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Daniel
Sep7-04, 02:53 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 understand that space and time are quantized in certain quantum\ngravity theories, but what about string/m theories?\n\ni understand that string length is at planck length, and that units\nsmaller than this may not be physically meaningful, but it\'s not\nobvious that this translates to quantized space and time.\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 understand that space and time are quantized in certain quantum
gravity theories, but what about string/m theories?

i understand that string length is at planck length, and that units
smaller than this may not be physically meaningful, but it's not
obvious that this translates to quantized space and time.
Lubos Motl
Sep7-04, 10:04 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, 7 Sep 2004, Daniel wrote:\n\n&gt; i understand that space and time are quantized in certain quantum\n&gt; gravity theories, but what about string/m theories?\n\nDear Daniel,\n\ngeometric quantities - such as the length and area, and especially time -\ncannot have quantized eigenvalues in a theory that locally respects the\npostulates of special relativity, and simultaneously has local physical\nexcitations.\n\nSeveral rather speculative approaches to quantum gravity, such as "loop\nquantum gravity", propose that the areas (and perhaps other spatial\nquantities) have discrete eigenvalues, but they rarely do the same thing\nfor time. In the Hamiltonian treatment of anything, including loop quantum\ngravity, time is always continuous. Incidentally, this means that if the\nlength is quantized, special relativity is severely broken because\nspecial relativity requires time to follow the same rules as space, so to\nsay.\n\nThe path-integral approach to loop quantum gravity - namely the spin foam\nmodels - should in principle be equivalent to the Hamiltonian treatment,\nbut in reality it is more subtle because spin foam sort of naturally\nallows even time to be quantized.\n\nBut the situation in string theory is different in details:\n\n&gt; i understand that string length is at planck length, and that units\n&gt; smaller than this may not be physically meaningful, but it\'s not\n&gt; obvious that this translates to quantized space and time.\n\nPlanck length is not quite the same thing as the string length - they can\ndiffer by a multiplicative factor that is a power of the string coupling\nconstant. If the coupling constant is small (weak coupling), the Planck\nlength is much shorter than the string length. Strings are light and\nloose, but true quantum gravity only starts to be relevant at very short\ndistance scales. In the 1980s people used to assume that the Planck length\nmust be comparable to the string length - and the coupling constant and\nother quantities (the volume of the hidden dimensions) must be of order\none. But the 1990s have made us think about many possibilities of large\ndimensions and very weakly coupled theories and many other scenarios where\nvarious numbers are very different from one - and the Planck length can be\nvery different from the string length.\n\nThe "full" physical string theory does not predict strict quantization of\nspace and time. Although it is true that the usual geometric concepts\nabout continuous space and time become wrong according to string theory as\nsoon as we look at very short distances, the precise modification of\ngeometry is different. String theory predicts a lot of new physics - new\nobjects, their vibrations, and new terms in the equations of motion, so to\nsay, caused by these objects, as well as "fuzziness" and mixing of\ngeometry with other physics. String theory seems to be much richer than a\nsimple picture of "atoms" of space and time - the spectrum of the object\nis large, but they are not arbitrary - all of them and their properties\nseem to follow from mathematics of string theory.\n\nThere are some intriguing observations in topological string theory - a\nmathematically interesting truncation (or topological twist) of string\ntheory that has no local degrees of freedom. The spacetime physics (or\nperhaps mathematics) is described by a sort of "melting crystal". It\nremains unlikely that a very similar discrete picture of spacetime can be\ngeneralized to the full "physical" string theory, but several smart people\nare trying to get as much from the crystal as possible.\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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>On Tue, 7 Sep 2004, Daniel wrote:

> i understand that space and time are quantized in certain quantum
> gravity theories, but what about string/m theories?

Dear Daniel,

geometric quantities - such as the length and area, and especially time -
cannot have quantized eigenvalues in a theory that locally respects the
postulates of special relativity, and simultaneously has local physical
excitations.

Several rather speculative approaches to quantum gravity, such as "loop
quantum gravity", propose that the areas (and perhaps other spatial
quantities) have discrete eigenvalues, but they rarely do the same thing
for time. In the Hamiltonian treatment of anything, including loop quantum
gravity, time is always continuous. Incidentally, this means that if the
length is quantized, special relativity is severely broken because
special relativity requires time to follow the same rules as space, so to
say.

The path-integral approach to loop quantum gravity - namely the spin foam
models - should in principle be equivalent to the Hamiltonian treatment,
but in reality it is more subtle because spin foam sort of naturally
allows even time to be quantized.

But the situation in string theory is different in details:

> i understand that string length is at planck length, and that units
> smaller than this may not be physically meaningful, but it's not
> obvious that this translates to quantized space and time.

Planck length is not quite the same thing as the string length - they can
differ by a multiplicative factor that is a power of the string coupling
constant. If the coupling constant is small (weak coupling), the Planck
length is much shorter than the string length. Strings are light and
loose, but true quantum gravity only starts to be relevant at very short
distance scales. In the 1980s people used to assume that the Planck length
must be comparable to the string length - and the coupling constant and
other quantities (the volume of the hidden dimensions) must be of order
one. But the 1990s have made us think about many possibilities of large
dimensions and very weakly coupled theories and many other scenarios where
various numbers are very different from one - and the Planck length can be
very different from the string length.

The "full" physical string theory does not predict strict quantization of
space and time. Although it is true that the usual geometric concepts
about continuous space and time become wrong according to string theory as
soon as we look at very short distances, the precise modification of
geometry is different. String theory predicts a lot of new physics - new
objects, their vibrations, and new terms in the equations of motion, so to
say, caused by these objects, as well as "fuzziness" and mixing of
geometry with other physics. String theory seems to be much richer than a
simple picture of "atoms" of space and time - the spectrum of the object
is large, but they are not arbitrary - all of them and their properties
seem to follow from mathematics of string theory.

There are some intriguing observations in topological string theory - a
mathematically interesting truncation (or topological twist) of string
theory that has no local degrees of freedom. The spacetime physics (or
perhaps mathematics) is described by a sort of "melting crystal". It
remains unlikely that a very similar discrete picture of spacetime can be
generalized to the full "physical" string theory, but several smart people
are trying to get as much from the crystal as possible.

All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Moshe Rozali
Sep8-04, 02:43 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,\n\nThere is some tension in my mind in the sentence "areas and volumes\nare quantized in Planckian units", since naively areas and volumes\nseem to be semiclassical concepts. Is there a way in your mind to\ndefine this sentence operationally, as a statement about a result of\nmeasurment (there may well be, did not think about it too much...)? is\nthe famous statement in LQG about the quantization of geometry to be\nunderstood along these lines? (again, this is really a question, I\ndon\'t know enough about LQG to guess an answer).\n\nthanks,\n\nMoshe\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>Lubos,

There is some tension in my mind in the sentence "areas and volumes
are quantized in Planckian units", since naively areas and volumes
seem to be semiclassical concepts. Is there a way in your mind to
define this sentence operationally, as a statement about a result of
measurment (there may well be, did not think about it too much...)? is
the famous statement in LQG about the quantization of geometry to be
understood along these lines? (again, this is really a question, I
don't know enough about LQG to guess an answer).

thanks,

Moshe
Lubos Motl
Sep8-04, 03:58 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>Hi Moshe!\n\n&gt; There is some tension in my mind in the sentence "areas and volumes ...\n\nMy feeling is that you wrote this posting to make me happier! ;-) I\ntotally agree. In a general enough UV complete theory, the areas and the\nvolumes are semiclassical concepts - the whole geometry is replaced by\nsomething more general in the UV. If one only considers geometry, it\ninvolves integrating out the other UV physics, and the effective action\nand measurements of the areas will depend on the chosen scale (different\nscales are related by renormalization, integrating out loops), and so on.\n\nSo in a general enough theory - and string theory is certainly general\nenough in this sense - I see no operational way to measure the geometric\nquantities with a subPlanckian precision. Of course, we do have indirect\nexact ways to measure the areas, but only the minimal areas of some\ncycles, e.g. the supersymmetric cycles. Well, you can wrap a brane on a\ncycle and measure its mass - this number divided by the tension is the\narea/volume of the cycle including the quantum corrections. I don\'t see a\ngeneral string-theoretical way to define such areas and volumes - with a\nsubPlanckian precision - for a general surface or volume. The very idea of\na surface (or volume) embedded in spacetime is based on the assumption\nthat "sharp", point-like geometry exists at all scales, at least in some\nway - otherwise the embedding cannot be determined accurately.\n\nLoop quantum gravity is a very different framework. It is not "general\nenough" according to the definitions above, and it is based on the\nassumption that the metric is a good variable in all regimes, including -\nor especially - the Planckian regime, and one can construct\ngauge-invariant operators out of the metric and its dual variables that\nhave well-defined finite eigenvalues, much like the Hamiltonian of the\nHydrogen atom. Pre-geometry exists at all scales, and the volumes are\ncomputed as the integrals of sqrt(g). It just happens that these integrals\n(at least for the areas) are supported on discrete points (intersections\nof a spin network with your surface), but nevertheless the usual geometric\ndefinitions of the proper areas are still assumed to hold.\n\nOne can then derive that the area eigenvalues are quantized. Of course,\nthis area quantization is not really a consequence of quantum gravity. It\nis a consequent of the specific change of the variables proposed by\nAshtekar. If we rewrite the metric using the "new variables" (the gauge\nfield), it may look as a legitimate operation locally on the configuration\nspace, but it is not faithful globally exactly because it introduces new\nperiodicities and/or quantization rules. The specific LQG area\nquantization rules follow from the exact choice of the variables, and\ndifferent variables would lead to different quantization rules for\ndifferent quantities. In this sense, the area quantization may be\ninterpreted as the assumption for taking exactly this type of LQG, but\nthere are infinitely many other LQG-like theories. Neither of them is\nexactly dual to gravity because all of them introduce new unphysical\nquantization rules that can neither be derived from the quantized metric\nnor from string theory.\n\nAnother point you are probably saying, too.\n\nEven if the areas had the discrete spectrum of eigenvalues, I don\'t\nunderstand how they would like to test it experimentally, even in\nprinciple. If there are no positive areas smaller than the Planck area,\nthen nothing in the world can measure other areas with a subPlanckian\nprecision. To make the statement about the quantized areas physical, our\nLQG friends would have to translate the area spectrum into a testable\nprediction of a particular (probably scattering) experiment. I am afraid\nthat there is no evidence that the framework of LQG could do anything like\nthat, and obviously no physical prediction can be extracted until the\nstructure of the vacuum in LQG is understood (and most likely, LQG does\nnot admit the physical - Minkowski, Lorentz-invariant - vacuum).\n\nIn string theory we often use some intermediate concepts, but all exact\nand meaningful statements about the short-distance physics can really be\nextracted from the S-matrix, which is why we compute it. The S-matrix\nworks and it is even necessary for certain purposes, but at the same\nmoment, the S-matrix is exactly what the relativists will always want to\navoid.\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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>Hi Moshe!

> There is some tension in my mind in the sentence "areas and volumes ...

My feeling is that you wrote this posting to make me happier! ;-) I
totally agree. In a general enough UV complete theory, the areas and the
volumes are semiclassical concepts - the whole geometry is replaced by
something more general in the UV. If one only considers geometry, it
involves integrating out the other UV physics, and the effective action
and measurements of the areas will depend on the chosen scale (different
scales are related by renormalization, integrating out loops), and so on.

So in a general enough theory - and string theory is certainly general
enough in this sense - I see no operational way to measure the geometric
quantities with a subPlanckian precision. Of course, we do have indirect
exact ways to measure the areas, but only the minimal areas of some
cycles, e.g. the supersymmetric cycles. Well, you can wrap a brane on a
cycle and measure its mass - this number divided by the tension is the
area/volume of the cycle including the quantum corrections. I don't see a
general string-theoretical way to define such areas and volumes - with a
subPlanckian precision - for a general surface or volume. The very idea of
a surface (or volume) embedded in spacetime is based on the assumption
that "sharp", point-like geometry exists at all scales, at least in some
way - otherwise the embedding cannot be determined accurately.

Loop quantum gravity is a very different framework. It is not "general
enough" according to the definitions above, and it is based on the
assumption that the metric is a good variable in all regimes, including -
or especially - the Planckian regime, and one can construct
gauge-invariant operators out of the metric and its dual variables that
have well-defined finite eigenvalues, much like the Hamiltonian of the
Hydrogen atom. Pre-geometry exists at all scales, and the volumes are
computed as the integrals of \sqrt(g). It just happens that these integrals
(at least for the areas) are supported on discrete points (intersections
of a spin network with your surface), but nevertheless the usual geometric
definitions of the proper areas are still assumed to hold.

One can then derive that the area eigenvalues are quantized. Of course,
this area quantization is not really a consequence of quantum gravity. It
is a consequent of the specific change of the variables proposed by
Ashtekar. If we rewrite the metric using the "new variables" (the gauge
field), it may look as a legitimate operation locally on the configuration
space, but it is not faithful globally exactly because it introduces new
periodicities and/or quantization rules. The specific LQG area
quantization rules follow from the exact choice of the variables, and
different variables would lead to different quantization rules for
different quantities. In this sense, the area quantization may be
interpreted as the assumption for taking exactly this type of LQG, but
there are infinitely many other LQG-like theories. Neither of them is
exactly dual to gravity because all of them introduce new unphysical
quantization rules that can neither be derived from the quantized metric
nor from string theory.

Another point you are probably saying, too.

Even if the areas had the discrete spectrum of eigenvalues, I don't
understand how they would like to test it experimentally, even in
principle. If there are no positive areas smaller than the Planck area,
then nothing in the world can measure other areas with a subPlanckian
precision. To make the statement about the quantized areas physical, our
LQG friends would have to translate the area spectrum into a testable
prediction of a particular (probably scattering) experiment. I am afraid
that there is no evidence that the framework of LQG could do anything like
that, and obviously no physical prediction can be extracted until the
structure of the vacuum in LQG is understood (and most likely, LQG does
not admit the physical - Minkowski, Lorentz-invariant - vacuum).

In string theory we often use some intermediate concepts, but all exact
and meaningful statements about the short-distance physics can really be
extracted from the S-matrix, which is why we compute it. The S-matrix
works and it is even necessary for certain purposes, but at the same
moment, the S-matrix is exactly what the relativists will always want to
avoid.

All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Moshe Rozali
Sep9-04, 02:54 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>Lubos,\n\nI write ALL my postings to make you happier, and now I finally\nacheived success... in retrospect, I should have mentioned LQG\nearlier.\n\nI guess we are in agreement on the main issue, which is a shame, I\nwas hoping I am missing something, thanks for the explanation of the\nLQG result.\n\nbest,\n\nMoshe\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>Lubos,

I write ALL my postings to make you happier, and now I finally
acheived success... in retrospect, I should have mentioned LQG
earlier.

I guess we are in agreement on the main issue, which is a shame, I
was hoping I am missing something, thanks for the explanation of the
LQG result.

best,

Moshe