Inonu-wigner contraction [Archive]

Paul math Albeirt

Sep4-04, 08:11 AM

Lubos Motl

Sep4-04, 08:51 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>My feeling from reading\n\nhttp://www.physics.umd.edu/robot/wigner/inonu.pdf\n\nis that Wigner and Inonu were really the first people who defined the\nwell-known (and rather simple) notion of a group contraction, or at least\na rather general special case of it. Therefore I think that the\nInonu-Wigner contraction is nothing else than "group contraction". Well,\ncontractions are rather important in mathematics and physics and they are\nfound in string theory, too, although I am not sure why exactly this\nexample was chosen.\n\nLet me tell you a simple example what a contraction is.\n\nSpecial relativity implies that the spacetime has a Lorentz symmetry\nSO(3,1) - symmetry under rotations and boosts. Let\'s only consider the\noperations that preserve the origin of spacetime.\n\nNewtonian physics is a limit of special relativity (for the speed of light\ngoing to infinity). It also has a symmetry including the democracy between\ndifferent inertial observers - the Galilean symmetry. A Galilean\ntransformation acts on spacetime much like a Lorentz transformation, but\nit always preserves the time coordinate (the universal time in\nnonrelativistic physics). On the other hand, the coordinate "x" may be\nmapped to "x-vt".\n\nHow do we take the "limit" of the Lorentz group to obtain the Galilean\ngroup? This limiting procedure is called "contraction". Let us choose a\nsubgroup of the Lorentz group that will be preserved. Well, the rotational\ngroup SO(3) is the good choice because the rotations do not care about the\nspeed of light because they do not affect the time coordinate.\n\nBut the transformations from one inertial observer to another do care.\nNewtonian physics has a sort of symmetry where all speeds are multiplied\nby a constant - i.e. the times are scaled by a number but distances are\nnot. But special relativity does not have this symmetry. A boost is\ndefined by a velocity. A Galilean transformation can be thought of as a\nspecial case of a Lorentz transformation in which the velocities\ngenerating the boost are much smaller than the speed of light. Setting the\nspeed of light c=1, they are infinitesimal.\n\nAny Lie group has an associated Lie algebra, and the "curved" structure of\nmultiplication rules on the group are encoded in the commutators of the\ngenerators of Lie algebra.\n\n[T^i, T^j] = \\sum_k c^{ij}_k T^k\n\nIf we want to contract this group, we don\'t change the generators T^i of a\nchosen subgroup, but we rescale the other generators as\n\nT^i = T^i (new) / \\epsilon\n\nwhere \\epsilon goes to zero. If you look at the commutators above, some of\nthem must be multiplied by \\epsilon^2 to get [T^i (new), T^j (new)] on the\nleft hand side. One of the \\epsilon factors may be canceled because the\nT^k on the right hand side is written as T^k (new) / \\epsilon, but one of\nthem survives. It means that in the limit \\epsilon\\to 0, some terms in the\ncommutators simply disappear.\n\nNote that it was important that the preserved, unscaled group was a\nsubgroup - otherwise we would find infinite (rescaled) quantities\nT^k (new) / \\epsilon on the right hand side of the commutators of these\npreserved generators.\n\nContractions produce new groups that are not as simple as the starting\ngroups. The Galilean example above is actually a prototype of a class of\nthe most important examples. Another way to describe a representative is\nto see that the rotational group SO(d) can be contracted to the full\nEuclidean affine group E_{d-1} - the rotations of SO(d) affecting a chosen\ndirection (x^d) are reinterpreted as translations.\n\nBest wishes\nLubos\n___________________________________ ___________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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>My feeling from reading

http://www.physics.umd.edu/robot/wigner/inonu.pdf

is that Wigner and Inonu were really the first people who defined the
well-known (and rather simple) notion of a group contraction, or at least
a rather general special case of it. Therefore I think that the
Inonu-Wigner contraction is nothing else than "group contraction". Well,
contractions are rather important in mathematics and physics and they are
found in string theory, too, although I am not sure why exactly this
example was chosen.

Let me tell you a simple example what a contraction is.

Special relativity implies that the spacetime has a Lorentz symmetry
SO(3,1) - symmetry under rotations and boosts. Let's only consider the
operations that preserve the origin of spacetime.

Newtonian physics is a limit of special relativity (for the speed of light
going to infinity). It also has a symmetry including the democracy between
different inertial observers - the Galilean symmetry. A Galilean
transformation acts on spacetime much like a Lorentz transformation, but
it always preserves the time coordinate (the universal time in
nonrelativistic physics). On the other hand, the coordinate "x" may be
mapped to "x-vt".

How do we take the "limit" of the Lorentz group to obtain the Galilean
group? This limiting procedure is called "contraction". Let us choose a
subgroup of the Lorentz group that will be preserved. Well, the rotational
group SO(3) is the good choice because the rotations do not care about the
speed of light because they do not affect the time coordinate.

But the transformations from one inertial observer to another do care.
Newtonian physics has a sort of symmetry where all speeds are multiplied
by a constant - i.e. the times are scaled by a number but distances are
not. But special relativity does not have this symmetry. A boost is
defined by a velocity. A Galilean transformation can be thought of as a
special case of a Lorentz transformation in which the velocities
generating the boost are much smaller than the speed of light. Setting the
speed of light c=1, they are infinitesimal.

Any Lie group has an associated Lie algebra, and the "curved" structure of
multiplication rules on the group are encoded in the commutators of the
generators of Lie algebra.

[T^i, T^j] = \sum_k c^{ij}_k T^k

If we want to contract this group, we don't change the generators T^i of a
chosen subgroup, but we rescale the other generators as

T^i = T^i[/itex] (new) [itex]/ \epsilon

where \epsilon goes to zero. If you look at the commutators above, some of
them must be multiplied by \epsilon^2 to get [T^i (new), T^j (new)] on the
left hand side. One of the \epsilon factors may be canceled because the
T^k on the right hand side is written as T^k (new) / \epsilon, but one of
them survives. It means that in the limit \epsilon\to 0, some terms in the
commutators simply disappear.

Note that it was important that the preserved, unscaled group was a
subgroup - otherwise we would find infinite (rescaled) quantities
T^k (new) / \epsilon on the right hand side of the commutators of these
preserved generators.

Contractions produce new groups that are not as simple as the starting
groups. The Galilean example above is actually a prototype of a class of
the most important examples. Another way to describe a representative is
to see that the rotational group SO(d) can be contracted to the full
Euclidean affine group E_{d-1} - the rotations of SO(d) affecting a chosen
direction (x^d) are reinterpreted as translations.

Best wishes
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^