View Full Version : the phi^4 theory in 2-dimensional spacetime
=?gbk?q?=C2=BD=C8=BB?=
Apr7-04, 09:22 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>Hi all,\n\nRecently I\'ve read a old thread in spr("unsolved problems in QED"), There Mr.\nJohn Baez mentioned that the phi^4 theory in 2-dimensional spacetime can be\nconstructed rigorously. Is there any reference about this. I think this\nshould be very interesting and want to know more about it. Thank you in\nadvance\n\n--\nBest Regards,\nLR\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>Hi all,
Recently I've read a old thread in spr("unsolved problems in QED"), There Mr.
John Baez mentioned that the \phi^4 theory in 2-dimensional spacetime can be
constructed rigorously. Is there any reference about this. I think this
should be very interesting and want to know more about it. Thank you in
advance
--
Best Regards,
LR
Arnold Neumaier
Apr8-04, 06:33 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>?? wrote:\n> Hi all,\n>=20\n> Recently I\'ve read a old thread in spr("unsolved problems in QED"), The=\nre Mr.=20\n> John Baez mentioned that the phi^4 theory in 2-dimensional spacetime ca=\nn be=20\n> constructed rigorously. Is there any reference about this. I think this=\n=20\n> should be very interesting and want to know more about it. Thank you in=\n=20\n> advance\n>=20\n\nThere is a thick book by Glimm and Jaffe about it, with all the\nbackground needed and all the rigor you may ever want.\nNot bedtime reading, though.\n\nArnold Neumaier\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>?? wrote:
> Hi all,
>=20
> Recently I've read a old thread in spr("unsolved problems in QED"), The=
re Mr.=20
> John Baez mentioned that the \phi^4 theory in 2-dimensional spacetime ca=
n be=20
> constructed rigorously. Is there any reference about this. I think this=
=20
> should be very interesting and want to know more about it. Thank you in=
=20
> advance
>=20
There is a thick book by Glimm and Jaffe about it, with all the
background needed and all the rigor you may ever want.
Not bedtime reading, though.
Arnold Neumaier
<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>Hi,\n\n\n> There is a thick book by Glimm and Jaffe about it, with all the\n> background needed and all the rigor you may ever want.\n> Not bedtime reading, though.\n\nSorry for the late reply, I do tried to understand their book, but I\njust cannot follow it, to me they just proved a lot of theorems that I\ndon\'t know what they are good for. I found their interest usually\nconnect to "renomalize" and "cutoff", which I cannot understand. Can\nanyone scratch a genenal process of this kind of "construct a field\ntheory" at a layman level so I can get a big picture? And I\'m still\nnot clear the difference between "solve" and "construct", are they\njust the two sides of a story? I mean if I can construct a field\ntheory, found its Lagrangian, can I just invert the process to solve\nthe theory from the Lagrangian? I guess it may have some difficult\nbecause no one ever mentioned this, at least for the papers and books\nI have read, but I cannot see why, Can anyone give a explanation?\nthanks.\n\n--\nBest Regards,\nLR\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>Hi,
> There is a thick book by Glimm and Jaffe about it, with all the
> background needed and all the rigor you may ever want.
> Not bedtime reading, though.
Sorry for the late reply, I do tried to understand their book, but I
just cannot follow it, to me they just proved a lot of theorems that I
don't know what they are good for. I found their interest usually
connect to "renomalize" and "cutoff", which I cannot understand. Can
anyone scratch a genenal process of this kind of "construct a field
theory" at a layman level so I can get a big picture? And I'm still
not clear the difference between "solve" and "construct", are they
just the two sides of a story? I mean if I can construct a field
theory, found its Lagrangian, can I just invert the process to solve
the theory from the Lagrangian? I guess it may have some difficult
because no one ever mentioned this, at least for the papers and books
I have read, but I cannot see why, Can anyone give a explanation?
thanks.
--
Best Regards,
LR
Arnold Neumaier
Apr28-04, 06:12 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>\nL.R. wrote:\n>\n>>There is a thick book by Glimm and Jaffe about it, with all the\n>>background needed and all the rigor you may ever want.\n>>Not bedtime reading, though.\n>\n>\n> Sorry for the late reply, I do tried to understand their book, but I\n\n\'do tried\' should be \'did try\' or \'tried\'\n\n> just cannot follow it, to me they just proved a lot of theorems that I\n> don\'t know what they are good for. I found their interest usually\n> connect to "renomalize" and "cutoff", which I cannot understand. Can\n> anyone scratch a general process of this kind of "construct a field\n> theory" at a layman level so I can get a big picture?\n\nIt _is_ difficult to understand. Maybe you\'d read the book by\nStreater and Wightman ("PCT, spin, statistics and all that") first.\n\nBasically, QFT assumes the existence of interacting (operator\ndistribution valued) fields Phi(x) with certain properties, which\nimply the existence of distributions\nW(x_1,...,x_n)=<0|Phi(x_1)...Phi(x_ n)|0>.\nBut the right hand side makes no rigorous sense in traditional QFT\nas found in most text books, except for free fields. Axiomatic QFT\ntherefore tries to construct the W\'s directly such that they have the\nproperties needed to get an S-matrix, whose perturbative expansion\ncan be compared with the nonrigorous mainstream computations.\nThis can be done successfully for many 2D theories and for some 3D\ntheories, but not, so far, in the physically relevant case of 4D.\n\n\n> And I\'m still\n> not clear the difference between "solve" and "construct", are they\n> just the two sides of a story?\n\nconstruct = show its existence as a mathematically well-defined object\n\nsolve = compute numerical properties (often approximately, occasionally\n- for simple problems - in closed analytic form)\n\n> I mean if I can construct a field\n> theory, found its Lagrangian, can I just invert the process to solve\n> the theory from the Lagrangian?\n\nI don\'t understand your question.\nGlimm and Jaffe\'s appoach is not via the Lagrangian.\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>L.R. wrote:
>
>>There is a thick book by Glimm and Jaffe about it, with all the
>>background needed and all the rigor you may ever want.
>>Not bedtime reading, though.
>
>
> Sorry for the late reply, I do tried to understand their book, but I
'do tried' should be 'did try' or 'tried'
> just cannot follow it, to me they just proved a lot of theorems that I
> don't know what they are good for. I found their interest usually
> connect to "renomalize" and "cutoff", which I cannot understand. Can
> anyone scratch a general process of this kind of "construct a field
> theory" at a layman level so I can get a big picture?
It _is_ difficult to understand. Maybe you'd read the book by
Streater and Wightman ("PCT, spin, statistics and all that") first.
Basically, QFT assumes the existence of interacting (operator
distribution valued) fields \Phi(x) with certain properties, which
imply the existence of distributions
W(x_1,...,x_n)=<0|\Phi(x_1)...\Phi(x_n)|0>.
But the right hand side makes no rigorous sense in traditional QFT
as found in most text books, except for free fields. Axiomatic QFT
therefore tries to construct the W's directly such that they have the
properties needed to get an S-matrix, whose perturbative expansion
can be compared with the nonrigorous mainstream computations.
This can be done successfully for many 2D theories and for some 3D
theories, but not, so far, in the physically relevant case of 4D.
> And I'm still
> not clear the difference between "solve" and "construct", are they
> just the two sides of a story?
construct = show its existence as a mathematically well-defined object
solve = compute numerical properties (often approximately, occasionally
- for simple problems - in closed analytic form)
> I mean if I can construct a field
> theory, found its Lagrangian, can I just invert the process to solve
> the theory from the Lagrangian?
I don't understand your question.
Glimm and Jaffe's appoach is not via the Lagrangian.
Arnold Neumaier
<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\nHi,\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<408F79FE.6080108@univie.ac.at>...\n> >\n> > Sorry for the late reply, I do tried to understand their book, but I\n>\n> \'do tried\' should be \'did try\' or \'tried\'\n\nSeems I need to improve my English :)\n\n> Basically, QFT assumes the existence of interacting (operator\n> distribution valued) fields Phi(x) with certain properties, which\n> imply the existence of distributions\n> W(x_1,...,x_n)=<0|Phi(x_1)...Phi(x_n)|0>.\n> But the right hand side makes no rigorous sense in traditional QFT\n> as found in most text books, except for free fields. Axiomatic QFT\n> therefore tries to construct the W\'s directly such that they have the\n> properties needed to get an S-matrix, whose perturbative expansion\n> can be compared with the nonrigorous mainstream computations.\n> This can be done successfully for many 2D theories and for some 3D\n> theories, but not, so far, in the physically relevant case of 4D.\n\nSo by "construct the W\'s" you mean just prove they can exist or give a\nsome kind of (analytic?) form of W\'s?\n\n> construct = show its existence as a mathematically well-defined object\n>\n> solve = compute numerical properties (often approximately, occasionally\n> - for simple problems - in closed analytic form)\n>\n\nSo I think the idea is: construct means prove the existence of the\ntheory but don\'t care about finding all the properties of it, so even\nif we constructed a theory, we cannot say that we solved it, am I\nright?\n\n> I don\'t understand your question.\n> Glimm and Jaffe\'s appoach is not via the Lagrangian.\n\nSorry, I thought the construction process should eventually give all\nthe infomation about the theory, like spectrum, S-matrix, etc. Now I\nthink maybe I was wrong.\n\n--\nBest Regards,\nLR\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>Hi,
Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<408F79FE.6080108@univie.ac.at>...
> >
> > Sorry for the late reply, I do tried to understand their book, but I
>
> 'do tried' should be 'did try' or 'tried'
Seems I need to improve my English :)
> Basically, QFT assumes the existence of interacting (operator
> distribution valued) fields \Phi(x) with certain properties, which
> imply the existence of distributions
> W(x_1,...,x_n)=<0|\Phi(x_1)...\Phi(x_n)|0>.
> But the right hand side makes no rigorous sense in traditional QFT
> as found in most text books, except for free fields. Axiomatic QFT
> therefore tries to construct the W's directly such that they have the
> properties needed to get an S-matrix, whose perturbative expansion
> can be compared with the nonrigorous mainstream computations.
> This can be done successfully for many 2D theories and for some 3D
> theories, but not, so far, in the physically relevant case of 4D.
So by "construct the W's" you mean just prove they can exist or give a
some kind of (analytic?) form of W's?
> construct = show its existence as a mathematically well-defined object
>
> solve = compute numerical properties (often approximately, occasionally
> - for simple problems - in closed analytic form)
>
So I think the idea is: construct means prove the existence of the
theory but don't care about finding all the properties of it, so even
if we constructed a theory, we cannot say that we solved it, am I
right?
> I don't understand your question.
> Glimm and Jaffe's appoach is not via the Lagrangian.
Sorry, I thought the construction process should eventually give all
the infomation about the theory, like spectrum, S-matrix, etc. Now I
think maybe I was wrong.
--
Best Regards,
LR
Arnold Neumaier
Apr30-04, 03:01 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>L.R. wrote:\n\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<408F79FE.6080108@univie.ac.at>...\n>\n>>Basi cally, QFT assumes the existence of interacting (operator\n>>distribution valued) fields Phi(x) with certain properties, which\n>>imply the existence of distributions\n>> W(x_1,...,x_n)=<0|Phi(x_1)...Phi(x_n)|0>.\n>>But the right hand side makes no rigorous sense in traditional QFT\n>>as found in most text books, except for free fields. Axiomatic QFT\n>>therefore tries to construct the W\'s directly such that they have the\n>>properties needed to get an S-matrix, whose perturbative expansion\n>>can be compared with the nonrigorous mainstream computations.\n>>This can be done successfully for many 2D theories and for some 3D\n>>theories, but not, so far, in the physically relevant case of 4D.\n>\n> So by "construct the W\'s" you mean just prove they can exist or give a\n> some kind of (analytic?) form of W\'s?\n\n.... prove they do exist. Usually this is done by giving a construction\nas a sort of limit, and proving that the limit is well-defined.\n\n\n>>construct = show its existence as a mathematically well-defined object\n>>\n>>solve = compute numerical properties (often approximately, occasionally\n>>- for simple problems - in closed analytic form)\n>>\n\n> So I think the idea is: construct means prove the existence of the\n> theory but don\'t care about finding all the properties of it, so even\n> if we constructed a theory, we cannot say that we solved it, am I\n> right?\n\nYes, though often one finds in the process of construction some of the\nproperties.\n\nTo compare it to something simpler: In mathematics one constructs the\nRiemann integral of a continuous function over a finite interval by\nsome kind of limit, and later the solution of an initial value problem\nordinary differential equations by using this and a fixed point theorem.\nThis shows that each (nice enough) initial value problem is uniquely\nsolvable. But it tells very little of its properties, and in practice\nno one uses this construction to calculate anything. But it is important\nas a mathematical tool since it shows that calculus is logically\nconsistent.\n\nSuch a logical consistence proof of 4D interacting QFT is presently\nstill missing. Since logical consistency of a theory is important,\nthe first person who finds such a proof will become famous - it means\ninventing new conceptual tools that can handle this currently\nintractable problem.\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>L.R. wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<408F79FE.6080108@univie.ac.at>...
>
>>Basically, QFT assumes the existence of interacting (operator
>>distribution valued) fields \Phi(x) with certain properties, which
>>imply the existence of distributions
>> W(x_1,...,x_n)=<0|\Phi(x_1)...\Phi(x_n)|0>.
>>But the right hand side makes no rigorous sense in traditional QFT
>>as found in most text books, except for free fields. Axiomatic QFT
>>therefore tries to construct the W's directly such that they have the
>>properties needed to get an S-matrix, whose perturbative expansion
>>can be compared with the nonrigorous mainstream computations.
>>This can be done successfully for many 2D theories and for some 3D
>>theories, but not, so far, in the physically relevant case of 4D.
>
> So by "construct the W's" you mean just prove they can exist or give a
> some kind of (analytic?) form of W's?
.... prove they do exist. Usually this is done by giving a construction
as a sort of limit, and proving that the limit is well-defined.
>>construct = show its existence as a mathematically well-defined object
>>
>>solve = compute numerical properties (often approximately, occasionally
>>- for simple problems - in closed analytic form)
>>
> So I think the idea is: construct means prove the existence of the
> theory but don't care about finding all the properties of it, so even
> if we constructed a theory, we cannot say that we solved it, am I
> right?
Yes, though often one finds in the process of construction some of the
properties.
To compare it to something simpler: In mathematics one constructs the
Riemann integral of a continuous function over a finite interval by
some kind of limit, and later the solution of an initial value problem
ordinary differential equations by using this and a fixed point theorem.
This shows that each (nice enough) initial value problem is uniquely
solvable. But it tells very little of its properties, and in practice
no one uses this construction to calculate anything. But it is important
as a mathematical tool since it shows that calculus is logically
consistent.
Such a logical consistence proof of 4D interacting QFT is presently
still missing. Since logical consistency of a theory is important,
the first person who finds such a proof will become famous - it means
inventing new conceptual tools that can handle this currently
intractable problem.
Arnold Neumaier
Gentil Correa
Apr30-04, 11:30 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>hephooey@fastmail.fm (L.R.) wrote in message news:<4d470e8b.0404262100.1994f428@posting.google. com>...\n> Hi,\n>\n>\n> > There is a thick book by Glimm and Jaffe about it, with all the\n> > background needed and all the rigor you may ever want.\n> > Not bedtime reading, though.\n>\n> Sorry for the late reply, I do tried to understand their book, but I\n> just cannot follow it, to me they just proved a lot of theorems that I\n> don\'t know what they are good for. I found their interest usually\n> connect to "renomalize" and "cutoff", which I cannot understand. Can\n> anyone scratch a genenal process of this kind of "construct a field\n> theory" at a layman level so I can get a big picture? And I\'m still\n> not clear the difference between "solve" and "construct", are they\n> just the two sides of a story? I mean if I can construct a field\n> theory, found its Lagrangian, can I just invert the process to solve\n> the theory from the Lagrangian? I guess it may have some difficult\n> because no one ever mentioned this, at least for the papers and books\n> I have read, but I cannot see why, Can anyone give a explanation?\n> thanks.\n\n\nWell, Glimm and Jaffe\'s book is the end of the road. Begin from the\nbeginning,\nas Lewis Carroll used to recommend. Maybe you should first understand,\nin\na lighter context, why is it that there are many soluble models of QFT\nin 2-dimensions (one of them being time). These things were learnt\ngradually. I believe the first soluble model was Schwinger\'s, then\nThirring\'s, and so on.\nPerhaps you should look at the book by Abdalla, Abdalla and Rothe,\nwhose title\nis something like "Quantum Field Theory in 2-Dimensions". This is a\nhuge tome, but begins rather lightly, and explains everything. I am\nassuming you are able to understand a standard text of QFT, like\nBjorken and Drell and know, for instance, what Feynman diagrams and\nperturbative renormalizarion are, at the level of a graduate student.\nIf this is the case, you are ready for the Abdallas book. After that\nyou could think of more mathematical things, like existence proofs and\nall that. But, perhaps, you\'ll then be interested in more recent\nthings, like those at the beautiful book by Haag, "Local Quantum\nPhysics".\n\n\nBest wishes,\n\nGentil\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>hephooey@fastmail.fm (L.R.) wrote in message news:<4d470e8b.0404262100.1994f428@posting.google.com>...
> Hi,
>
>
> > There is a thick book by Glimm and Jaffe about it, with all the
> > background needed and all the rigor you may ever want.
> > Not bedtime reading, though.
>
> Sorry for the late reply, I do tried to understand their book, but I
> just cannot follow it, to me they just proved a lot of theorems that I
> don't know what they are good for. I found their interest usually
> connect to "renomalize" and "cutoff", which I cannot understand. Can
> anyone scratch a genenal process of this kind of "construct a field
> theory" at a layman level so I can get a big picture? And I'm still
> not clear the difference between "solve" and "construct", are they
> just the two sides of a story? I mean if I can construct a field
> theory, found its Lagrangian, can I just invert the process to solve
> the theory from the Lagrangian? I guess it may have some difficult
> because no one ever mentioned this, at least for the papers and books
> I have read, but I cannot see why, Can anyone give a explanation?
> thanks.
Well, Glimm and Jaffe's book is the end of the road. Begin from the
beginning,
as Lewis Carroll used to recommend. Maybe you should first understand,
in
a lighter context, why is it that there are many soluble models of QFT
in 2-dimensions (one of them being time). These things were learnt
gradually. I believe the first soluble model was Schwinger's, then
Thirring's, and so on.
Perhaps you should look at the book by Abdalla, Abdalla and Rothe,
whose title
is something like "Quantum Field Theory in 2-Dimensions". This is a
huge tome, but begins rather lightly, and explains everything. I am
assuming you are able to understand a standard text of QFT, like
Bjorken and Drell and know, for instance, what Feynman diagrams and
perturbative renormalizarion are, at the level of a graduate student.
If this is the case, you are ready for the Abdallas book. After that
you could think of more mathematical things, like existence proofs and
all that. But, perhaps, you'll then be interested in more recent
things, like those at the beautiful book by Haag, "Local Quantum
Physics".
Best wishes,
Gentil
Arnold Neumaier
May1-04, 08:52 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>Gentil Correa wrote:\n\n> Well, Glimm and Jaffe\'s book is the end of the road.\n\n[giving rigorous existence proofs for Phi^4 QFT in 2D]\n\n> Begin from the beginning,\n> as Lewis Carroll used to recommend. Maybe you should first understand,\n> in\n> a lighter context, why is it that there are many soluble models of QFT\n> in 2-dimensions (one of them being time). These things were learnt\n> gradually. I believe the first soluble model was Schwinger\'s, then\n> Thirring\'s, and so on.\n> Perhaps you should look at the book by Abdalla, Abdalla and Rothe,\n> whose title\n> is something like "Quantum Field Theory in 2-Dimensions". This is a\n> huge tome, but begins rather lightly, and explains everything.\n\nElcio Abdalla, M. Christina Abdalla, Klaus D. Rothe\nNon-Perturbative Methods in 2 Dimensional Quantum Field Theory\nWorld Scientific, 1991\nhttp://www.wspc.com/books/physics/4678.html\n\n\nNote that \'solvable\' means in this context \'being able to\nfind a closed analytic expression for all S-matrix elements\'.\nThese solvable models are to QFT what the hydrogen atom is to\nquantum mechanics. The helium atom is no longer \'solvable\' in the\npresent sense, though of course very accurate approximate calculations\nare possible.\n\n\nUnfortunately, solvable models appear to be restricted to 2 dimensions.\nThe deeper reason for the observation that dimension d=2 is special\nseems to be that in 2D the line cone is just a pair of lines,\nand by a change of variables (light cone quantization), one can\ndisentangle things nicely.\n\nThis is no longer possible in higher dimensions, though people still\nuse light cone quantization as an approximate technique, e.g., to get\nnumerical results from QCD.\n\nThus, while 2D solvable models pave the way to get some rigorous\nunderstanding of the concepts, they are no substitute for the\nfunctional analytic techniques needed to handle the non-solvable\nmodels such as Phi^4 theory. If Glimm and Jaffe\'s functional analysis\nis too demanding, Volume 3 of Thirring\'s Course in Mathematical Physics\n(which only deals with nonrelativistic QM but in a reasonably\nrigorous way) might be a good preparation.\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>Gentil Correa wrote:
> Well, Glimm and Jaffe's book is the end of the road.
[giving rigorous existence proofs for \Phi^4 QFT in 2D]
> Begin from the beginning,
> as Lewis Carroll used to recommend. Maybe you should first understand,
> in
> a lighter context, why is it that there are many soluble models of QFT
> in 2-dimensions (one of them being time). These things were learnt
> gradually. I believe the first soluble model was Schwinger's, then
> Thirring's, and so on.
> Perhaps you should look at the book by Abdalla, Abdalla and Rothe,
> whose title
> is something like "Quantum Field Theory in 2-Dimensions". This is a
> huge tome, but begins rather lightly, and explains everything.
Elcio Abdalla, M. Christina Abdalla, Klaus D. Rothe
Non-Perturbative Methods in 2 Dimensional Quantum Field Theory
World Scientific, 1991
http://www.wspc.com/books/http://www.arxiv.org/abs/physics/4678.html
Note that 'solvable' means in this context 'being able to
find a closed analytic expression for all S-matrix elements'.
These solvable models are to QFT what the hydrogen atom is to
quantum mechanics. The helium atom is no longer 'solvable' in the
present sense, though of course very accurate approximate calculations
are possible.
Unfortunately, solvable models appear to be restricted to 2 dimensions.
The deeper reason for the observation that dimension d=2 is special
seems to be that in 2D the line cone is just a pair of lines,
and by a change of variables (light cone quantization), one can
disentangle things nicely.
This is no longer possible in higher dimensions, though people still
use light cone quantization as an approximate technique, e.g., to get
numerical results from QCD.
Thus, while 2D solvable models pave the way to get some rigorous
understanding of the concepts, they are no substitute for the
functional analytic techniques needed to handle the non-solvable
models such as \Phi^4 theory. If Glimm and Jaffe's functional analysis
is too demanding, Volume 3 of Thirring's Course in Mathematical Physics
(which only deals with nonrelativistic QM but in a reasonably
rigorous way) might be a good preparation.
Arnold Neumaier
<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>Hi,\n\n> Unfortunately, solvable models appear to be restricted to 2 dimensions.\n> The deeper reason for the observation that dimension d=2 is special\n> seems to be that in 2D the line cone is just a pair of lines,\n> and by a change of variables (light cone quantization), one can\n> disentangle things nicely.\n>\n> This is no longer possible in higher dimensions, though people still\n> use light cone quantization as an approximate technique, e.g., to get\n> numerical results from QCD.\n>\n> Thus, while 2D solvable models pave the way to get some rigorous\n> understanding of the concepts, they are no substitute for the\n> functional analytic techniques needed to handle the non-solvable\n> models such as Phi^4 theory. If Glimm and Jaffe\'s functional analysis\n> is too demanding, Volume 3 of Thirring\'s Course in Mathematical Physics\n> (which only deals with nonrelativistic QM but in a reasonably\n> rigorous way) might be a good preparation.\n>\n\nOne of my colleagues told me that, since QFT have infinite degrees of\nfreedom, we need infinite kinds of symmetry to solve it, and in 2D\nthere happens to have that kind of symmetry like conformal symmetry.\nso if we are lucky, we can use conformal invariant to solve the\nproblem. But in higher dimension the comformal symmetry group is\nfinite dimension, so we cannot apply it in the same way as 2D.\n\nI think this argument is OK, but I think there should be other\ninfinity dimensional symmetry group we can find in higher dimension.\nAny comments?\n\n--\nBest Regards,\nLR\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>Hi,
> Unfortunately, solvable models appear to be restricted to 2 dimensions.
> The deeper reason for the observation that dimension d=2 is special
> seems to be that in 2D the line cone is just a pair of lines,
> and by a change of variables (light cone quantization), one can
> disentangle things nicely.
>
> This is no longer possible in higher dimensions, though people still
> use light cone quantization as an approximate technique, e.g., to get
> numerical results from QCD.
>
> Thus, while 2D solvable models pave the way to get some rigorous
> understanding of the concepts, they are no substitute for the
> functional analytic techniques needed to handle the non-solvable
> models such as \Phi^4 theory. If Glimm and Jaffe's functional analysis
> is too demanding, Volume 3 of Thirring's Course in Mathematical Physics
> (which only deals with nonrelativistic QM but in a reasonably
> rigorous way) might be a good preparation.
>
One of my colleagues told me that, since QFT have infinite degrees of
freedom, we need infinite kinds of symmetry to solve it, and in 2D
there happens to have that kind of symmetry like conformal symmetry.
so if we are lucky, we can use conformal invariant to solve the
problem. But in higher dimension the comformal symmetry group is
finite dimension, so we cannot apply it in the same way as 2D.
I think this argument is OK, but I think there should be other
infinity dimensional symmetry group we can find in higher dimension.
Any comments?
--
Best Regards,
LR
Igor Khavkine
May3-04, 02:00 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>NNTP-Posting-Host: perimeterinstitute.ca\nDate: 3 May 2004 05:52:52 -0400\nX-Trace: news.sentex.net 1083577972 perimeterinstitute.ca (3 May 2004 05:52:52 -0400)\nLines: 37\nPath: news.easynews.com!core-easynews!newsfeed1.easynews.com!easynews.com!easyn ews!newshosting.com!nx02.iad01.newshosting.com!meg anewsservers.com!feeder2.on.meganewsservers.com!ne ws.uwaterloo.ca!news.sentex.net!not-for-mail\nXref: core-easynews sci.physics.research:55863\nX-Received-Date: Mon, 03 May 2004 02:59:36 MST (news.easynews.com)\n\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4090FEF4.7060403@univie.ac.at>...\n> L.R. wrote:\n>\n> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<408F79FE.6080108@univie.ac.at>...\n\n[...stuff about Wightman functions...]\n\n> > So by "construct the W\'s" you mean just prove they can exist or give a\n> > some kind of (analytic?) form of W\'s?\n>\n> ... prove they do exist. Usually this is done by giving a construction\n> as a sort of limit, and proving that the limit is well-defined.\n\n> > So I think the idea is: construct means prove the existence of the\n> > theory but don\'t care about finding all the properties of it, so even\n> > if we constructed a theory, we cannot say that we solved it, am I\n> > right?\n>\n> Yes, though often one finds in the process of construction some of the\n> properties.\n\nI\'m not familiar with the existence results of the phi^4 theory in 2D.\nBut I am curious, does the proof introduce any new information about the\ntheory that wasn\'t know from the perturbative treatment? Obviously,\nwe didn\'t have existence in the perturbative treatment, but I\'m wondering\nabout something non-trivial.\n\nAlso, I presume that, as you mentioned, that the proof is done as some\nsort of well defined limit. Is there any possibility to use this limiting\nprocedure to do calculations with this theory and compare them to usual\nperturbative calculations in terms of Feynman diagrams? If both give\nsimilar answers, which one is more efficient?\n\nThanks.\n\nIgor\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>NNTP-Posting-Host: perimeterinstitute.ca
Date: 3 May 2004 05:52:52 -0400
X-Trace: news.sentex.net 1083577972 perimeterinstitute.ca (3 May 2004 05:52:52 -0400)
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Xref: core-easynews sci.physics.research:55863
X-Received-Date: Mon, 03 May 2004 02:59:36 MST (news.easynews.com)
Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4090FEF4.7060403@univie.ac.at>...
> L.R. wrote:
>
> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<408F79FE.6080108@univie.ac.at>...
[...stuff about Wightman functions...]
> > So by "construct the W's" you mean just prove they can exist or give a
> > some kind of (analytic?) form of W's?
>
> ... prove they do exist. Usually this is done by giving a construction
> as a sort of limit, and proving that the limit is well-defined.
> > So I think the idea is: construct means prove the existence of the
> > theory but don't care about finding all the properties of it, so even
> > if we constructed a theory, we cannot say that we solved it, am I
> > right?
>
> Yes, though often one finds in the process of construction some of the
> properties.
I'm not familiar with the existence results of the \phi^4 theory in 2D.
But I am curious, does the proof introduce any new information about the
theory that wasn't know from the perturbative treatment? Obviously,
we didn't have existence in the perturbative treatment, but I'm wondering
about something non-trivial.
Also, I presume that, as you mentioned, that the proof is done as some
sort of well defined limit. Is there any possibility to use this limiting
procedure to do calculations with this theory and compare them to usual
perturbative calculations in terms of Feynman diagrams? If both give
similar answers, which one is more efficient?
Thanks.
Igor
Thomas Larsson
May3-04, 06:26 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n> One of my colleagues told me that, since QFT have infinite degrees of\n> freedom, we need infinite kinds of symmetry to solve it, and in 2D\n> there happens to have that kind of symmetry like conformal symmetry.\n> so if we are lucky, we can use conformal invariant to solve the\n> problem. But in higher dimension the comformal symmetry group is\n> finite dimension, so we cannot apply it in the same way as 2D.\n>\n> I think this argument is OK, but I think there should be other\n> infinity dimensional symmetry group we can find in higher dimension.\n> Any comments?\n>\n\nThe obvious infinite dimensional symmetry groups that do appear in\n4D physics are the groups of diffeomorphisms and gauge transformations,\nwhich are relevant to gravity and Yang-Mills theory, respectively.\n\nHowever, it is not really proper conformal symmetry that is of\nphysical interest in 2D, but rather its central extension known as\nthe Virasoro algebra (more precisely, it is Vir+Vir). So one must\nfirst find the analogous extensions of the diffeo and gauge algebras\nin higher dimensions, then work out their representation theory, and\nfinally convince people that this is of physical interest. It turns\nout that the last task is most difficult.\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>One of my colleagues told me that, since QFT have infinite degrees of
> freedom, we need infinite kinds of symmetry to solve it, and in 2D
> there happens to have that kind of symmetry like conformal symmetry.
> so if we are lucky, we can use conformal invariant to solve the
> problem. But in higher dimension the comformal symmetry group is
> finite dimension, so we cannot apply it in the same way as 2D.
>
> I think this argument is OK, but I think there should be other
> infinity dimensional symmetry group we can find in higher dimension.
> Any comments?
>
The obvious infinite dimensional symmetry groups that do appear in
4D physics are the groups of diffeomorphisms and gauge transformations,
which are relevant to gravity and Yang-Mills theory, respectively.
However, it is not really proper conformal symmetry that is of
physical interest in 2D, but rather its central extension known as
the Virasoro algebra (more precisely, it is Vir+Vir). So one must
first find the analogous extensions of the diffeo and gauge algebras
in higher dimensions, then work out their representation theory, and
finally convince people that this is of physical interest. It turns
out that the last task is most difficult.
Arnold Neumaier
May3-04, 06:48 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>Igor Khavkine wrote:\n\n> I\'m not familiar with the existence results of the phi^4 theory in 2D.\n> But I am curious, does the proof introduce any new information about the\n> theory that wasn\'t know from the perturbative treatment? Obviously,\n> we didn\'t have existence in the perturbative treatment, but I\'m wondering\n> about something non-trivial.\n\nYou obtain certain correlation inequalities that tell you how quickly\ncertain matrix elements decay. This cannot be obtained by perturbation\ntheory. But of course one can compute it approximately by perturbation\ntheory in a few instances and then look at the plot generated and say,\nof course, there is such and such a decay.\n\n> Also, I presume that, as you mentioned, that the proof is done as some\n> sort of well defined limit. Is there any possibility to use this limiting\n> procedure to do calculations with this theory and compare them to usual\n> perturbative calculations in terms of Feynman diagrams? If both give\n> similar answers, which one is more efficient?\n\nApproximate methods are almost always more efficient.\nYou can see this from the way integrals are calculated in\nnumerical analysis. No one uses the \'constructive proof\' by\nRiemann sums. But for logical coherence of a theory, the\nconstructive approach is important.\n\nTo prove that a long, complicated expression in a single variable is\nmonotone may be quite hard and exceed the capacity of a typical\nmathematician or phycisist, but to evaluate it at a few hundred points\nand look at the plot generated is easy. If you are satisfied with the\nlatter, never try to understand mathematical physics - it will be\na waste of your time.\n\nBut if you want to have physics in general look like classical\nHamiltonian mechanics - a beautiful piece of mathematically rich\nand powerful theory, then you should not be satisfied with the way\ncurrent QFT is done, and keep looking for a better, more solid,\nfoundation.\n\n\nArnold Neumaier\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>Igor Khavkine wrote:
> I'm not familiar with the existence results of the \phi^4 theory in 2D.
> But I am curious, does the proof introduce any new information about the
> theory that wasn't know from the perturbative treatment? Obviously,
> we didn't have existence in the perturbative treatment, but I'm wondering
> about something non-trivial.
You obtain certain correlation inequalities that tell you how quickly
certain matrix elements decay. This cannot be obtained by perturbation
theory. But of course one can compute it approximately by perturbation
theory in a few instances and then look at the plot generated and say,
of course, there is such and such a decay.
> Also, I presume that, as you mentioned, that the proof is done as some
> sort of well defined limit. Is there any possibility to use this limiting
> procedure to do calculations with this theory and compare them to usual
> perturbative calculations in terms of Feynman diagrams? If both give
> similar answers, which one is more efficient?
Approximate methods are almost always more efficient.
You can see this from the way integrals are calculated in
numerical analysis. No one uses the 'constructive proof' by
Riemann sums. But for logical coherence of a theory, the
constructive approach is important.
To prove that a long, complicated expression in a single variable is
monotone may be quite hard and exceed the capacity of a typical
mathematician or phycisist, but to evaluate it at a few hundred points
and look at the plot generated is easy. If you are satisfied with the
latter, never try to understand mathematical physics - it will be
a waste of your time.
But if you want to have physics in general look like classical
Hamiltonian mechanics - a beautiful piece of mathematically rich
and powerful theory, then you should not be satisfied with the way
current QFT is done, and keep looking for a better, more solid,
foundation.
Arnold Neumaier
Igor Khavkine
May6-04, 08:52 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\n\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<c76i7k\\$bul\\$1@lfa222122.richmond.edu>...\ n> Igor Khavkine wrote:\n\n> > Also, I presume that, as you mentioned, that the proof is done as some\n> > sort of well defined limit. Is there any possibility to use this limiting\n> > procedure to do calculations with this theory and compare them to usual\n> > perturbative calculations in terms of Feynman diagrams? If both give\n> > similar answers, which one is more efficient?\n>\n> Approximate methods are almost always more efficient.\n> You can see this from the way integrals are calculated in\n> numerical analysis. No one uses the \'constructive proof\' by\n> Riemann sums. But for logical coherence of a theory, the\n> constructive approach is important.\n\nSo this is the idea that I\'m getting at here. Pure analysis and\nnumerical analysis often walk hand in hand. For example, if we know\nthat a certain function is, say, smooth, then we know that it is Riemann\nintegrable on a bounded interval. But Riemann integration is defined in terms\nof partial sums of some rectangles that fit under the graph of this function.\nSo we know that as long as we take enough of these rectangles, we\'ll\nget a good enough approximation to the exact integral. How many rectangles\nwe need to take and how good is good enough is up to discussion, but\nnonetheless these partial sums give us a sensible algorithm to numerically\nevaluate the integral. It may not be the best, but it is sensible.\n\nThe moral of the story is that constructive existence proofs often\ngive computational algorithms as a by product.\n\nSo, say we have a 2D QFT that has been shown to exist rigorously.\nSome properties of this field theory can be calculated using an exansion\nin Feynman diagrams. But perhaps this expansion is only asymptotic\nand we cannot get answers with arbitrary precision since the series\nterms start behaving badly as you sum more and more of them. But,\nif the theory was proven to exist in some well defined limiting procedure,\nperhaps this limit can be turned into a computational algorithm that\ncan give answers to arbitrary precision and even give bounds on the\nerror. In this case the latter method is superior to the former (maybe\n"more efficient" wasn\'t the best choice of words) for REALLY precise\ncalculations.\n\nSo my question is whether this is the case with the existence proofs\nyou\'ve mentioned for some 2D and 3D field theories.\n\nThanks.\n\nIgor\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:<c76i7k$bul$1@lfa222122.richmond.edu>...
> Igor Khavkine wrote:
> > Also, I presume that, as you mentioned, that the proof is done as some
> > sort of well defined limit. Is there any possibility to use this limiting
> > procedure to do calculations with this theory and compare them to usual
> > perturbative calculations in terms of Feynman diagrams? If both give
> > similar answers, which one is more efficient?
>
> Approximate methods are almost always more efficient.
> You can see this from the way integrals are calculated in
> numerical analysis. No one uses the 'constructive proof' by
> Riemann sums. But for logical coherence of a theory, the
> constructive approach is important.
So this is the idea that I'm getting at here. Pure analysis and
numerical analysis often walk hand in hand. For example, if we know
that a certain function is, say, smooth, then we know that it is Riemann
integrable on a bounded interval. But Riemann integration is defined in terms
of partial sums of some rectangles that fit under the graph of this function.
So we know that as long as we take enough of these rectangles, we'll
get a good enough approximation to the exact integral. How many rectangles
we need to take and how good is good enough is up to discussion, but
nonetheless these partial sums give us a sensible algorithm to numerically
evaluate the integral. It may not be the best, but it is sensible.
The moral of the story is that constructive existence proofs often
give computational algorithms as a by product.
So, say we have a 2D QFT that has been shown to exist rigorously.
Some properties of this field theory can be calculated using an exansion
in Feynman diagrams. But perhaps this expansion is only asymptotic
and we cannot get answers with arbitrary precision since the series
terms start behaving badly as you sum more and more of them. But,
if the theory was proven to exist in some well defined limiting procedure,
perhaps this limit can be turned into a computational algorithm that
can give answers to arbitrary precision and even give bounds on the
error. In this case the latter method is superior to the former (maybe
"more efficient" wasn't the best choice of words) for REALLY precise
calculations.
So my question is whether this is the case with the existence proofs
you've mentioned for some 2D and 3D field theories.
Thanks.
Igor
<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>Hi,\n\nthomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0405030224.54284d38@posting.google. com>...\n\n> The obvious infinite dimensional symmetry groups that do appear in\n> 4D physics are the groups of diffeomorphisms and gauge transformations,\n> which are relevant to gravity and Yang-Mills theory, respectively.\n\nI think there is some different, conformal symmetry is a kind of\nglobal symmetry, I don\'t know diffeomorphism but gauge transformations\nbelong to, as the name, gauge symmetry. And I think gauge symmetry is\nnot really a kind of symmetry, for example they do not introduce new\nconservation law (but I also have not heard about the conservation law\ncorrespond to conformal symmetry...).\n\n--\nBest Regards,\nLR\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>Hi,
thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0405030224.54284d38@posting.google.com>...
> The obvious infinite dimensional symmetry groups that do appear in
> 4D physics are the groups of diffeomorphisms and gauge transformations,
> which are relevant to gravity and Yang-Mills theory, respectively.
I think there is some different, conformal symmetry is a kind of
global symmetry, I don't know diffeomorphism but gauge transformations
belong to, as the name, gauge symmetry. And I think gauge symmetry is
not really a kind of symmetry, for example they do not introduce new
conservation law (but I also have not heard about the conservation law
correspond to conformal symmetry...).
--
Best Regards,
LR
Arnold Neumaier
May7-04, 07:41 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>Igor Khavkine wrote:\n\n> The moral of the story is that constructive existence proofs often\n> give computational algorithms as a by product.\n\nYes, but generally poor algorithms.\n>\n> So, say we have a 2D QFT that has been shown to exist rigorously.\n> Some properties of this field theory can be calculated using an exansion\n> in Feynman diagrams. But perhaps this expansion is only asymptotic\n> and we cannot get answers with arbitrary precision since the series\n> terms start behaving badly as you sum more and more of them. But,\n> if the theory was proven to exist in some well defined limiting procedure,\n> perhaps this limit can be turned into a computational algorithm that\n> can give answers to arbitrary precision and even give bounds on the\n> error. In this case the latter method is superior to the former (maybe\n> "more efficient" wasn\'t the best choice of words) for REALLY precise\n> calculations.\n>\n> So my question is whether this is the case with the existence proofs\n> you\'ve mentioned for some 2D and 3D field theories.\n\nI don\'t think people have refined the estimates much beyond what was\nneeded for the existence proofs. What you want looks indeed possible\nin principle, but it is going to be hard analytic work.\n\nEven in the numerical analysis of finite dimensional problems, getting\nerror bounds (as opposed to asymptotic error statements) is much, much\nharger than devising even excellent approximation schemes.\n\nFor example, people solve routinely dynamical systems with thousands\nof variables to high but unproven accuracy (that can be checked by\nrepeating calculations with shorter steps and higher precision).\nBut algorithms for bounding the solutions jam for problems with more\nthan a handful of variables, typically due to excessive overestimation\nin the estimates. Those interested find papers at\nhttp://bt.pa.msu.edu/pub/papers/\n\nI don\'t know of anything related for field theories.\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>Igor Khavkine wrote:
> The moral of the story is that constructive existence proofs often
> give computational algorithms as a by product.
Yes, but generally poor algorithms.
>
> So, say we have a 2D QFT that has been shown to exist rigorously.
> Some properties of this field theory can be calculated using an exansion
> in Feynman diagrams. But perhaps this expansion is only asymptotic
> and we cannot get answers with arbitrary precision since the series
> terms start behaving badly as you sum more and more of them. But,
> if the theory was proven to exist in some well defined limiting procedure,
> perhaps this limit can be turned into a computational algorithm that
> can give answers to arbitrary precision and even give bounds on the
> error. In this case the latter method is superior to the former (maybe
> "more efficient" wasn't the best choice of words) for REALLY precise
> calculations.
>
> So my question is whether this is the case with the existence proofs
> you've mentioned for some 2D and 3D field theories.
I don't think people have refined the estimates much beyond what was
needed for the existence proofs. What you want looks indeed possible
in principle, but it is going to be hard analytic work.
Even in the numerical analysis of finite dimensional problems, getting
error bounds (as opposed to asymptotic error statements) is much, much
harger than devising even excellent approximation schemes.
For example, people solve routinely dynamical systems with thousands
of variables to high but unproven accuracy (that can be checked by
repeating calculations with shorter steps and higher precision).
But algorithms for bounding the solutions jam for problems with more
than a handful of variables, typically due to excessive overestimation
in the estimates. Those interested find papers at
http://bt.pa.msu.edu/pub/papers/
I don't know of anything related for field theories.
Arnold Neumaier
Thomas Larsson
May10-04, 06:02 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nhephooey@fastmail.fm (L.R.) wrote in message news:<4d470e8b.0405061736.19b8cb46@posting.google. com>...\n> Hi,\n>\n> thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0405030224.54284d38@posting.google. com>...\n>\n> > The obvious infinite dimensional symmetry groups that do appear in\n> > 4D physics are the groups of diffeomorphisms and gauge transformations,\n> > which are relevant to gravity and Yang-Mills theory, respectively.\n>\n> I think there is some different, conformal symmetry is a kind of\n> global symmetry, I don\'t know diffeomorphism but gauge transformations\n> belong to, as the name, gauge symmetry. And I think gauge symmetry is\n> not really a kind of symmetry, for example they do not introduce new\n> conservation law (but I also have not heard about the conservation law\n> correspond to conformal symmetry...).\n\n\nBut understanding representations of gauge groups are of some\nimportance, too. E.g., the Yang-Mills gauge group (and even its\nsubgroup of constant gauge transformations) implies that states in the\nstandard model can be characterized by their color, weak SU(2) charge\nand electric charge. I think this is useful information.\n\nSimilarly, one may view tensor calculus and more generally differential\ngeometry as the classical representation theory of the diffeomorphism\ngroup, since the proper irreps are tensor fields. Tensor calculus is\nperhaps of some value in general relativity - it is often taught as\npart of a first course in GR - so one may expect its quantum analogue\nmay be of similar use in quantum gravity.\n\nBut if you are looking for global infinite-dimensional symmetries, you\nare still constrained by classification theorems. E.g., the only\nsimple infinite-dimensional spacetime Lie groups are the groups of\nall diffeomorphisms, volume-preserving diffeos, canonical transformations,\nand contact transformations. That was conjectured by Sophus Lie himself\nand proven by Elie Cartan almost a century ago. Of course, nothing\nexcludes non-simple groups, such as the conformal group in 2D, but the\nsimple groups are the possible building blocks.\n\nThe distinction between global and gauge symmetries is evidently not\nintrinsic - conformal symmetry in 2D can be either global (in soft\ncondensed matter) or gauge (in string theory). So one could look for\nmodels with e.g. global (anomalous) diffeomorphism symmetry in\narbitrary dimension. Such models can be readily constructed - any\nlowest-energy rep can be rephrased as such a model. Nobody claims that\nthese models have anything to do with reality, but their construction\nand quantization is very straightforward.\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>hephooey@fastmail.fm (L.R.) wrote in message news:<4d470e8b.0405061736.19b8cb46@posting.google.com>...
> Hi,
>
> thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0405030224.54284d38@posting.google.com>...
>
> > The obvious infinite dimensional symmetry groups that do appear in
> > 4D physics are the groups of diffeomorphisms and gauge transformations,
> > which are relevant to gravity and Yang-Mills theory, respectively.
>
> I think there is some different, conformal symmetry is a kind of
> global symmetry, I don't know diffeomorphism but gauge transformations
> belong to, as the name, gauge symmetry. And I think gauge symmetry is
> not really a kind of symmetry, for example they do not introduce new
> conservation law (but I also have not heard about the conservation law
> correspond to conformal symmetry...).
But understanding representations of gauge groups are of some
importance, too. E.g., the Yang-Mills gauge group (and even its
subgroup of constant gauge transformations) implies that states in the
standard model can be characterized by their color, weak SU(2) charge
and electric charge. I think this is useful information.
Similarly, one may view tensor calculus and more generally differential
geometry as the classical representation theory of the diffeomorphism
group, since the proper irreps are tensor fields. Tensor calculus is
perhaps of some value in general relativity - it is often taught as
part of a first course in GR - so one may expect its quantum analogue
may be of similar use in quantum gravity.
But if you are looking for global infinite-dimensional symmetries, you
are still constrained by classification theorems. E.g., the only
simple infinite-dimensional spacetime Lie groups are the groups of
all diffeomorphisms, volume-preserving diffeos, canonical transformations,
and contact transformations. That was conjectured by Sophus Lie himself
and proven by Elie Cartan almost a century ago. Of course, nothing
excludes non-simple groups, such as the conformal group in 2D, but the
simple groups are the possible building blocks.
The distinction between global and gauge symmetries is evidently not
intrinsic - conformal symmetry in 2D can be either global (in soft
condensed matter) or gauge (in string theory). So one could look for
models with e.g. global (anomalous) diffeomorphism symmetry in
arbitrary dimension. Such models can be readily constructed - any
lowest-energy rep can be rephrased as such a model. Nobody claims that
these models have anything to do with reality, but their construction
and quantization is very straightforward.
Igor Khavkine
May10-04, 06:02 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<409BF7FF.2060508@univie.ac.at>...\n> Igor Khavkine wrote:\n>\n> > The moral of the story is that constructive existence proofs often\n> > give computational algorithms as a by product.\n>\n> Yes, but generally poor algorithms.\n\n> I don\'t think people have refined the estimates much beyond what was\n> needed for the existence proofs. What you want looks indeed possible\n> in principle, but it is going to be hard analytic work.\n\nIt is the principles that I am interested in at the moment.\nWhat I would like to understand is whether these proofs introduce\nany computational (read constructive) tools other than expansion\nin Feynman diagrams. I would also be happy to just understand\nthe simplest idea behind these existence proofs. I think it is\nconceptually fruitful to look at this idea as a computational\ntechnique. But perhaps, I\'m totally off the mark and these proofs\nare not constructive in the way I suspect they are.\n\nLet me convert my broad curiosity into a more specific question.\nSay I\'m given a QFT that has been proven to exist by one of the\nmethods you\'ve alluded to. And I want to calculate the scattering\namplitude for a pair of particles (or any other observable quantity).\nNaively, I write down an expression for this quantity as a series\nof Feynman diagrams. Using regularization and renormalization I may\neven get all the terms in this series to be finite. But even after\nthat, the convergence of this series is questionable. But, since\nthis QFT has been proven to exist, the real answer should be finite.\nNow, here\'s my question: does the existence proof give a fundamentally\ndifferent way to compute this quantity that is garanteed to converge\nin some (perhaps in practice intractible) limit, or does the proof\nplay tricks with the Feynman series that extract a finite answer\n(Borel summation or some such)?\n\nIf the technique used by the proof is fundamentally different from\nthe Feynman series, then could you describe it broad terms?\n\nThanks.\n\nIgor\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:<409BF7FF.2060508@univie.ac.at>...
> Igor Khavkine wrote:
>
> > The moral of the story is that constructive existence proofs often
> > give computational algorithms as a by product.
>
> Yes, but generally poor algorithms.
> I don't think people have refined the estimates much beyond what was
> needed for the existence proofs. What you want looks indeed possible
> in principle, but it is going to be hard analytic work.
It is the principles that I am interested in at the moment.
What I would like to understand is whether these proofs introduce
any computational (read constructive) tools other than expansion
in Feynman diagrams. I would also be happy to just understand
the simplest idea behind these existence proofs. I think it is
conceptually fruitful to look at this idea as a computational
technique. But perhaps, I'm totally off the mark and these proofs
are not constructive in the way I suspect they are.
Let me convert my broad curiosity into a more specific question.
Say I'm given a QFT that has been proven to exist by one of the
methods you've alluded to. And I want to calculate the scattering
amplitude for a pair of particles (or any other observable quantity).
Naively, I write down an expression for this quantity as a series
of Feynman diagrams. Using regularization and renormalization I may
even get all the terms in this series to be finite. But even after
that, the convergence of this series is questionable. But, since
this QFT has been proven to exist, the real answer should be finite.
Now, here's my question: does the existence proof give a fundamentally
different way to compute this quantity that is garanteed to converge
in some (perhaps in practice intractible) limit, or does the proof
play tricks with the Feynman series that extract a finite answer
(Borel summation or some such)?
If the technique used by the proof is fundamentally different from
the Feynman series, then could you describe it broad terms?
Thanks.
Igor
Arnold Neumaier
May12-04, 03:54 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>Igor Khavkine wrote:\n\n> It is the principles that I am interested in at the moment.\n> What I would like to understand is whether these proofs introduce\n> any computational (read constructive) tools other than expansion\n> in Feynman diagrams.\n\nYes. They are based on rigorously defined functional integrals in\nEuclidean spacetime, proving there certain correlation inequalities\nand then using analytic continuation (Osterwalder-Schrader theory)\nto obtain Wightman functions, which are renormalized vacuum\nexpectation values of operator products. Having these is having\ntime-ordered vacuum expectation values, which gives S-matrix elements\nby standard theory.\n\n> I would also be happy to just understand\n> the simplest idea behind these existence proofs. I think it is\n> conceptually fruitful to look at this idea as a computational\n> technique. But perhaps, I\'m totally off the mark and these proofs\n> are not constructive in the way I suspect they are.\n\nThey _are_ constructive, but it requires a significant effort to\nmaster the proofs.\n\n> If the technique used by the proof is fundamentally different from\n> the Feynman series, then could you describe it broad terms?\n\nThe proof is completely independent of Feynman graphs.\n\nThe existence proof does not prove the existence of the S-matrix,\nbut the existence of Wightman functions satisfying the Wightman\naxioms. From there to the S-matrix is another step, called Haag-Ruelle\ntheory. This is also rigorous, but has nothing to do with the\nexistence problem, which is the hard part.\n\nFor an overview, read:\nA.S. Wightman, Hilbert\'s sixth problem: Mathematical treatment of the axioms of physics, in: Mathematical Developments\nArising From Hilbert Problems, edited by F. Browder, (American Mathematical Society, Providence, R.I.) 1976, pp. 147--240.\n\n\nArnold Neumaier\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>Igor Khavkine wrote:
> It is the principles that I am interested in at the moment.
> What I would like to understand is whether these proofs introduce
> any computational (read constructive) tools other than expansion
> in Feynman diagrams.
Yes. They are based on rigorously defined functional integrals in
Euclidean spacetime, proving there certain correlation inequalities
and then using analytic continuation (Osterwalder-Schrader theory)
to obtain Wightman functions, which are renormalized vacuum
expectation values of operator products. Having these is having
time-ordered vacuum expectation values, which gives S-matrix elements
by standard theory.
> I would also be happy to just understand
> the simplest idea behind these existence proofs. I think it is
> conceptually fruitful to look at this idea as a computational
> technique. But perhaps, I'm totally off the mark and these proofs
> are not constructive in the way I suspect they are.
They _are_ constructive, but it requires a significant effort to
master the proofs.
> If the technique used by the proof is fundamentally different from
> the Feynman series, then could you describe it broad terms?
The proof is completely independent of Feynman graphs.
The existence proof does not prove the existence of the S-matrix,
but the existence of Wightman functions satisfying the Wightman
axioms. From there to the S-matrix is another step, called Haag-Ruelle
theory. This is also rigorous, but has nothing to do with the
existence problem, which is the hard part.
For an overview, read:
A.S. Wightman, Hilbert's sixth problem: Mathematical treatment of the axioms of physics, in: Mathematical Developments
Arising From Hilbert Problems, edited by F. Browder, (American Mathematical Society, Providence, R.I.) 1976, pp. 147--240.
Arnold Neumaier
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