## OSFT star and Moyal star

<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\'ve been reading\n\nM. Douglas, H. Liu. G. Moore, B. Zwiebach,\nOpen String Star as a Continuous Moyal Product\nhep-th/0202087\n\nwherein it is shown that the star product of open string field theory can be\nrewritten as a continuous sum of Moyal star products of certain combinations\nof modes of the string\'s coordinates and momenta.\n\nOver on\n\nhttp://golem.ph.utexas.edu/string/archives/000350.html#c000987\n\nEric and I are trying to find a heuristic interpretation of this\nequivalence.\n\nIt seems suggestive that the Moyal star product can be interpreted as a\nconcatenation of \'dipoles\', as described below formula (2.13) of the above\npaper. This formula says that using suitable coordinates x_l and x_r the\nMoyal star of two functions f and g can be rewritten as\n\n(f*g)(x_l,x_r) \\propto \\int dz f(x_l, z) g(z, x_r) .\n\nIf you think of x_l and x_r as the coordinates of two ends of a dipole, this\ngives a very nice interpretation of the Moyal star.\n\nIt then follows from section 3 of the above paper that if you rearrange the\nstring\'s oscillator modes into the particulr linear combinations given in\nequations (4.11)-(4.14), that then the string field star product is\nequivalent to a sum of Moyal star products on each of these new linear\ncombinations of coordinate and momentum modes.\n\nOne natural question hence seems to be: What are the \'dipoles\' described by\n(4.11)+(2.12)? That\'s because according to the results of that paper we can\nunderstand the string field star as a certain way to pairwise concatenate\nall these continuously many \'dipoles\' on the string, in the sense of that\npaper.\n\nOf course in order to answer this question I should sit down and work out,\nusing equations (3.4) and (4.11), what the modes x_\\kappa and q_kappa in\n(4.11) and (4.13) explicitly look like - straightforward but possibly\ntedious. But I was wondering if maybe somebody knows the answer to this\nquestion and can help me get a heuristic understanding of how, physically,\nthe Moyal star describes the OSFT star.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>I've been reading

M. Douglas, H. Liu. G. Moore, B. Zwiebach,
Open String Star as a Continuous Moyal Product
http://www.arxiv.org/abs/hep-th/0202087

wherein it is shown that the star product of open string field theory can be
rewritten as a continuous sum of Moyal star products of certain combinations
of modes of the string's coordinates and momenta.

Over on

http://golem.ph.utexas.edu/string/ar...0.html#c000987

Eric and I are trying to find a heuristic interpretation of this
equivalence.

It seems suggestive that the Moyal star product can be interpreted as a
concatenation of 'dipoles', as described below formula (2.13) of the above
paper. This formula says that using suitable coordinates $x_l$ and $x_r$ the
Moyal star of two functions f and g can be rewritten as

$$(f*g)(x_l,x_r) \propto \int dz f(x_l, z) g(z, x_r) .$$

If you think of $x_l$ and $x_r$ as the coordinates of two ends of a dipole, this
gives a very nice interpretation of the Moyal star.

It then follows from section 3 of the above paper that if you rearrange the
string's oscillator modes into the particulr linear combinations given in
equations (4.$11)-(4$.14), that then the string field star product is
equivalent to a sum of Moyal star products on each of these new linear
combinations of coordinate and momentum modes.

One natural question hence seems to be: What are the 'dipoles' described by
(4.$11)+(2$.12)? That's because according to the results of that paper we can
understand the string field star as a certain way to pairwise concatenate
all these continuously many 'dipoles' on the string, in the sense of that
paper.

Of course in order to answer this question I should sit down and work out,
using equations (3.4) and (4.11), what the modes $x_\kappa$ and $q_{kappa}$ in
(4.11) and (4.13) explicitly look like - straightforward but possibly
tedious. But I was wondering if maybe somebody knows the answer to this
question and can help me get a heuristic understanding of how, physically,
the Moyal star describes the OSFT star.

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