inonu-wigner contraction [Archive]

math

Aug24-04, 04:55 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>Dear physicists, as you know, recently Prof. Erdal Inonu from Sabanci\nUniversity in Istanbul, Turkey, won the Wigner medal for his\ncontribution to Inonu-Wigner contraction. The media mentioned the\nevent but no one really explained what this inonu-wigner contraction\nwas. Prof. Inonu himself only said that this was a work he did jointly\nwith Wigner about 50 years ago in Princeton and recently it acquired\nnew importance because string theorists made new use of it.\n\nI am planning to write a popular magazine article explaining\nInonu-Wigner contraction to laymen who know nothing about group theory\nand Lie algebras. I myself don\'t know much about group theory, except\nthat physicists like to use it to study symmetries in rotations.\n\nI was wondering if anybody here could explain Inonu-wigner contraction\nin a way that people who do not know group theory, Lie algebras and\nQuantum mechanics can understand.\n\nI did a google search but could not find an elementary explaination.\n\nThanks in advance.\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>Dear physicists, as you know, recently Prof. Erdal Inonu from Sabanci
University in Istanbul, Turkey, won the Wigner medal for his
contribution to Inonu-Wigner contraction. The media mentioned the
event but no one really explained what this inonu-wigner contraction
was. Prof. Inonu himself only said that this was a work he did jointly
with Wigner about 50 years ago in Princeton and recently it acquired
new importance because string theorists made new use of it.

I am planning to write a popular magazine article explaining
Inonu-Wigner contraction to laymen who know nothing about group theory
and Lie algebras. I myself don't know much about group theory, except
that physicists like to use it to study symmetries in rotations.

I was wondering if anybody here could explain Inonu-wigner contraction
in a way that people who do not know group theory, Lie algebras and
Quantum mechanics can understand.

I did a google search but could not find an elementary explaination.

Thanks in advance.

Arnold Neumaier

Sep24-04, 08:11 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 had sent this already on August 24,\nbut apparently it did not make it into spr.]\n\nmath wrote:\n\n> I was wondering if anybody here could explain Inonu-wigner contraction\n> in a way that people who do not know group theory, Lie algebras and\n> Quantum mechanics can understand.\n>\n> I did a google search but could not find an elementary explaination.\n\nThis is not something to be explained to laymen. One can only talk\naround the subject. If you use the following, please send me a copy of\nyour article!\n\n\nLie groups can be illustrated by continuous rigid motion of a ball\nwith painted patterns on it in 3-dimensional space. The Lie group ISO(3)\nconsists of all rigid transformations.\n\nA rigid transfromation is essentially the act of picking the ball and\nplacing it somewhere else, ignoring the detailed motion in between and\nthe location one started.\nSpecial transformations are for example a translation in northern direction\nby 1 meter, or a rotation by one quarter around the vertical axis at some\nparticular point (think of a ball with a string attached).\n\'Rigid\' means that the distances between marked points on the ball\nremains the same; the mathematician talks about \'preserving distances\',\nand the distances are therefore labeled \'invariants\'.\n\nOne can repeat the same transformation several times, or two different\ntransformations and get another one - This is called the product of\nthese transformations. For example, the product of a translations\nby 1 meter and another one by 2 meters in the same direction gives one\nof 1+2=3 meters in the same direction. In this case, the distances add,\nbut if one combines rotations about different axes the result is no\nlonger intuitive. To make this more tractable for calculations,\none needs to take some kind of logarithms of transformations - these\nbehave again additively and make up the corresponding Lie algebra\niso(3) [same letters but in lower case]. The elements of the Lie algebra\ncan be visualized as very small, or \'infinitesimal\', motions.\n\n\n\nGeneral Lie groups and Lie algebras extend these notions to to more\ngeneral manifolds. A manifold is just a higher-dimensional version\nof space, and transformations are generalized motions preserving\ninvariants that are important in the manifold. The transformations\npreserving these invariants are also called \'symmetries\', and the\nLie group consisting of all symmetries is called a \'symmetry group\'.\nThe elements of the corresponding Lie algebra are \'infinitesimal\nsymmetries\'.\n\nFor example, physical laws are invariant under rotations and translations,\nand hence unter all rigid motions. But not only these: If one includes\ntime explicitly, the resulting 4-dimensional space has more invariants.\nThe Lie group of all transformations preserving these is called the\nPoincar\'e group, and plays a basic role in the theory of relativity.\nThe transformations are now about balls in uniform motion.\nApart from translations and rotations there are symmetries called\n\'boosts\' that accelerate a ball in a certain direction, and combinations\nobtained by taking products. All infinitesimal symmetries together\nmake up the Poincar\'e algebra.\n\nHowever, before Einstein invented the theory of relativity,\nphysics was believed to follow Newton\'s laws, and these have a\ndifferent group of symmetries - the Galilei group, and its\ninfinitesimal symmetries form the Galilei algebra.\n\nNow Newton\'s physics is just a special case of the theory of relativity\nin which all motions are very slow compared to the speed of light.\nPhysicists speak of the \'nonrelativisitic limit\'.\nThus one would expect that the Galilei group is a kind of\nnonrelativistic limit of the Poincar\'e group.\n\nThis notion has been made precise by Inonu. He looked at the\nPoincar\'e algebra and \'contracted\' it in an ingenious way\nto the Galilei algebra. The construction could then be lifted to\nthe corresponding groups. Not only that, it turned out to be a\ngeneral machinery applicable to all Lie algebras and Lie groups,\nand therefore has found many applications far beyond that for which\nit was originally developed.\n\n\nArnold Neumaier\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 had sent this already on August 24,
but apparently it did not make it into spr.]

math wrote:

> I was wondering if anybody here could explain Inonu-wigner contraction
> in a way that people who do not know group theory, Lie algebras and
> Quantum mechanics can understand.
>
> I did a google search but could not find an elementary explaination.

This is not something to be explained to laymen. One can only talk
around the subject. If you use the following, please send me a copy of
your article!

Lie groups can be illustrated by continuous rigid motion of a ball
with painted patterns on it in 3-dimensional space. The Lie group ISO(3)
consists of all rigid transformations.

A rigid transfromation is essentially the act of picking the ball and
placing it somewhere else, ignoring the detailed motion in between and
the location one started.
Special transformations are for example a translation in northern direction
by 1 meter, or a rotation by one quarter around the vertical axis at some
particular point (think of a ball with a string attached).
'Rigid' means that the distances between marked points on the ball
remains the same; the mathematician talks about 'preserving distances',
and the distances are therefore labeled 'invariants'.

One can repeat the same transformation several times, or two different
transformations and get another one - This is called the product of
these transformations. For example, the product of a translations
by 1 meter and another one by 2 meters in the same direction gives one
of 1+2=3 meters in the same direction. In this case, the distances add,
but if one combines rotations about different axes the result is no
longer intuitive. To make this more tractable for calculations,
one needs to take some kind of logarithms of transformations - these
behave again additively and make up the corresponding Lie algebra
iso(3) [same letters but in lower case]. The elements of the Lie algebra
can be visualized as very small, or 'infinitesimal', motions.

General Lie groups and Lie algebras extend these notions to to more
general manifolds. A manifold is just a higher-dimensional version
of space, and transformations are generalized motions preserving
invariants that are important in the manifold. The transformations
preserving these invariants are also called 'symmetries', and the
Lie group consisting of all symmetries is called a 'symmetry group'.
The elements of the corresponding Lie algebra are 'infinitesimal
symmetries'.

For example, physical laws are invariant under rotations and translations,
and hence unter all rigid motions. But not only these: If one includes
time explicitly, the resulting 4-dimensional space has more invariants.
The Lie group of all transformations preserving these is called the
Poincar'e group, and plays a basic role in the theory of relativity.
The transformations are now about balls in uniform motion.
Apart from translations and rotations there are symmetries called
'boosts' that accelerate a ball in a certain direction, and combinations
obtained by taking products. All infinitesimal symmetries together
make up the Poincar'e algebra.

However, before Einstein invented the theory of relativity,
physics was believed to follow Newton's laws, and these have a
different group of symmetries - the Galilei group, and its
infinitesimal symmetries form the Galilei algebra.

Now Newton's physics is just a special case of the theory of relativity
in which all motions are very slow compared to the speed of light.
Physicists speak of the 'nonrelativisitic limit'.
Thus one would expect that the Galilei group is a kind of
nonrelativistic limit of the Poincar'e group.

This notion has been made precise by Inonu. He looked at the
Poincar'e algebra and 'contracted' it in an ingenious way
to the Galilei algebra. The construction could then be lifted to
the corresponding groups. Not only that, it turned out to be a
general machinery applicable to all Lie algebras and Lie groups,
and therefore has found many applications far beyond that for which
it was originally developed.

Arnold Neumaier

James Dolan

Sep27-04, 03:31 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>\nvery roughly, it\'s the kind of "conceptual unflattening" of a geometry\nthat you perform in your mind when you make a conceptual leap and\nrealize that the geometry that you thought was "flat" is actually\n"round" (in a certain very loose sense), but with such a big "radius\nof curvature" that it naively seems flat.\n\nso for example the process of realizing that the earth is round was an\ninonu-wigner contraction. in this example the role of the "radius of\ncurvature" is played by the radius of curvature (of the earth).\n\nanother example is the process of realizing that the space-time\ngeometry of our universe is special-relativistic (as lorentz and\neinstein proposed) instead of galilean-relativistic (as galileo had\nthought it was). in this example the role of the "radius of\ncurvature" is played by the maximum signaling speed of the universe,\nnamely the speed of light, which galileo had thought to be infinite,\nbut which later turned out to be finite. special-relativistic\nspace-time geometry is "round" in the sense that a combination of two\n"boosts" (the kind of shift in perspective that you experience when\nyou step onto a fast-moving train) is itself not necessarily just a\nboost anymore.\n\nanother example is the process of realizing that the physical quantity\ncalled "action" comes in discrete lumps (as in quantum physics)\ninstead of in a smooth flow (as in pre-quantum "classical" physics).\nin this example the role of the "radius of curvature" is played by the\ninverse of planck\'s constant (measuring the size of "the elementary\nquantum of action").\n\n--\n\n\n[e-mail address jdolan@math.ucr.edu]\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>very roughly, it's the kind of "conceptual unflattening" of a geometry
that you perform in your mind when you make a conceptual leap and
realize that the geometry that you thought was "flat" is actually
"round" (in a certain very loose sense), but with such a big "radius
of curvature" that it naively seems flat.

so for example the process of realizing that the earth is round was an
inonu-wigner contraction. in this example the role of the "radius of
curvature" is played by the radius of curvature (of the earth).

another example is the process of realizing that the space-time
geometry of our universe is special-relativistic (as lorentz and
einstein proposed) instead of galilean-relativistic (as galileo had
thought it was). in this example the role of the "radius of
curvature" is played by the maximum signaling speed of the universe,
namely the speed of light, which galileo had thought to be infinite,
but which later turned out to be finite. special-relativistic
space-time geometry is "round" in the sense that a combination of two
"boosts" (the kind of shift in perspective that you experience when
you step onto a fast-moving train) is itself not necessarily just a
boost anymore.

another example is the process of realizing that the physical quantity
called "action" comes in discrete lumps (as in quantum physics)
instead of in a smooth flow (as in pre-quantum "classical" physics).
in this example the role of the "radius of curvature" is played by the
inverse of planck's constant (measuring the size of "the elementary
quantum of action").

--

[e-mail address jdolan@math.ucr.edu]

Arnold Neumaier

Sep28-04, 02:26 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>James Dolan wrote:\n> very roughly, it\'s the kind of "conceptual unflattening" of a geometry\n> that you perform in your mind when you make a conceptual leap and\n> realize that the geometry that you thought was "flat" is actually\n> "round" (in a certain very loose sense), but with such a big "radius\n> of curvature" that it naively seems flat.\n>\n> so for example the process of realizing that the earth is round was an\n> inonu-wigner contraction. in this example the role of the "radius of\n> curvature" is played by the radius of curvature (of the earth).\n\nActually, the \'contraction\' is always the \'flattening\', not the\n\'unflattenting\'. One can get the flat object as a limit of the curved\none, but not the other way around.\n\nHere it is taking the limiting case of regarding the earth as flat\nsince it locally is.\n\n\n> another example is the process of realizing that the physical quantity\n> called "action" comes in discrete lumps (as in quantum physics)\n> instead of in a smooth flow (as in pre-quantum "classical" physics).\n> in this example the role of the "radius of curvature" is played by the\n> inverse of planck\'s constant (measuring the size of "the elementary\n> quantum of action").\n\nDo you mean this figuratively, or how does this fit the framework of\nInonu-Wigner contractions?\n\n\nArnold Neumaier\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>James Dolan wrote:
> very roughly, it's the kind of "conceptual unflattening" of a geometry
> that you perform in your mind when you make a conceptual leap and
> realize that the geometry that you thought was "flat" is actually
> "round" (in a certain very loose sense), but with such a big "radius
> of curvature" that it naively seems flat.
>
> so for example the process of realizing that the earth is round was an
> inonu-wigner contraction. in this example the role of the "radius of
> curvature" is played by the radius of curvature (of the earth).

Actually, the 'contraction' is always the 'flattening', not the
'unflattenting'. One can get the flat object as a limit of the curved
one, but not the other way around.

Here it is taking the limiting case of regarding the earth as flat
since it locally is.

> another example is the process of realizing that the physical quantity
> called "action" comes in discrete lumps (as in quantum physics)
> instead of in a smooth flow (as in pre-quantum "classical" physics).
> in this example the role of the "radius of curvature" is played by the
> inverse of planck's constant (measuring the size of "the elementary
> quantum of action").

Do you mean this figuratively, or how does this fit the framework of
Inonu-Wigner contractions?

Arnold Neumaier

Oz

Sep28-04, 10:19 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\nJames Dolan <jdolan@math-rs-n03.math.ucr.edu> writes\n>very roughly, it\'s the kind of "conceptual unflattening" of a geometry\n>that you perform in your mind when you make a conceptual leap and\n>realize that the geometry that you thought was "flat" is actually\n>"round" (in a certain very loose sense), but with such a big "radius\n>of curvature" that it naively seems flat.\n\nPenrose wrote a \'popular\' piece on twistors in New Scientist the other\nweek. Unfortunately it was a tad too popular in that it pretty well said\njust three things:\n\n1) Twistor space is associated with the concept that light beams are\nzero length and zero time.\n\n2) One needed to add a couple of other (unstated) criteria.\n\n3) The resultant space included calabi-yau and 4D GR spacetime and was a\n3-complex-D spacetime.\n\nThe implication being that this was a space where QM & GR could co-\nexist.\n\nUnfortunately there wasn\'t enough meat for me to see exactly what he was\nmeaning. I will admit to having tried (frequently) to see light as\ninstantly connecting two points, but never to be a physical description\nof two points being \'connected\'.\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<<\nozacoohdb@despammed.com still functions.\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>James Dolan <jdolan@math-rs-n03.math.ucr.edu> writes
>very roughly, it's the kind of "conceptual unflattening" of a geometry
>that you perform in your mind when you make a conceptual leap and
>realize that the geometry that you thought was "flat" is actually
>"round" (in a certain very loose sense), but with such a big "radius
>of curvature" that it naively seems flat.

Penrose wrote a 'popular' piece on twistors in New Scientist the other
week. Unfortunately it was a tad too popular in that it pretty well said
just three things:

1) Twistor space is associated with the concept that light beams are
zero length and zero time.

2) One needed to add a couple of other (unstated) criteria.

3) The resultant space included calabi-yau and 4D GR spacetime and was a
3-complex-D spacetime.

The implication being that this was a space where QM & GR could co-
exist.

Unfortunately there wasn't enough meat for me to see exactly what he was
meaning. I will admit to having tried (frequently) to see light as
instantly connecting two points, but never to be a physical description
of two points being 'connected'.

--
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<<
ozacoohdb@despammed.com still functions.