quantum quandaries - a category-theoretic perspective [Archive]

John Baez

Apr7-04, 03:12 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Some of you may enjoy this paper, or at least be infuriated by it:\n\nhttp://math.ucr.edu/home/baez/quantum/\n\nQuantum Quandaries: A Category-Theoretic Perspective\n\nJohn C. Baez\n\nTo appear in _Structural Foundations of Quantum Gravity_,\neds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press.\n\nAbstract:\n\nGeneral relativity may seem very different from quantum theory, but work\non quantum gravity has revealed a deep analogy between the two. General\nrelativity makes heavy use of the category nCob, whose objects are\n(n-1)-dimensional manifolds representing "space" and whose morphisms\nare n-dimensional cobordisms representing "spacetime". Quantum theory\nmakes heavy use of the category Hilb, whose objects are Hilbert spaces\nused to describe "states", and whose morphisms are bounded linear operators\nused to describe "processes". Moreover, the categories nCob and Hilb\nresemble each other far more than either resembles Set, the category\nwhose objects are sets and whose morphisms are functions. In particular,\nboth Hilb and nCob but not Set are *-categories with a noncartesian\nmonoidal structure. We show how this accounts for many of the famously\npuzzling features of quantum theory: the failure of local realism, the\nimpossibility of duplicating quantum information, and so on. We argue\nthat these features only seem puzzling when we try to treat Hilb as\nanalogous to Set rather than nCob, so that quantum theory will make\nmore sense when regarded as part of a theory of spacetime.\n\nThis will probably show up at http://www.arxiv.org/abs/quant-ph/0404040\npretty soon. (Yay! I got the coolest arxiv number this year!)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Some of you may enjoy this paper, or at least be infuriated by it:

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

Quantum Quandaries: A Category-Theoretic Perspective

John C. Baez

To appear in _Structural Foundations of Quantum Gravity_,
eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press.

Abstract:

General relativity may seem very different from quantum theory, but work
on quantum gravity has revealed a deep analogy between the two. General
relativity makes heavy use of the category nCob, whose objects are
(n-1)-dimensional manifolds representing "space" and whose morphisms
are n-dimensional cobordisms representing "spacetime". Quantum theory
makes heavy use of the category Hilb, whose objects are Hilbert spaces
used to describe "states", and whose morphisms are bounded linear operators
used to describe "processes". Moreover, the categories nCob and Hilb
resemble each other far more than either resembles Set, the category
whose objects are sets and whose morphisms are functions. In particular,
both Hilb and nCob but not Set are *-categories with a noncartesian
monoidal structure. We show how this accounts for many of the famously
puzzling features of quantum theory: the failure of local realism, the
impossibility of duplicating quantum information, and so on. We argue
that these features only seem puzzling when we try to treat Hilb as
analogous to Set rather than nCob, so that quantum theory will make
more sense when regarded as part of a theory of spacetime.

This will probably show up at http://www.arxiv.org/abs/quant-ph/0404040
pretty soon. (Yay! I got the coolest arxiv number this year!)

rof@maths.tcd.ie

Apr13-04, 05:42 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>baez@math-ws-n09.math.ucr.edu (John Baez) writes:\n\n>Some of you may enjoy this paper, or at least be infuriated by it:\n\n>http://math.ucr.edu/home/baez/quantum/\n\nIt makes very entertaining and educating reading. The last time I\nlooked at categories I gave up after a while because it seemed\ncute but useless. Maybe I\'ll have another look.\n\n>In particular,\n>both Hilb and nCob but not Set are *-categories with a noncartesian\n>monoidal structure. We show how this accounts for many of the famously\n>puzzling features of quantum theory: the failure of local realism, the\n>impossibility of duplicating quantum information, and so on. We argue\n>that these features only seem puzzling when we try to treat Hilb as\n>analogous to Set rather than nCob, so that quantum theory will make\n>more sense when regarded as part of a theory of spacetime.\n\nThat claim is rather ambitious - from what I can see your solution\nto the puzzles is merely to say just think about Hilbert spaces\nand it\'ll be fine, which is the "shut up and calculate" approach\nin disguise. You have definitely pinpointed one of the surprising\nand perhaps disturbing aspects of quantum mechanics with the observation\nthat the product structure is noncartesian, although I think this\nproduct discrepancy is known, if not understood so clearly, to anybody\nwho thinks about quantum mechanics.\n\nThe similarity of Hilb to relations rather than functions is\nphilosophically interesting as well, but I would say that overall,\nthe most puzzling features of quantum mechanics do not come from\nits mathematical structures, but from from the thing which is not\nexpressed anywhere in the mathematics - the fact that individual\nmeasurements have individual results, rather than mere amplitudes\nof results.\n\n"It is as if classical logic continued to apply to us, while the\nmysterious rules of quantum theory apply only to the physical systems\nwe are studying. But of course this is not true: we are part of the\nworld being studied."\n\nHere\'s a comment that most physicists won\'t like and will consider\nuseless philosophical rubbish, but which is true nonetheless: our\nbodies are physical systems - parts of the world being studied, but\nour minds are not.\n\nR.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>baez@math-ws-n09.math.ucr.edu (John Baez) writes:

>Some of you may enjoy this paper, or at least be infuriated by it:

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

It makes very entertaining and educating reading. The last time I
looked at categories I gave up after a while because it seemed
cute but useless. Maybe I'll have another look.

>In particular,
>both Hilb and nCob but not Set are *-categories with a noncartesian
>monoidal structure. We show how this accounts for many of the famously
>puzzling features of quantum theory: the failure of local realism, the
>impossibility of duplicating quantum information, and so on. We argue
>that these features only seem puzzling when we try to treat Hilb as
>analogous to Set rather than nCob, so that quantum theory will make
>more sense when regarded as part of a theory of spacetime.

That claim is rather ambitious - from what I can see your solution
to the puzzles is merely to say just think about Hilbert spaces
and it'll be fine, which is the "shut up and calculate" approach
in disguise. You have definitely pinpointed one of the surprising
and perhaps disturbing aspects of quantum mechanics with the observation
that the product structure is noncartesian, although I think this
product discrepancy is known, if not understood so clearly, to anybody
who thinks about quantum mechanics.

The similarity of Hilb to relations rather than functions is
philosophically interesting as well, but I would say that overall,
the most puzzling features of quantum mechanics do not come from
its mathematical structures, but from from the thing which is not
expressed anywhere in the mathematics - the fact that individual
measurements have individual results, rather than mere amplitudes
of results.

"It is as if classical logic continued to apply to us, while the
mysterious rules of quantum theory apply only to the physical systems
we are studying. But of course this is not true: we are part of the
world being studied."

Here's a comment that most physicists won't like and will consider
useless philosophical rubbish, but which is true nonetheless: our
bodies are physical systems - parts of the world being studied, but
our minds are not.

R.

Danny Ross Lunsford

Apr13-04, 05:44 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>John Baez wrote:\n\n> Some of you may enjoy this paper, or at least be infuriated by it:\n>\n> http://math.ucr.edu/home/baez/quantum/\n>\n> Quantum Quandaries: A Category-Theoretic Perspective\n>\n> John C. Baez\n\nThis is very nice!\n\nA thing that always bugs me - quantum mechanics is really projective at\nbase, but using it requires positing an isometry. Now in ordinary\nprojective geometry this is accomplished by specifying a quadratic form\n(a metric). If one considers all the projective transformations that\npreserve this quadratic form, one gets a way to form projective\ninvariants that behave like metric invariants. One forms the cross ratio\nof 4 points, two of which are given, and two of which are defined by the\nintersection points of the line through the given points and the\nquadratic form. One now has four points on a line and forms the "Klein\nangle" as the imaginary log of their cross-ratio:\n\nW = i log XR(A,S;B,S\')\n\nThis is the closest you can get to physically sensing i :)\n\nThis allows a completely consistent definition of an isometry group and\nassociated metric geometry. The additive aspects of metric geometry come\nfrom the additivity of exponents in a product!\n\nNow people unconsciously apply just this process when orienting\nthemselves in space. They pick a quadratic form - the things that are\nfarthest away. One ignores the logical sense which says the tracks will\nnever converge and instead redefines the world so that convergence is\npossible in an "ideal domain".\n\nThere must be an analogy in QM to "establishing the invariant quadratic\nform". Something like insisting on the probability that SOMETHING\nhappens is 1.\n\n-drl\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>John Baez wrote:

> Some of you may enjoy this paper, or at least be infuriated by it:
>
> http://math.ucr.edu/home/baez/quantum/
>
> Quantum Quandaries: A Category-Theoretic Perspective
>
> John C. Baez

This is very nice!

A thing that always bugs me - quantum mechanics is really projective at
base, but using it requires positing an isometry. Now in ordinary
projective geometry this is accomplished by specifying a quadratic form
(a metric). If one considers all the projective transformations that
preserve this quadratic form, one gets a way to form projective
invariants that behave like metric invariants. One forms the cross ratio
of 4 points, two of which are given, and two of which are defined by the
intersection points of the line through the given points and the
quadratic form. One now has four points on a line and forms the "Klein
angle" as the imaginary log of their cross-ratio:

W = i[/itex] log XR(A,S;B,S')

This is the closest you can get to physically sensing i :)

This allows a completely consistent definition of an isometry group and
associated metric geometry. The additive aspects of metric geometry come
from the additivity of exponents in a product!

Now people unconsciously apply just this process when orienting
themselves in space. They pick a quadratic form - the things that are
farthest away. One ignores the logical sense which says the tracks will
never converge and instead redefines the world so that convergence is
possible in an "ideal domain".

There must be an analogy in QM to "establishing the invariant quadratic
form". Something like insisting on the probability that SOMETHING
happens is 1.

[itex]-drl

Mike Stay

Apr15-04, 07:02 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nYou compared Hilb and nCob in this paper, but it looks like any of the\nmatrix-mechanics-over-rigs structures from your fall 2003 qg notes\nought to work in the same way. Is that right?\n\nToday I went to a lecture by V.S. Sunder, since the abstract sounded\nso similar to what you wrote in this paper. Here it is:\n\n"In recent work with my colleague Vijay Kodiyalam, we showed that\nthere is a bijective correspondence between Vaughan Jones\' `subfactor\nplanar algebras\' on the one hand, and what may be called `unitary\ntopological quantum field theories\' defined on a category `D\' on the\nother, where the objects of `D\' are suitably `decorated closed\noriented 1-manifolds\' and the morphisms are similarly decorated\nclasses of cobordisms between a pair of objects.\n\nSince the subject is slightly technical, it will help to give the talk\nin two parts, with the first part devoted to a discussion of Vaughan\'s\nplanar algebras, and the second part to our work."\n\n(I had to giggle at "slightly technical" after reading the first\nparagraph, but he was right. It was mostly drawing nice pictures of\ntangles.) I got it down to the end, but I missed the punchline. His\npaper isn\'t online, so I\'ll have to see if I can figure it out during\nthe second lecture.\n\nAnyway, here\'s what I got:\n\nA tangle T has\n\n1) An outer disk D0 minus an ordered (possibly empty) list of\nsubdisks.\n2) A bunch of curves that divide up the interior into\ncheckerboard-colorable regions (equivalently, the boundaries of the\ndisks have an even number of curves ending on them).\n3) A set of distinguished points: each disk boundary with at least one\ncurve intersecting it has one point, where white goes to black when\ngoing clockwise around the disk, that is distinguished (denoted * in\nthe diagram)\n4) "color": take the number of curves intersecting the outside edge\nand divide by 2.\n\nAnd a few other things I\'ll get to below.\n\nSo here is an example of a tangle:\n\n---------------\n-----...............-----\n*--.........................--\\\nD0 ///|............................---\\\\\n// |......................../--/ \\\\\n// |....................---/ \\\\\n// |................/--/ \\\\\n/ /-----\\........---/ \\\n/ // \\\\..---/ \\\n/ | +- \\\n/-------+ D1 | \\\n|........| | |\n|.........| | |\n|......../-*\\ // ----------- |\n|......../ \\---+-/ //...........\\\\ |\n|.......| |.| //...............\\\\ |\n|.........\\ /..| /...................\\ |\n|..........\\---/...| |.....................| |\n|..................| |........-----........| |\n|..................| |.......// \\\\.......| |\n|..................| |......| |......| |\n|..................| |......| D3 |......| |\n|..................| |......| |......| |\n|............./----+\\ |......\\\\ //......| |\n|..........// \\\\ |........-----........| |\n|.........| | \\.................../ |\n+--------* | \\\\...............// |\n| | D2 | \\\\...........// |\n| | | ----------- |\n\\ \\\\ // /\n\\ \\---+-/..\\\\ /\n\\ |......\\\\ /\n\\ |........\\\\ /\n\\\\ |..........\\\\ //\n\\\\ |............\\\\ //\n\\\\ |..............\\\\ //\n\\\\\\ |................\\\\ ///\n\\+-.................\\\\ --/\n-----..............\\-----\n---------------\n\nSubdisks are "inputs" and the outer boundary is the "output" of the\ntangle. There\'s a natural way to compose tangles: if the input and\noutput are colored the same, match up the *\'s and the curves.\n\nHere is a tangle M(3):\n\n---*--+--+---\n----- |..| |...-----\n///- |..| |........-\\\\\\\n// |..| |............\\\\\n// |..| |..............\\\\\n// |..| |................\\\\\n/ *--+--+-.................\\\n// // \\\\................\\\\\n/ / \\.................\\\n/ | |.................\\\n| | D1 |................|\n| | |.................|\n| | |..................|\n| | |...................|\n| \\ /.....................|\n| \\\\ //......................|\n| +--+--+-........................|\n| |..| |.........................|\n| |..| |.........................|\n| -*--+--|.........................|\n| //- -\\\\.......................|\n| // \\\\.....................|\n| / \\...................|\n| | |...................|\n| | |.................|\n| | D2 |................|\n\\ | |................/\n\\ | |.............../\n\\\\ | |..............//\n\\ \\ /............./\n\\\\ \\\\ //............//\n\\\\ \\\\- -//............//\n\\\\ +-+-+--.............//\n\\\\\\- |.| |...........-///\n----- |.| |......-----\n--+-+-+------\n\n\nIt takes two 3-colored tangles X, Y as input and outputs a 3-colored\ntangle. We can call it multiplication and denote the output as XY.\n\n\nAnnular tangles have one subdisk. An annular tangle A(m,n) is a\ntangle with an m-colored input and an n-colored output. Here is the\nidentity(3,3) tangle:\n\n-*--+--+---\n/--- |..| |...---\\\n// |..| |.......\\\\\n// |..| |.........\\\\\n/ |..| |...........\\\n/ *--+--+-...........\\\n/ // \\\\..........\\\n| // \\\\.........|\n| / \\........|\n| | |.........|\n| | |........|\n| | |........|\n| | |........|\n| | |.........|\n| \\ /........|\n| \\\\ //.........|\n\\ \\\\ //........../\n\\ +---+--+.........../\n\\ |...| |........../\n\\\\ |...| |........//\n\\\\ |...| |......//\n\\--- |...| |..---/\n-+---+--+--\n\nThen there are tangles with no subdisks. A function from a\nzero-dimensional vector space to an n-dimensional one is really just\nscalar multiplication. So here\'s 1(3):\n\n----+----+\n//*-....| |--\\\\\n// |.....| |....\\\\\n/ |.....| |......\\\n/ |.....| |.......\\\n/ |.....| |........\\\n| |.....| |.........|\n| |.....| |..........|\n| |.....| |..........|\n| |.....| |..........|\n| |.....| |..........|\n| |.....| |..........|\n| |.....| |..........|\n| |.....| |.........|\n\\ |.....| |......../\n\\ |.....| |......./\n\\ |.....| |....../\n\\\\ |.....| |....//\n\\\\+-....| |--//\n----+----+\n\n\n\nThere\'s a conjugation operator * that\'s the following steps: reflect,\nthen move all the *\'s counterclockwise (in the original drawing,\nclockwise in the reflected one) one position on the disk boundary. So\nM*(3) is (note the subdisk labels)\n\n\n---*--+--+---\n----- |..| |...-----\n///- |..| |........-\\\\\\\n// |..| |............\\\\\n// |..| |..............\\\\\n// |..| |................\\\\\n/ *--+--+-.................\\\n// // \\\\................\\\\\n/ / \\.................\\\n/ | |.................\\\n| | D2 |................|\n| | |.................|\n| | |..................|\n| | |...................|\n| \\ /.....................|\n| \\\\ //......................|\n| +--+--+-........................|\n| |..| |.........................|\n| |..| |.........................|\n| -*--+--|.........................|\n| //- -\\\\.......................|\n| // \\\\.....................|\n| / \\...................|\n| | |...................|\n| | |.................|\n| | D1 |................|\n\\ | |................/\n\\ | |.............../\n\\\\ | |..............//\n\\ \\ /............./\n\\\\ \\\\ //............//\n\\\\ \\\\- -//............//\n\\\\ +-+-+--.............//\n\\\\\\- |.| |...........-///\n----- |.| |......-----\n--+-+-+------\n\nI.e. (XY)* = Y*X*.\n\nWe get an algebra out of tangles with no subdisks by making the disks\ninto squares with the * in the upper left, and half the curve\nendpoints on top, half on bottom. So 1(3) also looks like this:\n\n\n| | |\n+------*------+-------+--------+\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n| |......| |........|\n+------+------+-------+--------+\n| | |\n\n\nMultiplication is just stacking these; inputs are on top, outputs on\nbottom.\n\nSometimes you get loops:\n\n\n| | |\n+-*----+------+------+\n| |....| |......|\n| \\__/ |......|\n| /......|\n| /.......|\n| /........|\n| /.........|\n| /..........|\n| /...__......|\n| /.../ \\.....|\n| |...| |....|\n+------+---+----+----+\n| | | <----- like this\n+------*---+----+----+\n| |...| |....|\n| \\...\\__/.....|\n| \\...........|\n| \\..........|\n| \\.........|\n| \\........|\n| \\.......|\n| ___ \\......|\n| /...\\ |.....|\n| |.....| |.....|\n+--+-----+-----+-----+\n| | |\n\nWhen you do, you multiply by a constant, delta. This was the\nimportant part that I missed. Something special happens when delta is\nof the form\n\ndelta = cos 4pi/n (I think)\n\nwhich has something to do with Vaughan Jones\' subfactor planar\nalgebras. I didn\'t get all the details, and now I can\'t remember.\nDoes anyone know?\n\nNext week I\'ll see how this works with cobordisms and TQFT\'s.\n\nP.S. ASCII art courtesy of Email Effects. Great stuff, even includes\nfiglet fonts. http://www.sigsoftware.com/emaileffects/\n--\nMike Stay\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>You compared Hilb and nCob in this paper, but it looks like any of the
matrix-mechanics-over-rigs structures from your fall 2003 qg notes
ought to work in the same way. Is that right?

Today I went to a lecture by V.S. Sunder, since the abstract sounded
so similar to what you wrote in this paper. Here it is:

"In recent work with my colleague Vijay Kodiyalam, we showed that
there is a bijective correspondence between Vaughan Jones' `subfactor
planar algebras' on the one hand, and what may be called `unitary
topological quantum field theories' defined on a category `D' on the
other, where the objects of `D' are suitably `decorated closed
oriented 1-manifolds' and the morphisms are similarly decorated
classes of cobordisms between a pair of objects.

Since the subject is slightly technical, it will help to give the talk
in two parts, with the first part devoted to a discussion of Vaughan's
planar algebras, and the second part to our work."

(I had to giggle at "slightly technical" after reading the first
paragraph, but he was right. It was mostly drawing nice pictures of
tangles.) I got it down to the end, but I missed the punchline. His
paper isn't online, so I'll have to see if I can figure it out during
the second lecture.

Anyway, here's what I got:

A tangle T has

1) An outer disk D0 minus an ordered (possibly empty) list of
subdisks.
2) A bunch of curves that divide up the interior into
checkerboard-colorable regions (equivalently, the boundaries of the
disks have an even number of curves ending on them).
3) A set of distinguished points: each disk boundary with at least one
curve intersecting it has one point, where white goes to black when
going clockwise around the disk, that is distinguished (denoted * in
the diagram)
4) "color": take the number of curves intersecting the outside edge
and divide by 2.

And a few other things I'll get to below.

So here is an example of a tangle:

---------------
-----...............-----
*--.........................--\
D0 ///|............................---\\
// |......................../--/ \\
// |....................---/ \\
// |................/--/ \\
/ /-----\........---/ \
/ // \\..---/ \
/ | +- \
/-------+ D1 | \
|........| | |
|.........| | |
|......../-*\ // ----------- |
|......../ \---+-/ //...........\\ |
|.......| |.| //...............\\ |
|.........\ /..| /...................\ |
|..........\---/...| |.....................| |
|..................| |........-----........| |
|..................| |.......// \\.......| |
|..................| |......| |......| |
|..................| |......| D3 |......| |
|..................| |......| |......| |
|............./----+\ |......\\ //......| |
|..........// \\ |........-----........| |
|.........| | \.................../ |
+--------* | \\...............// |
| | D2 | \\...........// |
| | | ----------- |
\ \\ // /
\ \---+-/..\\ /
\ |......\\ /
\ |........\\ /
\\ |..........\\ //
\\ |............\\ //
\\ |..............\\ //
\\\ |................\\ ///
\+-.................\\ --/
-----..............\-----
---------------

Subdisks are "inputs" and the outer boundary is the "output" of the
tangle. There's a natural way to compose tangles: if the input and
output are colored the same, match up the *'s and the curves.

Here is a tangle M(3):

---*--+--+---
----- |..| |...-----
///- |..| |........-\\\
// |..| |............\\
// |..| |..............\\
// |..| |................\\
/ *--+--+-.................\
// // \\................\\
/ / \.................\
/ | |.................\
| | D1 |................|
| | |.................|
| | |..................|
| | |...................|
| \ /.....................|
| \\ //......................|
| +--+--+-........................|
| |..| |.........................|
| |..| |.........................|
| -*--+--|.........................|
| //- -\\.......................|
| // \\.....................|
| / \...................|
| | |...................|
| | |.................|
| | D2 |................|
\ | |................/
\ | |.............../
\\ | |..............//
\ \ /............./
\\ \\ //............//
\\ \\- -//............//
\\ +-+-+--.............//
\\\- |.| |...........-///
----- |.| |......-----
--+-+-+------

It takes two 3-colored tangles X, Y as input and outputs a 3-colored
tangle. We can call it multiplication and denote the output as XY.

Annular tangles have one subdisk. An annular tangle A(m,n) is a
tangle with an m-colored input and an n-colored output. Here is the
identity(3,3) tangle:

-*--+--+---
/--- |..| |...---\
// |..| |.......\\
// |..| |.........\\
/ |..| |...........\
/ *--+--+-...........\
/ // \\..........\
| // \\.........|
| / \........|
| | |.........|
| | |........|
| | |........|
| | |........|
| | |.........|
| \ /........|
| \\ //.........|
\ \\ //........../
\ +---+--+.........../
\ |...| |........../
\\ |...| |........//
\\ |...| |......//
\--- |...| |..---/
-+---+--+--

Then there are tangles with no subdisks. A function from a
zero-dimensional vector space to an n-dimensional one is really just
scalar multiplication. So here's 1(3):

----+----+
//*-....| |--\\
// |.....| |....\\
/ |.....| |......\
/ |.....| |.......\
/ |.....| |........\
| |.....| |.........|
| |.....| |..........|
| |.....| |..........|
| |.....| |..........|
| |.....| |..........|
| |.....| |..........|
| |.....| |..........|
| |.....| |.........|
\ |.....| |......../
\ |.....| |......./
\ |.....| |....../
\\ |.....| |....//
\\+-....| |--//
----+----+

There's a conjugation operator * that's the following steps: reflect,
then move all the *'s counterclockwise (in the original drawing,
clockwise in the reflected one) one position on the disk boundary. So
M*(3) is (note the subdisk labels)

---*--+--+---
----- |..| |...-----
///- |..| |........-\\\
// |..| |............\\
// |..| |..............\\
// |..| |................\\
/ *--+--+-.................\
// // \\................\\
/ / \.................\
/ | |.................\
| | D2 |................|
| | |.................|
| | |..................|
| | |...................|
| \ /.....................|
| \\ //......................|
| +--+--+-........................|
| |..| |.........................|
| |..| |.........................|
| -*--+--|.........................|
| //- -\\.......................|
| // \\.....................|
| / \...................|
| | |...................|
| | |.................|
| | D1 |................|
\ | |................/
\ | |.............../
\\ | |..............//
\ \ /............./
\\ \\ //............//
\\ \\- -//............//
\\ +-+-+--.............//
\\\- |.| |...........-///
----- |.| |......-----
--+-+-+------

I.e. (XY)* = Y*X*.

We get an algebra out of tangles with no subdisks by making the disks
into squares with the * in the upper left, and half the curve
endpoints on top, half on bottom. So 1(3) also looks like this:

| | |[/itex]
+------*------+-------+--------+
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
| |......| |........|
+------+------+-------+--------+
| | |

Multiplication is just stacking these; inputs are on top, outputs on
bottom.

Sometimes you get loops:

| | |
+-*----+------+------+
| |....| |......|
| \__/ |......|
| /......|
| /.......|
| /........|
| /.........|
| /..........|
| /...__......|
| /.../ \.....|
| |...| |....|
+------+---+----+----+
| | | <----- like this
+------*---+----+----+
| |...| |....|
| \...\__/.....|
| \...........|
| \..........|
| \.........|
| \........|
| \.......|
| ___ \......|
| /...\ |.....|
| |.....| |.....|
+--+-----+-----+-----+
[itex]| | |

When you do, you multiply by a constant, \delta. This was the
important part that I missed. Something special happens when \delta is
of the form

\delta = cos 4pi/n (I think)

which has something to do with Vaughan Jones' subfactor planar
algebras. I didn't get all the details, and now I can't remember.
Does anyone know?

Next week I'll see how this works with cobordisms and TQFT's.

P.S. ASCII art courtesy of Email Effects. Great stuff, even includes
figlet fonts. http://www.sigsoftware.com/emaileffects/
--
Mike Stay

Matthew Donald

Apr19-04, 02:14 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nJohn Baez wrote\n> Some of you may enjoy this paper, or at least be infuriated by\n> it:\n\n> http://math.ucr.edu/home/baez/quantum/\n\n> Quantum Quandaries: A Category-Theoretic Perspective\n\nIt has now also appeared as quant-ph/0404040\n\nI both did enjoy it and I was infuriated by it.\n\n\nAt\n\nhttp://www.poco.phy.cam.ac.uk/~mjd1014/readings.html\n\nI keep a page with comments on some of the papers available\nfrom the physics e-print archive which are relevant, or\nsignificant, or recommended in the broad context of my\nmany-minds interpretation of quantum theory. Here\'s what I\'ve\nwritten for that page about John\'s paper:\n\n> J.C. Baez, ``Quantum Quandaries: a Category-Theoretic\n> Perspective\'\' quant-ph/0404040.\n>\n> Baez describes similarities between a category significant for\n> quantum theory and one significant for general relativity. The\n> similarities are at the level of abstract mathematical structure.\n> Baez claims that the mathematics ``accounts for many of the\n> famously puzzling features of quantum theory\'\'. This is correct\n> only in the sense that the mathematics provides a framework\n> within which those puzzling features somehow arise. It does not\n> address the most puzzling questions: specific questions like\n> ``Precisely what might we see happening next?\'\' and ``Precisely\n> how is what we might see restricted by what we are, or by what\n> we do?\'\'.\n\nYou\'ll see from this that I am largely in agreement with the\ncomments made by R (rof@maths.tcd.ie)\n\n-- but only largely -- in as far as R wrote\n> I would say that overall, the most puzzling features of\n> quantum mechanics do not come from its mathematical\n> structures, but from the thing which is not expressed\n> anywhere in the mathematics - the fact that individual\n> measurements have individual results, rather than mere\n> amplitudes of results.\n\nI would have said ``the fact that individual measurements appear\nto us to have individual results\'\'.\n\nR also quoted some of the passage in which John most\n``infuriated\'\' me:\n\n> the famously counter-intuitive behavior of the microworld\n> suggests that not only set theory but even classical logic is not\n> optimized for understanding quantum systems. While there are\n> no real paradoxes, and one can compute everything to one\'s\n> heart\'s content, one often feels that one is grasping these\n> systems `indirectly\', like a nuclear power plant operator\n> handling radioactive material behind a plate glass window with\n> robot arms. This sense of distance is reflected in the endless\n> literature on `interpretations of quantum mechanics\', and also in\n> the constant invocation of the split between `observer\' and\n> `system\'. It is as if classical logic continued to apply to us,\n> while the mysterious rules of quantum theory apply only to the\n> physical systems we are studying. But of course this is not\n> true: we are part of the world being studied.\n\nThe trouble with statements like this is knowing what it could\nmean for classical logic not to apply to us.\n\nSaying that some sort of quantum logic applies hardly answers\nthe question. If you try to unpack an answer, then I think you\nwill end up contributing to the ``endless literature\'\'.\n\nOriginally, quantum logic suggested that the Boolean algebra of\nconventional logic ought to be replaced by the non-Boolean\nalgebra of projections on a Hilbert space. This is an interesting\nand insightful analogy. But what exactly does it do for us?\n\nThe part of the endless literature called ``consistent\nhistories\'\' tries to analyse classes of sequences of projections\nconstituting situations in which we could continue to apply\nclassical logic. It fails because there are far too many\nsuch situations. This means, in particular, that there is no way\nof knowing what the possible continuations of a given history\nshould be, unless a specific set of future possibilities is\nsomehow given a priori.\n\nIn this context, invoking quantum logic, which will widen, rather\nthan restrict, the set of possible histories, seems to me to be\nunhelpful.\n\nIn his paper, Baez\n> suggests that the interpretation of quantum theory will\n> become easier, not harder, when we finally succeed in\n> merging it with general relativity.\n\nMy take is a bit different. I suspect that the interpretative\nproblems of quantum gravity will only become addressable if\nwe first address the problems of the interpretation of\nconventional quantum theory.\n\nOne of the ways that this might happen is that we might come to\naccept that the interpretation of quantum theory has to be\napplied to entities (``observers\'\') which are local both in space\nand in time. This might allow us to accept that conventional\nideas about space and time only need to be applied to the\nobservations of individual local observers. (If we accept, for\nexample, that the big bang didn\'t ``happen\'\', because there was no\nwhen for it to happen in and that it exists only as a singularity in\nextrapolations from our observations, then it might become\neasier to contemplate superpositions of ``initial\'\' temporal\nsingularities.)\n\nIn Baez\'s paper, there is a suggestion that locality might\namount to triviality:\n> a passage of time in which no topology change occurs has no\n> effect at all on the state of the universe. This seems\n> paradoxical at first, since it seems we regularly observe things\n> happening even in the absence of topology change. However,\n> this paradox is easily resolved: a topological quantum field\n> theory describes a world without local degrees of freedom. In\n> such a world, nothing local happens, so the state of the universe\n> can only change when the topology of space itself changes.\n\nAlthough John goes on to describe this as a ``peculiarity of\ntopological quantum field theory\'\', in fact, his paper doesn\'t talk\nabout ``observation\'\' at all. And in a flat space many-worlds\ntheory also, nothing happens without observation. The\nHeisenberg state of the universe just is.\n\nLater Baez says\n> unitary time evolution is not a built-in feature of quantum\n> theory but rather the consequence of specific assumptions\n> about the nature of spacetime\n\nAm I right in thinking that natural assumptions (Hadamard\nstates, for example) imply local unitarity? (I\'ll read that as\nlocally ``nothing is happening without observation\'\'.)\n\nIt seems to me that the analogy between nCob and Hilb is mainly\nlikely to be helpful in talking about global structure. But the\npuzzling features of quantum theory are local.\n\n\nMatthew Donald (matthew.donald@phy.cam.ac.uk)\nweb site:\nhttp://www.poco.phy.cam.ac.uk/~mjd1014\n``a many-minds interpretation of quantum theory\'\'\n************************************** ***\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>John Baez wrote
> Some of you may enjoy this paper, or at least be infuriated by
> it:

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

> Quantum Quandaries: A Category-Theoretic Perspective

It has now also appeared as http://www.arxiv.org/abs/quant-ph/0404040

I both did enjoy it and I was infuriated by it.

At

http://www.poco.phy.cam.ac.uk/~mjd1014/readings.html

I keep a page with comments on some of the papers available
from the physics e-print archive which are relevant, or
significant, or recommended in the broad context of my
many-minds interpretation of quantum theory. Here's what I've
written for that page about John's paper:

> J.C. Baez, ``Quantum Quandaries: a Category-Theoretic
> Perspective'' http://www.arxiv.org/abs/quant-ph/0404040.
>
> Baez describes similarities between a category significant for
> quantum theory and one significant for general relativity. The
> similarities are at the level of abstract mathematical structure.
> Baez claims that the mathematics ``accounts for many of the
> famously puzzling features of quantum theory''. This is correct
> only in the sense that the mathematics provides a framework
> within which those puzzling features somehow arise. It does not
> address the most puzzling questions: specific questions like
> ``Precisely what might we see happening next?'' and ``Precisely
> how is what we might see restricted by what we are, or by what
> we do?''.

You'll see from this that I am largely in agreement with the
comments made by R (rof@maths.tcd.ie)

-- but only largely -- in as far as R wrote
> I would say that overall, the most puzzling features of
> quantum mechanics do not come from its mathematical
> structures, but from the thing which is not expressed
> anywhere in the mathematics - the fact that individual
> measurements have individual results, rather than mere
> amplitudes of results.

I would have said ``the fact that individual measurements appear
to us to have individual results''.

R also quoted some of the passage in which John most
``infuriated'' me:

> the famously counter-intuitive behavior of the microworld
> suggests that not only set theory but even classical logic is not
> optimized for understanding quantum systems. While there are
> no real paradoxes, and one can compute everything to one's
> heart's content, one often feels that one is grasping these
> systems `indirectly', like a nuclear power plant operator
> handling radioactive material behind a plate glass window with
> robot arms. This sense of distance is reflected in the endless
> literature on `interpretations of quantum mechanics', and also in
> the constant invocation of the split between `observer' and
> `system'. It is as if classical logic continued to apply to us,
> while the mysterious rules of quantum theory apply only to the
> physical systems we are studying. But of course this is not
> true: we are part of the world being studied.

The trouble with statements like this is knowing what it could
mean for classical logic not to apply to us.

Saying that some sort of quantum logic applies hardly answers
the question. If you try to unpack an answer, then I think you
will end up contributing to the ``endless literature''.

Originally, quantum logic suggested that the Boolean algebra of
conventional logic ought to be replaced by the non-Boolean
algebra of projections on a Hilbert space. This is an interesting
and insightful analogy. But what exactly does it do for us?

The part of the endless literature called ``consistent
histories'' tries to analyse classes of sequences of projections
constituting situations in which we could continue to apply
classical logic. It fails because there are far too many
such situations. This means, in particular, that there is no way
of knowing what the possible continuations of a given history
should be, unless a specific set of future possibilities is
somehow given a priori.

In this context, invoking quantum logic, which will widen, rather
than restrict, the set of possible histories, seems to me to be
unhelpful.

In his paper, Baez
> suggests that the interpretation of quantum theory will
> become easier, not harder, when we finally succeed in
> merging it with general relativity.

My take is a bit different. I suspect that the interpretative
problems of quantum gravity will only become addressable if
we first address the problems of the interpretation of
conventional quantum theory.

One of the ways that this might happen is that we might come to
accept that the interpretation of quantum theory has to be
applied to entities (``observers'') which are local both in space
and in time. This might allow us to accept that conventional
ideas about space and time only need to be applied to the
observations of individual local observers. (If we accept, for
example, that the big bang didn't ``happen'', because there was no
when for it to happen in and that it exists only as a singularity in
extrapolations from our observations, then it might become
easier to contemplate superpositions of ``initial'' temporal
singularities.)

In Baez's paper, there is a suggestion that locality might
amount to triviality:
> a passage of time in which no topology change occurs has no
> effect at all on the state of the universe. This seems
> paradoxical at first, since it seems we regularly observe things
> happening even in the absence of topology change. However,
> this paradox is easily resolved: a topological quantum field
> theory describes a world without local degrees of freedom. In
> such a world, nothing local happens, so the state of the universe
> can only change when the topology of space itself changes.

Although John goes on to describe this as a ``peculiarity of
topological quantum field theory'', in fact, his paper doesn't talk
about ``observation'' at all. And in a flat space many-worlds
theory also, nothing happens without observation. The
Heisenberg state of the universe just is.

Later Baez says
> unitary time evolution is not a built-in feature of quantum
> theory but rather the consequence of specific assumptions
> about the nature of spacetime

Am I right in thinking that natural assumptions (Hadamard
states, for example) imply local unitarity? (I'll read that as
locally ``nothing is happening without observation''.)

It seems to me that the analogy between nCob and Hilb is mainly
likely to be helpful in talking about global structure. But the
puzzling features of quantum theory are local.

Matthew Donald (matthew.donald@phy.cam.ac.uk)
web site:
http://www.poco.phy.cam.ac.uk/~mjd1014
``a many-minds interpretation of quantum theory''
*****************************************

John Baez

Apr22-04, 04:29 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <d6255a14.0404150226.3145a805@posting.google.com>, \nMike Stay <staym@datawest.net> wrote:\n\n>You compared Hilb and nCob in this paper, but it looks like any of the\n>matrix-mechanics-over-rigs structures from your fall 2003 qg notes\n>ought to work in the same way. Is that right?\n\nRight. Since this paper was written for philosophers of\nphysics, I figured I\'d better not bombard them with *too*\nmuch math, like matrix mechanics over an arbitrary *-rig.\n\n*-rig?\n\nWell, a rig is just a "ring without negatives", like the\nnatural numbers or {F,T} with "or" as plus and "and" as times.\nWe can do a lot of matrix mechanics using matrices with\ncoefficients in an arbitrary rig, and in this paper I note\nthat if we use {F,T} our matrices are just what people usually\ncall "binary relations". These matrices say whether or not a\ntransition is *possible*, instead of giving a transition *amplitude*.\n\nHowever, for the formalism to nicely include complex matrix\nmechanics - the example that actually comes up in quantum theory! -\nwe really want a *-structure on our rig, i.e. an operation satisfying\n\n(a+b)* = a* + b* 0* = 0\n(ab)* = b* a* 1* = 1\n\nThis fills the role played by complex conjugation in complex\nmatrix mechanics. I forget if I got around to talking about\nthis in the fall 2003 quantum gravity seminar:\n\nhttp://www.math.ucr.edu/home/baez/qg-fall2003/\n\n.... but I should have.\n\nAny commutative rig becomes a *-rig if we define a* = a;\nthe real numbers and natural numbers and {F,T} should be thought\nof as *-rigs of this degenerate sort. It\'s interesting that\nNature has chosen a *-rig of a more interesting sort.\n\n>Today I went to a lecture by V.S. Sunder, since the abstract sounded\n>so similar to what you wrote in this paper. Here it is:\n>\n>"In recent work with my colleague Vijay Kodiyalam, we showed that\n>there is a bijective correspondence between Vaughan Jones\' `subfactor\n>planar algebras\' on the one hand, and what may be called `unitary\n>topological quantum field theories\' defined on a category `D\' on the\n>other, where the objects of `D\' are suitably `decorated closed\n>oriented 1-manifolds\' and the morphisms are similarly decorated\n>classes of cobordisms between a pair of objects.\n\nCool! I\'ve been interested in Jones\' planar algebras for a while,\nbut perhaps because he doesn\'t use enough n-category theory, his\ndefinition of a "planar algebra" sounds rather ad hoc and contrived.\nUnfortunately, I\'ve never had the energy to translate his definition\ninto the language of n-category theory to see exactly how much modification\nit needs (if any) to appear beautiful and "inevitable".\n\nJones is a very good mathematician, so presumably planar algebras\n*are* beautiful and inevitable, at least after a little tweaking or\ngeneralization, if one looks at them the right way.\n\nSunder sounds like he\'s struggling to do this.\n\nBut....\n\n>Anyway, here\'s what I got:\n>\n>A tangle T has\n>\n>1) An outer disk D0 minus an ordered (possibly empty) list of\n>subdisks.\n>2) A bunch of curves that divide up the interior into\n>checkerboard-colorable regions (equivalently, the boundaries of the\n>disks have an even number of curves ending on them).\n>3) A set of distinguished points: each disk boundary with at least one\n>curve intersecting it has one point, where white goes to black when\n>going clockwise around the disk, that is distinguished (denoted * in\n>the diagram)\n>4) "color": take the number of curves intersecting the outside edge\n>and divide by 2.\n>\n>And a few other things I\'ll get to below.\n\nUgh... these rather complex "decorations" are precisely the sort\nof things that seem ad hoc and complicated about planar algebras!\nI think there has to be some better way to understand all this stuff\nusing a bit categories or n-categories. But I\'d need to understand\nwhat all these decorations *accomplish* to figure this out.\n\n>So here is an example of a tangle:\n\n---------------\n-----...............-----\n*--.........................--\\\nD0 ///|............................---\\\\\n// |......................../--/ \\\\\n// |....................---/ \\\\\n// |................/--/ \\\\\n/ /-----\\........---/ \\\n/ // \\\\..---/ \\\n/ | +- \\\n/-------+ D1 | \\\n|........| | |\n|.........| | |\n|......../-*\\ // ----------- |\n|......../ \\---+-/ //...........\\\\ |\n|.......| |.| //...............\\\\ |\n|.........\\ /..| /...................\\ |\n|..........\\---/...| |.....................| |\n|..................| |........-----........| |\n|..................| |.......// \\\\.......| |\n|..................| |......| |......| |\n|..................| |......| D3 |......| |\n|..................| |......| |......| |\n|............./----+\\ |......\\\\ //......| |\n|..........// \\\\ |........-----........| |\n|.........| | \\.................../ |\n+--------* | \\\\...............// |\n| | D2 | \\\\...........// |\n| | | ----------- |\n\\ \\\\ // /\n\\ \\---+-/..\\\\ /\n\\ |......\\\\ /\n\\ |........\\\\ /\n\\\\ |..........\\\\ //\n\\\\ |............\\\\ //\n\\\\ |..............\\\\ //\n\\\\\\ |................\\\\ ///\n\\+-.................\\\\ --/\n-----..............\\-----\n---------------\n\nCool! It looks like a Yin-Yang symbol on steroids!\n\n>When you do, you multiply by a constant, delta. This was the\n>important part that I missed. Something special happens when delta is\n>of the form\n>\n>delta = cos 4pi/n (I think)\n>\n>which has something to do with Vaughan Jones\' subfactor planar\n>algebras. I didn\'t get all the details, and now I can\'t remember.\n>Does anyone know?\n\nThe "hyperfinite type II_1 factor" is an incredibly cool algebra\nthat shows up in von Neumann\'s classification of operator algebras\n(see below). Just as modules of the complex numbers are called\n"complex vector spaces" and these can have dimension 0,1,2,3,...,\nthe hyperfinite type II_1 finite factor has modules which have some\nsort of "dimension" that can be any nonnegative real number! This\nwas shown by von Neumann and Murray back in the 1930\'s or 1940\'s.\n\nThe hyperfinite type II_1 factor has copies of itself sitting inside\nitself, and one of Jones\' great achievements (for which he won the\nFields medal) was to figure out what dimensions these could have.\nThe allowed dimensions are numbers pretty much like your numbers\ncos(4 pi / n) - though like you I forget the exact formula. I\'m\nfuzzy about the details, but I\'m sure this is what\'s lurking behind\nthe appearance of that number delta!\n\nBest,\njb\n\n........................... .................................................\ n\nAlso available at http://math.ucr.edu/home/baez/week175.html\n\nDecember 29, 2001\nThis Week\'s Finds in Mathematical Physics (Week 175)\nJohn Baez\n\n[stuff deleted]\n\nIn case you don\'t know: Alain Connes is a Fields medalist, who won the\nprize mainly for two things: his work on Von Neumann algebras, and his\nwork on noncommutative geometry. Now I\'ll talk a bit about von Neumann\nalgebras, since you\'ll need to understand a bit about them to follow the\nrest of my description of the paper by Michael Mueger that I have\nbeen slowly explaining throughout "week173" and "week174".\n\nSo: what\'s a von Neumann algebra? Before I get technical and you all\nleave, I should just say that von Neumann designed these algebras to be\ngood "algebras of observables" in quantum theory. The simplest example\nconsists of all n x n complex matrices: these become an algebra if you\nadd and multiply them the usual way. So, the subject of von Neumann\nalgebras is really just a grand generalization of the theory of matrix\nmultiplication.\n\nBut enough beating around the bush! For starters, a von Neumann algebra\nis a *-algebra of bounded operators on some Hilbert space of countable\ndimension - that is, a bunch of bounded operators closed under addition,\nmultiplication, scalar multiplication, and taking adjoints: that\'s the *\nbusiness. However, to be a von Neumann algebra, our *-algebra needs one\nextra property! This extra property is cleverly chosen so that we can\napply functions to observables and get new observables, which is\nsomething we do all the time in physics.\n\nMore precisely, given any self-adjoint operator A in our von Neumann\nalgebra and any measurable function f: R -> R, we want there to be a\nself-adjoint operator f(A) that again lies in our von Neumann algebra.\nTo make sure this works, we need our von Neumann algebra to be "closed"\nin a certain sense. The nice thing is that we can state this closure\nproperty either algebraically or topologically.\n\nIn the algebraic approach, we define the "commutant" of a bunch of\noperators to be the set of operators that commute with all of them.\nWe then say a von Neumann algebra is a *-algebra of operators that\'s\nthe commutant of its commutant.\n\nIn the topological approach, we say a bunch of operators T_i converges\n"weakly" to an operator T if their expectation values converge to that\nof T in every state, that is,\n\n<psi, T_i psi> -> <psi, T psi>\n\nfor all unit vectors psi in the Hilbert space. We then say a von\nNeumann algebra is an *-algebra of operators that is closed in the\nweak topology.\n\nIt\'s a nontrivial theorem that these two definitions agree!\n\nWhile classifying all *-algebras of operators is an utterly hopeless\ntask, classifying von Neumann algebras is almost within reach - close\nenough to be tantalizing, anyway. Every von Neumann algebra can be\nbuilt from so-called "simple" ones as a direct sum, or more generally a\n"direct integral", which is a kind of continuous version of a direct\nsum. As usual in algebra, the "simple" von Neumann algebras are defined\nto be those without any nontrivial ideals. This turns out to be\nequivalent to saying that only scalar multiples of the identity commute\nwith everything in the von Neumann algebra.\n\nPeople call simple von Neumann algebras "factors" for short. Anyway,\nthe point is that we just need to classify the factors: the process\nof sticking these together to get the other von Neumann algebras is\nnot tricky.\n\nThe first step in classifying factors was done by von Neumann and\nMurray, who divided them into types I, II, and III. This classification\ninvolves the concept of a "trace", which is a generalization of the\nusual trace of a matrix.\n\nHere\'s the definition of a trace on a von Neumann algebra. First, we say\nan element of a von Neumann algebra is "nonnegative" if it\'s of the form\nxx* for some element x. The nonnegative elements form a "cone": they\nare closed under addition and under multiplication by nonnegative\nscalars. Let C be the cone of nonnegative elements. Then a "trace" is\na function\n\ntr: C -> [0, +infinity]\n\nwhich is linear in the obvious sense and satisfies\n\ntr(xy) = tr(yx)\n\nwhenever both xy and yx are nonnegative.\n\nNote: we allow the trace to be infinite, since the interesting von\nNeumann algebras are infinite-dimensional. This is why we define\nthe trace only on nonnegative elements; otherwise we get "infinity minus\ninfinity" problems. The same thing shows up in the measure theory,\nwhere we start by integrating nonnegative functions, possibly getting\nthe answer +infinity, and worry later about other functions.\n\nIndeed, a trace very much like an integral, so we\'re really studying a\nnoncommutative version of the theory of integration. On the other hand,\nin the matrix case, the trace of a projection operator is just the\ndimension of the space it\'s the projection onto. We can define a\n"projection" in any von Neumann algebra to be an operator with p* = p\nand p^2 = p. If we study the trace of such a thing, we\'re studying a\nGENERALIZATION OF THE CONCEPT OF DIMENSION. It turns out this can be\ninfinite, or even nonintegral!\n\nWe say a factor is "type I" if it admits a nonzero trace for which the\ntrace of a projection lies in the set {0,1,2,...,+infinity}. We say it\'s\n"type I_n" if we can normalize the trace so we get the values {0,1,...,n}.\nOtherwise, we say it\'s "type I_infinity", and we can normalize the trace\nto get all the values {0,1,2,...,+infinity}.\n\nIt turn out that every type I_n factor is isomorphic to the algebra of\nn x n matrices. Also, every type I_infinity factor is isomorphic to the\nalgebra of all bounded operators on a Hilbert space of countably infinite\ndimension.\n\nType I factors are the algebras of observables that we learn to love in\nquantum mechanics. So, the real achievement of von Neumann was to begin\nexploring the other factors, which turned out to be important in quantum\nfield theory.\n\nWe say a factor is "type II_1" if it admits a trace whose values on\nprojections are all the numbers in the unit interval [0,1]. We say it\nis "type II_infinity" if it admits a trace whose value on projections\nis everything in [0,infinity].\n\nPlaying with type II factors amounts to letting dimension be a\ncontinuous rather than discrete parameter!\n\nWeird as this seems, it\'s easy to construct a type II_1 factor. Start\nwith the algebra of 1 x 1 matrices, and stuff it into the algebra of\n2 x 2 matrices as follows:\n\n( x 0 )\nx |-> ( )\n( 0 x )\n\nThis doubles the trace, so define a new trace on the algebra of 2 x 2\nmatrices which is half the usual one. Now keep doing this, doubling the\ndimension each time, using the above formula to define a map from the\n2^n x 2^n matrices into the 2^{n+1} x 2^{n+1} matrices, and normalizing\nthe trace on each of these matrix algebras so that all the maps are\ntrace-preserving. Then take the UNION of all these algebras... and\nfinally, with a little work, complete this and get a von Neumann algebra!\n\nOne can show this von Neumann algebra is a factor. It\'s pretty\nobvious that the trace of a projection can be any fraction in the\ninterval [0,1] whose denominator is a power of two. But actually,\n*any* number from 0 to 1 is the trace of some projection in this\nalgebra - so we\'ve got our paws on a type II_1 factor.\n\nThis isn\'t the only II_1 factor, but it\'s the only one that contains a\nsequence of finite-dimensional von Neumann algebras whose union is dense\nin the weak topology. A von Neumann algebra like that is called\n"hyperfinite", so this guy is called "the hyperfinite II_1 factor".\n\nIt may sound like something out of bad science fiction, but the\nhyperfinite II_1 factor shows up all over the place in physics!\n\nFirst of all, the algebra of 2^n x 2^n matrices is a Clifford algebra,\nso the hyperfinite II_1 factor is a kind of infinite-dimensional\nClifford algebra. But the Clifford algebra of 2^n x 2^n matrices is\nsecretly just another name for the algebra generated by creation and\nannihilation operators on the fermionic Fock space over C^{2n}.\nPondering this a bit, you can show that the hyperfinite II_1 factor is\nthe smallest von Neumann algebra containing the creation and\nannihilation operators on a fermionic Fock space of countably infinite\ndimension.\n\nIn less technical lingo - I\'m afraid I\'m starting to assume you know\nquantum field theory! - the hyperfinite II_1 factor is the right algebra\nof observables for a free quantum field theory with only fermions.\nFor bosons, you want the type I_infinity factor.\n\nThere is more than one type II_infinity factor, but again there is\nonly one that is hyperfinite. You can get this by tensoring the type\nI_infinity factor and the hyperfinite II_1 factor. Physically, this\nmeans that the hyperfinite II_infinity factor is the right algebra of\nobservables for a free quantum field theory with both bosons and fermions.\n\nThe most mysterious factors are those of type III. These can be simply\ndefined as "none of the above"! Equivalently, they are factors for\nwhich any nonzero trace takes values in {0,infinity}. In a type III\nfactor, all projections other than 0 have infinite trace. In other\nwords, the trace is a useless concept for these guys.\n\nAs far as I\'m concerned, the easiest way to construct a type III factor\nuses physics. Now, I said that free quantum field theories had\ndifferent kinds of type I or type II factors as their algebras of\nobservables. This is true if you consider the algebra of *all*\nobservables. However, if you consider a free quantum field theory on\n(say) Minkowski spacetime, and look only at the observables that you can\ncook from the field operators on some bounded open set, you get a\nsubalgebra of observables which turns out to be a type III factor!\n\nIn fact, this isn\'t just true for free field theories. According to a\ntheorem of axiomatic quantum field theory, pretty much all the usual\nfield theories on Minkowski spacetime have type III factors as their\nalgebras of "local observables" - observables that can be measured in\na bounded open set.\n\nOkay, so much for the crash course on von Neumann algebras! Next time\nI\'ll hook this up to Mueger\'s work on 2-categories.\n\nIn the meantime, here are some references on von Neumann algebras in\ncase you want to dig deeper. For the math, try these:\n\n5) Masamichi Takesaki, Theory of Operator Algebras I, Springer,\nBerlin, 1979.\n\n6) Richard V. Kadison and John Ringrose, Fundamentals of the\nTheory of Operator Algebras, 4 volumes, Academic Press, New York,\n1983-1992.\n\n7) Shoichiro Sakai, C*-algebras and W*-algebras, Springer, Berlin,\n1971.\n\nA W*-algebra is basically just a von Neumann algebra, but defined\n"intrinsically", in a way that doesn\'t refer to a particular\nrepresentation as operators on a Hilbert space.\n\nFor applications to physics, try these:\n\n8) Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum\nField Theory, Wiley-Interscience, New York, 1972.\n\n9) Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras,\nSpringer, Berlin, 1992.\n\n10) Ola Bratelli and Derek W. Robinson, Operator Algebras and Quantum\nStatistical Mechanics, 2 volumes, Springer, Berlin, 1987-1997.\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <d6255a14.0404150226.3145a805@posting.google.com>,
Mike Stay <staym@datawest.net> wrote:

>You compared Hilb and nCob in this paper, but it looks like any of the
>matrix-mechanics-over-rigs structures from your fall 2003 qg notes
>ought to work in the same way. Is that right?

Right. Since this paper was written for philosophers of
physics, I figured I'd better not bombard them with *too*
much math, like matrix mechanics over an arbitrary *-rig.

*-rig[/itex]?

Well, a rig is just a "ring without negatives", like the
natural numbers or {F,T} with "or" as plus and "and" as times.
We can do a lot of matrix mechanics using matrices with
coefficients in an arbitrary rig, and in this paper I note
that if we use {F,T} our matrices are just what people usually
call "binary relations". These matrices say whether or not a
transition is *possible*, instead of giving a transition *amplitude*.

However, for the formalism to nicely include complex matrix
mechanics - the example that actually comes up in quantum theory! -
we really want a *-structure on our rig, i.e. an operation satisfying

(a+b)* = a* + b* 0* =(ab)* = b* a* 1* = 1

This fills the role played by complex conjugation in complex
matrix mechanics. I forget if I got around to talking about
this in the fall 2003 quantum gravity seminar:

http://www.math.ucr.edu/home/baez/qg-fall2003/

.... but I should have.

Any commutative rig becomes a *-rig if we define a* = a;
the real numbers and natural numbers and {F,T} should be thought
of as *-rigs of this degenerate sort. It's interesting that
Nature has chosen a *-rig of a more interesting sort.

>Today I went to a lecture by V.S. Sunder, since the abstract sounded
>so similar to what you wrote in this paper. Here it is:
>
>"In recent work with my colleague Vijay Kodiyalam, we showed that
>there is a bijective correspondence between Vaughan Jones' `subfactor
>planar algebras' on the one hand, and what may be called `unitary
>topological quantum field theories' defined on a category `D' on the
>other, where the objects of `D' are suitably `decorated closed
>oriented 1-manifolds' and the morphisms are similarly decorated
>classes of cobordisms between a pair of objects.

Cool! I've been interested in Jones' planar algebras for a while,
but perhaps because he doesn't use enough n-category theory, his
definition of a "planar algebra" sounds rather ad hoc and contrived.
Unfortunately, I've never had the energy to translate his definition
into the language of n-category theory to see exactly how much modification
it needs (if any) to appear beautiful and "inevitable".

Jones is a very good mathematician, so presumably planar algebras
*are* beautiful and inevitable, at least after a little tweaking or
generalization, if one looks at them the right way.

Sunder sounds like he's struggling to do this.

But....

>Anyway, here's what I got:
>
>A tangle T has
>
>1) An outer disk D0 minus an ordered (possibly empty) list of
>subdisks.
>2) A bunch of curves that divide up the interior into
>checkerboard-colorable regions (equivalently, the boundaries of the
>disks have an even number of curves ending on them).
>3) A set of distinguished points: each disk boundary with at least one
>curve intersecting it has one point, where white goes to black when
>going clockwise around the disk, that is distinguished (denoted * in
>the diagram)
>4) "color": take the number of curves intersecting the outside edge
>and divide by 2.
>
>And a few other things I'll get to below.

Ugh... these rather complex "decorations" are precisely the sort
of things that seem ad hoc and complicated about planar algebras!
I think there has to be some better way to understand all this stuff
using a bit categories or n-categories. But I'd need to understand
what all these decorations *accomplish* to figure this out.

>So here is an example of a tangle:

---------------
-----...............-----
*--.........................--\
D0 ///|............................---\\// |......................../--/ \\
// |....................---/ \\
// |................/--/ \\
/ /-----\........---/ \
/ // \\..---/ \
/ | +- \
/-------+ D1 | \
|........| | |
|.........| | |
|......../-*\ // ----------- |
|......../ \---+-/ //...........\\ |
|.......| |.| //...............\\ |
|.........\ /..| /...................\ |
|..........\---/...| |.....................| |
|..................| |........-----........| |
|..................| |.......// \\.......| |
|..................| |......| |......| |
|..................| |......| D3 |......| |
|..................| |......| |......| |
|............./----+\ |......\\ //......| |
|..........// \\ |........-----........| |
|.........| | \.................../ |
+--------* | \\...............// || | D2 | \\...........// || | | ----------- |
\ \\ // /
\ \---+-/..\\ /\ |......\\ /\ |........\\ /\\ |..........\\ //\\ |............\\ //\\ |..............\\ //\\\ |................\\ ///
\+-.................\\ --/
-----..............\-----
---------------

Cool! It looks like a Yin-Yang symbol on steroids!

>When you do, you multiply by a constant, [itex]\delta. This was the
>important part that I missed. Something special happens when \delta is
>of the form
>
>\delta = cos 4pi/n (I think)
>
>which has something to do with Vaughan Jones' subfactor planar
>algebras. I didn't get all the details, and now I can't remember.
>Does anyone know?

The "hyperfinite type II_1 factor" is an incredibly cool algebra
that shows up in von Neumann's classification of operator algebras
(see below). Just as modules of the complex numbers are called
"complex vector spaces" and these can have dimension 0,1,2,3,...,
the hyperfinite type II_1 finite factor has modules which have some
sort of "dimension" that can be any nonnegative real number! This
was shown by von Neumann and Murray back in the 1930's or 1940's.

The hyperfinite type II_1 factor has copies of itself sitting inside
itself, and one of Jones' great achievements (for which he won the
Fields medal) was to figure out what dimensions these could have.
The allowed dimensions are numbers pretty much like your numbers
cos(4 \pi / n) - though like you I forget the exact formula. I'm
fuzzy about the details, but I'm sure this is what's lurking behind
the appearance of that number \delta!

Best,
jb

.................................................. ..........................

Also available at http://math.ucr.edu/home/baez/week175.html

December 29, 2001
This Week's Finds in Mathematical Physics (Week 175)
John Baez

[stuff deleted]

In case you don't know: Alain Connes is a Fields medalist, who won the
prize mainly for two things: his work on Von Neumann algebras, and his
work on noncommutative geometry. Now I'll talk a bit about von Neumann
algebras, since you'll need to understand a bit about them to follow the
rest of my description of the paper by Michael Mueger that I have
been slowly explaining throughout "week173" and "week174".

So: what's a von Neumann algebra? Before I get technical and you all
leave, I should just say that von Neumann designed these algebras to be
good "algebras of observables" in quantum theory. The simplest example
consists of all n x n complex matrices: these become an algebra if you
add and multiply them the usual way. So, the subject of von Neumann
algebras is really just a grand generalization of the theory of matrix
multiplication.

But enough beating around the bush! For starters, a von Neumann algebra
is a *-algebra of bounded operators on some Hilbert space of countable
dimension - that is, a bunch of bounded operators closed under addition,
multiplication, scalar multiplication, and taking adjoints: that's the *
business. However, to be a von Neumann algebra, our *-algebra needs one
extra property! This extra property is cleverly chosen so that we can
apply functions to observables and get new observables, which is
something we do all the time in physics.

More precisely, given any self-adjoint operator A in our von Neumann
algebra and any measurable function f: R -> R, we want there to be a
self-adjoint operator f(A) that again lies in our von Neumann algebra.
To make sure this works, we need our von Neumann algebra to be "closed"
in a certain sense. The nice thing is that we can state this closure
property either algebraically or topologically.

In the algebraic approach, we define the "commutant" of a bunch of
operators to be the set of operators that commute with all of them.
We then say a von Neumann algebra is a *-algebra of operators that's
the commutant of its commutant.

In the topological approach, we say a bunch of operators T_i converges
"weakly" to an operator T if their expectation values converge to that
of T in every state, that is,

<\psi, T_i \psi> -> <\psi, T \psi>

for all unit vectors \psi in the Hilbert space. We then say a von
Neumann algebra is an *-algebra of operators that is closed in the
weak topology.

It's a nontrivial theorem that these two definitions agree!

While classifying all *-algebras of operators is an utterly hopeless
task, classifying von Neumann algebras is almost within reach - close
enough to be tantalizing, anyway. Every von Neumann algebra can be
built from so-called "simple" ones as a direct sum, or more generally a
"direct integral", which is a kind of continuous version of a direct
sum. As usual in algebra, the "simple" von Neumann algebras are defined
to be those without any nontrivial ideals. This turns out to be
equivalent to saying that only scalar multiples of the identity commute
with everything in the von Neumann algebra.

People call simple von Neumann algebras "factors" for short. Anyway,
the point is that we just need to classify the factors: the process
of sticking these together to get the other von Neumann algebras is
not tricky.

The first step in classifying factors was done by von Neumann and
Murray, who divided them into types I, II, and III. This classification
involves the concept of a "trace", which is a generalization of the
usual trace of a matrix.

Here's the definition of a trace on a von Neumann algebra. First, we say
an element of a von Neumann algebra is "nonnegative" if it's of the form
xx* for some element x. The nonnegative elements form a "cone": they
are closed under addition and under multiplication by nonnegative
scalars. Let C be the cone of nonnegative elements. Then a "trace" is
a function

tr: C -> [0, +infinity]

which is linear in the obvious sense and satisfies

tr(xy) = tr(yx)

whenever both xy and yx are nonnegative.

Note: we allow the trace to be infinite, since the interesting von
Neumann algebras are infinite-dimensional. This is why we define
the trace only on nonnegative elements; otherwise we get "infinity minus
infinity" problems. The same thing shows up in the measure theory,
where we start by integrating nonnegative functions, possibly getting
the answer +infinity, and worry later about other functions.

Indeed, a trace very much like an integral, so we're really studying a
noncommutative version of the theory of integration. On the other hand,
in the matrix case, the trace of a projection operator is just the
dimension of the space it's the projection onto. We can define a
"projection" in any von Neumann algebra to be an operator with p* = p
and p^2 = p. If we study the trace of such a thing, we're studying a
GENERALIZATION OF THE CONCEPT OF DIMENSION. It turns out this can be
infinite, or even nonintegral!

We say a factor is "type I" if it admits a nonzero trace for which the
trace of a projection lies in the set {0,1,2,...,+infinity}. We say it's
"type I_n" if we can normalize the trace so we get the values {0,1,...,n}.
Otherwise, we say it's "type I_{infinity}", and we can normalize the trace
to get all the values {0,1,2,...,+infinity}.

It turn out that every type I_n factor is isomorphic to the algebra of
n x n matrices. Also, every type I_{infinity} factor is isomorphic to the
algebra of all bounded operators on a Hilbert space of countably infinite
dimension.

Type I factors are the algebras of observables that we learn to love in
quantum mechanics. So, the real achievement of von Neumann was to begin
exploring the other factors, which turned out to be important in quantum
field theory.

We say a factor is "type II_1" if it admits a trace whose values on
projections are all the numbers in the unit interval [0,1]. We say it
is "type II_infinity" if it admits a trace whose value on projections
is everything in [0,infinity].

Playing with type II factors amounts to letting dimension be a
continuous rather than discrete parameter!

Weird as this seems, it's easy to construct a type II_1 factor. Start
with the algebra of 1 x 1 matrices, and stuff it into the algebra of
2 x 2 matrices as follows:

( x )
x |-> ( )( x )

This doubles the trace, so define a new trace on the algebra of 2 x 2
matrices which is half the usual one. Now keep doing this, doubling the
dimension each time, using the above formula to define a map from the
2^n x 2^n matrices into the 2^{n+1} x 2^{n+1} matrices, and normalizing
the trace on each of these matrix algebras so that all the maps are
trace-preserving. Then take the UNION of all these algebras... and
finally, with a little work, complete this and get a von Neumann algebra!

One can show this von Neumann algebra is a factor. It's pretty
obvious that the trace of a projection can be any fraction in the
interval [0,1] whose denominator is a power of two. But actually,
*any* number from to 1 is the trace of some projection in this
algebra - so we've got our paws on a type II_1 factor.

This isn't the only II_1 factor, but it's the only one that contains a
sequence of finite-dimensional von Neumann algebras whose union is dense
in the weak topology. A von Neumann algebra like that is called
"hyperfinite", so this guy is called "the hyperfinite II_1 factor".

It may sound like something out of bad science fiction, but the
hyperfinite II_1 factor shows up all over the place in physics!

First of all, the algebra of 2^n x 2^n matrices is a Clifford algebra,
so the hyperfinite II_1 factor is a kind of infinite-dimensional
Clifford algebra. But the Clifford algebra of 2^n x 2^n matrices is
secretly just another name for the algebra generated by creation and
annihilation operators on the fermionic Fock space over C^{2n}.
Pondering this a bit, you can show that the hyperfinite II_1 factor is
the smallest von Neumann algebra containing the creation and
annihilation operators on a fermionic Fock space of countably infinite
dimension.

In less technical lingo - I'm afraid I'm starting to assume you know
quantum field theory! - the hyperfinite II_1 factor is the right algebra
of observables for a free quantum field theory with only fermions.
For bosons, you want the type I_{infinity} factor.

There is more than one type II_infinity factor, but again there is
only one that is hyperfinite. You can get this by tensoring the type
I_{infinity} factor and the hyperfinite II_1 factor. Physically, this
means that the hyperfinite II_infinity factor is the right algebra of
observables for a free quantum field theory with both bosons and fermions.

The most mysterious factors are those of type III. These can be simply
defined as "none of the above"! Equivalently, they are factors for
which any nonzero trace takes values in {0,infinity}. In a type III
factor, all projections other than have infinite trace. In other
words, the trace is a useless concept for these guys.

As far as I'm concerned, the easiest way to construct a type III factor
uses physics. Now, I said that free quantum field theories had
different kinds of type I or type II factors as their algebras of
observables. This is true if you consider the algebra of *all*
observables. However, if you consider a free quantum field theory on
(say) Minkowski spacetime, and look only at the observables that you can
cook from the field operators on some bounded open set, you get a
subalgebra of observables which turns out to be a type III factor!

In fact, this isn't just true for free field theories. According to a
theorem of axiomatic quantum field theory, pretty much all the usual
field theories on Minkowski spacetime have type III factors as their
algebras of "local observables" - observables that can be measured in
a bounded open set.

Okay, so much for the crash course on von Neumann algebras! Next time
I'll hook this up to Mueger's work on 2-categories.

In the meantime, here are some references on von Neumann algebras in
case you want to dig deeper. For the math, try these:

5) Masamichi Takesaki, Theory of Operator Algebras I, Springer,
Berlin, 1979.

6) Richard V. Kadison and John Ringrose, Fundamentals of the
Theory of Operator Algebras, 4 volumes, Academic Press, New York,
1983-1992.

7) Shoichiro Sakai, C*-algebras and W*-algebras, Springer, Berlin,
1971.

A W*-algebra is basically just a von Neumann algebra, but defined
"intrinsically", in a way that doesn't refer to a particular
representation as operators on a Hilbert space.

For applications to physics, try these:

8) Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum
Field Theory, Wiley-Interscience, New York, 1972.

9) Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras,
Springer, Berlin, 1992.

10) Ola Bratelli and Derek W. Robinson, Operator Algebras and Quantum
Statistical Mechanics, 2 volumes, Springer, Berlin, 1987-1997.