started teaching at University
    College, London. I met him there this summer, in front of the
    philosopher Jeremy Bentham, who had himself mummified and stuck
    in a wooden cabinet near the school's entrance.

    If you're not into number theory, maybe you should read this:

    5) Minhyong Kim, Why everyone should know number theory,
    available at http://www.ucl.ac.uk/~ucahmki/numbers.pdf

    Personally I never liked the subject until I realized it was
    a form of *geometry*. For example, when we take an equation like
    this

    x^2 + y^3 = 1

    and look at the real solutions, we get a curve in the plane -
    a "real curve". If we look at the complex solutions, we get
    something bigger. People call it a "complex curve", because
    it's analogous to a real curve. But topologically, it's
    2-dimensional. This will be important in a few minutes, so
    don't forget it!

    If we use polynomial equations with more variables, we get
    higher-dimensional shapes called "algebraic varieties" - either
    real or complex. Either way, we can study these shapes using
    geometry and topology.

    But in number theory, we might study the solutions of these
    equations in some other number system - for example in Z/p,
    meaning the integers modulo some prime p. At first glance there's
    no geometry involved anymore. After all, there's just a *finite
    set* of solutions! However, algebraic geometers have figured
    out how to apply ideas from geometry and topology, mimicking
    tricks that work for the real and complex numbers.

    All this is very fun and mind-blowing - especially when we reach
    Grothendieck's idea of "etale topology", developed around 1958.
    This is a way of studying "holes" in things like algebraic
    varieties over finite fields. Amazingly, it gives results that
    nicely match the results we get for the corresponding complex
    algebraic varieties! That's part of what the "Weil conjectures"
    say.

    You can learn the details here:

    6) J. S. Milne, Lectures on Etale Cohomology, available at
    http://www.jmilne.org/math/CourseNotes/math732.html

    Anyway, I quizzed about Minhyong about one of the big mysteries
    that's been puzzling me lately. I want to know why the integers
    resemble a 3-dimensional space - and how prime numbers are like
    "knots" in this space!

    Let me try to explain this in a very sketchy way, without getting
    into any technical details. I'll still make mistakes... but this
    stuff is just too cool to keep secret - so if the experts don't
    explain it, nonexperts like me have to try.

    You can think of Z/p as giving a very simple sort of curve.
    Naively you could imagine it as shaped like a ring, for example
    the integers mod 7 here:

    0
    6 1

    5 2
    4 3


    But now it's better to think of Z/p as a "line". After
    all, a line is defined by one variable and no equations. Here
    we have one variable in Z/p.

    But remember: a curve defined in a field like Z/p acts a lot
    like a complex curve. And, a complex curve is topologically
    2-dimensional!

    So, the "line" associated to Z/p seems 2-dimensional from the
    viewpoint of etale topology. In other words, it's really more
    like a "plane" - just like the complex numbers are topologically a
    plane.

    This is true for each prime p. But the integers, Z, are more
    complicated than any of these Z/p's. To be precise, we have maps

    Z -> Z/p

    for each p. So, if we think of Z as a kind of space, it's a big
    space that contains all the "planes" corresponding to the Z/p's.
    So, it's 3-dimensonal!

    In short: from the viewpoint of etale topology, the integers have
    one dimension that says which prime you're at, and two more coming
    from the plane-like nature of each individual Z/p.

    Naively you might imagine a stack of planes, one for each prime.
    But that's a very crude picture, and it misses a crucial fact: the
    primes get "tangled up" with each other. In fact, each "plane" has
    a specially nice circle in it, and these circles are *linked*.

    I've been fascinated by this ever since I heard about it, but I
    got even more interested when I saw a draft of a paper by
    Kapranov and some coauthor. I got it from Thomas Riepe, who got
    it from Yuri Manin. I don't have it right here with me, so I'll
    add a reference later... but I don't think it's available yet,
    so the reference won't do you much good anyway.

    In this paper, the authors explain how the "Legendre symbol" of
    primes is analogous to the "linking number" of knots.

    The Legendre symbol depends on two primes: it's 1 or -1 depending
    on whether or not the first is a square modulo the second. The
    linking number depends on two knots: it says how many times the
    first winds around the second.

    The linking number stays the same when you switch the two knots.
    The Legendre symbol has a subtler symmetry when you switch the
    two primes: this symmetry is called "quadratic reciprocity", and
    it has lots of proofs, starting with a bunch by Gauss - all a bit
    tricky.

    I'd feel very happy if I truly understood why quadratic reciprocity
    reduces to the symmetry of the linking number when we think of
    primes as analogous to knots. Unfortunately, I'll need to think a
    lot more before I really get the idea. I got into number theory
    late in life, so I'm pretty slow at it.

    This paper studies subtler ways in which primes can be "linked":

    7) Masanori Morigarbagea, Milnor invariants and Massey products for
    prime numbers, Compositio Mathematica 140 (2004), 69-83.

    You may know the Borromean rings, a design where no two rings are
    linked in isolation, but all three are when taken together. Here
    the linking numbers are zero, but the linking can be detected by
    something called the "Massey triple product". Morigarbagea
    generalizes this to primes!

    But I want to understand the basics...

    The secret 3-dimensional nature of the integers and certain other
    "rings of algebraic integers" seems to go back at least to the work
    of Artin and Verdier:

    8) Michael Artin and Jean-Louis Verdier, Seminar on etale cohomology
    of number fields, Woods Hole, 1964.

    You can see it clearly here, starting in section 2:

    9) Barry Mazur, Notes on the etale cohomology of number fields,
    Annales Scientifiques de l'Ecole Normale Superieure Ser. 4,
    6 (1973), 521-552. Also available at
    http://www.numdam.org/numdam-bin/fitem?id=ASENS_1973_4_6_4_521_0

    By now, a big "dictionary" relating knots to primes has been
    developed by Kapranov, Mazur, Morigarbagea, and Reznikov. This
    seems like a readable introduction:

    10) Adam S. Sikora, Analogies between group actions on 3-manifolds
    and number fields, available as arXiv:math/0107210.

    I need to study it. These might also be good - I haven't looked
    at them yet:

    11) Masanori Morigarbagea, On certain analogies between knots and
    primes, J. Reine Angew. Math. 550 (2002), 141-167.

    Masanori Morigarbagea, On analogies between knots and primes,
    Sugaku 58 (2006), 40-63.

    After giving a talk on 2-Hilbert spaces at University College, I went
    to dinner with Minhyong and some folks including Ray Streater. Ray
    Streater and Arthur Wightman wrote the book "PCT, Spin, Statistics and
    All That". Like almost every mathematician who has seriously tried to
    understand quantum field theory, I've learned a lot from this book.
    So, it was fun meeting Streater, talking with him - and finding out
    he'd once been made an honorary colonel of the US Army to get a free
    plane trip to the Rochester Conference! This was a big important
    particle physics conference, back in the good old days.

    He also described Geoffrey Chew's Rochester conference talk on the
    analytic S-matrix, given at the height of the bootstrap theory fad.
    Wightman asked Chew: why assume from the start that the S-matrix was
    analytic? Why not try to derive it from simpler principles? Chew
    replied that "everything in physics is smooth". Wightman asked about
    smooth functions that aren't analytic. Chew thought a moment and
    replied that there weren't any.

    Ha-ha-ha...

    What's the joke? Well, first of all, Wightman had already succeeded
    in deriving the analyticity of the S-matrix from simpler principles.
    Second, any good mathematician - but not necessarily every physicist,
    like Chew - will know examples of smooth functions that aren't
    analytic.

    Anyway, Streater has just finished an interesting book on "lost
    causes" in physics: ideas that sounded good, but never panned out.
    Of course it's hard to know when a cause is truly lost. But a
    good pragmatic definition of a lost cause in physics is a topic
    that shouldn't be given as a thesis problem.

    So, if you're a physics grad student and some professor wants you to
    work on hidden variable theories, or octonionic quantum mechanics,
    or deriving laws of physics from Fisher information, you'd better
    read this:

    11) Ray F. Streater, Lost Causes in and Beyond Physics, Springer
    Verlag, Berlin, 2007.

    (I like octonions - but I agree with Streater about not inflicting
    them on physics grad students! Even though all my students are in
    the math department, I still wouldn't want them working mainly on
    something like that. There's a lot of more general, clearly useful
    stuff that students should learn.)

    I also spoke to Andreas Doering and Chris Isham about their work
    on topos theory and quantum physics. Andreas Doering lives near
    Greenwich, while Isham lives across the Thames in London proper.
    So, I talked to Doering a couple times, and once we visited Isham
    at his house.

    I mainly mention this because Isham is one of the gurus of quantum
    gravity, profoundly interested in philosophy... so I was surprised,
    at the end of our talk, when he showed me into a room with a huge
    rack of computers hooked up to a bank of about 8 video monitors,
    and controls reminiscent of an airplane cockpit.

    It turned out to be his homemade flight simulator! He's been a
    hobbyist electrical engineer for years - the kind of guy who
    loves nothing more than a soldering iron in his hand. He'd just
    gotten a big 750-watt power supply, since he'd blown out his
    previous one.

    Anyway, he and Doering have just come out with a series of papers:

    11) Andreas Doering and Christopher Isham, A topos foundation
    for theories of physics: I. Formal languages for physics,
    available as arXiv:quant-ph/0703060.

    II. Daseinisation and the liberation of quantum theory,
    available as arXiv:quant-ph/0703062.

    III. The representation of physical quantities with arrows,
    available as arXiv:quant-ph/0703064.

    IV. Categories of systems, available as arXiv:quant-ph/0703066.

    Though they probably don't think of it this way, you can think
    of their work as making precise Bohr's ideas on seeing the quantum
    world through classical eyes. Instead of talking about all
    observables at once, they consider collections of observables that
    you can measure simultaneously without the uncertainty principle
    kicking in. These collections are called "commutative subalgebras".

    You can think of a commutative subalgebra as a classical snapshot
    of the full quantum reality. Each snapshot only shows part of the
    reality. One might show an electron's position; another might show
    its momentum.

    Some commutative subalgebras contain others, just like some open
    sets of a topological space contain others. The analogy is a good
    one, except there's no one commutative subalgebra that contains
    *all* the others.

    Topos theory is a kind of "local" version of logic, but where the
    concept of locality goes way beyond the ordinary notion from
    topology. In topology, we say a property makes sense "locally"
    if it makes sense for points in some particular open set.
    In the Doering-Isham setup, a property makes sense "locally" if
    it makes sense "within a particular classical snapshot of reality" -
    that is, relative to a particular commutative subalgebra.

    (Speaking of topology and its generalizations, this work on topoi and
    physics is related to the "etale topology" idea I mentioned a while
    back - but technically it's much simpler. The etale topology lets
    you define a topos of "sheaves" on a certain category. The
    Doering-Isham work just uses the topos of "presheaves" on the poset
    of commutative subalgebras. Trust me - while this may sound scary,
    it's much easier.)

    Doering and Isham set up a whole program for doing physics
    "within a topos", based on existing ideas on how to do math in
    a topos. You can do vast amounts of math inside any topos just
    as if you were in the ordinary world of set theory - but using
    intuitionistic logic instead of classical logic. Intuitionistic
    logic denies the principle of excluded middle, namely:

    "For any statement P, either P is true or not(P) is true."

    In Doering and Isham's setup, if you pick a commutative subalgebra
    that contains the position of an electron as one of its observables,
    it can't contain the electron's momentum. That's because these
    observables don't commute: you can't measure them both simultaneously.
    So, working "locally" - that is, relative to this particular
    subalgebra - the statement

    P = "the momentum of the electron is zero"

    is neither true nor false! It's just not defined.

    Their work has inspired this very nice paper:

    12) Chris Heunen and Bas Spitters, A topos for algebraic quantum
    theory, available as arXiv:0709.4364.

    so let me explain that too.

    I said you can do a lot of math inside a topos. In particular,
    you can define an algebra of observables - or technically, a
    "C*-algebra".

    By the Isham-Doering work I just sketched, any C*-algebra of
    observables gives a topos. Heunen and Spitters show that
    the original C*-algebra gives rise to a commutative
    C*-algebra in this topos, even if the original one was
    noncommutative!

    That actually makes sense, since in this setup, each "local view"
    of the full quantum reality is classical. What's really neat is
    that the Gelfand-Naimark theorem, saying commutative C*-algebras
    are always algebras of continuous functions on compact Hausdorff
    spaces, can be generalized to work within any topos. So, we get
    a space *in our topos* such that observables of the C*-algebra
    *in the topos* are just functions on this space.

    I know this sounds technical if you're not into this stuff. But
    it's really quite wonderful. It basically means this: using topos
    logic, we can talk about a classical space of states for a quantum
    system! However, this space typically has "no global points". In
    other words, there's no overall classical reality that matches all
    the classical snapshots.

    As you can probably tell, category theory is gradually seeping
    into this post, though I've been doing my best to keep it
    hidden. Now I want to say what Eugenia Cheng explained on
    that train to Sheffield. But at this point, I'll break down and
    assume you know some category theory - for example, monads.

    If you don't know about monads, never fear! I defined them in
    "week89", and studied them using string diagrams in "week92".
    Even better, Eugenia Cheng and Simon Willerton have formed a
    little group called the Catsters - and under this name, they've
    put some videos about monads and string diagrams onto YouTube!
    This is a really great new use of technology. So, you should
    also watch these:

    14) The Catsters, Monads,
    http://youtube.com/view_play_list?p=0E91279846EC843E

    The Catsters, Adjunctions,
    http://youtube.com/view_play_list?p=54B49729E5102248

    The Catsters, String diagrams, monads and adjunctions,
    http://youtube.com/view_play_list?p=50ABC4792BD0A086

    A very famous monad is the "free abelian group" monad

    F: Set -> Set

    which eats any set X and spits out the free abelian group on X,
    say F(X). A guy in F(X) is just a formal linear combination
    of guys in X, with integer coefficients.

    Another famous monad is the "free monoid" monad

    G: Set -> Set

    This eats any set X and spits out the free monoid on X, namely
    G(X). A guy in G(X) is just a formal product of guys in X.

    Now, there's yet another famous monad, called the "free
    ring" monad, which eats any set X and spits out the free ring on
    this set. But, it's easy to see that this is just F(G(X))!
    After all, F(G(X)) consists of formal linear combinations of
    formal products of guys in X. But that's precisely what you find
    in the free ring on X.

    But why is FG a monad? There's more to a monad than just a
    functor. A monad is really a kind of *monoid* in the world of
    functors from our category (here Set) to itself. In particular,
    since F is a monad, it comes with a natural transformation called
    a "multiplication":

    m: FF => F

    which sends formal linear combinations of formal linear combinations
    to formal linear combinations, in the obvious way. Similarly,
    since G is a monad, it comes with a natural transformation

    n: GG => G

    sending formal products of formal products to formal products.
    But how does FG get to be a monad? For this, we need some
    natural transformation from FGFG to FG!

    There's an obvious thing to try, namely

    mn
    FGFG ======> FFGG ======> FG

    where in the first step we switch G and F somehow, and in the
    second step we use m and n. But, how do we do the first step?

    We need a natural transformation

    d: GF => FG

    which sends formal products of formal linear combinations
    to formal linear combinations of formal products. Such a
    thing obviously exists; for example, it sends

    (x + 2y)(x - 3z)

    to

    xx + 2yx - 3xz - 6yz

    It's just the distributive law!

    Quite generally, to make the composite of monads F and G
    into a new monad FG, we need something that people call a
    "distributive law", which is a natural transformation

    d: GF => FG

    This must satisfy some equations - but you can work out
    those yourself. For example, you can demand that

    FdG mn
    FGFG ======> FFGG ======> FG

    make FG into a monad, and see what that requires. Besides the
    "multiplication" in our monad, we also need the "unit", so you
    should also think about that - I'm ignoring it here because it's
    less sexy than the multiplication, but it's equally essential.

    However: all this becomes more fun with string diagrams!
    As the Catsters explain, and I explained in "week89", the
    multiplication m: FF => F can be drawn like this:

    \ /
    \ /
    F\ F/
    \ /
    \ /
    \ /
    \ /
    \ /
    |m
    |
    |
    |
    |
    |
    F|
    |

    And, it has to satisfy the associative law, which says we
    get the same answer either way when we multiply three things:

    \ / / \ \ /
    \ / / \ \ /
    F\ /F F/ F\ F\ /F
    \/ / \ \/
    m\ / \ /m
    \ / \ /
    F\ / \ /F
    \ / \ /
    |m |m
    | |
    | = |
    | |
    | |
    | |
    F| F|
    | |

    The multiplication n: GG => G looks similar to m, and it too has
    to satisfy the associative law.

    How do we draw the distributive law d: FG => GF? Since it's a
    process of switching two things, we draw it as a *braiding*:

    F\ /G
    \ /
    /
    / \
    G/ \F

    I hope you see how incredibly cool this is: the good old
    distributive law is now a *braiding*, which pushes our diagrams
    into the third dimension!

    Given this, let's draw the multiplication for our would-be
    monad FG, namely

    FdG mn
    FGFG ======> FFGG ======> FG

    It looks like this:

    \ \ / /
    \ \ / /
    F\ G\ F/ /G
    \ \ / /
    \ \ / /
    \ \ / /
    \ / /
    \ / \ /
    |m |n
    | |
    | |
    | |
    | |
    | |
    F| |G
    | |

    Now, we want *this* multiplication to be associative! So,
    we need to draw an equation like this:

    \ / / \ \ /
    \ / / \ \ /
    \ / / \ \ /
    \/ / \ \/
    \ / \ /
    \ / \ /
    \ / \ /
    \ / \ /
    | |
    | |
    | = |
    | |
    | |
    | |
    | |
    | |

    but with the strands *doubled*, as above - I'm too lazy to draw
    this here. And then we need to find some nice conditions that
    make this associative law true. Clearly we should use the
    associative laws for m and n, but the "braiding" - the
    distributive law d: FG => GF - also gets into the act.

    I'll leave this as a pleasant exercise in string diagram
    manipulation. If you get stuck, you can peek in the back of
    the book:

    14) Wikipedia, Distibutive law between monads,
    http://en.wikipedia.org/wiki/Distributive_law_between_monads

    The two scary commutative rectangles on this page are the
    "nice conditions" you need. They look nicer as string
    diagrams. One looks like this:

    F\ G\ /G F\ G/ /G
    \ \ / \ / /
    \ |n \ / /
    \ / / /
    \ / = / \ /
    / / /
    / \ / /\
    / \ \ / \
    / \ \ / \
    G/ \F |n \F
    / \ G| \

    In words:

    "multiply two G's and slide the result over an F" =
    "slide both the G's over the F and then multiply them"

    If the pictures were made of actual string, this would be obvious!

    The other condition is very similar. I'm too lazy to draw it,
    but it says

    "multiply two F's and slide the result under a G" =
    "slide both the F's under a G and then multiply them"

    All this is very nice, and it goes back to a paper by Beck:

    15) Jon Beck, Distributive laws, Lecture Notes in Mathematics
    80, Springer, Berlin, pp. 119–140.

    This isn't what Eugenia explained to me, though - I already knew
    this stuff. She started out by explaining something in a paper
    by Street:

    16) Ross Street, The formal theory of monads, J. Pure Appl. Alg.
    2 (1972), 149-168.

    which is reviewed at the beginning here:

    17) Steve Lack and Ross Street, The formal theory of monads II,
    J. Pure Appl. Alg. 175 (2002), 243-265. Also available at
    http://www.maths.usyd.edu.au/u/stevel/papers/ftm2.html

    (Check out the cool string diagrams near the end!)

    Street noted that for any category C, there's a category Mnd(C)
    whose objects are monads on C and whose morphisms are "monad
    transforms": functors from C to C that make an obvious square
    commute.

    And, he noted that a monad on Mnd(C) is a pair of monads on C
    related by a distributive law!

    That's already mindbogglingly beautiful. According to Eugenia,
    it's in the last sentence of Street's paper. But in her new work:

    18) Eugenia Cheng, Iterated distributive laws, available as
    arXiv:0710.1120.

    she goes a bit further: she considers monads in Mnd(Mnd(C)),
    and so on. Here's the punchline, at least for today: she shows
    that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related
    by distributive laws satisfying the Yang-Baxter equation:

    \F G/ |H F| G\ /H
    \ / | | \ /
    / | | /
    / \ | | / \
    / \ | \ / \
    | \ / \ / |
    | / = / |
    | / \ / \ |
    | / \ / \ |
    \ / | | \ /
    \ / | | \ /
    / | | /
    / \ | | / \
    /H \G |F H| G/ \F

    This is also just what you need to make the composite FGH
    into a monad!

    By the way, the pathetic piece of ASCII art above is lifted
    from "week1", where I first explained the Yang-Baxter equation.
    That was back in 1993. So, it's only taken me 14 years to learn
    that you can derive this equation from considering monads on
    the category of monads on the category of monads on a category.

    You may wonder if this counts as progress - but Eugenia
    studies lots of *examples* of this sort of thing, so it's far
    from pointless.

    Okay... finally, the Tale of Groupoidification. I'm a bit tired
    now, so instead of telling you more of the tale, let me just say
    the big news.

    Starting this fall, James Dolan and I are running a seminar on
    geometric representation theory, which will discuss:

    Actions and representations of groups, especially symmetric groups
    Hecke algebras and Hecke operators
    Young diagrams
    Schubert cells for flag varieties
    q-deformation
    Spans of groupoids and groupoidification

    This is the Tale of Groupoidification in another guise.

    Moreover, the Catsters have inspired me to make videos of this
    seminar! You can already find some here, along with course
    notes and blog entries where you can ask questions and talk about
    the material:

    19) John Baez and James Dolan, Geometric representation theory seminar,
    http://math.ucr.edu/home/baez/qg-fall2007/

    More will show up in due course. I hope you join the fun.

    -----------------------------------------------------------------------

    Quote of the Week:

    It is a glorious feeling to discover the unity of a set of phenomena
    that at first seem completely separate. - Albert Einstein

    -----------------------------------------------------------------------
    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at

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

    For a table of contents of all the issues of This Week's Finds, try

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

    A simple jumping-off point to the old issues is available at

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

    If you just want the latest issue, go to

    http://math.ucr.edu/home/baez/this.week.html
     
  2. jcsd
  3. >... lost cause in physics is a topic that shouldn't be given as a >thesis problem.

    >So, if you're a physics grad student and some professor wants you >to work on hidden variable theories, ..., you'd better read this:

    Maybe grad students shouldn't be assigned to work on hidden variable theories, but that doesn't mean mature researchers shouldn't look at them. While the conventional wisdom always seems to relegate these efforts to the 'crack-pot' realm, I find it interesting that J. S. Bell's book - The Speakable and Unspeakable in QM, is largely an ode to the de Broglie/Bohm pilot wave interpretation of QM, which is a (non-local) hidden variable theory. Yes, yes, I know - there are issues with a covariant formulation. But Bell seemed optimistic there, also. I don't think he would call it a lost cause. Unfortunately, we no longer have his input.
     
  4. > After giving a talk on 2-Hilbert spaces at University College, I went
    > to dinner with Minhyong and some folks including Ray Streater. Ray
    > Streater and Arthur Wightman wrote the book "PCT, Spin, Statistics and
    > All That". Like almost every mathematician who has seriously tried to
    > understand quantum field theory, I've learned a lot from this book.
    > So, it was fun meeting Streater, talking with him - and finding out
    > he'd once been made an honorary colonel of the US Army to get a free
    > plane trip to the Rochester Conference! This was a big important
    > particle physics conference, back in the good old days.
    >
    > He also described Geoffrey Chew's Rochester conference talk on the
    > analytic S-matrix, given at the height of the bootstrap theory fad.
    > Wightman asked Chew: why assume from the start that the S-matrix was
    > analytic? Why not try to derive it from simpler principles?


    There is an variant of this anecdote in Streater's site, arguing to use
    positivity there, and in any case explaining why the bootstrap as
    defined by S-matrix practitioners is a lost cause.

    Yet, by reading non technical accounts such as
    www.journals.uchicago.edu/cgi-bin/resolve?id=doi:10.1086/344960 or
    http://www.slac.stanford.edu/spires/find/hep/www?r=LBL-18372 one gets
    the feeling that Chewish program of "nuclear democracy" was wider than
    its implementation by S-matrix practitioners. And also that the
    interpretation of the program by non-practitioners was even wider.

    It is interesting that that remark of Streater was about the positivity
    axiom. Bootstrap theory never knew of supersymmetry, and here it come to
    come back with a revenge.

    Alejandro
     
  5. In sci.math John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

    > I also spoke to Andreas Doering and Chris Isham about their work
    > on topos theory and quantum physics.


    Is topos theory the next hot thing? Just yesterday I read about
    it in Lee Smolin's book "Three Roads to Quantum Gravity".

    P.S. I proved that the CPT theorem follows from knot theory.
    OK, "it follows" is a bit misleading, and "I proved x" would
    always be an outright lie, even for x="1+1=2" but you get
    the picture :-)

    --
    Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
    order stormed the surface where chaos set norm
    had there always been balance? ...surely not
    therein lies the beauty
     
  6. On Oct 17, 10:41 am, Hauke Reddmann <fc3a...@uni-hamburg.de> wrote:
    > In sci.math John Baez <b...@math.removethis.ucr.andthis.edu> wrote:
    >
    > > I also spoke to Andreas Doering and Chris Isham about their work
    > > on topos theory and quantum physics.

    >
    > Is topos theory the next hot thing? Just yesterday I read about
    > it in Lee Smolin's book "Three Roads to Quantum Gravity".


    its hard to call certain approaches
    that are more than 10 years old
    hot, new, or the next thing

    though i agree they are certainly important directions
    and may be gaining in popularity

    topoi are found now in several approaches
    each with their own important integration of the field

    along with isham's work
    you will find markopoulou's work on internal logics
    which
    although with a slightly different view
    than isham's main focus on consistent histories
    is strongly aligned to that idea
    (they have worked together)

    also
    there is the work of bob coecke
    and operationalist interpretations of quantum theory
    also bringing in topoi
    using galois adjunction of the standard
    (orthomodular)
    quantum logics

    i have made a number of posts on these topics in the past
    on various pieces of these theories
    and their relations to larger programmes
    if you are looking for some background
    though many of the source articles
    are now available on arXiv
    and do not require much more background

    -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
    galathaea: prankster, fablist, magician, liar
     
  7. In sci.math galathaea <galathaea@gmail.com> wrote:

    > its hard to call certain approaches
    > that are more than 10 years old
    > hot, new, or the next thing


    I disagree. If my memory serves me well, string theory was
    decades old before a major mathematical breakthrough sent
    it rocketing off. So, why should not the same happen to
    topos theory? I'm a complete layman, and consider a topic
    as "hot" if it appears in layman-directed books. Zounds,
    I had to learn about the whole quantum theory interpretation
    smeg from "Illuminatus!" :-)
    --
    Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
    order stormed the surface where chaos set norm
    had there always been balance? ...surely not
    therein lies the beauty
     
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