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Chaudhuri on the Hagedorn "myth" |
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| Jun19-05, 07:46 PM | #1 |
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Chaudhuri on the Hagedorn "myth"
<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>http://motls.blogspot.com/2005/06/chaudhuri-on-hagedorn-myth.html\n\nThis short note is closely related to a previous article about the work of\nDienes and Lennek. Tonight, it is Shyamoli Chaudhuri who is "dispelling\nthe Hagedorn myth" (incidentally, it was already in 1965 when Hagedorn\nsuggested that at high enough temperatures, open strings merge into a gas\nof chaotic long closed strings):\n\nhep-th/0506143\n\nShe calculates the thermal free energy - apparently in a different way\nthan we are used to (from Atick and Witten and related works) - to\nconclude that the exponential growth of the states with the energy does\nnot exist. In section 2.1 she argues that the growth of the number of\nstates with the level does not imply the same growth of free energy as a\nfunction of temperature (or the density of states with the total energy).\nThe true growth is slower, she says, making the full expression\nconvergent. Nevertheless, she finds a first order phase transition at the\nT-self-dual temperature.\n\nHer basic argument is the same as in the Dienes and Lennek\'s paper: the\ncorrect one-loop torus path integral only goes over the fundamental region\nof the modular group which removes the dangerous region with small\n"Im(tau)" and makes, according to her beliefs, the integral convergent for\nany temperature.\n\nI encourage everyone for whose research and thinking the Hagedorn behavior\nis important to decide about the fate of the transition without any\nprejudices. After checking various things, I personally believe that the\nHagedorn "folklore" will survive and both of the recent anti-Hagedorn\npapers are misled. (Chaudhuri is more radical because she seems to believe\nthat the transition would be absent even in type 0 and other strings.)\n\nThe integral over the fundamental region combined with the summation over\nthe two winding numbers that count how both circles of the worldsheet\ntorus wind around the thermal circle in spacetime may be replaced by a\nfull integral over the upper "tau" half-plane, which re-introduces the\ndangerous region with small "Im(tau)" and revives the "Hagedorn myth".\n\nTechnically, I think that her error is the step from (15) to (16) in her\npaper where she uses the Hardy-Ramanujan formula, assuming that the\nexcitation of the string is very large, which removes by hand the actual\ndivergence that would, in this calculational procedure, emerge from the\nthermal tachyon (the ground state of the winding sector "w=1" around the\nthermal circle in spacetime - in this sector the GSO projection is\nreversed) - a contribution that she neglects because the Hardy-Ramanujan\nformula is definitely not applicable for low-lying states such as this\nthermal tachyon.\n\nNote that once you admit that the relevant CFT has a thermal tachyon, the\ndiscussion simply ends. With a thermal tachyon, the Hagedorn divergence\narises from the region with large values of "Im(tau)", not small ones. And\nthis "infrared" region is definitely not removed in string theory. To\nsummarize, I now believe that if one defines the thermal stringy\namplitudes in the most obvious stringy extension of the thermal\npath-integral rules of QFT, one finds the thermally wound tachyon whose\nmass determines the Hagedorn temperature, and the new critical papers\nfail.\n\nFeel free to disagree.\n____________________________________________________________ __________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\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>http://motls.blogspot.com/2005/06/ch...dorn-myth.html
This short note is closely related to a previous article about the work of Dienes and Lennek. Tonight, it is Shyamoli Chaudhuri who is "dispelling the Hagedorn myth" (incidentally, it was already in 1965 when Hagedorn suggested that at high enough temperatures, open strings merge into a gas of chaotic long closed strings): http://www.arxiv.org/abs/hep-th/0506143 She calculates the thermal free energy - apparently in a different way than we are used to (from Atick and Witten and related works) - to conclude that the exponential growth of the states with the energy does not exist. In section 2.1 she argues that the growth of the number of states with the level does not imply the same growth of free energy as a function of temperature (or the density of states with the total energy). The true growth is slower, she says, making the full expression convergent. Nevertheless, she finds a first order phase transition at the T-self-dual temperature. Her basic argument is the same as in the Dienes and Lennek's paper: the correct one-loop torus path integral only goes over the fundamental region of the modular group which removes the dangerous region with small "[itex]Im(\tau)[/itex]" and makes, according to her beliefs, the integral convergent for any temperature. I encourage everyone for whose research and thinking the Hagedorn behavior is important to decide about the fate of the transition without any prejudices. After checking various things, I personally believe that the Hagedorn "folklore" will survive and both of the recent anti-Hagedorn papers are misled. (Chaudhuri is more radical because she seems to believe that the transition would be absent even in type and other strings.) The integral over the fundamental region combined with the summation over the two winding numbers that count how both circles of the worldsheet torus wind around the thermal circle in spacetime may be replaced by a full integral over the upper "[itex]\tau[/itex]" half-plane, which re-introduces the dangerous region with small "[itex]Im(\tau)[/itex]" and revives the "Hagedorn myth". Technically, I think that her error is the step from (15) to (16) in her paper where she uses the Hardy-Ramanujan formula, assuming that the excitation of the string is very large, which removes by hand the actual divergence that would, in this calculational procedure, emerge from the thermal tachyon (the ground state of the winding sector "w=1" around the thermal circle in spacetime - in this sector the GSO projection is reversed) - a contribution that she neglects because the Hardy-Ramanujan formula is definitely not applicable for low-lying states such as this thermal tachyon. Note that once you admit that the relevant CFT has a thermal tachyon, the discussion simply ends. With a thermal tachyon, the Hagedorn divergence arises from the region with large values of "[itex]Im(\tau)[/itex]", not small ones. And this "infrared" region is definitely not removed in string theory. To summarize, I now believe that if one defines the thermal stringy amplitudes in the most obvious stringy extension of the thermal path-integral rules of QFT, one finds the thermally wound tachyon whose mass determines the Hagedorn temperature, and the new critical papers fail. Feel free to disagree. __{____________________________________________________________________ ________} E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/ eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/384-9488[/itex] home: [itex]+1-617/868-4487[/itex] (call) Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^ |
| Jun20-05, 01:17 PM | #2 |
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<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 believe that the error is in (25), which is the first equation that\ndiffers from the standard understanding, by the unmotivated introduction\nof the phase factor e^{i \\pi w}. This manifestly destroys modular\ninvariance, as it breaks the symmetry between w and the Poisson\nresummation variable m.\n\nSimilar equations appeared in several earlier and now-withdrawn preprints\nby this author, in each case with the assertion that it was the unique\nmodular-invariant partition function. In fact, the unique\nmodular-invariant partition function is the usual one, with Hagedorn\ndivergence.\n\nLegitimately Anonymous Physicist\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 believe that the error is in (25), which is the first equation that
differs from the standard understanding, by the unmotivated introduction of the phase factor [itex]e^{i \pi w}[/itex]. This manifestly destroys modular invariance, as it breaks the symmetry between w and the Poisson resummation variable m. Similar equations appeared in several earlier and now-withdrawn preprints by this author, in each case with the assertion that it was the unique modular-invariant partition function. In fact, the unique modular-invariant partition function is the usual one, with Hagedorn divergence. Legitimately Anonymous Physicist |
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