View Full Version : Query - on the definition of 'OPE'
Katie Russell
Jun5-04, 01:19 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Evening all,\n\nAnyone got any idea what an OPE is? It keeps being mentioned around the\nsame time as the Virosoro algebra and vertex operators but there is no\ndefinition anywhere...not in GSW, any of the papers I\'ve skimmed which\nmention it (the main source I\'m reading is by Hoker - IASSNS-HEP-97/72)\nand Google yields very little that is helpful.\n\nI\'m thinking maybe Operator Product Expansion (my only other guess, One\nParticle Excitation, doesn\'t seem to make sense in this context.)\n\nThanks in advance to any older and wiser who can help.\nKatie\n\n[Moderator\'s note: Yes, Operator Product Expansion, a way\nto write the product of operators O_1(x)O_2(y)\nas a linear combination \\sum_i f_i(x-y) O_i(y);\nthe most important terms in an OPE are those that diverge\nas x approaches y, for example 1/|x-y|^n where n is a positive\npower, and these functions of x,y can be multiplied\nby an operator, or a c-number (ordinary commuting number). LM]\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>Evening all,
Anyone got any idea what an OPE is? It keeps being mentioned around the
same time as the Virosoro algebra and vertex operators but there is no
definition anywhere...not in GSW, any of the papers I've skimmed which
mention it (the main source I'm reading is by Hoker - IASSNS-HEP-97/72)
and Google yields very little that is helpful.
I'm thinking maybe Operator Product Expansion (my only other guess, One
Particle Excitation, doesn't seem to make sense in this context.)
Thanks in advance to any older and wiser who can help.
Katie
[Moderator's note: Yes, Operator Product Expansion, a way
to write the product of operators O_1(x)O_2(y)
as a linear combination \sum_i f_i(x-y) O_i(y);
the most important terms in an OPE are those that diverge
as x approaches y, for example 1/|x-y|^n where n is a positive
power, and these functions of x,y can be multiplied
by an operator, or a c-number (ordinary commuting number). LM]
Katie Russell
Jun6-04, 02:02 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>Lubos Motl wrote:\n\n> [Moderator\'s note: Yes, Operator Product Expansion, a way\n> to write the product of operators O_1(x)O_2(y)\n> as a linear combination \\sum_i f_i(x-y) O_i(y);\n> the most important terms in an OPE are those that diverge\n> as x approaches y, for example 1/|x-y|^n where n is a positive\n> power, and these functions of x,y can be multiplied\n> by an operator, or a c-number (ordinary commuting number). LM]\n\nThank you. I should add my signature seems to have a default setting -\nI\'m actually from the other Cambridge, and not from Harvard.\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>Lubos Motl wrote:
> [Moderator's note: Yes, Operator Product Expansion, a way
> to write the product of operators O_1(x)O_2(y)
> as a linear combination \sum_i f_i(x-y) O_i(y);
> the most important terms in an OPE are those that diverge
> as x approaches y, for example 1/|x-y|^n where n is a positive
> power, and these functions of x,y can be multiplied
> by an operator, or a c-number (ordinary commuting number). LM]
Thank you. I should add my signature seems to have a default setting -
I'm actually from the other Cambridge, and not from Harvard.
Charlie Stromeyer Jr.
Jun6-04, 09:21 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Katie Russell <kmr35@cam.ac.uk> wrote in message news:\n\n> (...) Thank you. (...)\n\nOf course, Lubos is correct but if you might be more interested in the\nmathematical rather than the physical apsects of string theory then\nyou may wish to take a look at OPE-algebras [1] which were first\ndescribed by the string theorists Anton Kapustin and Dmitri Orlov.\n\nSometimes, you might also find that the mathematics of string theory\nseems easier to understand than the physics because either some\nmathematical patterns or constructs will fit together well or they\nwill not, and because the physics often enough involves more\nspeculative ideas or intuitions which I myself have found to be\n(sometimes fiendishly) difficult to properly understand in a more\nrigorous way. Furthermore, I get the impression that the mathematical\nphysicist Peter Woit believes that some of these physics ideas may\neven be impossibly difficult to understand.\n\nHowever, I truly do not think that you, I or anyone else should become\ntoo upset by the difficulties of trying to understand string theory\nbecause I suspect that Woit\'s criticisms are too premature for very\nimportant reasons which I will now illustrate by using the example of\nAlbert Einstein:\n\nEinstein was able to use his kinesthetic intuition to conceive of his\nfamous elevator analogy which helped inspire and guide him with\nfurther developing GTR.\n\nHowever, the realm of quantum phenomena is too far removed from the\nrealm of experience of the human body which is one reason why string\ntheorists often have to turn more to mathematics than to what some\nmight think of as more physical inuitions. (Although, I did see a\nnewspaper article on the bulletin board of the MIT Science Library in\nwhich Brian Greene says that he thinks many of the concepts of string\ntheory should be translatable into intuitions which do not depend too\nmuch upon a prior knowledge of the underlying mathematical\nformalisms).\n\nIn addition, Einstein had two major advantages which string theorists\ndo not have:\n\n1) The mathematics that Einstein needed for making GTR rigorous had\nalready been developed previously by mathematicians such as Riemann\nand Minkowski. By constrast, it is quite obvious that the mathematics\nof string theory remains incomplete. I don\'t want to burden you with\ntoo many details but one example would be how to further extend the\nconcept of democracy for p-branes, e.g. if a string theorist wanted to\nknow how to parallel transport an arbitrarily given p-brane then which\nmathematical framework would he or she use.\n\n2) By the time Einstein was busy further developing GTR, he had\nalready established that he was the greatest theoretical physicist to\nthink about the subjects of gravity, space, time and particles since\nIsaac Newton.\n\n> I should add my signature seems to have a default setting -\n> I\'m actually from the other Cambridge, and not from Harvard.\n\nWell, in that case then please tell Bertrand Russell that I said "Hi"\n:-)\n\n\nhttp://arXiv.org/abs/math/0312313\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>Katie Russell <kmr35@cam.ac.uk> wrote in message news:
> (...) Thank you. (...)
Of course, Lubos is correct but if you might be more interested in the
mathematical rather than the physical apsects of string theory then
you may wish to take a look at OPE-algebras [1] which were first
described by the string theorists Anton Kapustin and Dmitri Orlov.
Sometimes, you might also find that the mathematics of string theory
seems easier to understand than the physics because either some
mathematical patterns or constructs will fit together well or they
will not, and because the physics often enough involves more
speculative ideas or intuitions which I myself have found to be
(sometimes fiendishly) difficult to properly understand in a more
rigorous way. Furthermore, I get the impression that the mathematical
physicist Peter Woit believes that some of these physics ideas may
even be impossibly difficult to understand.
However, I truly do not think that you, I or anyone else should become
too upset by the difficulties of trying to understand string theory
because I suspect that Woit's criticisms are too premature for very
important reasons which I will now illustrate by using the example of
Albert Einstein:
Einstein was able to use his kinesthetic intuition to conceive of his
famous elevator analogy which helped inspire and guide him with
further developing GTR.
However, the realm of quantum phenomena is too far removed from the
realm of experience of the human body which is one reason why string
theorists often have to turn more to mathematics than to what some
might think of as more physical inuitions. (Although, I did see a
newspaper article on the bulletin board of the MIT Science Library in
which Brian Greene says that he thinks many of the concepts of string
theory should be translatable into intuitions which do not depend too
much upon a prior knowledge of the underlying mathematical
formalisms).
In addition, Einstein had two major advantages which string theorists
do not have:
1) The mathematics that Einstein needed for making GTR rigorous had
already been developed previously by mathematicians such as Riemann
and Minkowski. By constrast, it is quite obvious that the mathematics
of string theory remains incomplete. I don't want to burden you with
too many details but one example would be how to further extend the
concept of democracy for p-branes, e.g. if a string theorist wanted to
know how to parallel transport an arbitrarily given p-brane then which
mathematical framework would he or she use.
2) By the time Einstein was busy further developing GTR, he had
already established that he was the greatest theoretical physicist to
think about the subjects of gravity, space, time and particles since
Isaac Newton.
> I should add my signature seems to have a default setting -
> I'm actually from the other Cambridge, and not from Harvard.
Well, in that case then please tell Bertrand Russell that I said "Hi"
:-)
http://arXiv.org/abs/math/0312313
Charlie Stromeyer Jr.
Jun6-04, 10:36 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Katie Russell <kmr35@cam.ac.uk> wrote in message news:\n\n> (...) Thank you. (...)\n\nOf course, Lubos is correct but if you might be more interested in the\nmathematical rather than the physical apsects of string theory then\nyou may wish to take a look at OPE-algebras [1] which were first\ndescribed by the string theorists Anton Kapustin and Dmitri Orlov.\n\nSometimes, you might also find that the mathematics of string theory\nseems easier to understand than the physics because either some\nmathematical patterns or constructs will fit together well or they\nwill not, and because the physics often enough involves more\nspeculative ideas or intuitions which I myself have found to be\n(sometimes fiendishly) difficult to properly understand in a more\nrigorous way. Furthermore, I get the impression that the mathematical\nphysicist Peter Woit believes that some of these physics ideas may\neven be impossibly difficult to understand.\n\nHowever, I truly do not think that you, I or anyone else should become\ntoo upset by the difficulties of trying to understand string theory\nbecause I suspect that Woit\'s criticisms are too premature for very\nimportant reasons which I will now illustrate by using the example of\nAlbert Einstein:\n\nEinstein was able to use his kinesthetic intuition to conceive of his\nfamous elevator analogy which helped inspire and guide him with\nfurther developing GTR.\n\nHowever, the realm of quantum phenomena is too far removed from the\nrealm of experience of the human body which is one reason why string\ntheorists often have to turn more to mathematics than to what some\nmight think of as more physical inuitions. (Although, I did see a\nnewspaper article on the bulletin board of the MIT Science Library in\nwhich Brian Greene says that he thinks many of the concepts of string\ntheory should be translatable into intuitions which do not depend too\nmuch upon a prior knowledge of the underlying mathematical\nformalisms).\n\nIn addition, Einstein had two major advantages which string theorists\ndo not have:\n\n1) The mathematics that Einstein needed for making GTR rigorous had\nalready been developed previously by mathematicians such as Riemann\nand Minkowski. By constrast, it is quite obvious that the mathematics\nof string theory remains incomplete. I don\'t want to burden you with\ntoo many details but one example would be how to further extend the\nconcept of democracy for p-branes, e.g. if a string theorist wanted to\nknow how to parallel transport an arbitrarily given p-brane then which\nmathematical framework would he or she use.\n\n2) By the time Einstein was busy further developing GTR, he had\nalready established that he was the greatest theoretical physicist to\nthink about the subjects of gravity, space, time and particles since\nIsaac Newton.\n\n> I should add my signature seems to have a default setting -\n> I\'m actually from the other Cambridge, and not from Harvard.\n\nWell, in that case then please tell Bertrand Russell that I said "Hi"\n:-)\n\n\nhttp://arXiv.org/abs/math/0312313\n\n\n[Posted for the 2nd time because of continuing problems with the\nFAS newsserver. LM]\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>Katie Russell <kmr35@cam.ac.uk> wrote in message news:
> (...) Thank you. (...)
Of course, Lubos is correct but if you might be more interested in the
mathematical rather than the physical apsects of string theory then
you may wish to take a look at OPE-algebras [1] which were first
described by the string theorists Anton Kapustin and Dmitri Orlov.
Sometimes, you might also find that the mathematics of string theory
seems easier to understand than the physics because either some
mathematical patterns or constructs will fit together well or they
will not, and because the physics often enough involves more
speculative ideas or intuitions which I myself have found to be
(sometimes fiendishly) difficult to properly understand in a more
rigorous way. Furthermore, I get the impression that the mathematical
physicist Peter Woit believes that some of these physics ideas may
even be impossibly difficult to understand.
However, I truly do not think that you, I or anyone else should become
too upset by the difficulties of trying to understand string theory
because I suspect that Woit's criticisms are too premature for very
important reasons which I will now illustrate by using the example of
Albert Einstein:
Einstein was able to use his kinesthetic intuition to conceive of his
famous elevator analogy which helped inspire and guide him with
further developing GTR.
However, the realm of quantum phenomena is too far removed from the
realm of experience of the human body which is one reason why string
theorists often have to turn more to mathematics than to what some
might think of as more physical inuitions. (Although, I did see a
newspaper article on the bulletin board of the MIT Science Library in
which Brian Greene says that he thinks many of the concepts of string
theory should be translatable into intuitions which do not depend too
much upon a prior knowledge of the underlying mathematical
formalisms).
In addition, Einstein had two major advantages which string theorists
do not have:
1) The mathematics that Einstein needed for making GTR rigorous had
already been developed previously by mathematicians such as Riemann
and Minkowski. By constrast, it is quite obvious that the mathematics
of string theory remains incomplete. I don't want to burden you with
too many details but one example would be how to further extend the
concept of democracy for p-branes, e.g. if a string theorist wanted to
know how to parallel transport an arbitrarily given p-brane then which
mathematical framework would he or she use.
2) By the time Einstein was busy further developing GTR, he had
already established that he was the greatest theoretical physicist to
think about the subjects of gravity, space, time and particles since
Isaac Newton.
> I should add my signature seems to have a default setting -
> I'm actually from the other Cambridge, and not from Harvard.
Well, in that case then please tell Bertrand Russell that I said "Hi"
:-)
http://arXiv.org/abs/math/0312313
[Posted for the 2nd time because of continuing problems with the
FAS newsserver. LM]
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