View Full Version : Some questions in QFT
<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\nHi all,\n\nI noticed that in QFT there is two kinds of commutators in QFT, some\ncommutators like [x,p]=i or [phi, phi]=0 give you a number, and some\ncommutators give you another operator like [J_i, J_j] = c_ijk J_k, I\noften feel perplexed, when should I expect a number and when should I\nexpect a operator? Do they have some kind of connections so we can\nhandle them in a uniform way? I heard about that people can thinks\n[x,p]=i as a central extension of the abelian group [x,p]=0, and there\nis some weird group called Heisenberg group, but I have not seen any\nvirtue of this kind of viewpoint yet.\n\nanother question is Weinberg\'s cluster decomposition principle.\nWeinberg seems feel very pleased that he replaced causality with CDP,\nI wonder why he is so upset about causality, my guess is that maybe\nbecause grativity. Is there some more practical reasons for this?\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 all,
I noticed that in QFT there is two kinds of commutators in QFT, some
commutators like [x,p]=i or [\phi, \phi]=0 give you a number, and some
commutators give you another operator like [J_i, J_j] = c_{ijk} J_k, I
often feel perplexed, when should I expect a number and when should I
expect a operator? Do they have some kind of connections so we can
handle them in a uniform way? I heard about that people can thinks
[x,p]=i as a central extension of the abelian group [x,p]=0, and there
is some weird group called Heisenberg group, but I have not seen any
virtue of this kind of viewpoint yet.
another question is Weinberg's cluster decomposition principle.
Weinberg seems feel very pleased that he replaced causality with CDP,
I wonder why he is so upset about causality, my guess is that maybe
because grativity. Is there some more practical reasons for this?
--
Best Regards,
LR
Arnold Neumaier
Oct13-04, 08:44 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> Hi all,\n>\n> I noticed that in QFT there is two kinds of commutators in QFT, some\n> commutators like [x,p]=i or [phi, phi]=0 give you a number, and some\n> commutators give you another operator like [J_i, J_j] = c_ijk J_k, I\n> often feel perplexed, when should I expect a number and when should I\n> expect a operator?\n\nCommutators of operators are always operators.\nA number is just the special case of an operator which multiplies each\nwave function by a constant.\n\n\n> another question is Weinberg\'s cluster decomposition principle.\n> Weinberg seems feel very pleased that he replaced causality with CDP,\n> I wonder why he is so upset about causality, my guess is that maybe\n> because grativity. Is there some more practical reasons for this?\n\nCausality in QFT is a postulate, and Weinberg shows why this postulate\nis necessary. Without such a connection, it would be quite reasonable to\nattempt a theory with small violations of microcausality. But these\nwould wrack cluster decomposition, which is macroscopically observable\nand hence cannot be dispensed with.\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:
> Hi all,
>
> I noticed that in QFT there is two kinds of commutators in QFT, some
> commutators like [x,p]=i or [\phi, \phi]=0 give you a number, and some
> commutators give you another operator like [J_i, J_j] = c_{ijk} J_k, I
> often feel perplexed, when should I expect a number and when should I
> expect a operator?
Commutators of operators are always operators.
A number is just the special case of an operator which multiplies each
wave function by a constant.
> another question is Weinberg's cluster decomposition principle.
> Weinberg seems feel very pleased that he replaced causality with CDP,
> I wonder why he is so upset about causality, my guess is that maybe
> because grativity. Is there some more practical reasons for this?
Causality in QFT is a postulate, and Weinberg shows why this postulate
is necessary. Without such a connection, it would be quite reasonable to
attempt a theory with small violations of microcausality. But these
would wrack cluster decomposition, which is macroscopically observable
and hence cannot be dispensed with.
Arnold Neumaier
Charles Torre
Oct13-04, 02:13 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> I noticed that in QFT there is two kinds of commutators in QFT, some\n> commutators like [x,p]=i or [phi, phi]=0 give you a number, and some\n> commutators give you another operator like [J_i, J_j] = c_ijk J_k, I\n> often feel perplexed, when should I expect a number and when should I\n> expect a operator? Do they have some kind of connections so we can\n> handle them in a uniform way?\n\nThe commutator of two linear operators is always another linear operator.\nWhen you see something like [x,p]=i, what is really meant is [x,p]=i 1, where\n1 is the identity operator.\n\ncharlie\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I noticed that in QFT there is two kinds of commutators in QFT, some
> commutators like [x,p]=i or [\phi, \phi]=0 give you a number, and some
> commutators give you another operator like [J_i, J_j] = c_{ijk} J_k, I
> often feel perplexed, when should I expect a number and when should I
> expect a operator? Do they have some kind of connections so we can
> handle them in a uniform way?
The commutator of two linear operators is always another linear operator.
When you see something like [x,p]=i, what is really meant is [x,p]=i 1, where
1 is the identity operator.
charlie
<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\nHi,\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<416D1981.1060904@univie.ac.at>...\n> L.R. wrote:\n>\n> Commutators of operators are always operators.\n> A number is just the special case of an operator which multiplies each\n> wave function by a constant.\n\nYes, it should be so in mathematics, but I still feel some distinction\nbetween [x,p] = 1 and [J_x, J_y]=J_z, but I don\'t know why ;)\n\n> Causality in QFT is a postulate, and Weinberg shows why this postulate\n> is necessary. Without such a connection, it would be quite reasonable to\n> attempt a theory with small violations of microcausality. But these\n> would wrack cluster decomposition, which is macroscopically observable\n> and hence cannot be dispensed with.\n>\n\nYou\'re telling me that the cluster decompostion is some\nobservable/practical realization of causality? I think the requirment\nof cluster decomposition is weaker than causality, but I\'m not sure,\nis there any proof of this kind of things?\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:<416D1981.1060904@univie.ac.at>...
> L.R. wrote:
>
> Commutators of operators are always operators.
> A number is just the special case of an operator which multiplies each
> wave function by a constant.
Yes, it should be so in mathematics, but I still feel some distinction
between [x,p] = 1 and [J_x, J_y]=J_z, but I don't know why ;)
> Causality in QFT is a postulate, and Weinberg shows why this postulate
> is necessary. Without such a connection, it would be quite reasonable to
> attempt a theory with small violations of microcausality. But these
> would wrack cluster decomposition, which is macroscopically observable
> and hence cannot be dispensed with.
>
You're telling me that the cluster decompostion is some
observable/practical realization of causality? I think the requirment
of cluster decomposition is weaker than causality, but I'm not sure,
is there any proof of this kind of things?
--
Best Regards,
LR
Blake Winter
Oct14-04, 11:07 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> Causality in QFT is a postulate, and Weinberg shows why this postulate\n> is necessary. Without such a connection, it would be quite reasonable to\n> attempt a theory with small violations of microcausality. But these\n> would wrack cluster decomposition, which is macroscopically observable\n> and hence cannot be dispensed with.\n\n\nWhat exactly is cluster decomposition, and is there a good\nintroduction to how it is equivalent to requiring microcausality?\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>Causality in QFT is a postulate, and Weinberg shows why this postulate
> is necessary. Without such a connection, it would be quite reasonable to
> attempt a theory with small violations of microcausality. But these
> would wrack cluster decomposition, which is macroscopically observable
> and hence cannot be dispensed with.
What exactly is cluster decomposition, and is there a good
introduction to how it is equivalent to requiring microcausality?
<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\nHi,\n\n> What exactly is cluster decomposition, and is there a good\n> introduction to how it is equivalent to requiring microcausality?\n\nYou can find it in Weinberg\'s "The Quantum Theory of Fields", volume\nI, and there is some reference to the original paper created the word\n"cluster decompostion". But I cannot find other books talk about this.\nIf cluster decomposition is a very important principle, why there are\nso few books mentioned it, even in those qft books full of mathematics\nyou can hardly find this principle. So I began to wounder if it is\nWeinberg\'s personal favor...\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,
> What exactly is cluster decomposition, and is there a good
> introduction to how it is equivalent to requiring microcausality?
You can find it in Weinberg's "The Quantum Theory of Fields", volume
I, and there is some reference to the original paper created the word
"cluster decompostion". But I cannot find other books talk about this.
If cluster decomposition is a very important principle, why there are
so few books mentioned it, even in those qft books full of mathematics
you can hardly find this principle. So I began to wounder if it is
Weinberg's personal favor...
--
Best Regards,
LR
Arnold Neumaier
Oct15-04, 01:55 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\nBlake Winter wrote:\n\nActually, I wrote this...\n\n>>Causality in QFT is a postulate, and Weinberg shows why this postulate\n>>is necessary. Without such a connection, it would be quite reasonable to\n>>attempt a theory with small violations of microcausality. But these\n>>would wrack cluster decomposition, which is macroscopically observable\n>>and hence cannot be dispensed with.\n>\n> What exactly is cluster decomposition, and is there a good\n> introduction to how it is equivalent to requiring microcausality?\n\nIt is not quite equivalent, but closely related.\n\nCluster decomposition means essentally that you cannot scatter a particle\nat a very distant particle. So what happens on the moon is irrelevant\nto experiments on the earth (except those depending on moonshine).\nCluster decomposition is the basis of all physics.\n\nSee Chapter 4 of Weinberg\'s book, which is quite readable after a first\nexposure to QFT by other books.\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>Blake Winter wrote:
Actually, I wrote this...
>>Causality in QFT is a postulate, and Weinberg shows why this postulate
>>is necessary. Without such a connection, it would be quite reasonable to
>>attempt a theory with small violations of microcausality. But these
>>would wrack cluster decomposition, which is macroscopically observable
>>and hence cannot be dispensed with.
>
> What exactly is cluster decomposition, and is there a good
> introduction to how it is equivalent to requiring microcausality?
It is not quite equivalent, but closely related.
Cluster decomposition means essentally that you cannot scatter a particle
at a very distant particle. So what happens on the moon is irrelevant
to experiments on the earth (except those depending on moonshine).
Cluster decomposition is the basis of all physics.
See Chapter 4 of Weinberg's book, which is quite readable after a first
exposure to QFT by other books.
Arnold Neumaier
Arnold Neumaier
Oct15-04, 01:55 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\nL.R. wrote:\n\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<416D1981.1060904@univie.ac.at>...\n>\n>>L.R. wrote:\n>>\n>>Commutators of operators are always operators.\n>>A number is just the special case of an operator which multiplies each\n>>wave function by a constant.\n>\n> Yes, it should be so in mathematics, but I still feel some distinction\n> between [x,p] = 1 and [J_x, J_y]=J_z, but I don\'t know why ;)\n\nOf course there is a difference. In the first case, double commutators\nvanish. Different commutation rules produce different Lie algebras and\nhence different dynamical symmetry groups.\n\n\n>>Causality in QFT is a postulate, and Weinberg shows why this postulate\n>>is necessary. Without such a connection, it would be quite reasonable to\n>>attempt a theory with small violations of microcausality. But these\n>>would wrack cluster decomposition, which is macroscopically observable\n>>and hence cannot be dispensed with.\n\n> You\'re telling me that the cluster decomposition is some\n> observable/practical realization of causality?\n\nNo, just the opposite, if I understand your statement correctly.\nThe only known way to realize cluster decomposition is implementing\nit through microcausality.\n\n\n> I think the requirement\n> of cluster decomposition is weaker than causality, but I\'m not sure,\n\nThere is no conflict with what I said.\n\n\n> is there any proof of this kind of things?\n\nIt is proved in Weinberg\'s QFT theory book. Weaker means \'less restrictive\',\nand since cluster decomposition is experimentally verifiable directly\nwhile microcausality is not, the former is a safer starting point.\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:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<416D1981.1060904@univie.ac.at>...
>
>>L.R. wrote:
>>
>>Commutators of operators are always operators.
>>A number is just the special case of an operator which multiplies each
>>wave function by a constant.
>
> Yes, it should be so in mathematics, but I still feel some distinction
> between [x,p] = 1 and [J_x, J_y]=J_z, but I don't know why ;)
Of course there is a difference. In the first case, double commutators
vanish. Different commutation rules produce different Lie algebras and
hence different dynamical symmetry groups.
>>Causality in QFT is a postulate, and Weinberg shows why this postulate
>>is necessary. Without such a connection, it would be quite reasonable to
>>attempt a theory with small violations of microcausality. But these
>>would wrack cluster decomposition, which is macroscopically observable
>>and hence cannot be dispensed with.
> You're telling me that the cluster decomposition is some
> observable/practical realization of causality?
No, just the opposite, if I understand your statement correctly.
The only known way to realize cluster decomposition is implementing
it through microcausality.
> I think the requirement
> of cluster decomposition is weaker than causality, but I'm not sure,
There is no conflict with what I said.
> is there any proof of this kind of things?
It is proved in Weinberg's QFT theory book. Weaker means 'less restrictive',
and since cluster decomposition is experimentally verifiable directly
while microcausality is not, the former is a safer starting point.
Arnold Neumaier
Frank Hellmann
Oct15-04, 01:55 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\nblake.winter@houghton.edu (Blake Winter) wrote in message news:<87423d2a.0410131607.4570fa17@posting.google. com>...\n> > Causality in QFT is a postulate, and Weinberg shows why this postulate\n> > is necessary. Without such a connection, it would be quite reasonable to\n> > attempt a theory with small violations of microcausality. But these\n> > would wrack cluster decomposition, which is macroscopically observable\n> > and hence cannot be dispensed with.\n>\n>\n> What exactly is cluster decomposition, and is there a good\n> introduction to how it is equivalent to requiring microcausality?\n\nGo to the source, Weinbergs QFT book is the best introduction I have found.\n\nFrank.\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>blake.winter@houghton.edu (Blake Winter) wrote in message news:<87423d2a.0410131607.4570fa17@posting.google.com>...
> > Causality in QFT is a postulate, and Weinberg shows why this postulate
> > is necessary. Without such a connection, it would be quite reasonable to
> > attempt a theory with small violations of microcausality. But these
> > would wrack cluster decomposition, which is macroscopically observable
> > and hence cannot be dispensed with.
>
>
> What exactly is cluster decomposition, and is there a good
> introduction to how it is equivalent to requiring microcausality?
Go to the source, Weinbergs QFT book is the best introduction I have found.
Frank.
Frank Hellmann
Oct15-04, 01:55 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\nhephooey@hotmail.com (L.R.) wrote in message news:<231bb620.0410132113.3db8a2cf@posting.google. com>...\n> Hi,\n>\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<416D1981.1060904@univie.ac.at>...\n> > L.R. wrote:\n> >\n> > Commutators of operators are always operators.\n> > A number is just the special case of an operator which multiplies each\n> > wave function by a constant.\n>\n> Yes, it should be so in mathematics, but I still feel some distinction\n> between [x,p] = 1 and [J_x, J_y]=J_z, but I don\'t know why ;)\n>\n\nThere is indeed a difference significant difference. In the first case\nexponentiate to get the group elements and take the commutator.\nYou\'ll find that the group elements commute up to a factor.\nexp(A)exp(B) = exp(B)exp(A)exp([A,B]) for [A[A,B]]=0 as is the case\nhere. Thus the group looks somewhat close to abelian. exp(A)exp(B) =\nexp(B)exp(A)exp([A,B]) = exp(B)exp(A)exp(1). Now you wave your hands,\nredefine a couple of things, and actually keep track of required "i"s\nand the right hand and the left hand side differ by a phase only. This\nis known as the Weyl form of the CCR.\n\nhttp://planetmath.org/encyclopedia/BakerCampellHausdorffFormulae.html\nand\nhttp://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/Stone-von_Neumann_theorem#Weyl_form_of_the_canonical_com mutation_relations\n\nFrank\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@hotmail.com (L.R.) wrote in message news:<231bb620.0410132113.3db8a2cf@posting.google.com>...
> Hi,
>
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<416D1981.1060904@univie.ac.at>...
> > L.R. wrote:
> >
> > Commutators of operators are always operators.
> > A number is just the special case of an operator which multiplies each
> > wave function by a constant.
>
> Yes, it should be so in mathematics, but I still feel some distinction
> between [x,p] = 1 and [J_x, J_y]=J_z, but I don't know why ;)
>
There is indeed a difference significant difference. In the first case
exponentiate to get the group elements and take the commutator.
You'll find that the group elements commute up to a factor.
\exp(A)\exp(B) = \exp(B)\exp(A)\exp([A,B]) for [A[A,B]]=0 as is the case
here. Thus the group looks somewhat close to abelian. \exp(A)\exp(B) =\exp(B)\exp(A)\exp([A,B]) = \exp(B)\exp(A)\exp(1). Now you wave your hands,
redefine a couple of things, and actually keep track of required "i"s
and the right hand and the left hand side differ by a phase only. This
is known as the Weyl form of the CCR.
http://planetmath.org/encyclopedia/BakerCampellHausdorffFormulae.html
and
http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/Stone-von_Neumann_theorem#Weyl_form_of_the_canonical_com mutation_relations
Frank
Constantine
Oct15-04, 01:55 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"L.R." <hephooey@hotmail.com> wrote in message\nnews:231bb620.0410132113.3db8a2cf@posting .google.com...\n>\n>\n> Hi,\n>\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message\n> news:<416D1981.1060904@univie.ac.at>...\n>> L.R. wrote:\n>>\n>> Commutators of operators are always operators.\n>> A number is just the special case of an operator which multiplies each\n>> wave function by a constant.\n>\n> Yes, it should be so in mathematics, but I still feel some distinction\n> between [x,p] = 1 and [J_x, J_y]=J_z, but I don\'t know why ;)\n>\n\nA canonical system is quantised by promoting the canonical coordinates and\nmomenta to operators and then demanding that their commutator is i times the\nPoisson bracket of the coresponding classical quantities. The Poisson\nbracket of the canonical coordinates with their momenta is always a number,\nhence the quantisation condition reads [commutator] = (a number).\n\nRelations of the form [J_i, J_j] ~J_k are a different story. These operators\nform an algebra and this is related to symmetries of the system.\n\nThe difference is that [x,p]=1 is the quantisation condition and that [J_i,\nJ_j]~J_k are symmetry conditions. They are different things.\n\nHope that this helps.\n\nFriendly, Kostas.\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>"L.R." <hephooey@hotmail.com> wrote in message
news:231bb620.0410132113.3db8a2cf@posting.google.c om...
>
>
> Hi,
>
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message
> news:<416D1981.1060904@univie.ac.at>...
>> L.R. wrote:
>>
>> Commutators of operators are always operators.
>> A number is just the special case of an operator which multiplies each
>> wave function by a constant.
>
> Yes, it should be so in mathematics, but I still feel some distinction
> between [x,p] = 1 and [J_x, J_y]=J_z, but I don't know why ;)
>
A canonical system is quantised by promoting the canonical coordinates and
momenta to operators and then demanding that their commutator is i times the
Poisson bracket of the coresponding classical quantities. The Poisson
bracket of the canonical coordinates with their momenta is always a number,
hence the quantisation condition reads [commutator] = (a number).
Relations of the form [J_i, J_j] ~J_k are a different story. These operators
form an algebra and this is related to symmetries of the system.
The difference is that [x,p]=1 is the quantisation condition and that [J_i,J_j]~J_k are symmetry conditions. They are different things.
Hope that this helps.
Friendly, Kostas.
Arnold Neumaier
Oct19-04, 03:29 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: [on cluster decomposition]\n\n> You can find it in Weinberg\'s "The Quantum Theory of Fields", volume\n> I, and there is some reference to the original paper created the word\n> "cluster decompostion". But I cannot find other books talk about this.\n> If cluster decomposition is a very important principle, why there are\n> so few books mentioned it, even in those qft books full of mathematics\n> you can hardly find this principle. So I began to wounder if it is\n> Weinberg\'s personal favor...\n\nThe lack of references in most books of QFT is explained by the\nfact that local QFT automatically satisfies cluster decomposition.\nMost people start by taking QFT as starting point, without asking why.\nWeinberg\'s treatise is about the only book that asks this question and\nanswers it in some depth.\n\nBut when you look at the literature on phenomenological covariant\nmultiparticle models, cluster decomposition plays an essential role\nin that it is the main hurdle to overcome to get realistic models for\nsystems made of more than two unconfined particles. See, e.g.,\nB.D. Keister and W.N. Polyzou,\nRelativistic Hamiltonian Dynamics in Nuclear and Particle Physics,\nin: Advances in Nuclear Physics, Volume 20,\n(J. W. Negele and E.W. Vogt, eds.)\nPlenum Press 1991.\nand the references there.\n\nCluster decomposition for field theory is also discussed from a rigorous\npoint of view in the book by Glimm and Jaffe, where connections are made\nto multiparticle scattering. And indeed, books on (nonrelativistic)\nscattering theory are the ones where the cluster decomposition is\ndiscussed in detail, since it is needed to describe the result of the\nmost general multiparticle scattering experiments, and an understanding\nof it is essential for proving the asymptotic completeness of scattering\nstates.\n\nUnfortunately, most physicists tend to work in isolated fragments of the\nwhole edifice of physics, thus losing connections that may be important\nto understanding. Cluster decomposition would perhaps be more prominent\nin QFT if it were easier to calculate properties of bound states and\ntheir scattering or breaking up, since that is where one can see the\nprinciple at work. But such calculations are presently out of reach.\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: [on cluster decomposition]
> You can find it in Weinberg's "The Quantum Theory of Fields", volume
> I, and there is some reference to the original paper created the word
> "cluster decompostion". But I cannot find other books talk about this.
> If cluster decomposition is a very important principle, why there are
> so few books mentioned it, even in those qft books full of mathematics
> you can hardly find this principle. So I began to wounder if it is
> Weinberg's personal favor...
The lack of references in most books of QFT is explained by the
fact that local QFT automatically satisfies cluster decomposition.
Most people start by taking QFT as starting point, without asking why.
Weinberg's treatise is about the only book that asks this question and
answers it in some depth.
But when you look at the literature on phenomenological covariant
multiparticle models, cluster decomposition plays an essential role
in that it is the main hurdle to overcome to get realistic models for
systems made of more than two unconfined particles. See, e.g.,
B.D. Keister and W.N. Polyzou,
Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics,
in: Advances in Nuclear Physics, Volume 20,
(J. W. Negele and E.W. Vogt, eds.)
Plenum Press 1991.
and the references there.
Cluster decomposition for field theory is also discussed from a rigorous
point of view in the book by Glimm and Jaffe, where connections are made
to multiparticle scattering. And indeed, books on (nonrelativistic)
scattering theory are the ones where the cluster decomposition is
discussed in detail, since it is needed to describe the result of the
most general multiparticle scattering experiments, and an understanding
of it is essential for proving the asymptotic completeness of scattering
states.
Unfortunately, most physicists tend to work in isolated fragments of the
whole edifice of physics, thus losing connections that may be important
to understanding. Cluster decomposition would perhaps be more prominent
in QFT if it were easier to calculate properties of bound states and
their scattering or breaking up, since that is where one can see the
principle at work. But such calculations are presently out of reach.
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\nHi,\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4173C5E0.50209@univie.ac.at>...\n>\n> Unfortunately, most physicists tend to work in isolated fragments of the\n> whole edifice of physics, thus losing connections that may be important\n> to understanding. Cluster decomposition would perhaps be more prominent\n> in QFT if it were easier to calculate properties of bound states and\n> their scattering or breaking up, since that is where one can see the\n> principle at work. But such calculations are presently out of reach.\n>\n\nMay be a stupid question, can we get causality from the requirement of\nrelativistic? just use that space-like separated operation should\ncommute with each other, or are there some subtlety involved in the\nqft?\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:<4173C5E0.50209@univie.ac.at>...
>
> Unfortunately, most physicists tend to work in isolated fragments of the
> whole edifice of physics, thus losing connections that may be important
> to understanding. Cluster decomposition would perhaps be more prominent
> in QFT if it were easier to calculate properties of bound states and
> their scattering or breaking up, since that is where one can see the
> principle at work. But such calculations are presently out of reach.
>
May be a stupid question, can we get causality from the requirement of
relativistic? just use that space-like separated operation should
commute with each other, or are there some subtlety involved in the
qft?
--
Best Regards,
LR
Arnold Neumaier
Oct20-04, 10:48 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\nL.R. wrote:\n> Hi,\n>\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4173C5E0.50209@univie.ac.at>...\n>\n>>Unfort unately, most physicists tend to work in isolated fragments of the\n>>whole edifice of physics, thus losing connections that may be important\n>>to understanding. Cluster decomposition would perhaps be more prominent\n>>in QFT if it were easier to calculate properties of bound states and\n>>their scattering or breaking up, since that is where one can see the\n>>principle at work. But such calculations are presently out of reach.\n>>\n>\n>\n> May be a stupid question, can we get causality from the requirement of\n> relativistic? just use that space-like separated operation should\n> commute with each other,\n\nWhat you want to use here is precisely what people call \'causality\'\nor \'microcausality\'. It is this property which Weinberg justifies in\nterms of the cluster decomposition.\n\nThus your suggestion amounts to a tautology.\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:
> Hi,
>
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4173C5E0.50209@univie.ac.at>...
>
>>Unfortunately, most physicists tend to work in isolated fragments of the
>>whole edifice of physics, thus losing connections that may be important
>>to understanding. Cluster decomposition would perhaps be more prominent
>>in QFT if it were easier to calculate properties of bound states and
>>their scattering or breaking up, since that is where one can see the
>>principle at work. But such calculations are presently out of reach.
>>
>
>
> May be a stupid question, can we get causality from the requirement of
> relativistic? just use that space-like separated operation should
> commute with each other,
What you want to use here is precisely what people call 'causality'
or 'microcausality'. It is this property which Weinberg justifies in
terms of the cluster decomposition.
Thus your suggestion amounts to a tautology.
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\nHi,\n\n> What you want to use here is precisely what people call \'causality\'\n> or \'microcausality\'. It is this property which Weinberg justifies in\n> terms of the cluster decomposition.\n>\n> Thus your suggestion amounts to a tautology.\n>\n\nOk, I feel like begin to understand this: even we required relativity\nin qft, we still cannot get the microcausality because quantum field\ntheory propagation amplitude may not equal to zero even in the\nspace-like separation, so we need an extra rule so called cluster\ndecomposition to assure the microcausality, that two operation will\nnot affect each other in the space-like separation. am I right?\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,
> What you want to use here is precisely what people call 'causality'
> or 'microcausality'. It is this property which Weinberg justifies in
> terms of the cluster decomposition.
>
> Thus your suggestion amounts to a tautology.
>
Ok, I feel like begin to understand this: even we required relativity
in qft, we still cannot get the microcausality because quantum field
theory propagation amplitude may not equal to zero even in the
space-like separation, so we need an extra rule so called cluster
decomposition to assure the microcausality, that two operation will
not affect each other in the space-like separation. am I right?
--
Best Regards,
LR
Arnold Neumaier
Oct22-04, 12:40 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\nL.R. wrote:\n>\n> Ok, I feel like begin to understand this: even we required relativity\n> in qft, we still cannot get the microcausality because quantum field\n> theory propagation amplitude may not equal to zero even in the\n> space-like separation, so we need an extra rule so called cluster\n> decomposition to assure the microcausality, that two operation will\n> not affect each other in the space-like separation. am I right?\n\nMore or less yes. But the connection is the other way around.\nWe need cluster decomposition because it is observed. We need\nlocal fields and microcausality, mainly because it implies\n(modulo fine print involving contact terms) at least perturbatively\ncluster decomposition, and there is no other known way in QFT to\nensure the latter. But there are covariant N-particle models with\ncluster decomposition, discussed, e.g., in\nB.D. Keister and W.N. Polyzou,\nRelativistic Hamiltonian Dynamics in Nuclear and Particle Physics,\nin: Advances in Nuclear Physics, Volume 20,\n(J. W. Negele and E.W. Vogt, eds.)\nPlenum Press 1991.\n(The constructions are quite messy; they have, however, the\nadvantage that they do not need renormalization, and are useful\nphenomenological models.)\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:
>
> Ok, I feel like begin to understand this: even we required relativity
> in qft, we still cannot get the microcausality because quantum field
> theory propagation amplitude may not equal to zero even in the
> space-like separation, so we need an extra rule so called cluster
> decomposition to assure the microcausality, that two operation will
> not affect each other in the space-like separation. am I right?
More or less yes. But the connection is the other way around.
We need cluster decomposition because it is observed. We need
local fields and microcausality, mainly because it implies
(modulo fine print involving contact terms) at least perturbatively
cluster decomposition, and there is no other known way in QFT to
ensure the latter. But there are covariant N-particle models with
cluster decomposition, discussed, e.g., in
B.D. Keister and W.N. Polyzou,
Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics,
in: Advances in Nuclear Physics, Volume 20,
(J. W. Negele and E.W. Vogt, eds.)
Plenum Press 1991.
(The constructions are quite messy; they have, however, the
advantage that they do not need renormalization, and are useful
phenomenological models.)
Arnold Neumaier
Hendrik van Hees
Oct24-04, 09:04 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\nArnold Neumaier wrote:\n\n\n> More or less yes. But the connection is the other way around.\n> We need cluster decomposition because it is observed. We need\n> local fields and microcausality, mainly because it implies\n> (modulo fine print involving contact terms) at least perturbatively\n> cluster decomposition, and there is no other known way in QFT to\n> ensure the latter. But there are covariant N-particle models with\n> cluster decomposition, discussed, e.g., in\n> B.D. Keister and W.N. Polyzou,\n> Relativistic Hamiltonian Dynamics in Nuclear and Particle\n> Physics, in: Advances in Nuclear Physics, Volume 20,\n> (J. W. Negele and E.W. Vogt, eds.)\n> Plenum Press 1991.\n> (The constructions are quite messy; they have, however, the\n> advantage that they do not need renormalization, and are useful\n> phenomenological models.)\n\nWait a moment. Of course, microcausality is sufficient for the\ncluster-decomposition property (at least in the perturbative sense),\nbut I have never seen the proof that it is also necessary. Also\nWeinberg, for whom the cluster-decomposition property is one of the\nimportant ingredients for model building, does not claim this, but he\nsays, he is not aware of other models than microcausal field theory\nwhich fulfill the cluster-decomposition property.\n\nHave the above mentioned authors a model which is not microcausal but\nfulfill the cluster-decomposition property? This would be interesting.\n\nWhat do you mean by the statement, that they do not need\nrenormalisation?\n\nHow can this be in an interacting field theory? Are there no\nwave-function renormalisations? If so, is then Haag\'s theorem finally\nproven wrong?\n\n--\nHendrik van Hees Cyclotron Institute\nPhone: +1 979/845-1411 Texas A&M University\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\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 wrote:
> More or less yes. But the connection is the other way around.
> We need cluster decomposition because it is observed. We need
> local fields and microcausality, mainly because it implies
> (modulo fine print involving contact terms) at least perturbatively
> cluster decomposition, and there is no other known way in QFT to
> ensure the latter. But there are covariant N-particle models with
> cluster decomposition, discussed, e.g., in
> B.D. Keister and W.N. Polyzou,
> Relativistic Hamiltonian Dynamics in Nuclear and Particle
> Physics, in: Advances in Nuclear Physics, Volume 20,
> (J. W. Negele and E.W. Vogt, eds.)
> Plenum Press 1991.
> (The constructions are quite messy; they have, however, the
> advantage that they do not need renormalization, and are useful
> phenomenological models.)
Wait a moment. Of course, microcausality is sufficient for the
cluster-decomposition property (at least in the perturbative sense),
but I have never seen the proof that it is also necessary. Also
Weinberg, for whom the cluster-decomposition property is one of the
important ingredients for model building, does not claim this, but he
says, he is not aware of other models than microcausal field theory
which fulfill the cluster-decomposition property.
Have the above mentioned authors a model which is not microcausal but
fulfill the cluster-decomposition property? This would be interesting.
What do you mean by the statement, that they do not need
renormalisation?
How can this be in an interacting field theory? Are there no
wave-function renormalisations? If so, is then Haag's theorem finally
proven wrong?
--
Hendrik van Hees Cyclotron Institute
Phone: +1 979/845-1411 Texas A&M University
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366
Arnold Neumaier
Oct25-04, 10:42 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\nHendrik van Hees wrote:\n> Arnold Neumaier wrote:\n>\n>>We need cluster decomposition because it is observed. We need\n>>local fields and microcausality, mainly because it implies\n>>(modulo fine print involving contact terms) at least perturbatively\n>>cluster decomposition, and there is no other known way in QFT to\n>>ensure the latter. But there are covariant N-particle models with\n>>cluster decomposition, discussed, e.g., in\n>> B.D. Keister and W.N. Polyzou,\n>> Relativistic Hamiltonian Dynamics in Nuclear and Particle\n>> Physics, in: Advances in Nuclear Physics, Volume 20,\n>> (J. W. Negele and E.W. Vogt, eds.)\n>> Plenum Press 1991.\n>>(The constructions are quite messy; they have, however, the\n>>advantage that they do not need renormalization, and are useful\n>>phenomenological models.)\n>\n> Wait a moment. Of course, microcausality is sufficient for the\n> cluster-decomposition property (at least in the perturbative sense),\n> but I have never seen the proof that it is also necessary.\n\nThis is precisely what I state above. In fact it is _not_ necessary\nsince the constructions in Keister and Polyzou are not based on\nfield theory at all, so microcausality cannot even be formulated.\n\n\n> Also\n> Weinberg, for whom the cluster-decomposition property is one of the\n> important ingredients for model building, does not claim this, but he\n> says, he is not aware of other models than microcausal field theory\n> which fulfill the cluster-decomposition property.\n\nActually he says that there is no relativistic quantum mechanics for more\nthan 2 particles because of the cluster-decomposition problem, but here\nhe is wrong, as the above survey shows.\n\n\n> Have the above mentioned authors a model which is not microcausal but\n> fulfill the cluster-decomposition property? This would be interesting.\n\nYes.\n\n\n> What do you mean by the statement, that they do not need\n> renormalisation?\n\nWell, they only define effective few-particle models. There are no\nsingular interactions, hence no need for renormalization.\nIt is worth reading the survey (at least part of it - it is quite thick)\nThey construct useful phenomenological models, though somewhat limited;\nfor example, it is not clear how to incorporate external fields.\n\n\n> How can this be in an interacting field theory? Are there no\n> wave-function renormalisations? If so, is then Haag\'s theorem finally\n> proven wrong?\n\nNo. The models are _not_ field theories, only Poincare-invariant few-body\ndynamics with cluster decomposition and phenomenological terms\nwhich can be matched to approximate form factors from experiment or\nsome field theory. (Actually many-body dynamics also works, but the\nmany particle case is extremely messy.)\n\nThe papers by Klink at\nhttp://www.slac.stanford.edu/spires/find/hep/wwwcite?rawcmd=find+a+klink\nand work by Polyzou at\nhttp://www.physics.uiowa.edu/~wpolyzou/\ncontain lots of multiparticle relativistic quantum mechanics,\napplied to real particles.\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>Hendrik van Hees wrote:
> Arnold Neumaier wrote:
>
>>We need cluster decomposition because it is observed. We need
>>local fields and microcausality, mainly because it implies
>>(modulo fine print involving contact terms) at least perturbatively
>>cluster decomposition, and there is no other known way in QFT to
>>ensure the latter. But there are covariant N-particle models with
>>cluster decomposition, discussed, e.g., in
>> B.D. Keister and W.N. Polyzou,
>> Relativistic Hamiltonian Dynamics in Nuclear and Particle
>> Physics, in: Advances in Nuclear Physics, Volume 20,
>> (J. W. Negele and E.W. Vogt, eds.)
>> Plenum Press 1991.
>>(The constructions are quite messy; they have, however, the
>>advantage that they do not need renormalization, and are useful
>>phenomenological models.)
>
> Wait a moment. Of course, microcausality is sufficient for the
> cluster-decomposition property (at least in the perturbative sense),
> but I have never seen the proof that it is also necessary.
This is precisely what I state above. In fact it is _not_ necessary
since the constructions in Keister and Polyzou are not based on
field theory at all, so microcausality cannot even be formulated.
> Also
> Weinberg, for whom the cluster-decomposition property is one of the
> important ingredients for model building, does not claim this, but he
> says, he is not aware of other models than microcausal field theory
> which fulfill the cluster-decomposition property.
Actually he says that there is no relativistic quantum mechanics for more
than 2 particles because of the cluster-decomposition problem, but here
he is wrong, as the above survey shows.
> Have the above mentioned authors a model which is not microcausal but
> fulfill the cluster-decomposition property? This would be interesting.
Yes.
> What do you mean by the statement, that they do not need
> renormalisation?
Well, they only define effective few-particle models. There are no
singular interactions, hence no need for renormalization.
It is worth reading the survey (at least part of it - it is quite thick)
They construct useful phenomenological models, though somewhat limited;
for example, it is not clear how to incorporate external fields.
> How can this be in an interacting field theory? Are there no
> wave-function renormalisations? If so, is then Haag's theorem finally
> proven wrong?
No. The models are _not_ field theories, only Poincare-invariant few-body
dynamics with cluster decomposition and phenomenological terms
which can be matched to approximate form factors from experiment or
some field theory. (Actually many-body dynamics also works, but the
many particle case is extremely messy.)
The papers by Klink at
http://www.slac.stanford.edu/spires/find/hep/wwwcite?rawcmd=find+a+klink
and work by Polyzou at
http://www.physics.uiowa.edu/~wpolyzou/
contain lots of multiparticle relativistic quantum mechanics,
applied to real particles.
Arnold Neumaier
Arnold Neumaier
Oct26-04, 12:57 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>L.R. wrote:\n>\n> You can find it in Weinberg\'s "The Quantum Theory of Fields", volume\n> I, and there is some reference to the original paper created the word\n> "cluster decompostion". But I cannot find other books talk about this.\n> If cluster decomposition is a very important principle, why there are\n> so few books mentioned it, even in those qft books full of mathematics\n> you can hardly find this principle. So I began to wounder if it is\n> Weinberg\'s personal favor...\n\nYou may be interested in Weinberg\'s article hep-th/9702027,\nwhere he talks about the necessity of motivating QFT, and that he\npromoted cluster decomposition (which before was a special subject in\nscattering) to a didactical tool. The article is fairly self-contained\n(assuming only some general exposure to the QFT ideas such as free\nboson and fermion fields).\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:
>
> You can find it in Weinberg's "The Quantum Theory of Fields", volume
> I, and there is some reference to the original paper created the word
> "cluster decompostion". But I cannot find other books talk about this.
> If cluster decomposition is a very important principle, why there are
> so few books mentioned it, even in those qft books full of mathematics
> you can hardly find this principle. So I began to wounder if it is
> Weinberg's personal favor...
You may be interested in Weinberg's article http://www.arxiv.org/abs/hep-th/9702027,
where he talks about the necessity of motivating QFT, and that he
promoted cluster decomposition (which before was a special subject in
scattering) to a didactical tool. The article is fairly self-contained
(assuming only some general exposure to the QFT ideas such as free
boson and fermion fields).
Arnold Neumaier
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