View Full Version : Meaning of self-duality?
Danny Ross Lunsford
Apr24-04, 11:16 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>I\'ve been looking at a theory of trivectors on a 6-d space. The issue of\nself-duality naturally arises in this theory in a striking way.\n\nWhat is generally assumed to be the physical meaning of this condition\nimposed on a field? (One sees many "dualities" and it is hard to\ndetermine which ones are real and which are wishful thinking.)\n\n-drl\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>I've been looking at a theory of trivectors on a 6-d space. The issue of
self-duality naturally arises in this theory in a striking way.
What is generally assumed to be the physical meaning of this condition
imposed on a field? (One sees many "dualities" and it is hard to
determine which ones are real and which are wishful thinking.)
-drl
<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>Danny Ross Lunsford <antimatter33@yahoo.nose-pam.com> wrote in message news:<nrHhc.19787\\$Fz6.1453@newssvr22.news.prodig y.com>...\n> I\'ve been looking at a theory of trivectors on a 6-d space. The issue of\n> self-duality naturally arises in this theory in a striking way.\n>\n> What is generally assumed to be the physical meaning of this condition\n> imposed on a field? (One sees many "dualities" and it is hard to\n> determine which ones are real and which are wishful thinking.)\n>\n> -drl\n\nThere are many definitions of duality in mathematics and physics, but\nit appears that the one you\'re talking about involves objects in a\nClifford Algebra. In an n dimensional geometry, we can talk about a\nset of 2^n independent objects that make up a linear basis over the\ngeometry. These things have many different names, but the general\nnomenclature refers to them as 0-vectors,\n1-vectors,2-vectors,...,n-vectors, each number being known as the\ngrade of that particular set of objects. Usually these are called\nscalars, vectors, bi-vectors,...,psuedo-scalars. The number of\nindependent elements in each set of a particular grade is determined\nby using Pascal\'s triangle. In other words, the number of independent\nm-vector elements in n dimensions is always given by the combinations\nof n objects taken m at a time.\n\nThe notion of duality works such that, in n dimensions, two classes of\nobjects are said to be dual if their grades add up to n. A good\nexample is in 3 dimensional space, where objects of grade 0 (scalars)\nare dual to objects of grade 3 (psuedo-scalars), and grade 1 objects\n(vectors) are dual to grade 2 objects (bi-vectors or axial vectors).\nObviously no set of objects of a particular grade are self dual in n =\n3. In n = 4, however, it is a different picture. Scalars are still\ndual to pseudo-scalars (they always will be). Vectors are dual to\ngrade 3 objects (sometimes called pseudo-vectors). But grade 2\nobjects are self-dual.\n\nNow to answer your question, tri-vectors in n = 6 will be self dual\nfor the same reason bi-vectors are self-dual in n = 4. The grades add\nup to six. Just all there is to it. There is really nothing physical\nabout it. It\'s pure math, but extremely useful math.\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>Danny Ross Lunsford <antimatter33@yahoo.nose-pam.com> wrote in message news:<nrHhc.19787$Fz6.1453@newssvr22.news.prodigy.com>...
> I've been looking at a theory of trivectors on a 6-d space. The issue of
> self-duality naturally arises in this theory in a striking way.
>
> What is generally assumed to be the physical meaning of this condition
> imposed on a field? (One sees many "dualities" and it is hard to
> determine which ones are real and which are wishful thinking.)
>
> -drl
There are many definitions of duality in mathematics and physics, but
it appears that the one you're talking about involves objects in a
Clifford Algebra. In an n dimensional geometry, we can talk about a
set of 2^n independent objects that make up a linear basis over the
geometry. These things have many different names, but the general
nomenclature refers to them as 0-vectors,
1-vectors,2-vectors,...,n-vectors, each number being known as the
grade of that particular set of objects. Usually these are called
scalars, vectors, bi-vectors,...,psuedo-scalars. The number of
independent elements in each set of a particular grade is determined
by using Pascal's triangle. In other words, the number of independent
m-vector elements in n dimensions is always given by the combinations
of n objects taken m at a time.
The notion of duality works such that, in n dimensions, two classes of
objects are said to be dual if their grades add up to n. A good
example is in 3 dimensional space, where objects of grade (scalars)
are dual to objects of grade 3 (psuedo-scalars), and grade 1 objects
(vectors) are dual to grade 2 objects (bi-vectors or axial vectors).
Obviously no set of objects of a particular grade are self dual in n =
3. In n = 4, however, it is a different picture. Scalars are still
dual to pseudo-scalars (they always will be). Vectors are dual to
grade 3 objects (sometimes called pseudo-vectors). But grade 2
objects are self-dual.
Now to answer your question, tri-vectors in n = 6 will be self dual
for the same reason bi-vectors are self-dual in n = 4. The grades add
up to six. Just all there is to it. There is really nothing physical
about it. It's pure math, but extremely useful math.
Arkadiusz Jadczyk
Apr28-04, 02:28 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>On Sat, 24 Apr 2004 16:16:12 +0000 (UTC), Danny Ross Lunsford\n<antimatter33@yahoo.nose-pam.com> wrote:\n\n>What is generally assumed to be the physical meaning of this condition\n>imposed on a field?\n\nIt probably depends on the signature of your metric, more precisely, on\nthe sign of the determinant of the metric tensor.\n\nDuality is usually implemented by Hodge star operator *. Its physical\nproperties depends on its square.\n\nIn your case (3 vectors in 6 space) square of the Hodge is\n\n*^2 = - sign(det g)\n\nTherefore, if your metric is +++--- (like in Tifft\'s 3d time model) then\n*^2=1 like for 2 vectors in 4d Euclidean space\n\nBut if your metric is ++++-- (like in conformal group models) then\n*^2 = -1 - much like duality for 2 vectors in 4d Minkowski space-time\n\n\nark\n--\n\nArkadiusz Jadczyk\nhttp://www.cassiopaea.org/quantum_future/homepage.htm\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>On Sat, 24 Apr 2004 16:16:12 +0000 (UTC), Danny Ross Lunsford
<antimatter33@yahoo.nose-pam.com> wrote:
>What is generally assumed to be the physical meaning of this condition
>imposed on a field?
It probably depends on the signature of your metric, more precisely, on
the sign of the determinant of the metric tensor.
Duality is usually implemented by Hodge star operator *. Its physical
properties depends on its square.
In your case (3 vectors in 6 space) square of the Hodge is
*^2 = - sign(det g)
Therefore, if your metric is +++--- (like in Tifft's 3d time model) then
*^2=1 like for 2 vectors in 4d Euclidean space
But if your metric is ++++-- (like in conformal group models) then
*^2 = -1 - much like duality for 2 vectors in 4d Minkowski space-time
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
Danny Ross Lunsford
Apr30-04, 02:01 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>Arkadiusz Jadczyk wrote:\n\n>>What is generally assumed to be the physical meaning of this condition\n>>imposed on a field?\n>\n> It probably depends on the signature of your metric, more precisely, on\n> the sign of the determinant of the metric tensor.\n\nCertainly.\n\n> Therefore, if your metric is +++--- (like in Tifft\'s 3d time model) then\n> *^2=1 like for 2 vectors in 4d Euclidean space\n\nExactly. The trivector on SO(3,3) looks like electric and magnetic\ncurrents coupled to two bivectors. Self-duality amounts to explicitly\ndestroying parity invariance by equating a vector to a pseudovector, and\na bivector (on spacetime) to a dual bivector.\n\nWho is Tifft?\n\n-drl\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>Arkadiusz Jadczyk wrote:
>>What is generally assumed to be the physical meaning of this condition
>>imposed on a field?
>
> It probably depends on the signature of your metric, more precisely, on
> the sign of the determinant of the metric tensor.
Certainly.
> Therefore, if your metric is +++--- (like in Tifft's 3d time model) then
> *^2=1 like for 2 vectors in 4d Euclidean space
Exactly. The trivector on SO(3,3) looks like electric and magnetic
currents coupled to two bivectors. Self-duality amounts to explicitly
destroying parity invariance by equating a vector to a pseudovector, and
a bivector (on spacetime) to a dual bivector.
Who is Tifft?
-drl
Arkadiusz Jadczyk
May3-04, 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>On Fri, 30 Apr 2004 07:01:09 +0000 (UTC), Danny Ross Lunsford\n<antimatter33@yahoo.nose-pam.com> wrote:\n\n>Who is Tifft?\n\n\nhttp://www.science-frontiers.com/sf103/sf103a05.htm\n\nhttp://zwicky.as.arizona.edu/~rix/annrep97/annrep97.tifft.html\n\nSee also\n\nhttp://www.kingsu.ab.ca/~brian/astro/course/lectures/winter/chp20.htm\n\nark\n--\n\nArkadiusz Jadczyk\nhttp://www.cassiopaea.org/quantum_future/homepage.htm\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>On Fri, 30 Apr 2004 07:01:09 +0000 (UTC), Danny Ross Lunsford
<antimatter33@yahoo.nose-pam.com> wrote:
>Who is Tifft?
http://www.science-frontiers.com/sf103/sf103a05.htm
http://zwicky.as.arizona.edu/~rix/annrep97/annrep97.tifft.html
See also
http://www.kingsu.ab.ca/~brian/astro/course/lectures/winter/chp20.htm
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
<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>Igor <thoovler@excite.com> writes\n>There are many definitions of duality in mathematics and physics, but\n>it appears that the one you\'re talking about involves objects in a\n>Clifford Algebra. In an n dimensional geometry, we can talk about a\n>set of 2^n independent objects that make up a linear basis over the\n>geometry. These things have many different names, but the general\n>nomenclature refers to them as 0-vectors,\n>1-vectors,2-vectors,...,n-vectors, each number being known as the\n>grade of that particular set of objects. Usually these are called\n>scalars, vectors, bi-vectors,...,psuedo-scalars. The number of\n>independent elements in each set of a particular grade is determined\n>by using Pascal\'s triangle. In other words, the number of independent\n>m-vector elements in n dimensions is always given by the combinations\n>of n objects taken m at a time.\n\nI wonder if you wouldn\'t mind a brief discussion of this concept because\nit does seem very important and differentiating between grades seems to\nme an essential physical thing. Its also a huge gap in my worldview.\n\n=====\nNB Please read to the end before replying.\n=====\n\nI expect the category theorists would express it in two lines, but I\nwould like to see it more physically. In particular, from my position of\ninfinite ignorance, these seem to be related to differential forms and\ntensors (at least tend to be expressed as tensors).\n\nPerhaps I should start in 2-D. That would be a 1:2:1 thing?\nLength, vector (direction?) and area?\n\nInteresting. I think thats all the structures one can get in 2-D.\nI can see that area is scalar in manipulation, but isn\'t really a\nscalar, because its an area, a 2-D object. More on that later.\n\nOK 3-D. That should be a 1:3:3:1 thing.\nFortunately someone told me what the 3\'s were.\n\nLength, 3-d vector, rotation thingy, volume.\n\nHere we see immediately that elementary mechanics doesn\'t help the naive\nstudent (me). We are encouraged to see rotation thingies as vectors,\nbehaving like vectors, which they clearly do not. Therein lies the\npotential for confusion.\n\nWhen you think about it, what does the addition of pseudo-2-vectors\nactually mean, and how should it be executed? In general not in a\nvectorlike way unless you put them into a very special inter-\nrelationship first. I presume to handle them properly one should use\nspinors, or something similar.\n\n======\n\nThis must surely be related to tensors in the einstein form.\nSheesh, how to do this? Perhaps\n\n1:2:1 -> F^ab, F^a_b, F_ab is that right?\n\n1:3:3:1 -> F^abc, F^ab_c, F^a_bc, F_abc ????\n\nHmm, yes, that looks good, easily extendable to more dimensions.\n\nIt also suggests one standardised mathematics should be able to handle\nall the different forms.\n\n======\n\nFor the moment I think that is enough for this post.\n\nI have quite a few things to ask on this topic, and previous experience\nhas shown I am prone to exponentially growing post lengths, which I\nwould like to avoid. I think this can be done by very short answers to\nmy simple questions at this stage (almost yes-no).\n\nAssuming anyone replies at all of course.\n\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<\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>Igor <thoovler@excite.com> writes
>There are many definitions of duality in mathematics and physics, but
>it appears that the one you're talking about involves objects in a
>Clifford Algebra. In an n dimensional geometry, we can talk about a
>set of 2^n independent objects that make up a linear basis over the
>geometry. These things have many different names, but the general
>nomenclature refers to them as 0-vectors,
>1-vectors,2-vectors,...,n-vectors, each number being known as the
>grade of that particular set of objects. Usually these are called
>scalars, vectors, bi-vectors,...,psuedo-scalars. The number of
>independent elements in each set of a particular grade is determined
>by using Pascal's triangle. In other words, the number of independent
>m-vector elements in n dimensions is always given by the combinations
>of n objects taken m at a time.
I wonder if you wouldn't mind a brief discussion of this concept because
it does seem very important and differentiating between grades seems to
me an essential physical thing. Its also a huge gap in my worldview.
=====
NB Please read to the end before replying.
=====
I expect the category theorists would express it in two lines, but I
would like to see it more physically. In particular, from my position of
infinite ignorance, these seem to be related to differential forms and
tensors (at least tend to be expressed as tensors).
Perhaps I should start in 2-D. That would be a 1:2:1 thing?
Length, vector (direction?) and area?
Interesting. I think thats all the structures one can get in 2-D.
I can see that area is scalar in manipulation, but isn't really a
scalar, because its an area, a 2-D object. More on that later.
OK 3-D. That should be a 1:3:3:1 thing.
Fortunately someone told me what the 3's were.
Length, 3-d vector, rotation thingy, volume.
Here we see immediately that elementary mechanics doesn't help the naive
student (me). We are encouraged to see rotation thingies as vectors,
behaving like vectors, which they clearly do not. Therein lies the
potential for confusion.
When you think about it, what does the addition of pseudo-2-vectors
actually mean, and how should it be executed? In general not in a
vectorlike way unless you put them into a very special inter-
relationship first. I presume to handle them properly one should use
spinors, or something similar.
======
This must surely be related to tensors in the einstein form.
Sheesh, how to do this? Perhaps
1:2:1 -> F^{ab}, F^{a_b}, F_{ab} is that right?
1:3:3:1 -> F^{abc}, F^{ab_c}, F^{a_}{bc}, F_{abc} ????
Hmm, yes, that looks good, easily extendable to more dimensions.
It also suggests one standardised mathematics should be able to handle
all the different forms.
======
For the moment I think that is enough for this post.
I have quite a few things to ask on this topic, and previous experience
has shown I am prone to exponentially growing post lengths, which I
would like to avoid. I think this can be done by very short answers to
my simple questions at this stage (almost yes-no).
Assuming anyone replies at all of course.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<
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