View Full Version : Einstein & Quantum Gravity
Russell E. Rierson
Apr28-04, 03:17 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\n\n\nA physical system is described by a normalized vector[state vector] in\nHilbert space. All possible information can be known about the system,\nsince, for every physical observable there corresponds a self adjoint\noperator in Hilbert space.\n\nThe only allowed physical results of measurements of some observable\nU, are the elements of the spectrum of the operator which corresponds\nto U.\n\nSo all properties of a physical system may not be completely known,\nbut that which is known, must be necessarily true on logical or\nanalytic grounds.\n\nDistance is a property between objects in space. Space is a structure,\nwhich is constructed of discrete units. The structure of space is\npossibly a distributive lattice? A lattice is a partially ordered\nset, closed under least upper and greatest lower bounds. Any lattice\nwhich is isomorphic to a collection of sets, closed under\ncomplementation and intersection, is a Boolean algebra.\n\nIs it possible to derive Einstein\'s field equation strictly in terms\nof quantum mechanical operators? using n-dimensional cross sections of\ncotangent vector spaces? Near a massive object M, the *isobar* cross\nsections increase in density, as wave density gradients.\n\nCompression waves become a self embedding of surface integrals? This\ngives continuously increasing density gradients, as matter-energy is\nsequentially re-embedded from previous computations/iterations.\n\nIf the universe is closed, the "information" or entangled quantum\nstates cannot leak out of the closed system. So the density of\nentangled quantum states, continually increases, as the entropy must\nalways increase. While to us, it is interpreted as entropy or lost\ninformation, it is actually recombined information, to the universe.\nShannon entropy.\n\n\nWhat is needed is a tensor equation which is parallel to "wave"\nequations described in terms of a covariant d\'Alembertian operator.\nAn alternative description for the general relativistic space-time\ncontinuum that allows for "compressional" waves, rather than allowing\nonly "transverse" waves?\n\nThe gravitational action of matter/energy, etc. can be studied in\nterms of boundary value problems.\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>A physical system is described by a normalized vector[state vector] in
Hilbert space. All possible information can be known about the system,
since, for every physical observable there corresponds a self adjoint
operator in Hilbert space.
The only allowed physical results of measurements of some observable
U, are the elements of the spectrum of the operator which corresponds
to U.
So all properties of a physical system may not be completely known,
but that which is known, must be necessarily true on logical or
analytic grounds.
Distance is a property between objects in space. Space is a structure,
which is constructed of discrete units. The structure of space is
possibly a distributive lattice? A lattice is a partially ordered
set, closed under least upper and greatest lower bounds. Any lattice
which is isomorphic to a collection of sets, closed under
complementation and intersection, is a Boolean algebra.
Is it possible to derive Einstein's field equation strictly in terms
of quantum mechanical operators? using n-dimensional cross sections of
cotangent vector spaces? Near a massive object M, the *isobar* cross
sections increase in density, as wave density gradients.
Compression waves become a self embedding of surface integrals? This
gives continuously increasing density gradients, as matter-energy is
sequentially re-embedded from previous computations/iterations.
If the universe is closed, the "information" or entangled quantum
states cannot leak out of the closed system. So the density of
entangled quantum states, continually increases, as the entropy must
always increase. While to us, it is interpreted as entropy or lost
information, it is actually recombined information, to the universe.
Shannon entropy.
What is needed is a tensor equation which is parallel to "wave"
equations described in terms of a covariant d'Alembertian operator.
An alternative description for the general relativistic space-time
continuum that allows for "compressional" waves, rather than allowing
only "transverse" waves?
The gravitational action of matter/energy, etc. can be studied in
terms of boundary value problems.
Alfred Einstead
Apr29-04, 12:47 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>\n\n\n\nanalog57@yahoo.com (Russell E. Rierson) wrote:\n> Distance is a property between objects in space. Space is a structure,\n> which is constructed of discrete units. The structure of space is\n> possibly a distributive lattice?\n\nA by-product of Connes\' research in non-commutative geometry was the\ndiscovery of an expression for distance in a Riemannian space M that is\nentirely written in terms of the function space over M and is algebraic,\nmaking no explicit reference to the underlying point set of M. Because\nit\'s purely algebraic, the expression can also be used for non-commutative\nspaces where they may not even be a point set.\n\n> What is needed is a tensor equation which is parallel to "wave"\n> equations described in terms of a covariant d\'Alembertian operator.\n\nDifferential geometry is already established on a purely algebraic\nbasis. In particular, given a manifold M, all the items you normally\ndefine (covariant derivative, n-forms, exterior differential, etc.)\nare actually defined in reference to the function algebra E(M) of\nC^{infinity} functions over M. The point set M, itself, is actually\nnever used anywhere -- except to initially define E(M).\n\nThere is an algebraic expression for the Dirac operator D which\napplies to the manifold M, when it also carries a spin bundle\nstructure (i.e., something to allow you to set up the gamma matrices\non).\n\nThe square of this operator D is an algebraic expression that contains\nthe curvature scalar R in it; as well as the d\'Alembertian. Since it\'s\nalgebraic, then it too can be used for more general (non-commutative)\ngeometries where there may not be a point set..\n\nSo, there are actually 3 items you need: (1) the (function) algebra,\nwhich for manifolds is just E(M); (2) a spin structure (for the\ngammas); and (3) the Dirac operator D. Out of this you can do\neverything you need to do.\n\nSince you raised the topic with the subject header, it\'s both\ninstructive and revealing to see what Einstein, himself, had to\nsay on the subject of quantum gravity at the end of his life:\n\n"One can give good reasons why reality cannot at all be represented\nby a continuous field. From the quantum phenomena it appears to\nfollow with certainty that a finite system of finite energy can be\ncompletely described by a finite set of numbers (quantum numbers).\nThis does not seem to be in accordance with a continuum theory,\nand must lead to an attempt to find a purely algebraic theory for\nthe description of reality. But [sic] nobody knows how to find\nthe basis of such a theory."\n\nIn fact, at least as far as differential geometry goes, the\nwhole field had already been put in algebraic form by the 1950\'s;\nand the generalization to manifolds with torsion (and\nnon-symmetric connections -- the subject discussed just prior\nto the above-mentioned remark) was already established; the\nRiemannian-Cartan manifolds. It\'s on the latter that spin\nmanifolds are defined; and torsion, in particular, is needed to\ncouple to fermion spin.\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>analog57@yahoo.com (Russell E. Rierson) wrote:
> Distance is a property between objects in space. Space is a structure,
> which is constructed of discrete units. The structure of space is
> possibly a distributive lattice?
A by-product of Connes' research in non-commutative geometry was the
discovery of an expression for distance in a Riemannian space M that is
entirely written in terms of the function space over M and is algebraic,
making no explicit reference to the underlying point set of M. Because
it's purely algebraic, the expression can also be used for non-commutative
spaces where they may not even be a point set.
> What is needed is a tensor equation which is parallel to "wave"
> equations described in terms of a covariant d'Alembertian operator.
Differential geometry is already established on a purely algebraic
basis. In particular, given a manifold M, all the items you normally
define (covariant derivative, n-forms, exterior differential, etc.)
are actually defined in reference to the function algebra E(M) of
C^{infinity} functions over M. The point set M, itself, is actually
never used anywhere -- except to initially define E(M).
There is an algebraic expression for the Dirac operator D which
applies to the manifold M, when it also carries a spin bundle
structure (i.e., something to allow you to set up the \gamma matrices
on).
The square of this operator D is an algebraic expression that contains
the curvature scalar R in it; as well as the d'Alembertian. Since it's
algebraic, then it too can be used for more general (non-commutative)
geometries where there may not be a point set..
So, there are actually 3 items you need: (1) the (function) algebra,
which for manifolds is just E(M); (2) a spin structure (for the
gammas); and (3) the Dirac operator D. Out of this you can do
everything you need to do.
Since you raised the topic with the subject header, it's both
instructive and revealing to see what Einstein, himself, had to
say on the subject of quantum gravity at the end of his life:
"One can give good reasons why reality cannot at all be represented
by a continuous field. From the quantum phenomena it appears to
follow with certainty that a finite system of finite energy can be
completely described by a finite set of numbers (quantum numbers).
This does not seem to be in accordance with a continuum theory,
and must lead to an attempt to find a purely algebraic theory for
the description of reality. But [sic] nobody knows how to find
the basis of such a theory."
In fact, at least as far as differential geometry goes, the
whole field had already been put in algebraic form by the 1950's;
and the generalization to manifolds with torsion (and
non-symmetric connections -- the subject discussed just prior
to the above-mentioned remark) was already established; the
Riemannian-Cartan manifolds. It's on the latter that spin
manifolds are defined; and torsion, in particular, is needed to
couple to fermion spin.
Arnold Neumaier
Apr30-04, 10:36 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>Alfred Einstead wrote:\n>\n> A by-product of Connes\' research in non-commutative geometry was the\n> discovery of an expression for distance in a Riemannian space M that is\n> entirely written in terms of the function space over M and is algebraic,\n> making no explicit reference to the underlying point set of M. Because\n> it\'s purely algebraic, the expression can also be used for non-commutative\n> spaces where they may not even be a point set.\n\nDistance between what, if there are no points???\n\n\nArnold Neumaier\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>Alfred Einstead wrote:
>
> A by-product of Connes' research in non-commutative geometry was the
> discovery of an expression for distance in a Riemannian space M that is
> entirely written in terms of the function space over M and is algebraic,
> making no explicit reference to the underlying point set of M. Because
> it's purely algebraic, the expression can also be used for non-commutative
> spaces where they may not even be a point set.
Distance between what, if there are no points???
Arnold Neumaier
Suresh K Maran
May3-04, 04:52 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>\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote\n> Alfred Einstead wrote:\n> >\n> > A by-product of Connes\' research in non-commutative geometry was the\n> > discovery of an expression for distance in a Riemannian space M that is\n> > entirely written in terms of the function space over M and is algebraic,\n> > making no explicit reference to the underlying point set of M. Because\n> > it\'s purely algebraic, the expression can also be used for non-commutative\n> > spaces where they may not even be a point set.\n>\n> Distance between what, if there are no points???\n\nGreetings!.\nSpin foam models of quantum gravity has clear answers for this.\nPlease read the article by John Baez at the lanl archive\nhttp://xxx.lanl.gov/abs/gr-qc/9905087\n\nThanks\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote
> Alfred Einstead wrote:
> >
> > A by-product of Connes' research in non-commutative geometry was the
> > discovery of an expression for distance in a Riemannian space M that is
> > entirely written in terms of the function space over M and is algebraic,
> > making no explicit reference to the underlying point set of M. Because
> > it's purely algebraic, the expression can also be used for non-commutative
> > spaces where they may not even be a point set.
>
> Distance between what, if there are no points???
Greetings!.
Spin foam models of quantum gravity has clear answers for this.
Please read the article by John Baez at the lanl archive
http://xxx.lanl.gov/abs/http://www.arxiv.org/abs/gr-qc/9905087
Thanks
Ken S. Tucker
May3-04, 04:52 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c6trp4\\$127\\$1@lfa222122.richmond.edu>...\ n>Alfred Einstead wrote:\n>>\n>> A by-product of Connes\' research in non-commutative geometry was the\n>> discovery of an expression for distance in a Riemannian space M that is\n>> entirely written in terms of the function space over M and is algebraic,\n>> making no explicit reference to the underlying point set of M. Because\n>> it\'s purely algebraic, the expression can also be used for non-commutative\n>> spaces where they may not even be a point set.\n>\n>Distance between what, if there are no points???\n>Arnold Neumaier\n\nYeah, to me that\'s a dichotomy, seems there are two\nexclusive ideas, a continuum of points with infinitesmal\ndistances, or distances defined by quantum effects.\n\nTurning to the International Standard of Units, (ISU)\nfor some guidance, distance (length) is defined by\nL = cT.\nIf the "c" is taken literally, then L must be defined by\na photon, (or an EM-wave classically).\n\nIn the classical sense, the production of an EM-\nwave needs the relative motion of two discrete\ncharges, such as an electron changing orbital\nenergy in the presence of a nucleus. From the\nfrequency of the emitted radiation we infer the\nvariation of radius, within uncertainty limits.\n\nSo to Arnold\'s question, the distance would seem\nto require at least two discrete charges, naturally\nthese charges are associated with charged particles\nagain using the atomic example.\n\nPhysically the classical "electrical potential energy"\nis given by,\n\nE = qQ/L where q and Q are charges and\n\nL separates q and Q. The ISU defines L =cT so\n\nE = qQ/cT and ET = qQ (c=1).\n\nand ET in erg*seconds is Plancks\'s constant,\n\nh = 6,625*10^-27 erg*secs ( * integer )\n\nPrior to Plancks quantum hypothesis, "h" could\ntend to zero, and an infinite accuracy measuring\nthe continuum was thought possible.\nSubsequently experimentalists have found\nh=constant, and the measurement of the continuum\nis limited to adapting the nearest integer*h.\n\nClassical tensor analysis, involves a measurement\nof the continuum at a point, and the relation of that\npoint related to infinitesmally close points when\nthings like curvature are calculated.\n\nSo I think the queston is, can an operational tensor\nanalysis (that\'s Generally Covariant) be based on\n"h = constant"?\n\nIn light of Alfred\'s post, we might use metrics defined\nover finite lengths, defined by charges qQ, in accord\nwith QT, but with relative rotations that would yield\nthe non-symmetrical characteristics of a metric that\nappear ridiculous using a continuum.\n\nRegards\nKen S. Tucker\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c6trp4$127$1@lfa222122.richmond.edu>...
>Alfred Einstead wrote:
>>
>> A by-product of Connes' research in non-commutative geometry was the
>> discovery of an expression for distance in a Riemannian space M that is
>> entirely written in terms of the function space over M and is algebraic,
>> making no explicit reference to the underlying point set of M. Because
>> it's purely algebraic, the expression can also be used for non-commutative
>> spaces where they may not even be a point set.
>
>Distance between what, if there are no points???
>Arnold Neumaier
Yeah, to me that's a dichotomy, seems there are two
exclusive ideas, a continuum of points with infinitesmal
distances, or distances defined by quantum effects.
Turning to the International Standard of Units, (ISU)
for some guidance, distance (length) is defined by
L = cT.
If the "c" is taken literally, then L must be defined by
a photon, (or an EM-wave classically).
In the classical sense, the production of an EM-
wave needs the relative motion of two discrete
charges, such as an electron changing orbital
energy in the presence of a nucleus. From the
frequency of the emitted radiation we infer the
variation of radius, within uncertainty limits.
So to Arnold's question, the distance would seem
to require at least two discrete charges, naturally
these charges are associated with charged particles
again using the atomic example.
Physically the classical "electrical potential energy"
is given by,
E = qQ/L where q and Q are charges and
L separates q and Q. The ISU defines L =cT so
E = qQ/cT and ET = qQ (c=1).
and ET in erg*seconds is Plancks's constant,
h = 6,625*10^-27 erg*secs ( * integer )
Prior to Plancks quantum hypothesis, "h" could
tend to zero, and an infinite accuracy measuring
the continuum was thought possible.
Subsequently experimentalists have found
h=constant, and the measurement of the continuum
is limited to adapting the nearest integer*h.
Classical tensor analysis, involves a measurement
of the continuum at a point, and the relation of that
point related to infinitesmally close points when
things like curvature are calculated.
So I think the queston is, can an operational tensor
analysis (that's Generally Covariant) be based on
"h = constant"?
In light of Alfred's post, we might use metrics defined
over finite lengths, defined by charges qQ, in accord
with QT, but with relative rotations that would yield
the non-symmetrical characteristics of a metric that
appear ridiculous using a continuum.
Regards
Ken S. Tucker
Arnold Neumaier
May3-04, 08:22 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>\nSuresh K Maran wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote\n>\n>>Alfred Einstead wrote:\n>>\n>>>A by-product of Connes\' research in non-commutative geometry was the\n>>>discovery of an expression for distance in a Riemannian space M that is\n>>>entirely written in terms of the function space over M and is algebraic,\n>>>making no explicit reference to the underlying point set of M. Because\n>>>it\'s purely algebraic, the expression can also be used for non-commutative\n>>>spaces where they may not even be a point set.\n>>\n>>Distance between what, if there are no points???\n>\n>\n> Greetings!.\n> Spin foam models of quantum gravity has clear answers for this.\n> Please read the article by John Baez at the lanl archive\n> http://xxx.lanl.gov/abs/gr-qc/9905087\n\nPlease be more explicit about what you mean. I didnt\' question\ngeometries without points, but was worried about the meaning of\n\'distance\' in such a sitouation.\n\nBut the term \'distance\' is used only once in the paper you mention,\nand there is only says,\n\n\'\'Even worse, our ability to do computations with the theory is too poor to\nreally tell if it reduces to classical Riemannian general relativity\nin the large-scale limit, i.e. the limit of distances much larger\nthan the Planck length.\'\'\n\n\nArnold Neumaier\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>Suresh K Maran wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote
>
>>Alfred Einstead wrote:
>>
>>>A by-product of Connes' research in non-commutative geometry was the
>>>discovery of an expression for distance in a Riemannian space M that is
>>>entirely written in terms of the function space over M and is algebraic,
>>>making no explicit reference to the underlying point set of M. Because
>>>it's purely algebraic, the expression can also be used for non-commutative
>>>spaces where they may not even be a point set.
>>
>>Distance between what, if there are no points???
>
>
> Greetings!.
> Spin foam models of quantum gravity has clear answers for this.
> Please read the article by John Baez at the lanl archive
> http://xxx.lanl.gov/abs/http://www.arxiv.org/abs/gr-qc/9905087
Please be more explicit about what you mean. I didnt' question
geometries without points, but was worried about the meaning of
'distance' in such a sitouation.
But the term 'distance' is used only once in the paper you mention,
and there is only says,
''Even worse, our ability to do computations with the theory is too poor to
really tell if it reduces to classical Riemannian general relativity
in the large-scale limit, i.e. the limit of distances much larger
than the Planck length.''
Arnold Neumaier
Thomas Larsson
May4-04, 03:06 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c6trp4\\$127\\$1@lfa222122.richmond.edu>...\ n> Alfred Einstead wrote:\n> >\n> > A by-product of Connes\' research in non-commutative geometry was the\n> > discovery of an expression for distance in a Riemannian space M that is\n> > entirely written in terms of the function space over M and is algebraic,\n> > making no explicit reference to the underlying point set of M. Because\n> > it\'s purely algebraic, the expression can also be used for non-commutative\n> > spaces where they may not even be a point set.\n>\n> Distance between what, if there are no points???\n>\n\n\nA review of Connes\' ideas can be found in hep-th/0111236.\nThe geodesic distance is reconstructed in eq (86) on page 27.\nGiven a triple (A, H, D), where A is a commutative (function)\nalgebra, H a Hilbert space which carries a rep of A, and D the\nDirac operator (which I think specifies the polarization of H),\nthen the geodesic distance between two points x and y is\n\nsup {|delta_x(a) - delta_y(a)|}\n\nwhere the supremum is taken over a \\in A such that\n\n|| [D, \\rho(a)] || <= 1.\n\nHere delta_x(a) = a(x) is the Dirac distribution and \\rho is\nthe rep of A on H.\n\nBut I have never understood the Connes stuff well enough to\nreally get a feeling for what this means, so you have to look\nin the e-print. I am pretty sure that this approach is background\ndependent, since there is a background metric lurking in the\nDirac operator.\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c6trp4$127$1@lfa222122.richmond.edu>...
> Alfred Einstead wrote:
> >
> > A by-product of Connes' research in non-commutative geometry was the
> > discovery of an expression for distance in a Riemannian space M that is
> > entirely written in terms of the function space over M and is algebraic,
> > making no explicit reference to the underlying point set of M. Because
> > it's purely algebraic, the expression can also be used for non-commutative
> > spaces where they may not even be a point set.
>
> Distance between what, if there are no points???
>
A review of Connes' ideas can be found in http://www.arxiv.org/abs/hep-th/0111236.
The geodesic distance is reconstructed in eq (86) on page 27.
Given a triple (A, H, D), where A is a commutative (function)
algebra, H a Hilbert space which carries a rep of A, and D the
Dirac operator (which I think specifies the polarization of H),
then the geodesic distance between two points x and y is
sup {|\delta_x(a) - \delta_y(a)|}
where the supremum is taken over a \in A such that
|| [D, \rho(a)] || <= 1.
Here \delta_x(a) = a(x) is the Dirac distribution and \rho is
the rep of A on H.
But I have never understood the Connes stuff well enough to
really get a feeling for what this means, so you have to look
in the e-print. I am pretty sure that this approach is background
dependent, since there is a background metric lurking in the
Dirac operator.
Urs Schreiber
May6-04, 06:22 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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0405040319.76a39507@pos ting.google.com...\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message\nnews:<c6trp4\\$127\\$1@lfa222122.richmond .edu>...\n\n> > Distance between what, if there are no points???\n\nRight, the fact that you can reconstruct geodesic distance by knowing just\nthe algebra of functions on a manifold together with the Dirac operator has\nno application as soon as your are dealing with algebras that don\'t have\nlots of ideals.\n\nThe point of the theorem is rather the following: It serves to show that in\nthe continuum the "spectral triple" consisting of an algebra of functions,\nthe usual Dirac operator and the usual spinor Hilbert space on which both\nare represented encodes precisely all the geometric information about the\nmanifold. This is just to show that spectral triple are a consistent\ngeneralization of geometry, because in order for that to be true they have\nto be able to reproduce ordinary geometry as a special case.\n\nMore generally, somewhat heuristically, the inverse of the Dirac operator is\nsaid to be similar to the line element ds.\n\nThis is motivated from the fact that for a an object of grade 0\n\n[D,a] = gamma . grad a\n\n=> |[D,a]| = grad a ~ "da/ds"\n\nwhere the last expression on the right makes heuristic sense in the context\nof the formula for the geodesic distance\n\nd(x,y) = Sup { |a(x) - a(y)| } for |[D,a]| <= 1\n\nThinking of "D ~ 1/ds" also gives a nice interpretation of some of the trace\nformulas that are used in NCG as a generalization of the ordinary integral\nover the manifold.\n\n> But I have never understood the Connes stuff well enough to\n> really get a feeling for what this means, so you have to look\n> in the e-print. I am pretty sure that this approach is background\n> dependent, since there is a background metric lurking in the\n> Dirac operator.\n\nIt\'s the other way round: The Dirac operator encodes the geometry. But it is\nnot fixed background. Instead of varying the metric you think of varying the\nDirac operator in NCG.\n\nThe key to this kind of reasoning is the "Spectral Action Principle" e.g.\n\nA. Chamseddine & A. Connes: The Spectral Action Principle\nhep-th/9606001\n\nwhich essentially says that the trace over D^2 (something like the\nLaplace(-Beltrami) operator) has an expansion in terms of powers of the\nRiemann tensor which to lowest order reproduces the Einstein-Hilbert plus YM\naction. (The action of the fermions themselves comes as usual from\n<psi|D|psi>)\n\nOr does it? I was always confused about this claim because actually at the\norder at which the YM terms appear there are also already corrections to the\nEH action, e.g. equation (2.24) of the above paper. I would like to see a\ndiscussion why these correction terms would be consistent with experiment.\n\nThe big achievment of the spectral action principle is that by choosing an\nappropriate noncommutative algebra (in the Connes-Lott model of the standard\nmodel this is the usual algebra of smooth functions on spacetime times the\nalgebra C+H+M_3(C) (1.17 of the above paper)) we get the action of gravity\nas well as the gauge theory sector from the same generalized geometric\nprinciple.\n\nThe big drawback is that this so far only works for compact Riemannian\nmanifolds and that, without further work, it doesn\'t seem to help much in\nquantizing the resulting action.\n\nI have tried to discuss these two points in the past, e.g.\n\nhttp://golem.ph.utexas.edu/string/archives/000298.html#c000622 .\n\nI just come from DESY/Hamburg\n\nhttp://golem.ph.utexas.edu/string/archives/000358.html\n\nwhere, among other things, I talked about this stuff with Prof. Fredenhagen\nand others and Fredenhagen recalled that when he had asked Connes about the\nproblem of semi-Riemannian metrics in his formalism Connes replied that he\nwas aware that this is a problem but is postponing thinking about it until\nsome other aspects are better understood.\n\nAt the last DPG spring conference in Ulm there were several talks by people\nwho presented ideas how to circumvent the restriction to compact Riemannian\nspaces in NCG:\n\nhttp://golem.ph.utexas.edu/string/archives/000330.html#c000799\n\nBut all these approaches are lacking some of the elegance of the original\nformalism, I think. To my delight, Fredenhagen said that it would be nicer\nto have the Lorentzian metric drop out all by itself, as in the approach\nthat Eric and I have taken:\n\nhttp://www-stud.uni-essen.de/~sb0264/p4a.pdf .\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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0405040319.76a39507@posting.google.c om...
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message
news:<c6trp4$127$1@lfa222122.richmond.edu>...
> > Distance between what, if there are no points???
Right, the fact that you can reconstruct geodesic distance by knowing just
the algebra of functions on a manifold together with the Dirac operator has
no application as soon as your are dealing with algebras that don't have
lots of ideals.
The point of the theorem is rather the following: It serves to show that in
the continuum the "spectral triple" consisting of an algebra of functions,
the usual Dirac operator and the usual spinor Hilbert space on which both
are represented encodes precisely all the geometric information about the
manifold. This is just to show that spectral triple are a consistent
generalization of geometry, because in order for that to be true they have
to be able to reproduce ordinary geometry as a special case.
More generally, somewhat heuristically, the inverse of the Dirac operator is
said to be similar to the line element ds.
This is motivated from the fact that for a an object of grade
[D,a] = \gamma . grad a=> |[D,a]| = grad a ~ "da/ds"
where the last expression on the right makes heuristic sense in the context
of the formula for the geodesic distance
d(x,y) = Sup { |a(x) - a(y)| } for |[D,a]| <= 1
Thinking of "D ~ 1/ds" also gives a nice interpretation of some of the trace
formulas that are used in NCG as a generalization of the ordinary integral
over the manifold.
> But I have never understood the Connes stuff well enough to
> really get a feeling for what this means, so you have to look
> in the e-print. I am pretty sure that this approach is background
> dependent, since there is a background metric lurking in the
> Dirac operator.
It's the other way round: The Dirac operator encodes the geometry. But it is
not fixed background. Instead of varying the metric you think of varying the
Dirac operator in NCG.
The key to this kind of reasoning is the "Spectral Action Principle" e.g.
A. Chamseddine & A. Connes: The Spectral Action Principle
http://www.arxiv.org/abs/hep-th/9606001
which essentially says that the trace over D^2 (something like the
Laplace(-Beltrami) operator) has an expansion in terms of powers of the
Riemann tensor which to lowest order reproduces the Einstein-Hilbert plus YM
action. (The action of the fermions themselves comes as usual from
<\psi|D|\psi>)
Or does it? I was always confused about this claim because actually at the
order at which the YM terms appear there are also already corrections to the
EH action, e.g. equation (2.24) of the above paper. I would like to see a
discussion why these correction terms would be consistent with experiment.
The big achievment of the spectral action principle is that by choosing an
appropriate noncommutative algebra (in the Connes-Lott model of the standard
model this is the usual algebra of smooth functions on spacetime times the
algebra C+H+M_3(C) (1.17 of the above paper)) we get the action of gravity
as well as the gauge theory sector from the same generalized geometric
principle.
The big drawback is that this so far only works for compact Riemannian
manifolds and that, without further work, it doesn't seem to help much in
quantizing the resulting action.
I have tried to discuss these two points in the past, e.g.
http://golem.ph.utexas.edu/string/archives/000298.html#c000622 .
I just come from DESY/Hamburg
http://golem.ph.utexas.edu/string/archives/000358.html
where, among other things, I talked about this stuff with Prof. Fredenhagen
and others and Fredenhagen recalled that when he had asked Connes about the
problem of semi-Riemannian metrics in his formalism Connes replied that he
was aware that this is a problem but is postponing thinking about it until
some other aspects are better understood.
At the last DPG spring conference in Ulm there were several talks by people
who presented ideas how to circumvent the restriction to compact Riemannian
spaces in NCG:
http://golem.ph.utexas.edu/string/archives/000330.html#c000799
But all these approaches are lacking some of the elegance of the original
formalism, I think. To my delight, Fredenhagen said that it would be nicer
to have the Lorentzian metric drop out all by itself, as in the approach
that Eric and I have taken:
http://www-stud.uni-essen.de/~sb0264/p4a.pdf .
Alfred Einstead
May6-04, 07:52 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\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n> > A by-product of Connes\' research in non-commutative geometry was the\n> > discovery of an expression for distance in a Riemannian space M that is\n> > entirely written in terms of the function space over M and is algebraic,\n> > making no explicit reference to the underlying point set of M. Because\n> > it\'s purely algebraic, the expression can also be used for non-commutative\n> > spaces where they may not even be a point set.\n>\n> Distance between what, if there are no points???\n\nPoints in the function space of a manifold M are just prime 2-sided\nideals. So, in that case, it\'s a distance between prime 2-sided\nideals.\n\nIn the more general case, you\'re putting the cart before the horse.\nDistance comes first not points. It\'s not "distance between points",\nbut more like "points between distance". Everything\'s backwards in\nfunction spaces.\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
> > A by-product of Connes' research in non-commutative geometry was the
> > discovery of an expression for distance in a Riemannian space M that is
> > entirely written in terms of the function space over M and is algebraic,
> > making no explicit reference to the underlying point set of M. Because
> > it's purely algebraic, the expression can also be used for non-commutative
> > spaces where they may not even be a point set.
>
> Distance between what, if there are no points???
Points in the function space of a manifold M are just prime 2-sided
ideals. So, in that case, it's a distance between prime 2-sided
ideals.
In the more general case, you're putting the cart before the horse.
Distance comes first not points. It's not "distance between points",
but more like "points between distance". Everything's backwards in
function spaces.
Arnold Neumaier
May6-04, 12:20 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>Thomas Larsson wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c6trp4\\$127\\$1@lfa222122.richmond.edu>...\ n>\n>>Alfred Einstead wrote:\n>>\n>>>A by-product of Connes\' research in non-commutative geometry was the\n>>>discovery of an expression for distance in a Riemannian space M that is\n>>>entirely written in terms of the function space over M and is algebraic,\n>>>making no explicit reference to the underlying point set of M. Because\n>>>it\'s purely algebraic, the expression can also be used for non-commutative\n>>>spaces where they may not even be a point set.\n>>\n>>Distance between what, if there are no points???\n>>\n\n> A review of Connes\' ideas can be found in hep-th/0111236.\n> The geodesic distance is reconstructed in eq (86) on page 27.\n> Given a triple (A, H, D), where A is a commutative (function)\n> algebra, H a Hilbert space which carries a rep of A, and D the\n> Dirac operator (which I think specifies the polarization of H),\n> then the geodesic distance between two points x and y is\n>\n> sup {|delta_x(a) - delta_y(a)|}\n>\n> where the supremum is taken over a \\in A such that\n>\n> || [D, \\rho(a)] || <= 1.\n>\n> Here delta_x(a) = a(x) is the Dirac distribution and \\rho is\n> the rep of A on H.\n\nThanks.\n\n\nThis is about commutative algebras. But Alfred Einstead claimed that\n\n>>>the expression can also be used for non-commutative\n>>>spaces where they may not even be a point set.\n\nMy question referred to that. If there are no points, there are no\ndelta-functions either, so the formula becomes meaningless.\n\n\nArnold Neumaier\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>Thomas Larsson wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c6trp4$127$1@lfa222122.richmond.edu>...
>
>>Alfred Einstead wrote:
>>
>>>A by-product of Connes' research in non-commutative geometry was the
>>>discovery of an expression for distance in a Riemannian space M that is
>>>entirely written in terms of the function space over M and is algebraic,
>>>making no explicit reference to the underlying point set of M. Because
>>>it's purely algebraic, the expression can also be used for non-commutative
>>>spaces where they may not even be a point set.
>>
>>Distance between what, if there are no points???
>>
> A review of Connes' ideas can be found in http://www.arxiv.org/abs/hep-th/0111236.
> The geodesic distance is reconstructed in eq (86) on page 27.
> Given a triple (A, H, D), where A is a commutative (function)
> algebra, H a Hilbert space which carries a rep of A, and D the
> Dirac operator (which I think specifies the polarization of H),
> then the geodesic distance between two points x and y is
>
> sup {|\delta_x(a) - \delta_y(a)|}
>
> where the supremum is taken over a \in A such that
>
> || [D, \rho(a)] || <= 1.
>
> Here \delta_x(a) = a(x) is the Dirac distribution and \rho is
> the rep of A on H.
Thanks.
This is about commutative algebras. But Alfred Einstead claimed that
>>>the expression can also be used for non-commutative
>>>spaces where they may not even be a point set.
My question referred to that. If there are no points, there are no
\delta-functions either, so the formula becomes meaningless.
Arnold Neumaier
Alfred Einstead
May7-04, 06:36 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>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote:\n> > Distance between what, if there are no points???\n> sup {|delta_x(a) - delta_y(a)|}\n> || [D, \\rho(a)] || <= 1.\n\nRight. In a function space E(M) = C^{infinity}(M,R), delta_x,\nwhich resides in E(M)* is just the point, x, itself, for all\npractical purposes.\n\nThe distance function is a norm over E(M)*. That distance is also\na distance between points comes second, not first. It will always\nbe so when there are points (i.e. prime 2-sided ideals in E(M)),\nbut will STILL be there even when there aren\'t.\n\n.... which goes to prove that distance comes first, not points. It\'s\n"points between distance", not "distance between points". You\'ve\nall had it backwards all these years until now.\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>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote:
> > Distance between what, if there are no points???
> sup {|\delta_x(a) - \delta_y(a)|}
> || [D, \rho(a)] || <= 1.
Right. In a function space E(M) = C^{infinity}(M,R), \delta_x,
which resides in E(M)* is just the point, x, itself, for all
practical purposes.
The distance function is a norm over E(M)*. That distance is also
a distance between points comes second, not first. It will always
be so when there are points (i.e. prime 2-sided ideals in E(M)),
but will STILL be there even when there aren't.
.... which goes to prove that distance comes first, not points. It's
"points between distance", not "distance between points". You've
all had it backwards all these years until now.
Urs Schreiber
May13-04, 05:25 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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0405090202.17c36bb2@pos ting.google.com...\n\n> Well, I dont really know anything about NCG. For example, is it a quantum\n> theory or only classical? Quantization can be regarded as the introduction\nof\n> non-commutative geometry in phase space - von Neumann\'s pointless\ngeometry -\n> but NCG seems to posit another type of non-commutativity in spacetime. How\n> do these two types of non-commutativity work together?\n\nAs I said, as far as I am aware (having talked to some people who should\nknow, but certainly not to all who could know :-) quantization of "spectral\nactions" a la Connes, Lott et al. which include gravity and gauge fields is\nnot any better understood than the quantization of ordinary gravity, at\nleast as long as the gravitational part comes from a continuous manifold and\nonly the YM part is due to noncommutive geometry.\n\nThat\'s not too surprising, since as long as spacetime is modeled by a\ncommuting algebra the spectral action is nothing but the ordinary EH action\nplus higher order corrections.\n\nIn this context, all that "spectral" NCG gives you is a nice single\nprinciple which treats gravity and gauge fields on the same footing,\nspitting out their ordinary action functionals plus higher order\ncorrections.\n\nNote that on the other hand there is an entire industry concerned with what\nI would like to call "algebraic" NCG (quantum) field theories in which one\nis not concerned with the Dirac operator and its spectrum, but just with\nsome noncommutative algebra as such: Namely what is usually just called\nnoncommutative field theory, where Lagrangians are written down in terms of\nnoncommutative products of physial fields (e.g Moyal stars, etc.). The\nquantization of these theories is being studied in great detail. But I think\nin this context people just look at the YM sector, nothing gravitational\nbeing done here.\n\nSo the natural question seems to be: Could the spectral action principle\nhelp to quantize gravity as soon as we model spacetime _not_ by the usual\ncommutative algebra of functions? One could imagine that some noncommutative\neffects smear out the divergencies of naive perturbative quantum gravity.\n\nI know very little about the possible answer to this question and about\npeople who might know the answer. It _seems_ to me that papers like\n\nAlbuquerque et al.,\nFluctuating Dimension in a Discrete Model for Quantum Gravity Based on the\nSpectral Principle,\nhep-th/0305082\n\nhave something to say concerning this question, but I am not sure. See my\ndiscussion with Alejandro Rivero at\n\nhttp://golem.ph.utexas.edu/string/archives/000298.html#c000622 .\n\nSo far to me the main insight of the spectral action principle is that that\nin terms of the Dirac operator and its spectrum the different fields in\nnature can naturally be regarded as manifestations of a unique underlying\nprinciple, an idea which is nicely made precise by Connes "group koan" which\nsays that the exact sequence\n\n1 -> Int(A) -> Aut(A) -> Out(A) -> 1 ,\n\nwhere A is the (noncommutative) algebra, Aut(A) are its automorphisms and\nInt(A), Out(A) its inner and outer automorphisms, respectively, is in direct\ncorrespondence to the exact sequence\n\n1 -> U -> G -> Diff(M) -> 1 ,\n\nwhere now G is the full symmetry group of a given physical theory, U is the\n"internal" gauge group and Diff(M) the diffeomorphism group.\n\nSee our discussion at\n\nhttp://golem.ph.utexas.edu/string/archives/000298.html#c000695\n\nfor references - and please note that I am not really an expert on this\nstuff, so beware of mistakes that I might make! :-)\n\nAn similar concept of having all physical fields arise from a single\nprinciple is of course present in string theory. That\'s why I find it\nremarkable that when one uses the worldsheet supercharge of the superstring\n(the Dirac-Ramond operator) as the Dirac operator in a spectral triple, the\nresulting spectral action is indeed (at least to lowest order, where this\nhas been checked) the usual effective background action of string theory, as\nshown in\n\nChamseddine:\nAn Effective Superstring Spectral Action,\nhep-th/9705153 .\n\nBy the way, I haven\'t looked at it yet, but maybe you would like to read\n\nChamseddine,\nNoncommutative Gravity,\nhep-th/0301112 .\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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0405090202.17c36bb2@posting.google.c om...
> Well, I dont really know anything about NCG. For example, is it a quantum
> theory or only classical? Quantization can be regarded as the introduction
of
> non-commutative geometry in phase space - von Neumann's pointless
geometry -
> but NCG seems to posit another type of non-commutativity in spacetime. How
> do these two types of non-commutativity work together?
As I said, as far as I am aware (having talked to some people who should
know, but certainly not to all who could know :-) quantization of "spectral
actions" a la Connes, Lott et al. which include gravity and gauge fields is
not any better understood than the quantization of ordinary gravity, at
least as long as the gravitational part comes from a continuous manifold and
only the YM part is due to noncommutive geometry.
That's not too surprising, since as long as spacetime is modeled by a
commuting algebra the spectral action is nothing but the ordinary EH action
plus higher order corrections.
In this context, all that "spectral" NCG gives you is a nice single
principle which treats gravity and gauge fields on the same footing,
spitting out their ordinary action functionals plus higher order
corrections.
Note that on the other hand there is an entire industry concerned with what
I would like to call "algebraic" NCG (quantum) field theories in which one
is not concerned with the Dirac operator and its spectrum, but just with
some noncommutative algebra as such: Namely what is usually just called
noncommutative field theory, where Lagrangians are written down in terms of
noncommutative products of physial fields (e.g Moyal stars, etc.). The
quantization of these theories is being studied in great detail. But I think
in this context people just look at the YM sector, nothing gravitational
being done here.
So the natural question seems to be: Could the spectral action principle
help to quantize gravity as soon as we model spacetime _not_ by the usual
commutative algebra of functions? One could imagine that some noncommutative
effects smear out the divergencies of naive perturbative quantum gravity.
I know very little about the possible answer to this question and about
people who might know the answer. It _seems_ to me that papers like
Albuquerque et al.,
Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the
Spectral Principle,
http://www.arxiv.org/abs/hep-th/0305082
have something to say concerning this question, but I am not sure. See my
discussion with Alejandro Rivero at
http://golem.ph.utexas.edu/string/archives/000298.html#c000622 .
So far to me the main insight of the spectral action principle is that that
in terms of the Dirac operator and its spectrum the different fields in
nature can naturally be regarded as manifestations of a unique underlying
principle, an idea which is nicely made precise by Connes "group koan" which
says that the exact sequence
1 -> \Int(A) ->[/itex] Aut(A) -> Out(A) -> 1 ,
where A is the (noncommutative) algebra, Aut(A) are its automorphisms and
\Int(A), Out(A) its inner and outer automorphisms, respectively, is in direct
correspondence to the exact sequence
1 -> U -> G -> Diff(M) [itex]-> 1 ,
where now G is the full symmetry group of a given physical theory, U is the
"internal" gauge group and Diff(M) the diffeomorphism group.
See our discussion at
http://golem.ph.utexas.edu/string/archives/000298.html#c000695
for references - and please note that I am not really an expert on this
stuff, so beware of mistakes that I might make! :-)
An similar concept of having all physical fields arise from a single
principle is of course present in string theory. That's why I find it
remarkable that when one uses the worldsheet supercharge of the superstring
(the Dirac-Ramond operator) as the Dirac operator in a spectral triple, the
resulting spectral action is indeed (at least to lowest order, where this
has been checked) the usual effective background action of string theory, as
shown in
Chamseddine:
An Effective Superstring Spectral Action,
http://www.arxiv.org/abs/hep-th/9705153 .
By the way, I haven't looked at it yet, but maybe you would like to read
Chamseddine,
Noncommutative Gravity,
http://www.arxiv.org/abs/hep-th/0301112 .
Arnold Neumaier
May14-04, 04:08 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>Thomas Larsson wrote:\n\n> Well, I dont really know anything about NCG.\n\n[= noncommutative geometry]\n\n> For example, is it a quantum\n> theory or only classical?\n\nThis may depend on whether you call Dirac\'s or Maxwell\'s equation\nquantum or classical. Both positions are tenable - as a classical field theory\nor as a quantum single particle theory.\n\n> Quantization can be regarded as the introduction of\n> non-commutative geometry in phase space - von Neumann\'s pointless geometry -\n> but NCG seems to posit another type of non-commutativity in spacetime. How\n> do these two types of non-commutativity work together?\n\nStandard quantization makes phase space noncommutative, but leaves there\nlots of good commutative 3-spaces, one for each observer. This is the\ndeeper meaning of the locality requirement in QFT, that you have all these\n3-spaces. And that\'s indeed our daily experience - each observer sees\nhis/her own 3-space, which changes position as they move around. I believe\nthe psychological fact that we experience only 3 of the 4 space-time\ndimensions as space has its ultimate roots in this mathematical property.\n\nOne can also make position space noncommutative then things simply get\na little weirder. This happens when one looks as q-versions of everything,\nsuch as the noncommutative torus and quantum groups.\nWe don\'t observe such weirdness, so such studies are probably just a\nmathematical curiosity. Nature does not use every mathematical possibility.\n\nConnes\' use of noncommutativity makes only minor changes to 4-space, though;\none just has a product of ordinary 4-space with a pointless space carrying\nno position but spin and other internal quantum numbers only. Thus he\npreserves the important part of the picture, the 3-spaces needed for our\nexperience (and correspondence to experiment) to come out right. That\'s\nwhy he is able to recover the standard model, which is usually based on\ncommutative space.\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>Thomas Larsson wrote:
> Well, I dont really know anything about NCG.
[= noncommutative geometry]
> For example, is it a quantum
> theory or only classical?
This may depend on whether you call Dirac's or Maxwell's equation
quantum or classical. Both positions are tenable - as a classical field theory
or as a quantum single particle theory.
> Quantization can be regarded as the introduction of
> non-commutative geometry in phase space - von Neumann's pointless geometry -
> but NCG seems to posit another type of non-commutativity in spacetime. How
> do these two types of non-commutativity work together?
Standard quantization makes phase space noncommutative, but leaves there
lots of good commutative 3-spaces, one for each observer. This is the
deeper meaning of the locality requirement in QFT, that you have all these
3-spaces. And that's indeed our daily experience - each observer sees
his/her own 3-space, which changes position as they move around. I believe
the psychological fact that we experience only 3 of the 4 space-time
dimensions as space has its ultimate roots in this mathematical property.
One can also make position space noncommutative then things simply get
a little weirder. This happens when one looks as q-versions of everything,
such as the noncommutative torus and quantum groups.
We don't observe such weirdness, so such studies are probably just a
mathematical curiosity. Nature does not use every mathematical possibility.
Connes' use of noncommutativity makes only minor changes to 4-space, though;
one just has a product of ordinary 4-space with a pointless space carrying
no position but spin and other internal quantum numbers only. Thus he
preserves the important part of the picture, the 3-spaces needed for our
experience (and correspondence to experiment) to come out right. That's
why he is able to recover the standard model, which is usually based on
commutative space.
Arnold Neumaier
alejandro.rivero
May15-04, 02:45 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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<40a34cfd\\$1@news.sentex.net>...\n\n> I know very little about the possible answer to this question and about\n> people who might know the answer. It _seems_ to me that papers like\n>\n> Albuquerque et al.,\n> Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the\n> Spectral Principle,\n> hep-th/0305082\n>\n> have something to say concerning this question, but I am not sure. See my\n> discussion with Alejandro Rivero at\n>\n> http://golem.ph.utexas.edu/string/archives/000298.html#c000622 .\n>\n> So far to me the main insight...\n\nEr... sorry to be "out town", I got busy around hep-ph and nucl-th...\nlet me add here that NCG has a principle of "complex dimension" where\na spectral triple does not need to have a single integer number as\ndimension; it can be a whole set of the pure imaginary line (and\nperhaps the rest of the complex plane). This was worked out by\nMoscovici and Connes in the GAFA paper. It is useful, for instance, if\nyour spectral triple is the union of a plane and a line... then you\nwill have poles marking both dimensions or something so.\n\nTchuss\n\nAlejandro\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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<40a34cfd$1@news.sentex.net>...
> I know very little about the possible answer to this question and about
> people who might know the answer. It _seems_ to me that papers like
>
> Albuquerque et al.,
> Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the
> Spectral Principle,
> http://www.arxiv.org/abs/hep-th/0305082
>
> have something to say concerning this question, but I am not sure. See my
> discussion with Alejandro Rivero at
>
> http://golem.ph.utexas.edu/string/archives/000298.html#c000622 .
>
> So far to me the main insight...
Er... sorry to be "out town", I got busy around hep-ph and nucl-th...
let me add here that NCG has a principle of "complex dimension" where
a spectral triple does not need to have a single integer number as
dimension; it can be a whole set of the pure imaginary line (and
perhaps the rest of the complex plane). This was worked out by
Moscovici and Connes in the GAFA paper. It is useful, for instance, if
your spectral triple is the union of a plane and a line... then you
will have poles marking both dimensions or something so.
Tchuss
Alejandro
Doctordick
May15-04, 02:47 AM
Hi Russell,
I was just looking around ('cause I had some spare time) and noticed this thread posted by you. I just had to post after I read Einstead's response in message #3.
Since you raised the topic with the subject header, it's both instructive and revealing to see what Einstein, himself, had to
say on the subject of quantum gravity at the end of his life:
"One can give good reasons why reality cannot at all be represented
by a continuous field. From the quantum phenomena it appears to
follow with certainty that a finite system of finite energy can be
completely described by a finite set of numbers (quantum numbers).
This does not seem to be in accordance with a continuum theory,
and must lead to an attempt to find a purely algebraic theory for
the description of reality. But [sic] nobody knows how to find
the basis of such a theory."I just had to laugh when I saw that "But [sic] nobody knows how to find the basis of such a theory." I remember when he died as I had always wished I could have talked to him. This comment alone tells me he would have taken me seriously. Where are the intelligent physicists of today?
Have fun -- Dick
Thomas Larsson
May17-04, 06:19 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>Urs Schreiber <Urs.Schreiber@uni-essen.de> skrev i\ndiskussionsgruppsmeddelandet:40a34cfd\\$1@news. sentex.net...\n\n> Note that on the other hand there is an entire industry concerned with what\n> I would like to call "algebraic" NCG (quantum) field theories in which one\n> is not concerned with the Dirac operator and its spectrum, but just with\n> some noncommutative algebra as such: Namely what is usually just called\n> noncommutative field theory, where Lagrangians are written down in terms of\n> noncommutative products of physial fields (e.g Moyal stars, etc.). The\n> quantization of these theories is being studied in great detail. But I think\n> in this context people just look at the YM sector, nothing gravitational\n> being done here.\n\nAssociativity is of course always a nice thing, so it\'s is natural\nfor model-builders to use any associative product around. Hm. Are\nthere any associative products around except Moyal? I.e., is there\nany deformation of the function algebra C(R^n) which does not\nlocally look like the Moyal product?\n\n>\n> So the natural question seems to be: Could the spectral action principle\n> help to quantize gravity as soon as we model spacetime _not_ by the usual\n> commutative algebra of functions? One could imagine that some noncommutative\n> effects smear out the divergencies of naive perturbative quantum gravity.\n\nI have difficulties to visualize quantization on a noncommutative\nspacetime. In QM we have particles that live in spacetime, and I\nunderstand sort of how to make this noncommutative. But in QFT the\nspacetime points label the fields. It seems that we try to quantize\ndegrees of freedom that are labelled by a noncommutative index set.\nWhat does that mean?\n\n> So far to me the main insight of the spectral action principle is that that\n> in terms of the Dirac operator and its spectrum the different fields in\n> nature can naturally be regarded as manifestations of a unique underlying\n> principle, an idea which is nicely made precise by Connes "group koan" which\n> says that the exact sequence\n>\n> 1 -> Int(A) -> Aut(A) -> Out(A) -> 1 ,\n>\n> where A is the (noncommutative) algebra, Aut(A) are its automorphisms and\n> Int(A), Out(A) its inner and outer automorphisms, respectively, is in direct\n> correspondence to the exact sequence\n>\n> 1 -> U -> G -> Diff(M) -> 1 ,\n>\n> where now G is the full symmetry group of a given physical theory, U is the\n> "internal" gauge group and Diff(M) the diffeomorphism group.\n\nHm, I think I understand this. The last sequence basically says\nthat the symmetry group is the semidirect product of\ndiffeomorphisms and gauge transformations. If we work on a\nnontrivial fiber bundle things get a little more complicated (the\nsequence does not split), but Diff(M) are still horizontal\ntransformations and U vertical transformations.\n\nSo the upper exact sequence is a noncommutative generalization of\nthis situation. But is this really unification? If A is commutative\nand thus a function algebra, wouldn\'t G always locally look like a\nsemidirect product?\n\n> By the way, I haven\'t looked at it yet, but maybe you would like to read\n>\n> Chamseddine,\n> Noncommutative Gravity,\n> hep-th/0301112 .\n>\n\nIt ends with:\n\n"To conclude, it is clear from the above discussion that gauge\ntheories on noncommutative spaces are straightforward, but to\ndefine the gravitational action on such spaces is more difficult.\nAt present there are only partial answers, and more intensive\nresearch in this direction is needed."\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> skrev i
diskussionsgruppsmeddelandet:40a34cfd$1@news.sente x.net...
> Note that on the other hand there is an entire industry concerned with what
> I would like to call "algebraic" NCG (quantum) field theories in which one
> is not concerned with the Dirac operator and its spectrum, but just with
> some noncommutative algebra as such: Namely what is usually just called
> noncommutative field theory, where Lagrangians are written down in terms of
> noncommutative products of physial fields (e.g Moyal stars, etc.). The
> quantization of these theories is being studied in great detail. But I think
> in this context people just look at the YM sector, nothing gravitational
> being done here.
Associativity is of course always a nice thing, so it's is natural
for model-builders to use any associative product around. Hm. Are
there any associative products around except Moyal? I.e., is there
any deformation of the function algebra C(R^n) which does not
locally look like the Moyal product?
>
> So the natural question seems to be: Could the spectral action principle
> help to quantize gravity as soon as we model spacetime _not_ by the usual
> commutative algebra of functions? One could imagine that some noncommutative
> effects smear out the divergencies of naive perturbative quantum gravity.
I have difficulties to visualize quantization on a noncommutative
spacetime. In QM we have particles that live in spacetime, and I
understand sort of how to make this noncommutative. But in QFT the
spacetime points label the fields. It seems that we try to quantize
degrees of freedom that are labelled by a noncommutative index set.
What does that mean?
> So far to me the main insight of the spectral action principle is that that
> in terms of the Dirac operator and its spectrum the different fields in
> nature can naturally be regarded as manifestations of a unique underlying
> principle, an idea which is nicely made precise by Connes "group koan" which
> says that the exact sequence
>
> 1 -> \Int(A) -> Aut(A) -> Out(A) -> 1 ,
>
> where A is the (noncommutative) algebra, Aut(A) are its automorphisms and
> \Int(A), Out(A) its inner and outer automorphisms, respectively, is in direct
> correspondence to the exact sequence
>
> 1 -> U -> G -> Diff(M) -> 1 ,
>
> where now G is the full symmetry group of a given physical theory, U is the
> "internal" gauge group and Diff(M) the diffeomorphism group.
Hm, I think I understand this. The last sequence basically says
that the symmetry group is the semidirect product of
diffeomorphisms and gauge transformations. If we work on a
nontrivial fiber bundle things get a little more complicated (the
sequence does not split), but Diff(M) are still horizontal
transformations and U vertical transformations.
So the upper exact sequence is a noncommutative generalization of
this situation. But is this really unification? If A is commutative
and thus a function algebra, wouldn't G always locally look like a
semidirect product?
> By the way, I haven't looked at it yet, but maybe you would like to read
>
> Chamseddine,
> Noncommutative Gravity,
> http://www.arxiv.org/abs/hep-th/0301112 .
>
It ends with:
"To conclude, it is clear from the above discussion that gauge
theories on noncommutative spaces are straightforward, but to
define the gravitational action on such spaces is more difficult.
At present there are only partial answers, and more intensive
research in this direction is needed."
Thomas Larsson
May17-04, 06:21 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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<40a34cfd\\$1@news.sentex.net>...\n\n\n> An similar concept of having all physical fields arise from a single\n> principle is of course present in string theory. That\'s why I find it\n> remarkable that when one uses the worldsheet supercharge of the superstring\n> (the Dirac-Ramond operator) as the Dirac operator in a spectral triple, the\n> resulting spectral action is indeed (at least to lowest order, where this\n> has been checked) the usual effective background action of string theory, as\n\nWhen reading Bert Schroer\'s latest epos, hep-th/0405105, I learned\nabout two references that might be of interest: hep-th/0305093 and\nhep-th/0402212. Not unexpectedly, Schroer seems to think that these\npapers are nonsense, the former more than the latter.\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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message news:<40a34cfd$1@news.sentex.net>...
> An similar concept of having all physical fields arise from a single
> principle is of course present in string theory. That's why I find it
> remarkable that when one uses the worldsheet supercharge of the superstring
> (the Dirac-Ramond operator) as the Dirac operator in a spectral triple, the
> resulting spectral action is indeed (at least to lowest order, where this
> has been checked) the usual effective background action of string theory, as
When reading Bert Schroer's latest epos, http://www.arxiv.org/abs/hep-th/0405105, I learned
about two references that might be of interest: http://www.arxiv.org/abs/hep-th/0305093 and
http://www.arxiv.org/abs/hep-th/0402212. Not unexpectedly, Schroer seems to think that these
papers are nonsense, the former more than the latter.
Urs Schreiber
May18-04, 04:14 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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0405140010.3e2a2e0b@pos ting.google.com...\n\n> When reading Bert Schroer\'s latest epos, hep-th/0405105,\n\nI don\'t understand everything that Schroer discusses in this paper, but some\nthings I do understand, in particular the issues discussed on the first half\nof p. 19, and I\'d like to comment on these. Some of these arguments have\nalready been exchanged at\n\nhttp://golem.ph.utexas.edu/string/archives/000338.html\n\nwhich Bert Schroer cites as [48], but maybe it is worthwhile to repeat some\nof the points in the context of s.p.r. (I\'ll cc this to Bert Schroer):\n\nBert Schroer argues that the Pohlmeyer approach to string quantization has\nadvantages over the standard approach in that\n\n"there is no mysterious and unphysical distinction (resulting from the\nnon-intrinsic canonical quantization) of d=10,26 rather they exist as\nPoincarŽe\ninvariant theory in each dimension > 3." (hep-th/0405105, p. 19)\n\nBut as far as I am aware this claim is not known to be true. As Bert Schroer\nacknowledges himself a couple of lines above, the task of of demonstrating\nthe truth of this statement\n\n"has been almost completed" (hep-th/0405105, p. 19)\n\nand only almost, I must emphasize, namely up to the so-called "quadratic\ngeneration hypothesis". I am not saying that the converse can be proved, but\nas it stands the claim that there is a quantization of the Nambu-Goto action\nwhich does not exhibit the critical dimension is in fact a speculation. And\nnot everybody agrees on how likely it is that this speculation can be\nproven.\n\nIn fact, as I have said before, I claim that there is one known solution of\nthe Pohlmeyer program, as indicated in hep-th/0403260, and that this leads\nto the just the usual quantization and in particular the usual critical\ndimension. In the light of this the circumstantial evidence that there is no\nalternative quantization of the string seems to me to be rather stronger\nthan that (namely which?) pointing in the other direction.\n\nFurthermore, Bert Schroer argues that the fact that open string field theory\nin light cone gauge exhibits a notion of string localization which seems to\nviolate a certain intuition. Namely for two string fields to commute it is\napparently sufficient for the "center-of-mass" of the respective strings to\nbe spacelike seperated. He argues that this shows that there is no\n"intrinsic", as he calls it, notion of string (with spatial extension) in\nstandard string theory.\n\nSeveral comments to this point have been made at the String Coffee Table\nlinked above, but maybe one point has not been emphasized, namely that open\nstring field theory is indeed non-local, as can be seen explicitly by\nlooking at the OSFT action in component form (for the first couple of\nlevels). This is given for instance as equation (2.46) in the review\nhep-th/0102085, where it is crucial to note the appearance of the nonlocal\ntilded fields defined in equation (2.44).\n\n> I learned\n> about two references that might be of interest: hep-th/0305093 and\n> hep-th/0402212. Not unexpectedly, Schroer seems to think that these\n> papers are nonsense, the former more than the latter.\n\nI haven\'t read all that, but I would like to know what is concidered\nnonsense and for which reasons.\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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0405140010.3e2a2e0b@posting.google.c om...
> When reading Bert Schroer's latest epos, http://www.arxiv.org/abs/hep-th/0405105,
I don't understand everything that Schroer discusses in this paper, but some
things I do understand, in particular the issues discussed on the first half
of p. 19, and I'd like to comment on these. Some of these arguments have
already been exchanged at
http://golem.ph.utexas.edu/string/archives/000338.html
which Bert Schroer cites as [48], but maybe it is worthwhile to repeat some
of the points in the context of s.p.r. (I'll cc this to Bert Schroer):
Bert Schroer argues that the Pohlmeyer approach to string quantization has
advantages over the standard approach in that
"there is no mysterious and unphysical distinction (resulting from the
non-intrinsic canonical quantization) of d=10,26 rather they exist as
PoincarŽe
invariant theory in each dimension > 3." (http://www.arxiv.org/abs/hep-th/0405105, p. 19)
But as far as I am aware this claim is not known to be true. As Bert Schroer
acknowledges himself a couple of lines above, the task of of demonstrating
the truth of this statement
"has been almost completed" (http://www.arxiv.org/abs/hep-th/0405105, p. 19)
and only almost, I must emphasize, namely up to the so-called "quadratic
generation hypothesis". I am not saying that the converse can be proved, but
as it stands the claim that there is a quantization of the Nambu-Goto action
which does not exhibit the critical dimension is in fact a speculation. And
not everybody agrees on how likely it is that this speculation can be
proven.
In fact, as I have said before, I claim that there is one known solution of
the Pohlmeyer program, as indicated in http://www.arxiv.org/abs/hep-th/0403260, and that this leads
to the just the usual quantization and in particular the usual critical
dimension. In the light of this the circumstantial evidence that there is no
alternative quantization of the string seems to me to be rather stronger
than that (namely which?) pointing in the other direction.
Furthermore, Bert Schroer argues that the fact that open string field theory
in light cone gauge exhibits a notion of string localization which seems to
violate a certain intuition. Namely for two string fields to commute it is
apparently sufficient for the "center-of-mass" of the respective strings to
be spacelike seperated. He argues that this shows that there is no
"intrinsic", as he calls it, notion of string (with spatial extension) in
standard string theory.
Several comments to this point have been made at the String Coffee Table
linked above, but maybe one point has not been emphasized, namely that open
string field theory is indeed non-local, as can be seen explicitly by
looking at the OSFT action in component form (for the first couple of
levels). This is given for instance as equation (2.46) in the review
http://www.arxiv.org/abs/hep-th/0102085, where it is crucial to note the appearance of the nonlocal
tilded fields defined in equation (2.44).
> I learned
> about two references that might be of interest: http://www.arxiv.org/abs/hep-th/0305093 and
> http://www.arxiv.org/abs/hep-th/0402212. Not unexpectedly, Schroer seems to think that these
> papers are nonsense, the former more than the latter.
I haven't read all that, but I would like to know what is concidered
nonsense and for which reasons.
Urs Schreiber
May18-04, 04:37 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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0405130741.53fded94@pos ting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> skrev i\n> diskussionsgruppsmeddelandet:40a34cfd\\$1@news.sen tex.net...\n\n> Associativity is of course always a nice thing, so it\'s is natural\n> for model-builders to use any associative product around.\n\nI agree. On the other hand, as Eric Forgy likes to point out, given any\nnon-commutative but associative product a*b we obtain a commutative but\nnon-associative product by using the (graded, maybe) symmetrization\n\na x b := a*b + b*a .\n\n> Hm. Are\n> there any associative products around except Moyal? I.e., is there\n> any deformation of the function algebra C(R^n) which does not\n> locally look like the Moyal product?\n\nDon\'t know, maybe not. Even the Witten open string star is a continuous copy\nof Moyal stars, as has been discussed here before.\n\n> I have difficulties to visualize quantization on a noncommutative\n> spacetime. In QM we have particles that live in spacetime, and I\n> understand sort of how to make this noncommutative. But in QFT the\n> spacetime points label the fields. It seems that we try to quantize\n> degrees of freedom that are labelled by a noncommutative index set.\n> What does that mean?\n\nI am not sure what you are worried about. We have some configuration of the\nsystem and associate an action to it.. Using this action we can try to write\ndown a path integral or try to compute canonical momenta and do canonical\nquantization. In either case, this procedure works no matter how the action\nassociates a number to a certain configuration. If the action functional\ninvolves Moyal products of functions that coordinatize the configuration of\nthe system then that\'s how it is. It does not seem to interfere with the\nprescription of quantization to me.\n\n\n> > So far to me the main insight of the spectral action principle is that\nthat\n> > in terms of the Dirac operator and its spectrum the different fields in\n> > nature can naturally be regarded as manifestations of a unique\nunderlying\n> > principle, an idea which is nicely made precise by Connes "group koan"\nwhich\n> > says that the exact sequence\n> >\n> > 1 -> Int(A) -> Aut(A) -> Out(A) -> 1 ,\n> >\n> > where A is the (noncommutative) algebra, Aut(A) are its automorphisms\nand\n> > Int(A), Out(A) its inner and outer automorphisms, respectively, is in\ndirect\n> > correspondence to the exact sequence\n> >\n> > 1 -> U -> G -> Diff(M) -> 1 ,\n> >\n> > where now G is the full symmetry group of a given physical theory, U is\nthe\n> > "internal" gauge group and Diff(M) the diffeomorphism group.\n>\n> Hm, I think I understand this. The last sequence basically says\n> that the symmetry group is the semidirect product of\n> diffeomorphisms and gauge transformations. If we work on a\n> nontrivial fiber bundle things get a little more complicated (the\n> sequence does not split), but Diff(M) are still horizontal\n> transformations and U vertical transformations.\n\n> So the upper exact sequence is a noncommutative generalization of\n> this situation. But is this really unification? If A is commutative\n> and thus a function algebra, wouldn\'t G always locally look like a\n> semidirect product?\n\nYes, I guess so. Of course one would expect that fundamentally A is not\ncommutative. Still, the unifying idea is that feeding just the algebra and\nits Dirac operator into the spectral action the action for gravity plus YM\n(plus other stuff) drop out.\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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0405130741.53fded94@posting.google.c om...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> skrev i
> diskussionsgruppsmeddelandet:40a34cfd$1@news.sente x.net...
> Associativity is of course always a nice thing, so it's is natural
> for model-builders to use any associative product around.
I agree. On the other hand, as Eric Forgy likes to point out, given any
non-commutative but associative product a*b we obtain a commutative but
non-associative product by using the (graded, maybe) symmetrization
a x b := a*b + b*a .
> Hm. Are
> there any associative products around except Moyal? I.e., is there
> any deformation of the function algebra C(R^n) which does not
> locally look like the Moyal product?
Don't know, maybe not. Even the Witten open string star is a continuous copy
of Moyal stars, as has been discussed here before.
> I have difficulties to visualize quantization on a noncommutative
> spacetime. In QM we have particles that live in spacetime, and I
> understand sort of how to make this noncommutative. But in QFT the
> spacetime points label the fields. It seems that we try to quantize
> degrees of freedom that are labelled by a noncommutative index set.
> What does that mean?
I am not sure what you are worried about. We have some configuration of the
system and associate an action to it.. Using this action we can try to write
down a path integral or try to compute canonical momenta and do canonical
quantization. In either case, this procedure works no matter how the action
associates a number to a certain configuration. If the action functional
involves Moyal products of functions that coordinatize the configuration of
the system then that's how it is. It does not seem to interfere with the
prescription of quantization to me.
> > So far to me the main insight of the spectral action principle is that
that
> > in terms of the Dirac operator and its spectrum the different fields in
> > nature can naturally be regarded as manifestations of a unique
underlying
> > principle, an idea which is nicely made precise by Connes "group koan"
which
> > says that the exact sequence
> >
> > 1 -> \Int(A) -> Aut(A) -> Out(A) -> 1 ,
> >
> > where A is the (noncommutative) algebra, Aut(A) are its automorphisms
and
> > \Int(A), Out(A) its inner and outer automorphisms, respectively, is in
direct
> > correspondence to the exact sequence
> >
> > 1 -> U -> G -> Diff(M) -> 1 ,
> >
> > where now G is the full symmetry group of a given physical theory, U is
the
> > "internal" gauge group and Diff(M) the diffeomorphism group.
>
> Hm, I think I understand this. The last sequence basically says
> that the symmetry group is the semidirect product of
> diffeomorphisms and gauge transformations. If we work on a
> nontrivial fiber bundle things get a little more complicated (the
> sequence does not split), but Diff(M) are still horizontal
> transformations and U vertical transformations.
> So the upper exact sequence is a noncommutative generalization of
> this situation. But is this really unification? If A is commutative
> and thus a function algebra, wouldn't G always locally look like a
> semidirect product?
Yes, I guess so. Of course one would expect that fundamentally A is not
commutative. Still, the unifying idea is that feeding just the algebra and
its Dirac operator into the spectral action the action for gravity plus YM
(plus other stuff) drop out.
Urs Schreiber
May18-04, 10:27 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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag\nnews:24a23f36.0405140010.3e2a2e0b@pos ting.google.com...\n\n> When reading Bert Schroer\'s latest epos, hep-th/0405105, I learned\n> about two references that might be of interest: hep-th/0305093 and\n> hep-th/0402212. Not unexpectedly, Schroer seems to think that these\n> papers are nonsense, the former more than the latter.\n\nI am not sure how much room there is to question the viability of\nnoncommutative quantum field theories in general, since these seem to be\nrather well established. I currently happen to have the lecture notes\n\nI. Aref\'eva & Belov & Giryavets & Koshelev & Medvedev:\nNoncommutative Field Theories and (Super) String Field Theories\nhep-th/0111208\n\non my desk where in the first section all the standard stuff on diagrams,\netc. for the noncommutative theory is briefly reviewed.\n\nBut note that Bert Schroer has written a response at the String Coffee Table\n\nhttp://golem.ph.utexas.edu/string/archives/000338.html#c001097\n\nwhere he argues that the "interpretation" of the noncommutativity is\n"flawed". But I don\'t quite understand his point so far...\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>"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0405140010.3e2a2e0b@posting.google.c om...
> When reading Bert Schroer's latest epos, http://www.arxiv.org/abs/hep-th/0405105, I learned
> about two references that might be of interest: http://www.arxiv.org/abs/hep-th/0305093 and
> http://www.arxiv.org/abs/hep-th/0402212. Not unexpectedly, Schroer seems to think that these
> papers are nonsense, the former more than the latter.
I am not sure how much room there is to question the viability of
noncommutative quantum field theories in general, since these seem to be
rather well established. I currently happen to have the lecture notes
I. Aref'eva & Belov & Giryavets & Koshelev & Medvedev:
Noncommutative Field Theories and (Super) String Field Theories
http://www.arxiv.org/abs/hep-th/0111208
on my desk where in the first section all the standard stuff on diagrams,
etc. for the noncommutative theory is briefly reviewed.
But note that Bert Schroer has written a response at the String Coffee Table
http://golem.ph.utexas.edu/string/archives/000338.html#c001097
where he argues that the "interpretation" of the noncommutativity is
"flawed". But I don't quite understand his point so far...
Thomas Larsson
May20-04, 11:51 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>Urs Schreiber <Urs.Schreiber@uni-essen.de> skrev i\ndiskussionsgruppsmeddelandet:40a9d3f5\\$1@news. sentex.net...\n\nI wrote:\n> > I learned\n> > about two references that might be of interest: hep-th/0305093 and\n> > hep-th/0402212. Not unexpectedly, Schroer seems to think that these\n> > papers are nonsense, the former more than the latter.\n>\n> I haven\'t read all that, but I would like to know what is concidered\n> nonsense and for which reasons.\n>\n\nI only claimed that it is not unexpected for Bert Schroer to think\nthat it is nonsense - that\'s how I interpret the text around his\npage 6, anyway. Personally, I have no opinion and I don\'t\nunderstand what he says well enough to follow his argument, and it\nis not sufficiently interesting to me at this time to motivate a\nmajor effort.\n\nThere is one simple thing about hep-th/0402212 that I find very\nstrange, however. The same equation comes up in Schroer\'s article,\nso it has nothing to do with his argument. In the first equation\n(I.1), the authors postulate that the spacetime coordinates satisfy\n\n[x_u, x_v] = i theta_uv,\n\nwhere theta_uv is an antisymmetric constant matrix. From here it\nfollows that the functions should be multiplied using Moyal\nproducts, etc.\n\nHowever, the equation above posits that spacetime is a symplectic\nmanifold. In particular this would mean that the spatial directions\nare inequivalent. Namely, if you introduce Darboux coordinates, the\ncoordinates x_u are symplectically paired. I could imagine that we\nhave a distinguished direction in spacetime, namely time, although\nI find that unlikely, too. But this strong kind of non-commutativity\nseems to imply anisotropy in *space*, which must be ruled out\nexperimentally to almost arbitrary precision. Or is there some way\nto avoid this conclusion?\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> skrev i
diskussionsgruppsmeddelandet:40a9d3f5$1@news.sente x.net...
I wrote:
> > I learned
> > about two references that might be of interest: http://www.arxiv.org/abs/hep-th/0305093 and
> > http://www.arxiv.org/abs/hep-th/0402212. Not unexpectedly, Schroer seems to think that these
> > papers are nonsense, the former more than the latter.
>
> I haven't read all that, but I would like to know what is concidered
> nonsense and for which reasons.
>
I only claimed that it is not unexpected for Bert Schroer to think
that it is nonsense - that's how I interpret the text around his
page 6, anyway. Personally, I have no opinion and I don't
understand what he says well enough to follow his argument, and it
is not sufficiently interesting to me at this time to motivate a
major effort.
There is one simple thing about http://www.arxiv.org/abs/hep-th/0402212 that I find very
strange, however. The same equation comes up in Schroer's article,
so it has nothing to do with his argument. In the first equation
(I.1), the authors postulate that the spacetime coordinates satisfy
[x_u, x_v] = i \theta_uv,
where \theta_uv is an antisymmetric constant matrix. From here it
follows that the functions should be multiplied using Moyal
products, etc.
However, the equation above posits that spacetime is a symplectic
manifold. In particular this would mean that the spatial directions
are inequivalent. Namely, if you introduce Darboux coordinates, the
coordinates x_u are symplectically paired. I could imagine that we
have a distinguished direction in spacetime, namely time, although
I find that unlikely, too. But this strong kind of non-commutativity
seems to imply anisotropy in *space*, which must be ruled out
experimentally to almost arbitrary precision. Or is there some way
to avoid this conclusion?
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