View Full Version : What are the basics of membrane theory
<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\'m wondering, what are the basics\nof membrane theory. I.e., in string\ntheory you assign probabilities to\nworldsheets (histories) interpolating\nbetween two curves (strings), at different\ntimes. In membrane theory do you do the\nsame. Also, since the dimension of the\nmembranes vary do you assign probability\nto histories that begin say with a 2-brane\nand end up with a 3-brane.\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'm wondering, what are the basics
of membrane theory. I.e., in string
theory you assign probabilities to
worldsheets (histories) interpolating
between two curves (strings), at different
times. In membrane theory do you do the
same. Also, since the dimension of the
membranes vary do you assign probability
to histories that begin say with a 2-brane
and end up with a 3-brane.
Lubos Motl
May24-04, 08:42 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>On Mon, 24 May 2004, Mandro wrote:\n\n> I\'m wondering, what are the basics\n> of membrane theory. I.e., in string\n> theory you assign probabilities to\n> worldsheets (histories) interpolating\n> between two curves (strings), at different\n> times. In membrane theory do you do the\n> same.\n\nNaively, you can do the same with membranes of any dimension, indeed. One\nmight try to define the sum over all histories of p-dimensional branes\nthat split and join. However if p>1 (where p=1 corresponds to strings),\nthe internal worldvolume theory describing the branes has a lot of UV\n(ultraviolet i.e. short distance) divergences, much like quantum\ngravitational theories based on point-like particles.\n\nIt is not well-defined and leads to infinite (or negative) probabilities\nonce you calculate the quantum corrections. The strings (p=1) seem to be\nthe perfect compromise where both the spacetime *and* the worldsheet\nultraviolet divergences seem to be under control. (p=0, pointlike\nparticles, often lead to divergences in spacetime because - poetically\nspeaking - their interactions are concentrated to points which is too\nsingular.) This is why the strings behave better than other extended\nobjects, and why Heisenberg, Dirac, and others gave up when they tried to\nquantize membranes and higher-dimensional objects. They got all kinds of\ninfinities and negative probabilities, and violations of causality -\nbecause they missed the only straightforward example of working\nrelativistic fundamental objects, namely the strings.\n\nString theorists have been proud to have discovered this loophole - a\ntheory based on extended objects that nevertheless satisfies the basic\nrequirements of consistency - in fact, it satisfies them better than\npoint-like particles because it can describe finite observables in quantum\ngravity.\n\nHowever, string/M-theory today (since the middle 1990s) is known to\ncontain higher-dimensional objects, p-branes, and these branes are related\nto the fundamental strings via various dualities. Generally speaking, all\ntypes of such branes are equally good. String/M-theory is governed by\n"brane democracy".\n\nHowever, if the coupling constant is small in one of our string theories,\nthe fundamental strings are the lightest objects, and they are special. In\nfact, we describe all other branes in terms of strings as composites;\ntopological defects; solitons; monopoles - call it any way you want. Joe\nPolchinski from Santa Barbara is the person who developed a systematic\ntreatment of these Dp-branes - where "Dp" stands for\n"Dirichlet-Polchinski" while "p" also counts the spatial dimension.\n\nThe internal dynamics (oscillations) of D-branes of any dimension is in\nfact essentially as complicated as spacetime physics itself, and it needs\nto be described by string theory. There are infinitely many fields living\non such D-branes (not just the naive position-related fields that exist on\nstringy worldsheets, and that you are probably thinking about), and these\nfields arise from possible excitations of *open* strings (stretched\nbetween such D-branes) in the same way in which gravitational physics (and\nbeyond) in spacetime results from vibrating *closed* strings. The\ntransverse position of a point on the D-brane - imagine a membrane written\nas a function z(x,y) - is described by a scalar field "z", which is\nnothing else than one of the lowest-energy collective modes - namely the\nmode "z" - of an open string whose both endpoints are stuck at the\nD-brane.\n\nAlthough D-branes are very heavy (we also say: nonperturbative) objects,\ntheir dynamics is described by perturbative calculations analogous to the\nusual perturbative analysis of closed strings. The only difference is that\nwe use the open strings instead - but strings are behaving perfectly, and\ntherefore the D-brane dynamics is free of short distance divergences, too.\n\nM-theory in 11 dimensions is the only known limit whose physics we roughly\nunderstand, but which contains no fundamental string. Instead, M-theory\ncontains M2-branes and M5-branes, 2-dimensional and 5-dimensional objects,\nrespectively. There exist dualities relating M2 and M5, so at a very\nfundamental level, these two are equally fundamental. However in various\napproaches, M2 are more fundamental than M5 - membranes (M) are among the\njustifications of the letter "M" in M-theory. It is believed that\nneither M2-branes nor M5-branes contain infinitely many fields on their\nworldvolume, but no one knows a compact way how to calculate their\ninternal dynamics exactly at arbitrarily short distances.\n\n> Also, since the dimension of the\n> membranes vary do you assign probability\n> to histories that begin say with a 2-brane\n> and end up with a 3-brane.\n\nOne could imagine that the full string theory would be a path integral\ndescription that would describe histories with branes of arbitrary\ndimensions that transmute into one another, and this is indeed a correct\nqualitative way to look at many situations. The fact, however, is that no\nknown description of string/M-theory works in this way. Every description\nwe know treats one type of object as the fundamental one - for example,\nthe fundamental string - and others (like D-branes) are solitons or other\n"derived" configurations made out of the fundamental objects. There are\noften equivalent descriptions where the role of the fundamental object is\nplayed by an ingredient that is composite from the viewpoint of the\nprevious description, and this implies "democracy". However in every\nindividual description we only deal with 1 set of elementary degrees of\nfreedom, and others are viewed as "composites".\n\nLet me tell you another example. BFSS Matrix theory describes everything\nas quantum mechanics whose building blocks are NxN matrices where N is\nlarge - the entries of these matrices describe "non-commutative"\ncoordinates of 0-dimensional particles called the D0-branes. A\ntwo-dimensional membrane (M2-brane) is constructed as a very special\ncomposite of a large number of D0-branes. Because the D0-branes are finite\nat short distances (their dynamics follows from string theory of open\nstrings stretched between the D0-branes), the M2-branes constructed from\nD0-branes (like in Matrix theory) are equally well-behaved. Nevertheless\nthere is no room for "extra" fundamental objects in Matrix theory;\neverything that exists is (and must be) derivable from the limited set of\ndegrees of freedom that we start with.\n\nIn AdS/CFT, we describe physics of gravitons and other objects inside anti\nde Sitter space using a gauge theory defined on the boundary of the anti\nde Sitter space (at infinity). Everything is made of several fundamental\ngluon-like fields.\n\nBest wishes\nLubos\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 Mon, 24 May 2004, Mandro wrote:
> I'm wondering, what are the basics
> of membrane theory. I.e., in string
> theory you assign probabilities to
> worldsheets (histories) interpolating
> between two curves (strings), at different
> times. In membrane theory do you do the
> same.
Naively, you can do the same with membranes of any dimension, indeed. One
might try to define the sum over all histories of p-dimensional branes
that split and join. However if p>1 (where p=1 corresponds to strings),
the internal worldvolume theory describing the branes has a lot of UV
(ultraviolet i.e. short distance) divergences, much like quantum
gravitational theories based on point-like particles.
It is not well-defined and leads to infinite (or negative) probabilities
once you calculate the quantum corrections. The strings (p=1) seem to be
the perfect compromise where both the spacetime *and* the worldsheet
ultraviolet divergences seem to be under control. (p=0, pointlike
particles, often lead to divergences in spacetime because - poetically
speaking - their interactions are concentrated to points which is too
singular.) This is why the strings behave better than other extended
objects, and why Heisenberg, Dirac, and others gave up when they tried to
quantize membranes and higher-dimensional objects. They got all kinds of
infinities and negative probabilities, and violations of causality -
because they missed the only straightforward example of working
relativistic fundamental objects, namely the strings.
String theorists have been proud to have discovered this loophole - a
theory based on extended objects that nevertheless satisfies the basic
requirements of consistency - in fact, it satisfies them better than
point-like particles because it can describe finite observables in quantum
gravity.
However, string/M-theory today (since the middle 1990s) is known to
contain higher-dimensional objects, p-branes, and these branes are related
to the fundamental strings via various dualities. Generally speaking, all
types of such branes are equally good. String/M-theory is governed by
"brane democracy".
However, if the coupling constant is small in one of our string theories,
the fundamental strings are the lightest objects, and they are special. In
fact, we describe all other branes in terms of strings as composites;
topological defects; solitons; monopoles - call it any way you want. Joe
Polchinski from Santa Barbara is the person who developed a systematic
treatment of these Dp-branes - where "Dp" stands for
"Dirichlet-Polchinski" while "p" also counts the spatial dimension.
The internal dynamics (oscillations) of D-branes of any dimension is in
fact essentially as complicated as spacetime physics itself, and it needs
to be described by string theory. There are infinitely many fields living
on such D-branes (not just the naive position-related fields that exist on
stringy worldsheets, and that you are probably thinking about), and these
fields arise from possible excitations of *open* strings (stretched
between such D-branes) in the same way in which gravitational physics (and
beyond) in spacetime results from vibrating *closed* strings. The
transverse position of a point on the D-brane - imagine a membrane written
as a function z(x,y) - is described by a scalar field "z", which is
nothing else than one of the lowest-energy collective modes - namely the
mode "z" - of an open string whose both endpoints are stuck at the
D-brane.
Although D-branes are very heavy (we also say: nonperturbative) objects,
their dynamics is described by perturbative calculations analogous to the
usual perturbative analysis of closed strings. The only difference is that
we use the open strings instead - but strings are behaving perfectly, and
therefore the D-brane dynamics is free of short distance divergences, too.
M-theory in 11 dimensions is the only known limit whose physics we roughly
understand, but which contains no fundamental string. Instead, M-theory
contains M2-branes and M5-branes, 2-dimensional and 5-dimensional objects,
respectively. There exist dualities relating M2 and M5, so at a very
fundamental level, these two are equally fundamental. However in various
approaches, M2 are more fundamental than M5 - membranes (M) are among the
justifications of the letter "M" in M-theory. It is believed that
neither M2-branes nor M5-branes contain infinitely many fields on their
worldvolume, but no one knows a compact way how to calculate their
internal dynamics exactly at arbitrarily short distances.
> Also, since the dimension of the
> membranes vary do you assign probability
> to histories that begin say with a 2-brane
> and end up with a 3-brane.
One could imagine that the full string theory would be a path integral
description that would describe histories with branes of arbitrary
dimensions that transmute into one another, and this is indeed a correct
qualitative way to look at many situations. The fact, however, is that no
known description of string/M-theory works in this way. Every description
we know treats one type of object as the fundamental one - for example,
the fundamental string - and others (like D-branes) are solitons or other
"derived" configurations made out of the fundamental objects. There are
often equivalent descriptions where the role of the fundamental object is
played by an ingredient that is composite from the viewpoint of the
previous description, and this implies "democracy". However in every
individual description we only deal with 1 set of elementary degrees of
freedom, and others are viewed as "composites".
Let me tell you another example. BFSS Matrix theory describes everything
as quantum mechanics whose building blocks are NxN matrices where N is
large - the entries of these matrices describe "non-commutative"
coordinates of 0-dimensional particles called the D0-branes. A
two-dimensional membrane (M2-brane) is constructed as a very special
composite of a large number of D0-branes. Because the D0-branes are finite
at short distances (their dynamics follows from string theory of open
strings stretched between the D0-branes), the M2-branes constructed from
D0-branes (like in Matrix theory) are equally well-behaved. Nevertheless
there is no room for "extra" fundamental objects in Matrix theory;
everything that exists is (and must be) derivable from the limited set of
degrees of freedom that we start with.
In AdS/CFT, we describe physics of gravitons and other objects inside anti
de Sitter space using a gauge theory defined on the boundary of the anti
de Sitter space (at infinity). Everything is made of several fundamental
gluon-like fields.
Best wishes
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
May25-04, 02:10 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>On Mon, 24 May 2004, Lubos Motl wrote:\n\n> It is believed that\n> neither M2-branes nor M5-branes contain infinitely many fields on their\n> worldvolume, but no one knows a compact way how to calculate their\n> internal dynamics exactly at arbitrarily short distances.\n\nCould you explain what this problem with an infinity of worldvolume fields\nis? Aren\'t there just the embedding fields, the gauge field and their\nsuperpartners?\n\nA while ago you mentioned the "Automorphic membrane" by Pioline and\nWaldron, hep-th/0404018. I got the impression that some progress in\nquantizing the membrane has been made, but I don\'t understand this stuff.\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 Mon, 24 May 2004, Lubos Motl wrote:
> It is believed that
> neither M2-branes nor M5-branes contain infinitely many fields on their
> worldvolume, but no one knows a compact way how to calculate their
> internal dynamics exactly at arbitrarily short distances.
Could you explain what this problem with an infinity of worldvolume fields
is? Aren't there just the embedding fields, the gauge field and their
superpartners?
A while ago you mentioned the "Automorphic membrane" by Pioline and
Waldron, http://www.arxiv.org/abs/hep-th/0404018. I got the impression that some progress in
quantizing the membrane has been made, but I don't understand this stuff.
<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>So, Lubos, you are saying that how\nbranes came to be from ordinary string\ntheory is as composites of strings?\nYou mean that membranes are made of\nmany strings? Can you tell me an example\nof where they arise, in ordinary basic\nstring theory.\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>So, Lubos, you are saying that how
branes came to be from ordinary string
theory is as composites of strings?
You mean that membranes are made of
many strings? Can you tell me an example
of where they arise, in ordinary basic
string theory.
Lubos Motl
May25-04, 11:26 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>On Tue, 25 May 2004, mandro wrote:\n\n> So, Lubos, you are saying that how\n> branes came to be from ordinary string\n> theory is as composites of strings?\n> You mean that membranes are made of\n> many strings?\n\nNot quite. Even though there exists an example of this type. In Matrix\ntheory, M2-branes can be made of many "D0-branes" although I should not\nuse these two terms simultaneously - M2-branes only exist in M-theory\nwhile D0-branes exist in type IIA theory.\n\nIn perturbative string theory, D-branes are made out of strings\nindirectly, as solitons. What I want to say now is general enough - it\nalso describes the way in which the magnetic monopoles are "produced" from\nthe elementary electrically charged fields (and gauge fields). You first\nidentify various oscillations of a string with spacetime fields, and then\nyou can construct a solution (configuration of these fields that solves\nthe right equations of motion), and the soliton (e.g. a brane) is "made\nof" these fields. You don\'t need to add any new degrees of freedom by\nhand.\n\nNevertheless, there are many contexts in which a brane can be produced out\nof lower- or higher-dimensional branes. A Dp-brane is an instanton within\na D(p+4)-brane, for example, as Mike Douglas - as far as I know - was the\nfirst one who understood it. A D(p-2)-brane is a unit of magnetic flux\nwithin a Dp-brane, and the D0-M2 relation at the beginning of this post is\na closely related example.\n\nOn the other hand, higher-dimensional branes can be obtained (blown up)\nfrom lower-dimensional branes via the Myers dipole effect, and so forth.\nThere are many relations between various branes and I cannot enumerate all\nof them here.\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, 25 May 2004, mandro wrote:
> So, Lubos, you are saying that how
> branes came to be from ordinary string
> theory is as composites of strings?
> You mean that membranes are made of
> many strings?
Not quite. Even though there exists an example of this type. In Matrix
theory, M2-branes can be made of many "D0-branes" although I should not
use these two terms simultaneously - M2-branes only exist in M-theory
while D0-branes exist in type IIA theory.
In perturbative string theory, D-branes are made out of strings
indirectly, as solitons. What I want to say now is general enough - it
also describes the way in which the magnetic monopoles are "produced" from
the elementary electrically charged fields (and gauge fields). You first
identify various oscillations of a string with spacetime fields, and then
you can construct a solution (configuration of these fields that solves
the right equations of motion), and the soliton (e.g. a brane) is "made
of" these fields. You don't need to add any new degrees of freedom by
hand.
Nevertheless, there are many contexts in which a brane can be produced out
of lower- or higher-dimensional branes. A Dp-brane is an instanton within
a D(p+4)-brane, for example, as Mike Douglas - as far as I know - was the
first one who understood it. A D(p-2)-brane is a unit of magnetic flux
within a Dp-brane, and the D0-M2 relation at the beginning of this post is
a closely related example.
On the other hand, higher-dimensional branes can be obtained (blown up)
from lower-dimensional branes via the Myers dipole effect, and so forth.
There are many relations between various branes and I cannot enumerate all
of them here.
__{_______________________________________________ _____________________________}
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Lubos Motl
May25-04, 11:36 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>On Tue, 25 May 2004, Urs Schreiber wrote:\n\n> Could you explain what this problem with an infinity of worldvolume fields\n> is? Aren\'t there just the embedding fields, the gauge field and their\n> superpartners?\n\nI am afraid that I don\'t have anything intelligent to say. A worldvolume\ntheory for a M2-brane or a M5-brane has UV problems, much like quantum\ngravity itself in d>3. At very short distances, the membrane develops\n"quantum foam", where the topology "boils", much like the usual spacetime\nquantum foam, and there is no small dimensionless coupling that would\nsuppress the more complicated topologies (which happens for perturbative\nstringy diagrams).\n\nNeither membranes nor fivebranes can be used as fundamental objects to\nregulate the UV divergences - in this sense, the strings are the only\ngame in town.\n\nIn the more promising case of membranes, another problem encountered\nduring their quantization had been known: unlike the case of a string,\nwhose oscillation spectrum is discrete, the membranes also had a\ncontinuous spectrum. This subtlety was understood perfectly by BFSS in\nMatrix theory: the continuous spectrum corresponds to a membrane\ndeveloping a long thin throat, or - which is asymptotically equivalent -\nsplitting into several membranes - a process that can never be eliminated\n- and a correctly made quantization of a single membrane actually leads to\na description of an arbitrary number of membranes, as well as gravitons\nand fivebranes and other objects, and it is nothing else than Matrix\ntheory - where the Matrix structure also automatically regulates the\nshort-distance problems with the membrane.\n\n> A while ago you mentioned the "Automorphic membrane" by Pioline and\n> Waldron, hep-th/0404018. I got the impression that some progress in\n> quantizing the membrane has been made, but I don\'t understand this stuff.\n\nThat\'s right. But we need to learn p-adic and adelic numbers and\nautomorphic forms, functions on E_k group manifolds, and all this stuff,\nbefore we can follow all the details of their work (and beyond). So the\nidea that everything comes "from the embedding fields, gauge fields, and\ntheir superpartners" might be a bit naive.\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, 25 May 2004, Urs Schreiber wrote:
> Could you explain what this problem with an infinity of worldvolume fields
> is? Aren't there just the embedding fields, the gauge field and their
> superpartners?
I am afraid that I don't have anything intelligent to say. A worldvolume
theory for a M2-brane or a M5-brane has UV problems, much like quantum
gravity itself in d>3. At very short distances, the membrane develops
"quantum foam", where the topology "boils", much like the usual spacetime
quantum foam, and there is no small dimensionless coupling that would
suppress the more complicated topologies (which happens for perturbative
stringy diagrams).
Neither membranes nor fivebranes can be used as fundamental objects to
regulate the UV divergences - in this sense, the strings are the only
game in town.
In the more promising case of membranes, another problem encountered
during their quantization had been known: unlike the case of a string,
whose oscillation spectrum is discrete, the membranes also had a
continuous spectrum. This subtlety was understood perfectly by BFSS in
Matrix theory: the continuous spectrum corresponds to a membrane
developing a long thin throat, or - which is asymptotically equivalent -
splitting into several membranes - a process that can never be eliminated
- and a correctly made quantization of a single membrane actually leads to
a description of an arbitrary number of membranes, as well as gravitons
and fivebranes and other objects, and it is nothing else than Matrix
theory - where the Matrix structure also automatically regulates the
short-distance problems with the membrane.
> A while ago you mentioned the "Automorphic membrane" by Pioline and
> Waldron, http://www.arxiv.org/abs/hep-th/0404018. I got the impression that some progress in
> quantizing the membrane has been made, but I don't understand this stuff.
That's right. But we need to learn p-adic and adelic numbers and
automorphic forms, functions on E_k group manifolds, and all this stuff,
before we can follow all the details of their work (and beyond). So the
idea that everything comes "from the embedding fields, gauge fields, and
their superpartners" might be a bit naive.
__{_______________________________________________ _____________________________}
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Charlie Stromeyer Jr.
May26-04, 04:36 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>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:\n\n> I am afraid that I don\'t have anything intelligent to say.\n\nWell, I think what you say is quite intelligent. Lubos, IMHO, you\nshould remember not to sell yourself short or underestimate your own\nabilities because some experts believe that Einstein was not able to\naccomplish anything significant during the second half of his life\nbecause his ego had become too stubborn and perfectionistic.\n\nHowever, I will presume that your continuing signs of brilliance are\nnot also some kind of omen that the Czech Republic will win the next\nFIFA WorldCup :-)\n\nSpeaking of worldvolumes, since you are an expert on LSTs you might\nfind this new preprint to be interesting, however, I have not looked\nat this myself so I cannot really say if it will be interesting:\n\nhttp://arxiv.org/abs/hep-th/0405148\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 <motl@feynman.harvard.edu> wrote in message news:
> I am afraid that I don't have anything intelligent to say.
Well, I think what you say is quite intelligent. Lubos, IMHO, you
should remember not to sell yourself short or underestimate your own
abilities because some experts believe that Einstein was not able to
accomplish anything significant during the second half of his life
because his ego had become too stubborn and perfectionistic.
However, I will presume that your continuing signs of brilliance are
not also some kind of omen that the Czech Republic will win the next
FIFA WorldCup :-)
Speaking of worldvolumes, since you are an expert on LSTs you might
find this new preprint to be interesting, however, I have not looked
at this myself so I cannot really say if it will be interesting:
http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0405148
<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 saw in the Elegant Universe, that\nmembranes arise in string theory,\nwhen one makes the coupling constant\nlarge. In which way is this so. I.e.,\ncan you make this statement precise?\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 saw in the Elegant Universe, that
membranes arise in string theory,
when one makes the coupling constant
large. In which way is this so. I.e.,
can you make this statement precise?
<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>After asking this last question, I checked\nLubos\' firt reply and found that he said\nessentially the same thing that\'s stated\nin the "Elegant Universe" but which I had\nmissed when I read his reply. Armed with\nslightly more knowledge, I want to rephrase\nmy qestion as follows. It is obvious to me,\nat least from the point of view of the math,\nand common sense, that if I take a theory\n(perturbative theory) e.g., standard QED,\nand I just increase the coupling constant,\na new "space dimension" is not going to "pop\nout all of a sudden". Also, I thought a theory\njust has a fixed coupling constant, so it\'s\nalmost as if by M-theory, they really mean a\nfamily of theories, each with a different\nparameter no?\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>After asking this last question, I checked
Lubos' firt reply and found that he said
essentially the same thing that's stated
in the "Elegant Universe" but which I had
missed when I read his reply. Armed with
slightly more knowledge, I want to rephrase
my qestion as follows. It is obvious to me,
at least from the point of view of the math,
and common sense, that if I take a theory
(perturbative theory) e.g., standard QED,
and I just increase the coupling constant,
a new "space dimension" is not going to "pop
out all of a sudden". Also, I thought a theory
just has a fixed coupling constant, so it's
almost as if by M-theory, they really mean a
family of theories, each with a different
parameter no?
Urs Schreiber
May28-04, 12:58 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>On Fri, 28 May 2004, mandro wrote:\n\n> Also, I thought a theory\n> just has a fixed coupling constant, so it\'s\n> almost as if by M-theory, they really mean a\n> family of theories, each with a different\n> parameter no?\n\nLubos will probably answer this in more detail than I can, but let me just\nremark the following:\n\nMany things which are free parameters in ordinary field theories are\ndynamical fields in string theory. In particular the string coupling is\nrelated to the dilaton field. This essentially measures the size of the\ninternal 11th dimension of M-theory.\n\nTo see roughly how this works note that the dilaton Phi is a scalar\nspactime field and the natural way for it to couple to the worldsheet is\nby way of a term in the worldsheet action of the form\n\n\\int_{worldhseet} Phi(X(sigma,tau)) R vol\n\nwhere R is the worldsheet curvature scalar and vol the worldsheet volume\nform. Assume that Phi is constant in all of space, for simplicity, then\nit can be taken outside the integral and we get the action\n\nPhi \\int_{worldsheet} R vol .\n\nBut this integral is just the Euler number chi of the worldsheet, i.e. a\nnumber that depends on the topology of the worldsheet only and in\nparticular decreases by 2 when a handle is added to a worldsheet (e.g.\nwhen we go from the sphere to the torus).\n\nThis means that in the path integral the above term contributes just a\nconstant factor e^Phi taken to the power of the genus of the worldsheet.\nTherefore splitting/joining interaction of strings is suppressed precisely\nby this factor.\n\nThat\'s how the background field Phi determines the string coupling. But\nstill, Phi is a dynamical field in the theory and in particulaer need not\nbe constant. In this sense the \'coupling constant\' of strings is not a\nfree parameter but could in principle be determined by solving some\nappropriate equations of motion.\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 Fri, 28 May 2004, mandro wrote:
> Also, I thought a theory
> just has a fixed coupling constant, so it's
> almost as if by M-theory, they really mean a
> family of theories, each with a different
> parameter no?
Lubos will probably answer this in more detail than I can, but let me just
remark the following:
Many things which are free parameters in ordinary field theories are
dynamical fields in string theory. In particular the string coupling is
related to the dilaton field. This essentially measures the size of the
internal 11th dimension of M-theory.
To see roughly how this works note that the dilaton \Phi is a scalar
spactime field and the natural way for it to couple to the worldsheet is
by way of a term in the worldsheet action of the form
\int_{worldhseet} \Phi(X(\sigma,\tau)) R vol
where R is the worldsheet curvature scalar and vol the worldsheet volume
form. Assume that \Phi is constant in all of space, for simplicity, then
it can be taken outside the integral and we get the action
\Phi \int_{worldsheet} R vol .
But this integral is just the Euler number \chi of the worldsheet, i.e. a
number that depends on the topology of the worldsheet only and in
particular decreases by 2 when a handle is added to a worldsheet (e.g.
when we go from the sphere to the torus).
This means that in the path integral the above term contributes just a
constant factor e^\Phi taken to the power of the genus of the worldsheet.
Therefore splitting/joining interaction of strings is suppressed precisely
by this factor.
That's how the background field \Phi determines the string coupling. But
still, \Phi is a dynamical field in the theory and in particulaer need not
be constant. In this sense the 'coupling constant' of strings is not a
free parameter but could in principle be determined by solving some
appropriate equations of motion.
vBulletin® v3.7.6, Copyright ©2000-2009, Jelsoft Enterprises Ltd.