View Full Version : How to calculate the energy-momentum tensor of the superstring?
Rene Meyer
Apr27-04, 10:37 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello,\n\nGSW calculate the energy-momentum tensor of the bosonic string in\nchapter two of their book through the formula\n\nT_ab = -2/T 1/sqrt(g) delta S/ delta g^ab\n\nBut in chapter 4, p. 189, they use a reparametrization of the world-sheet\ncoordinates. Actually, it seems not very clear to me why one would use\ntwo different methods, and apparently get two different results.\nIf I use the ungauged Polyakov action\n\nS = -T/2 Int [d^2xi sqrt(g) ( g^ab d_ax d_ax -i psibar g^ab rho_a d_b psi)]\n\n(with spacetime indices not written), I get through the above formula\n\nT_ab = d_a x d_b x -1/2 g_ab d_cx d^c x\n-i/2 psibar (rho_a d_b + rho_b d_a) psi\n+i/2 psibar rho^a d_a psi\n\nI didn\'t check the result (4.1.14) of GSW yet, but it reads\n\nT_ab = d_axd_bx +i/4 psibar(rho_a d_b + rho_b d_a)psi - (trace)\n\nCan someone explain to me why my result is different?\n\nRené.\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\n\n_________________________________________ ______________________________________\nWeb page of SPS: http://schwinger.harvard.edu/~sps/\nPosted via: http://groups.google.com/groups?group=sci.physics.strings\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>Hello,
GSW calculate the energy-momentum tensor of the bosonic string in
chapter two of their book through the formula
T_{ab} = -2/T 1/\sqrt(g) \delta S/ \delta g^{ab}
But in chapter 4, p. 189, they use a reparametrization of the world-sheet
coordinates. Actually, it seems not very clear to me why one would use
two different methods, and apparently get two different results.
If I use the ungauged Polyakov action
S = -T/2 \Int [d^{2xi} \sqrt(g) ( g^{ab} d_{ax} d_{ax} -i[/itex] psibar g^{ab} \rho_a d_b \psi)]
(with spacetime indices not written), I get through the above formula
T_{ab} = d_a x d_b x -1/2 g_{ab} d_{cx} d^c x-i/2 psibar (\rho_a d_b + \rho_b d_a) \psi+i/2 psibar [itex]\rho^a d_a \psi
I didn't check the result (4.1.14) of GSW yet, but it reads
T_{ab} = d_{axd_bx} +i/4 psibar(\rho_a d_b + \rho_b d_a)\psi - (trace)
Can someone explain to me why my result is different?
René.
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
__{_______________________________________________ ______________________________}
Web page of SPS: http://schwinger.harvard.edu/~sps/
Posted via: http://groups.google.com/groups?group=sci.physics.strings
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Lubos Motl
Apr27-04, 10:58 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>On Tue, 27 Apr 2004, Rene Meyer wrote:\n\n> GSW calculate the energy-momentum tensor of the bosonic string in\n> chapter two of their book through the formula\n\nI think that the chapter 2 is the bosonic string, while the chapter 4 is\nthe superstring, is not it? The latter stress energy tensor seems to\ninclude some fermions psi, so it does not look like the bosonic string,\ndoes it? Well, two different theories gives rise to different stress\nenergy tensors.\n\n> T_ab = -2/T 1/sqrt(g) delta S/ delta g^ab (###)\n\nThis is the usual definition, but sometimes one can encounter other\ndefinitions of T_{ab} that differ by a tensor M_{ab} that is automatically\nconserved, d_a M^{ab}=0. Note that (###) is symmetric, because the metric\ng_{ab} is, but people sometimes work with asymmetric stress energy\ntensors.\n\nWhen I say that (###) is the usual definition, it does not mean that it is\nthe only way how to calculate it. Cheers, Lubos\n___________________________________________ ___________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\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>On Tue, 27 Apr 2004, Rene Meyer wrote:
> GSW calculate the energy-momentum tensor of the bosonic string in
> chapter two of their book through the formula
I think that the chapter 2 is the bosonic string, while the chapter 4 is
the superstring, is not it? The latter stress energy tensor seems to
include some fermions \psi, so it does not look like the bosonic string,
does it? Well, two different theories gives rise to different stress
energy tensors.
> T_{ab} = -2/T 1/\sqrt(g) \delta S/ \delta g^{ab} (###)
This is the usual definition, but sometimes one can encounter other
definitions of T_{ab} that differ by a tensor M_{ab} that is automatically
conserved, d_a M^{ab}=0. Note that (###) is symmetric, because the metric
g_{ab} is, but people sometimes work with asymmetric stress energy
tensors.
When I say that (###) is the usual definition, it does not mean that it is
the only way how to calculate it. Cheers, Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Rene Meyer
Apr27-04, 09:07 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>Dear Lubos,\n\n> I think that the chapter 2 is the bosonic string, while the chapter 4 is\n> the superstring, is not it? The latter stress energy tensor seems to\n> include some fermions psi, so it does not look like the bosonic string,\n> does it? Well, two different theories gives rise to different stress\n> energy tensors.\n\nI am aware of that. But as the definition (###) is linear, because of\nthe linearity of the functional derivative, one should expect, that\nthe bosonic part is the same, which is indeed the case. What worries\nme is the fermionic part, which I calculate differently from GSW.\n\n>> T_ab = -2/T 1/sqrt(g) delta S/ delta g^ab (###)\n> This is the usual definition, but sometimes one can encounter other\n> definitions of T_{ab} that differ by a tensor M_{ab} that is automatically\n> conserved, d_a M^{ab}=0. Note that (###) is symmetric, because the metric\n> g_{ab} is, but people sometimes work with asymmetric stress energy\n> tensors.\n\nOK, I\'ll bear this in mind, and again have a look at the formulas.\n\nThanks,\nRené.\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\n\n_________________________________________ ______________________________________\nWeb page of SPS: http://schwinger.harvard.edu/~sps/\nPosted via: http://groups.google.com/groups?group=sci.physics.strings\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>Dear Lubos,
> I think that the chapter 2 is the bosonic string, while the chapter 4 is
> the superstring, is not it? The latter stress energy tensor seems to
> include some fermions \psi, so it does not look like the bosonic string,
> does it? Well, two different theories gives rise to different stress
> energy tensors.
I am aware of that. But as the definition (###) is linear, because of
the linearity of the functional derivative, one should expect, that
the bosonic part is the same, which is indeed the case. What worries
me is the fermionic part, which I calculate differently from GSW.
>> T_{ab} = -2/T 1/\sqrt(g) \delta S/ \delta g^{ab} (###)
> This is the usual definition, but sometimes one can encounter other
> definitions of T_{ab} that differ by a tensor M_{ab} that is automatically
> conserved, d_a M^{ab}=0. Note that (###) is symmetric, because the metric
> g_{ab} is, but people sometimes work with asymmetric stress energy
> tensors.
OK, I'll bear this in mind, and again have a look at the formulas.
Thanks,
René.
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
__{_______________________________________________ ______________________________}
Web page of SPS: http://schwinger.harvard.edu/~sps/
Posted via: http://groups.google.com/groups?group=sci.physics.strings
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Rene Meyer
Apr28-04, 07:17 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 27 Apr 2004 10:37:19 -0400, Rene Meyer wrote:\n\n> I didn\'t check the result (4.1.14) of GSW yet, but it reads\n> T_ab = d_axd_bx +i/4 psibar(rho_a d_b + rho_b d_a)psi - (trace)\n\nI just tried to check the result given by GSW, and actually I still\ncan\'t reproduce the sign of the fermionic part of the stress energy\ntensor,\n\n+i/4....\n\nI get a negative sign, because in the Lagrangian d_axda^x -i psibar\ndslash psi, the fermionic contribution has a negative sign. Thus,\napplying a infinitesimal transformation delta sigma = a, one gets\nfor the fields\n\nx\'(sigma) - x(sigma) = -a^a d_a x(sigma)\npsi\' - psi = -a^a d_a psi\n\nThus, when calculating delta S, the terms from the bosonic and\nfermionic part of the action don\'t change their relative sign, and\nthus the bosonic and fermionic contributions to T_ab must be of\ndifferent sign, as there are no other integrations by part involved in\nmy calculation.\n\nDid anyone actually calculate the result of GSW himself and could tell\nme where I am wrong (assuming there is no typo in the book)?\n\n[Moderator\'s note: I don\'t have GSW here, but my guess is that the minus\nsign might come from the fact that the fermions anticommute, which you\nmight have neglected, or perhaps you neglected a sign from the\nintegration by parts, and some factors of two or four might arise from\nsome spinor identities. LM]\n\nRené.\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 27 Apr 2004 10:37:19 -0400, Rene Meyer wrote:
> I didn't check the result (4.1.14) of GSW yet, but it reads
> T_{ab} = d_{axd_bx} +i/4 psibar(\rho_a d_b + \rho_b d_a)\psi - (trace)
I just tried to check the result given by GSW, and actually I still
can't reproduce the sign of the fermionic part of the stress energy
tensor,
+i/4....
I get a negative sign, because in the Lagrangian d_{axda}^x -i psibar
dslash \psi, the fermionic contribution has a negative sign. Thus,
applying a infinitesimal transformation \delta \sigma = a, one gets
for the fields
x'(\sigma) - x(\sigma) = -a^a d_a x(\sigma)\psi' - \psi = -a^a d_a \psi
Thus, when calculating \delta S, the terms from the bosonic and
fermionic part of the action don't change their relative sign, and
thus the bosonic and fermionic contributions to T_{ab} must be of
different sign, as there are no other integrations by part involved in
my calculation.
Did anyone actually calculate the result of GSW himself and could tell
me where I am wrong (assuming there is no typo in the book)?
[Moderator's note: I don't have GSW here, but my guess is that the minus
sign might come from the fact that the fermions anticommute, which you
might have neglected, or perhaps you neglected a sign from the
integration by parts, and some factors of two or four might arise from
some spinor identities. LM]
René.
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Rene Meyer
Apr28-04, 07:27 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 27 Apr 2004 10:58:35 -0400, Lubos Motl wrote:\n\n>> T_ab = -2/T 1/sqrt(g) delta S/ delta g^ab (###)\n> This is the usual definition, but sometimes one can encounter other\n> definitions of T_{ab} that differ by a tensor M_{ab} that is automatically\n> conserved, d_a M^{ab}=0. Note that (###) is symmetric, because the metric\n> g_{ab} is, but people sometimes work with asymmetric stress energy\n> tensors.\n\nWhen working with such a shifted stress energy tensor, beside being\nallowed by the Noether procedure alone, will doesn\'t this have any\neffect on the super Virasoro constraints? These constraints consist in\nsetting T = J = 0, so wouldn\'t a shift of T contradict with this?\n\nRené.\n\n[Moderator\'s note: When I wrote that sometimes one allows different forms\nof the stress energy tensor, I was talking about a very general context,\nnot about the particular case of the Virasoro constraints in string theory.\n(For example, in classical electrodynamics one often has two choices, but\nin QED we don\'t require this tensor to vanish.\n\nIn string theory, we always uniquely know what the stress energy tensor\nis, and the correct Virasoro constraints are *always* saying that the\nvariation of the action with respect to the worldsheet metric must\nvanish - because the Virasoro constraint are nothing else than the\nequations of motion (Einstein\'s equations) that one can obtain by\nvariation of the action with respect to the worldsheet metric. The term\nR_{ab} - 1/2.R.g_{ab} can be omitted in two dimensions, because in\ntwo dimensions it vanishes identically. Nevertheless you can also include\nit - it is zero. The Virasoro constraints *are* Einstein\'s equations.\nIncidentally, this fact is one of the many reasons why Thiemann\'s paper\nabout the algebraic quantization of string theory is not physically\ncorrect: he imposes the Virasoro constraints even though there is no\nworldsheet metric, and therefore these constraints are not implied by\nanything.\n\nIf you required a modified tensor to vanish, it would generally change\nyour physics, and you certainly don\'t want to do it. In QED, an\nanalogous change could only affect Einstein\'s equations, once you\ncouple QED to gravity - because gravity *feels* the stress energy\ntensor. Such modified stress energy tensors can often be expressed as\nstandard stress energy tensors (variation with respect to metric)\narising from a modified action with additional couplings to gravity\n(the metric). LM]\n\nRené.\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\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>On Tue, 27 Apr 2004 10:58:35 -0400, Lubos Motl wrote:
>> T_{ab} = -2/T 1/\sqrt(g) \delta S/ \delta g^{ab} (###)
> This is the usual definition, but sometimes one can encounter other
> definitions of T_{ab} that differ by a tensor M_{ab} that is automatically
> conserved, d_a M^{ab}=0. Note that (###) is symmetric, because the metric
> g_{ab} is, but people sometimes work with asymmetric stress energy
> tensors.
When working with such a shifted stress energy tensor, beside being
allowed by the Noether procedure alone, will doesn't this have any
effect on the super Virasoro constraints? These constraints consist in
setting T = J = 0, so wouldn't a shift of T contradict with this?
René.
[Moderator's note: When I wrote that sometimes one allows different forms
of the stress energy tensor, I was talking about a very general context,
not about the particular case of the Virasoro constraints in string theory.
(For example, in classical electrodynamics one often has two choices, but
in QED we don't require this tensor to vanish.
In string theory, we always uniquely know what the stress energy tensor
is, and the correct Virasoro constraints are *always* saying that the
variation of the action with respect to the worldsheet metric must
vanish - because the Virasoro constraint are nothing else than the
equations of motion (Einstein's equations) that one can obtain by
variation of the action with respect to the worldsheet metric. The term
R_{ab} - 1/2.R.g_{ab} can be omitted in two dimensions, because in
two dimensions it vanishes identically. Nevertheless you can also include
it - it is zero. The Virasoro constraints *are* Einstein's equations.
Incidentally, this fact is one of the many reasons why Thiemann's paper
about the algebraic quantization of string theory is not physically
correct: he imposes the Virasoro constraints even though there is no
worldsheet metric, and therefore these constraints are not implied by
anything.
If you required a modified tensor to vanish, it would generally change
your physics, and you certainly don't want to do it. In QED, an
analogous change could only affect Einstein's equations, once you
couple QED to gravity - because gravity *feels* the stress energy
tensor. Such modified stress energy tensors can often be expressed as
standard stress energy tensors (variation with respect to metric)
arising from a modified action with additional couplings to gravity
(the metric). LM]
René.
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Rene Meyer
Apr28-04, 07:37 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 27 Apr 2004 10:58:35 -0400, Lubos Motl wrote:\n\n>> T_ab = -2/T 1/sqrt(g) delta S/ delta g^ab (###)\n> This is the usual definition, but sometimes one can encounter other\n> definitions of T_{ab} that differ by a tensor M_{ab} that is automatically\n> conserved, d_a M^{ab}=0.\n\nMy result obviously differs from that of GSW by a symmetric tensor\n\n3/4i { psibar(rho_a d_b + rho_b d_a)psi - g_ab psibar rho^c d_c psi }\n\nThe last term vanishes immediately because of the Dirac equation\nrho^c d_c psi = 0. But I have problems to see that the first term is\nconserved:\n\nd^a { psibar(rho_a d_b + rho_b d_a)psi }\n= (d^a psibar) rho_a d_b psi + psibar rho_a d^a d_b psi\n+ (d^a psibar) rho_b d_a psi + psibar rho_b d^a d_a psi\n\nThe first two terms vanish, but do the second two? The problem also\narises when showing that d^a T_ab = 0, and it doesn\'t look like the\nequations of motion rho^a d_a psi = 0 and (d_a psibar) rho^a = 0 would\nimply the last two terms being zero, just because rho_b having the\nwrong index. How to show this?\n\nRené.\n\n[Moderator\'s note: The last, 4th term actually also vanishes on-shell\nbecause satisfying Dirac\'s equation implies satisfying Klein-Gordon\nequation - "d slash" squared equals "box". It seems likely that the\nthird term in the conservation law does not vanish - the only possible\nreason for vanishing would be that it has the form "d_a psi" squared,\nwhich is the same fermion squared, which would vanish, but you should\ncheck the subtleties with the "bar" above "psi", and with "rho_b"\ninside. Well, if it does not vanish, then I think that you have just\nproved that you had to make an error. My guess is that GSW don\'t have\nthese "-1/4" errors in such fundamental equations, but I don\'t\nhave time to check it. LM]\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\n\n\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 27 Apr 2004 10:58:35 -0400, Lubos Motl wrote:
>> T_{ab} = -2/T 1/\sqrt(g) \delta S/ \delta g^{ab} (###)[/itex]
> This is the usual definition, but sometimes one can encounter other
> definitions of T_{ab} that differ by a tensor M_{ab} that is automatically
> conserved, d_a M^{ab}=0.
My result obviously differs from that of GSW by a symmetric tensor
3/4i { psibar(\rho_a d_b + \rho_b d_a)\psi - g_{ab} psibar \rho^c d_c \psi }
The last term vanishes immediately because of the Dirac equation
\rho^c d_c \psi = . But I have problems to see that the first term is
conserved:
d^a { psibar(\rho_a d_b + \rho_b d_a)\psi }= (d^a psibar) \rho_a d_b \psi + psibar \rho_a d^a d_b \psi+ (d^a psibar) \rho_b d_a \psi + psibar [itex]\rho_b d^a d_a \psi
The first two terms vanish, but do the second two? The problem also
arises when showing that d^a T_{ab} = 0, and it doesn't look like the
equations of motion \rho^a d_a \psi = and (d_a psibar) \rho^a = would
imply the last two terms being zero, just because \rho_b having the
wrong index. How to show this?
René.
[Moderator's note: The last, 4th term actually also vanishes on-shell
because satisfying Dirac's equation implies satisfying Klein-Gordon
equation - "d slash" squared equals "box". It seems likely that the
third term in the conservation law does not vanish - the only possible
reason for vanishing would be that it has the form "d_a \psi" squared,
which is the same fermion squared, which would vanish, but you should
check the subtleties with the "bar" above "\psi", and with "\rho_b"
inside. Well, if it does not vanish, then I think that you have just
proved that you had to make an error. My guess is that GSW don't have
these "-1/4" errors in such fundamental equations, but I don't
have time to check it. LM]
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Rene Meyer
Apr30-04, 09:36 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 28 Apr 2004 07:27:51 -0400, Rene Meyer wrote:\n> In string theory, we always uniquely know what the stress energy tensor\n> is, and the correct Virasoro constraints are *always* saying that the\n> variation of the action with respect to the worldsheet metric must\n> vanish because the Virasoro constraint are nothing else than the\n> equations of motion (Einstein\'s equations) that one can obtain by\n> variation of the action with respect to the worldsheet metric.\n\nDear Lubos,\n\nthank you for your answer.\n\nIsn\'t what you write above a contradiction to what we discussed\nbefore in this thread? The result calculated from the variation\nw.r.t. the metric is different than the one calculated with the\nmethod given by GSW, and actually GSW use the latter stress\nenergy tensor for the Virasoro constraints. I am somewhat confused,\nbecause I don\'t know what the right Einsteins equations are.\n\n> The term\n> R_{ab} - 1/2.R.g_{ab} can be omitted in two dimensions, because in\n> two dimensions it vanishes identically.\n\nSo Einsteins equations in two dimensions are equal to the Virasoro\nconstraints, just because the Einstein tensor vanishes in 2D. But I\ndon\'t see the connection to my problem.\n\n> Incidentally, this fact is one of the many reasons why Thiemann\'s paper\n> about the algebraic quantization of string theory is not physically\n> correct: he imposes the Virasoro constraints even though there is no\n> worldsheet metric, and therefore these constraints are not implied by\n> anything.\n\nWhat paper are you speaking of? I didn\'t read it.\n\n> Such modified stress energy tensors can often be expressed as\n> standard stress energy tensors (variation with respect to metric)\n> arising from a modified action with additional couplings to gravity\n> (the metric). LM]\n\nOK. But if it is like this, I don\'t understand why GSW don\'t mention\nthe modified action. Actually, I am not THIS firm within QFT that I\nwould be able to think of how this modified action would look like.\n\nRené.\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\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>On Wed, 28 Apr 2004 07:27:51 -0400, Rene Meyer wrote:
> In string theory, we always uniquely know what the stress energy tensor
> is, and the correct Virasoro constraints are *always* saying that the
> variation of the action with respect to the worldsheet metric must
> vanish because the Virasoro constraint are nothing else than the
> equations of motion (Einstein's equations) that one can obtain by
> variation of the action with respect to the worldsheet metric.
Dear Lubos,
thank you for your answer.
Isn't what you write above a contradiction to what we discussed
before in this thread? The result calculated from the variation
w.r.t. the metric is different than the one calculated with the
method given by GSW, and actually GSW use the latter stress
energy tensor for the Virasoro constraints. I am somewhat confused,
because I don't know what the right Einsteins equations are.
> The term
> R_{ab} - 1/2.R.g_{ab} can be omitted in two dimensions, because in
> two dimensions it vanishes identically.
So Einsteins equations in two dimensions are equal to the Virasoro
constraints, just because the Einstein tensor vanishes in 2D. But I
don't see the connection to my problem.
> Incidentally, this fact is one of the many reasons why Thiemann's paper
> about the algebraic quantization of string theory is not physically
> correct: he imposes the Virasoro constraints even though there is no
> worldsheet metric, and therefore these constraints are not implied by
> anything.
What paper are you speaking of? I didn't read it.
> Such modified stress energy tensors can often be expressed as
> standard stress energy tensors (variation with respect to metric)
> arising from a modified action with additional couplings to gravity
> (the metric). LM]
OK. But if it is like this, I don't understand why GSW don't mention
the modified action. Actually, I am not THIS firm within QFT that I
would be able to think of how this modified action would look like.
René.
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Urs Schreiber
Apr30-04, 10:40 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>"Rene Meyer" <meyr2@web.de> schrieb im Newsbeitrag\nnews:c6tf45\\$fit88\\$1-100000@ID-193047.news.uni-berlin.de...\n\n> > Incidentally, this fact is one of the many reasons why Thiemann\'s paper\n> > about the algebraic quantization of string theory is not physically\n> > correct: he imposes the Virasoro constraints even though there is no\n> > worldsheet metric, and therefore these constraints are not implied by\n> > anything.\n>\n> What paper are you speaking of? I didn\'t read it.\n\nLubos is referring to the paper discussed in some detail here:\n\nhttp://golem.ph.utexas.edu/string/archives/000299.html .\n\nIn this paper the classical Virasoro gauge group of the string (not its\ngenerators, though) is, rather arbitrarily, represented on some Hilbert\nspace. In this representation there is no sign left of the background that\nthe string propagates in, since only the abstract gauge group is considered.\n\nI believe that if we did the alaog of the LQG-string for the point particle\nwe would see that the point particle\'s single constraint (the Klein-Gordon\nconstraint) generates the group U(1) and that hence any rep of U(1) is a\n\'quantization\' of the Klein-Gordon particle. That\'s weird, and I bet people\nworking on LQG would not accept this as a valid quantization, but still it\nis precisely the same proedure used in the LQG-string and for the spatial\ndiffeo constraints in full 3+1 LQG\n\nSince Thomas Thiemann has announced a followup paper wherein strings in\nnontrivial background shall be treated similarly, I am wondering how this is\nsupposed to work:\n\nhttp://golem.ph.utexas.edu/string/archives/000330.html#c000786 .\n\nBe warned that the "LQG-string" is rather controversial. But if one is\ninterested in understanding what LQG is all about then I think it is an\nextremely valuable toy example of the methods used there. For instance, one\ncan see much clearer than in the full theory that it is not really about\ncanonical quantization in the usual sense:\n\nhttp://groups.google.de/groups?selm=c3p23d%242au24e%241%40ID-168578.news.uni-berlin.de&oe=UTF-8\n\nBut we are not supposed to talk about LQG here. I just pointed this out\nbecause Lubos mentioned it.\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>"Rene Meyer" <meyr2@web.de> schrieb im Newsbeitrag
news:c6tf45$fit88$1-100000@ID-193047.news.uni-berlin.de...
> > Incidentally, this fact is one of the many reasons why Thiemann's paper
> > about the algebraic quantization of string theory is not physically
> > correct: he imposes the Virasoro constraints even though there is no
> > worldsheet metric, and therefore these constraints are not implied by
> > anything.
>
> What paper are you speaking of? I didn't read it.
Lubos is referring to the paper discussed in some detail here:
http://golem.ph.utexas.edu/string/archives/000299.html .
In this paper the classical Virasoro gauge group of the string (not its
generators, though) is, rather arbitrarily, represented on some Hilbert
space. In this representation there is no sign left of the background that
the string propagates in, since only the abstract gauge group is considered.
I believe that if we did the alaog of the LQG-string for the point particle
we would see that the point particle's single constraint (the Klein-Gordon
constraint) generates the group U(1) and that hence any rep of U(1) is a
'quantization' of the Klein-Gordon particle. That's weird, and I bet people
working on LQG would not accept this as a valid quantization, but still it
is precisely the same proedure used in the LQG-string and for the spatial
diffeo constraints in full 3+1 LQG
Since Thomas Thiemann has announced a followup paper wherein strings in
nontrivial background shall be treated similarly, I am wondering how this is
supposed to work:
http://golem.ph.utexas.edu/string/archives/000330.html#c000786 .
Be warned that the "LQG-string" is rather controversial. But if one is
interested in understanding what LQG is all about then I think it is an
extremely valuable toy example of the methods used there. For instance, one
can see much clearer than in the full theory that it is not really about
canonical quantization in the usual sense:
http://groups.google.de/groups?selm=c3p23d%242au24e%241%40ID-168578.news.uni-berlin.de&oe=UTF-8
But we are not supposed to talk about LQG here. I just pointed this out
because Lubos mentioned it.
Urs Schreiber
Apr30-04, 11:45 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I wrote\n\n[...]\n\n> Be warned that the "LQG-string" is rather controversial. But if one is\n\nSince this here is a string discussion group I should say more on what this\nall _could_ have to do with string theory.\n\nAs you (Rene) know, the quantum effects in the usual quantization of the\nPolyakov action have drastic implications, such as fixing the allowed number\nof spacetime dimensions.\n\nNow, some people argue that first quantization, which takes you from a\nclassical action to a quantum theory, is not a procedure with unique\nresults. Mostly, the non-uniquenes of first quantization can be attributed\nto operator ordering ambiguities. However, these are actually fixed in\nstring theory, by consistency conditions: The ordering constant in L_0 is\nnot arbitrary.\n\nStill, some people believe that there might be other procedures possible for\nthe Polyakov/Nambu-Goto action that should be called \'quantization\' but\nwhich would yield different theories than the usual LCG or OCQ or BRST or\npath integral quantization of the string.\n\nFor instance when you look at the discussion at\n\nhttp://golem.ph.utexas.edu/string/archives/000338.html\n\nyou\'ll see some of the arguments and names mentioned.\n\nThe most prominent approach at an \'alternative\' quantization is that\ninitiated by K. Pohlmeyer:\n\nhttp://golem.ph.utexas.edu/string/archives/000300.html .\n\nPohlmeyer decided to study the Poisson algebra of _classical_ invariants of\nthe string (this are all the classical functions on phase space which\nPoisson commute with the classical Virasoro constraints) and to then\nquantize the string by finding a consistent deformation of this classical\nPoisson algebra to a commutator algebra of operators.\n\nIn order to do so, Pohlmeyer, and others working in this field, picked a\n(from the traditional point of view very unusual) basis of classical\ninvariants, which are obtained in terms of Wilson loops of large N constant\ngauge connections along the string. These are called the "Pohlmeyer\ninvariants".\n\nOne can work out the classical Poisson algebra of these invariants and it\nturns out to be very involved. For about 20 years Pohlmeyer, K.-H. Rehren\nand others have tried to find a consistent quantum deformation of this\nalgebra.\n\nThe hope of the people working in the field is, as far as I am aware at\nleast, that, if this consistent quantization could be found, it would _not_\nrequire the usual critical dimension for consistency.\n\nHowever, all attempts to find such a consistent algebra deformation had\nfailed. While K.-H. Reheren made a lot of progress, his approach still\ncontains an as yet unproven conjecture called the "quadratic generation\nhypothesis". I don\'t know how likely it is that this conjecture is true. But\nin any case, there is no proof yet.\n\nA while ago I noticed that (and how precisely) the Pohlmeyer invariants are\nclassically just a subset of all classical DDF invariants. DDF operators are\nused all over the place in string theory and are fully understood. My claim\nis that if you write down the classical Pohlmeyer invariants in terms of\nclassical DDF invariants and then quantize the latter the usual way, that\nyou actually solve the Pohlmeyer program (since the algebra of the DDF\ninvariants does close and everything is consistent) and that in particular\nthe usual critical dimension and everything shows up this way.\n\nPointers to the above mentioned literature as well as the Pohlmeyer/DDF\nconstruction can be found in\n\nhttp://arXiv.org/abs/hep-th/0403260 .\n\nIt has been argued\n\nhttp://golem.ph.utexas.edu/string/archives/000346.html\n\nthat my construction does not solve the Pohlmeyer program, due to an\napparent problem with Lorentz invariance, because of the fact that the\nconstruction of the DDF invariants requires to fix an arbitrary lightlike\nvector field on target space.\n\nI don\'t think that this is really an issue, as explained here:\n\nhttp://www-stud.uni-essen.de/~sb0264/p6.pdf .\n\nTo my mind the Pohlmeyer program of quantization of the string by means of\ndeforming its algebra of invariants can be solved in at least one way,\nnamely using DDF operators, which in particular does yield just the ordinary\nquantization, albeit from an unusual perspective.\n\nOn the other hand, one can always argue that a further, different,\nconsistent deformation of the algebra of invariants to a commutator algebra\nof operators is possible, one which does not have the critical dimension. I\ncannot prove that this is impossible, but I would be surprised if such an\nalternative does exist, in light of the fact that all other approaches\n(LCG/OCQ/BRST/PI/DDF) yield the same standard result.\n\n\n\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I wrote
[...]
> Be warned that the "LQG-string" is rather controversial. But if one is
Since this here is a string discussion group I should say more on what this
all _could_ have to do with string theory.
As you (Rene) know, the quantum effects in the usual quantization of the
Polyakov action have drastic implications, such as fixing the allowed number
of spacetime dimensions.
Now, some people argue that first quantization, which takes you from a
classical action to a quantum theory, is not a procedure with unique
results. Mostly, the non-uniquenes of first quantization can be attributed
to operator ordering ambiguities. However, these are actually fixed in
string theory, by consistency conditions: The ordering constant in L_0 is
not arbitrary.
Still, some people believe that there might be other procedures possible for
the Polyakov/Nambu-Goto action that should be called 'quantization' but
which would yield different theories than the usual LCG or OCQ or BRST or
path integral quantization of the string.
For instance when you look at the discussion at
http://golem.ph.utexas.edu/string/archives/000338.html
you'll see some of the arguments and names mentioned.
The most prominent approach at an 'alternative' quantization is that
initiated by K. Pohlmeyer:
http://golem.ph.utexas.edu/string/archives/000300.html .
Pohlmeyer decided to study the Poisson algebra of _classical_ invariants of
the string (this are all the classical functions on phase space which
Poisson commute with the classical Virasoro constraints) and to then
quantize the string by finding a consistent deformation of this classical
Poisson algebra to a commutator algebra of operators.
In order to do so, Pohlmeyer, and others working in this field, picked a
(from the traditional point of view very unusual) basis of classical
invariants, which are obtained in terms of Wilson loops of large N constant
gauge connections along the string. These are called the "Pohlmeyer
invariants".
One can work out the classical Poisson algebra of these invariants and it
turns out to be very involved. For about 20 years Pohlmeyer, K.-H. Rehren
and others have tried to find a consistent quantum deformation of this
algebra.
The hope of the people working in the field is, as far as I am aware at
least, that, if this consistent quantization could be found, it would _not_
require the usual critical dimension for consistency.
However, all attempts to find such a consistent algebra deformation had
failed. While K.-H. Reheren made a lot of progress, his approach still
contains an as yet unproven conjecture called the "quadratic generation
hypothesis". I don't know how likely it is that this conjecture is true. But
in any case, there is no proof yet.
A while ago I noticed that (and how precisely) the Pohlmeyer invariants are
classically just a subset of all classical DDF invariants. DDF operators are
used all over the place in string theory and are fully understood. My claim
is that if you write down the classical Pohlmeyer invariants in terms of
classical DDF invariants and then quantize the latter the usual way, that
you actually solve the Pohlmeyer program (since the algebra of the DDF
invariants does close and everything is consistent) and that in particular
the usual critical dimension and everything shows up this way.
Pointers to the above mentioned literature as well as the Pohlmeyer/DDF
construction can be found in
http://arXiv.org/abs/http://www.arxiv.org/abs/hep-th/0403260 .
It has been argued
http://golem.ph.utexas.edu/string/archives/000346.html
that my construction does not solve the Pohlmeyer program, due to an
apparent problem with Lorentz invariance, because of the fact that the
construction of the DDF invariants requires to fix an arbitrary lightlike
vector field on target space.
I don't think that this is really an issue, as explained here:
http://www-stud.uni-essen.de/~sb0264/p6.pdf .
To my mind the Pohlmeyer program of quantization of the string by means of
deforming its algebra of invariants can be solved in at least one way,
namely using DDF operators, which in particular does yield just the ordinary
quantization, albeit from an unusual perspective.
On the other hand, one can always argue that a further, different,
consistent deformation of the algebra of invariants to a commutator algebra
of operators is possible, one which does not have the critical dimension. I
cannot prove that this is impossible, but I would be surprised if such an
alternative does exist, in light of the fact that all other approaches
(LCG/OCQ/BRST/\PI/DDF) yield the same standard result.
Rene Meyer
May9-04, 06:59 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 Urs, hi to all,\n\nthanks for your remark on this paper, but actually that wasn\'t about my\noriginal question on the difference between the two different stress\nenergy tensors. I thought about this the whole holidays (in China\nthere is one week of holidays after labors day), but I still don\'t get\nit. The questions posted earlier still remain a mystery to me.\n\nRené.\n\n--\nRené Meyer\nStudent of Physics & Mathematics\nZhejiang University, Hangzhou, China\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 Urs, hi to all,
thanks for your remark on this paper, but actually that wasn't about my
original question on the difference between the two different stress
energy tensors. I thought about this the whole holidays (in China
there is one week of holidays after labors day), but I still don't get
it. The questions posted earlier still remain a mystery to me.
René.
--
René Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
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