View Full Version : Stringy naturalness
Lubos Motl
Sep30-04, 08:49 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 Michael,\n\nthanks for these interesting comments here. I\'ve read your constructively\nprovoking recent review\n\nhttp://www.arxiv.org/abs/hep-th/0409207\n\nWell, Peter Woit made some comments about the situation on his blog\n\nhttp://www.math.columbia.edu/~woit/blog/\n\non September 20th - his summary is that you say that string theory\npredicts that we can never see any physics related it. It would be too\ndifficult for me to pretend that I disagree with these Woit\'s remarks.\n\nDo I understand well that all these predictions of yours about the\nnonexistence of low energy SUSY and large dimensions critically rely on\nyour definition of stringy naturalness? You describe your notion of\nstringy naturalness very explicitly:\n\n(**) An effective field theory (or specific coupling or observable)\nT1 is more natural in string theory than T2, if the number\nof phenomenologically acceptable vacua leading to T1 is larger\nthan the number leading to T2.\n\nI could not disagree more. This very definition of naturalness already\nseems to contain - assume, in fact - Woit\'s result that the most typical\nprediction of this approach to string theory will be that there are no\npredictions. According to (**), the more ambiguous and unpredictive\nsomething is, the better.\n\nAlso, I don\'t think that this counting "the more vacua, the more natural"\ngeneralizes the notion of naturalness from physics "before" string theory\nin any natural way. I would say that naturalness means - and always meant\n- that the parameters that naturally appear in the description of physics\nshould be of order one. There are infinitely many more numbers (even among\nintegers!) :-) that are *not* of order one (for example 1235235236236236),\nbut this makes them *less* natural, not more, does not it?\n\nIf the notion of stringy naturalness were defined using the number of\nvacua, I would probably choose a definition which seems to be nearly the\nopposite of (**), namely\n\n(##) An effective field theory or physical mechanism - or a value\nof a coupling or another parameter - is natural from the\nstringy viewpoint if it can be expected to be reproduced\nin stringy backgrounds whose adjustable discrete parameters\nare of order one, i.e. backgrounds that are "simple".\n\nA more rigorous definition what is "simple" and what exactly should be of\norder one requires some deeper knowledge of physics than what we have, but\nthe rough philosophy difference seems clear, I think.\n\nNote that this definition more or less implies that the number of the\ndiscrete "natural" vacua with (approximately) the desired properties will\nalso be of order one, while your "natural" vacua are by definition members\nof huge families (unnatural families, in my language).\n\nI think that it is (##), not (**), that naturally generalizes the previous\nnotions of naturalness. Naturalness means that the properly defined\nparameters are of order one - not too small and not too large. The only\nopen question is "what it means a properly defined/parameterized\nparameter" i.e. "what is the measure", and deeper mechanisms such as those\nin string theory are here to answer the question.\n\nBecause of experimental observations, we simply know that some ratios in\nNature - such as m_{planck} / m_{electron} - are extremely large. These\nhierarchies of many types are simply a property of Nature and we can\'t do\nanything about them - except for trying to explain them (first\nqualitatively, and then perhaps quantitatively). The first obvious comment\nis that these large values seem *unnatural*. However, many such large\nratios are only unnatural until we learn and understand the physical\neffects that are underlying them. For example, it is not so shocking that\nm_{planck} / m_{QCD} is so large - once we realize that g_{strong} at the\nscale m_{planck} is a mildly small number of order one, and because\ng_{strong} only runs logarithmically, it is guaranteed to reach one at a\nmuch smaller energy scale.\n\nSimilarly, the large total mass of visible non-relativistic matter\n(expressed in Planck units) in the Universe today (it has not changed too\nmuch for billions of years, I think) can be explained by a reasonable\nnumber of e-foldings of inflation (which is able to produce mass "from\nnothing").\n\nThis is what I personally call a "natural" explanation of the large ratio\nm_{planck} / m_{QCD}, or a "natural" explanation of the large mass of the\nUniverse - and it is the kind of insights that we should be trying to\nfind. The role of string theory is to provide us with more reliable tools\nand mechanisms that can do this job. Do you think that you would agree\nwith this statement?\n\nOn the other hand, an explanation based on choosing some things very small\nor very large is unnatural, I think. An explanation based on a multiverse\n- or the conglomerate of all vacua in string theory that we can imagine -\nwhere all parameters can be very small or very large simply because there\nis a large number of such choices - it is a very unnatural (and\nunsatisfying) explanation.\n\nMoreover, we clearly know examples in which we must just find a scientific\nexplanation why something is small - the QCD theta-angle (strong CP\nproblem) could probably be of order one without spoiling life.\nNevertheless, it is very tiny for no good known (so far) reason, as we\nwere emphasized yesterday on a pheno talk by John Donoghue here.\n\nI would only claim that someone has understood why theta is small in\nstring theory if she had a simple realistic model with a calculation - at\nleast approximate one - that implies a small value of theta - or a class\nof models where the property holds universally. Finding 10^200 convoluted\nflux vacua is just not enough.\n\nThis debate may be philosophical today, but I believe that it will become\nvery scientific sometime in the future when people actually try to\nunderstand some cosmological (or other) vacuum selection mechanisms\nbecause whatever the mechanism will be, it will be favoring the choices\nwhere the "natural" parameters are of order one. This means, I believe,\nthat such a cosmological mechanism will produce vacua of the type (##)\nrather than the "generic convoluted" vacua of the type (**).\n\nA toy model: let\'s imagine that the number of Calabi-Yau topologies is\ninfinite, and chi goes to infinity, too. I think it is obvious that a\ncosmological "creation" will tend to produce Calabi-Yaus with reasonable\nchi\'s of order one, instead of some virtually infinite numbers - simply\nbecause it is also unnatural to create many handles of a Calabi-Yau. I\nthink that the reason why it\'s unnatural is the same like the reason why\nthe evolution theory is a more natural explanation than the Creator who\ncreated each species separately ad hoc (the analogy is simply 1 species =\n1 handle).\n\nMoreover, as the example above already indicates, it is conceivable that\nif one looks carefully enough, she can discover a discretely infinite\ntotal number of vacua in string theory, in which case the criterion (**)\nbreaks completely. On the other hand, there is nothing wrong with string\ntheory if it predicts a discrete infinite number of vacuum states (the\nharmonic oscillator has an infinite number of states, too). Any reasonable\nphysical mechanism that actually assigns "weights" or "probabilities" to\nthe vacua will not care whether the number of vacua is 10^300 or\ndiscretely infinite, which means that according to everything I can\nimagine, any reasoning that leads to very different results for these two\nchoices (10^300 vs. infinity) must be incorrect - which also means that\n(**) is incorrect. Don\'t you think that it should be legitimate to\napproximate 10^{300} by infinity? We can certainly do it for a harmonic\noscillator without getting too bad answers.\n\nThe number of vacua may be large (or discretely infinite), but they always\nhave some organization, hierarchy, and therefore include some "simple"\nvacua (analogy of the ground state of the harmonic oscillator and a few\nexcited states) where the discrete parameters are of order one - whatever\nit exactly means - and the "rest of the tower" which are unnatural states.\n\nWhat we should be interested in, I think, are mechanisms that illuminate\nhidden physics behind various numbers, and allows us to reparameterize\nthese (large or small) numbers as functions of natural numbers of order\none. Of course, this includes various mechanisms to generate numbers in\nthe exponential form. Inflation and the RG running of g_{strong} are\nexamples.\n\nAs John Donoghue has emphasized (also in his yesterday\'s talk at Harvard),\nthe "scale-invariant" form of fermion masses (i.e. the fact that they are\nmostly uniform on the logarithmic scale) has a nice explanation in\nintersecting brane models because the Yukawa couplings come from disks -\nand contain the exp(-A.tension) suppression. Assuming a uniform\ndistribution of the areas A (of the triangles - disks - stretched between\nthe three intersection points), we naturally obtain the qualitatively\ndesired spectrum of the fermion masses.\n\nEven if the number of stable (and so on) stringy vacua describing these\nmodels with intersecting branes turns out to be of order one, they will\nstill be natural, won\'t they? You can find some 10^200 of other vacua\nbased on complicated structures of large fluxes, among 10^300 of flux\nvacua in general, and it seems that according to your (**) rule, they will\nbe 10^200 times more interesting for you than the single intersecting\nbraneworld. Well, for me they will be much less interesting - even though,\nof course, they have a higher probability that they happen to describe the\nobservations accurately enough.\n\nIf one imagines that the "single" vacuum with intersecting branes and one\nof those 10^200 flux vacua will happen to agree with the experiments with\nthe desired accuracy, no doubt, I will prefer the "single intersecting\nbraneworld" - roughly with 10^{200} times bigger happiness than the flux\nvacuum. It\'s simply because this theory has a much smaller input and is\nmore natural.\n\nThere are other reasons why I think that (**) is obviously incorrect. Even\nif we imagine that the number of vacua is finite and exponentially large,\nthere is a subtlety. It is almost guaranteed that the exponents of\ndifferent types of vacua will be very different. There can be 10^{300}\ntype IIB flux vacua, but only 10^{280} vacua of M-theory on G2 manifolds\nwith fluxes. Does it mean that the IIB vacua are 10^{20} times more\nreasonable choice predicted by string theory? I hope not. It sounds\ncompletely irrational to me.\n\nMoreover, if the philosophy (**) were taken seriously, to more general\ntheories, the most natural theories according to this criterion would be\nnon-renormalizable theories because the number of the corresponding vacua\nis infinity^infinity - one can choose infinitely many numbers to be\nanything. These things are just the opposite of what I imagine to be\nnatural.\n\nBest regards\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Dear Michael,
thanks for these interesting comments here. I've read your constructively
provoking recent review
http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0409207
Well, Peter Woit made some comments about the situation on his blog
http://www.math.columbia.edu/~woit/blog/
on September 20th - his summary is that you say that string theory
predicts that we can never see any physics related it. It would be too
difficult for me to pretend that I disagree with these Woit's remarks.
Do I understand well that all these predictions of yours about the
nonexistence of low energy SUSY and large dimensions critically rely on
your definition of stringy naturalness? You describe your notion of
stringy naturalness very explicitly:
(**) An effective field theory (or specific coupling or observable)
T1 is more natural in string theory than T2, if the number
of phenomenologically acceptable vacua leading to T1 is larger
than the number leading to T2.
I could not disagree more. This very definition of naturalness already
seems to contain - assume, in fact - Woit's result that the most typical
prediction of this approach to string theory will be that there are no
predictions. According to (**), the more ambiguous and unpredictive
something is, the better.
Also, I don't think that this counting "the more vacua, the more natural"
generalizes the notion of naturalness from physics "before" string theory
in any natural way. I would say that naturalness means - and always meant
- that the parameters that naturally appear in the description of physics
should be of order one. There are infinitely many more numbers (even among
integers!) :-) that are *not* of order one (for example 1235235236236236),
but this makes them *less* natural, not more, does not it?
If the notion of stringy naturalness were defined using the number of
vacua, I would probably choose a definition which seems to be nearly the
opposite of (**), namely
(##) An effective field theory or physical mechanism - or a value
of a coupling or another parameter - is natural from the
stringy viewpoint if it can be expected to be reproduced
in stringy backgrounds whose adjustable discrete parameters
are of order one, i.e. backgrounds that are "simple".
A more rigorous definition what is "simple" and what exactly should be of
order one requires some deeper knowledge of physics than what we have, but
the rough philosophy difference seems clear, I think.
Note that this definition more or less implies that the number of the
discrete "natural" vacua with (approximately) the desired properties will
also be of order one, while your "natural" vacua are by definition members
of huge families (unnatural families, in my language).
I think that it is (##), not (**), that naturally generalizes the previous
notions of naturalness. Naturalness means that the properly defined
parameters are of order one - not too small and not too large. The only
open question is "what it means a properly defined/parameterized
parameter" i.e. "what is the measure", and deeper mechanisms such as those
in string theory are here to answer the question.
Because of experimental observations, we simply know that some ratios in
Nature - such as m_{planck} / m_{electron} - are extremely large. These
hierarchies of many types are simply a property of Nature and we can't do
anything about them - except for trying to explain them (first
qualitatively, and then perhaps quantitatively). The first obvious comment
is that these large values seem *unnatural*. However, many such large
ratios are only unnatural until we learn and understand the physical
effects that are underlying them. For example, it is not so shocking that
m_{planck} / m_{QCD} is so large - once we realize that g_{strong} at the
scale m_{planck} is a mildly small number of order one, and because
g_{strong} only runs logarithmically, it is guaranteed to reach one at a
much smaller energy scale.
Similarly, the large total mass of visible non-relativistic matter
(expressed in Planck units) in the Universe today (it has not changed too
much for billions of years, I think) can be explained by a reasonable
number of e-foldings of inflation (which is able to produce mass "from
nothing").
This is what I personally call a "natural" explanation of the large ratio
m_{planck} / m_{QCD}, or a "natural" explanation of the large mass of the
Universe - and it is the kind of insights that we should be trying to
find. The role of string theory is to provide us with more reliable tools
and mechanisms that can do this job. Do you think that you would agree
with this statement?
On the other hand, an explanation based on choosing some things very small
or very large is unnatural, I think. An explanation based on a multiverse
- or the conglomerate of all vacua in string theory that we can imagine -
where all parameters can be very small or very large simply because there
is a large number of such choices - it is a very unnatural (and
unsatisfying) explanation.
Moreover, we clearly know examples in which we must just find a scientific
explanation why something is small - the QCD \theta-angle (strong CP
problem) could probably be of order one without spoiling life.
Nevertheless, it is very tiny for no good known (so far) reason, as we
were emphasized yesterday on a pheno talk by John Donoghue here.
I would only claim that someone has understood why \theta is small in
string theory if she had a simple realistic model with a calculation - at
least approximate one - that implies a small value of \theta - or a class
of models where the property holds universally. Finding 10^200 convoluted
flux vacua is just not enough.
This debate may be philosophical today, but I believe that it will become
very scientific sometime in the future when people actually try to
understand some cosmological (or other) vacuum selection mechanisms
because whatever the mechanism will be, it will be favoring the choices
where the "natural" parameters are of order one. This means, I believe,
that such a cosmological mechanism will produce vacua of the type (##)
rather than the "generic convoluted" vacua of the type (**).
A toy model: let's imagine that the number of Calabi-Yau topologies is
infinite, and \chi goes to infinity, too. I think it is obvious that a
cosmological "creation" will tend to produce Calabi-Yaus with reasonable
\chi's of order one, instead of some virtually infinite numbers - simply
because it is also unnatural to create many handles of a Calabi-Yau. I
think that the reason why it's unnatural is the same like the reason why
the evolution theory is a more natural explanation than the Creator who
created each species separately ad hoc (the analogy is simply 1 species =
1 handle).
Moreover, as the example above already indicates, it is conceivable that
if one looks carefully enough, she can discover a discretely infinite
total number of vacua in string theory, in which case the criterion (**)
breaks completely. On the other hand, there is nothing wrong with string
theory if it predicts a discrete infinite number of vacuum states (the
harmonic oscillator has an infinite number of states, too). Any reasonable
physical mechanism that actually assigns "weights" or "probabilities" to
the vacua will not care whether the number of vacua is 10^300 or
discretely infinite, which means that according to everything I can
imagine, any reasoning that leads to very different results for these two
choices (10^300 vs. infinity) must be incorrect - which also means that
(**) is incorrect. Don't you think that it should be legitimate to
approximate 10^{300} by infinity? We can certainly do it for a harmonic
oscillator without getting too bad answers.
The number of vacua may be large (or discretely infinite), but they always
have some organization, hierarchy, and therefore include some "simple"
vacua (analogy of the ground state of the harmonic oscillator and a few
excited states) where the discrete parameters are of order one - whatever
it exactly means - and the "rest of the tower" which are unnatural states.
What we should be interested in, I think, are mechanisms that illuminate
hidden physics behind various numbers, and allows us to reparameterize
these (large or small) numbers as functions of natural numbers of order
one. Of course, this includes various mechanisms to generate numbers in
the exponential form. Inflation and the RG running of g_{strong} are
examples.
As John Donoghue has emphasized (also in his yesterday's talk at Harvard),
the "scale-invariant" form of fermion masses (i.e. the fact that they are
mostly uniform on the logarithmic scale) has a nice explanation in
intersecting brane models because the Yukawa couplings come from disks -
and contain the \exp(-A.tension) suppression. Assuming a uniform
distribution of the areas A (of the triangles - disks - stretched between
the three intersection points), we naturally obtain the qualitatively
desired spectrum of the fermion masses.
Even if the number of stable (and so on) stringy vacua describing these
models with intersecting branes turns out to be of order one, they will
still be natural, won't they? You can find some 10^200 of other vacua
based on complicated structures of large fluxes, among 10^300 of flux
vacua in general, and it seems that according to your (**) rule, they will
be 10^200 times more interesting for you than the single intersecting
braneworld. Well, for me they will be much less interesting - even though,
of course, they have a higher probability that they happen to describe the
observations accurately enough.
If one imagines that the "single" vacuum with intersecting branes and one
of those 10^200 flux vacua will happen to agree with the experiments with
the desired accuracy, no doubt, I will prefer the "single intersecting
braneworld" - roughly with 10^{200} times bigger happiness than the flux
vacuum. It's simply because this theory has a much smaller input and is
more natural.
There are other reasons why I think that (**) is obviously incorrect. Even
if we imagine that the number of vacua is finite and exponentially large,
there is a subtlety. It is almost guaranteed that the exponents of
different types of vacua will be very different. There can be 10^{300}
type IIB flux vacua, but only 10^{280} vacua of M-theory on G2 manifolds
with fluxes. Does it mean that the IIB vacua are 10^{20} times more
reasonable choice predicted by string theory? I hope not. It sounds
completely irrational to me.
Moreover, if the philosophy (**) were taken seriously, to more general
theories, the most natural theories according to this criterion would be
non-renormalizable theories because the number of the corresponding vacua
is infinity^infinity - one can choose infinitely many numbers to be
anything. These things are just the opposite of what I imagine to be
natural.
Best regards
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Michael R. Douglas
Oct4-04, 12:33 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\nThanks for the quick reply. I appreciate it, coming from someone who\nhas given real thought to the subject, as is evident in your paper\n0007206.\nYour message deserves a longer reply than I have time for right now,\nbut let me encourage anyone reading this to look at 0409207 and my\nother papers, especially the first two and last sections of 0303194,\nfor a real explanation of my point of view. But here is a brief\nresponse to your comments.\n\nFirst, it is very easy to get hung up in this context on definitions,\nphilosophical discussions of what it means to "explain" something,\netc. The best way to avoid this is to try to work towards some\nobjective claim. I have been trying to see if string theory can be\nfalsified, in the sense that we could show that some phenomenon which\nmight be observed could NOT be reproduced from string theory. Perhaps\nthe best candidate for this I know of is time variation\nof the fine structure constant, along lines argued in my\nhep-ph/0112059 with Banks and Dine. But it is interesting to consider\nother phenomena, even the standard predictions of low energy\nsupersymmetry, in this light.\n\nThere are various motivations for the definition (**) of "stringy\nnaturalness," as I explain in 0409207 and elsewhere. The main reason\nfor the name is just that it substitutes for and in some cases gives\ndifferent predictions from the traditional definitions of naturalness.\nAnother, as I discuss further below, is that it is "natural"\ninformation about the set of possibilities which can come out of\nstring theory, which is an important input into almost any candidate\nvacuum selection principle.\n\nBut perhaps the strongest motivation for the definition (**), as I\nexplain in 0409207 and elsewhere, is that under certain assumptions --\nmostly, that the number of vacua that we believe are candidates is in\nthis 10^100-10^300 range, (**) could lead to believable arguments that\ncertain possible physics could NOT be obtained from string/M theory,\nbecause there are just not enough vacua of that type to tune the\ncosmological constant and get the other parameters right.\nI believe that to falsify string theory, one really needs to argue\nthat there are NO consistent vacua (coming out of consistent\ncosmologies) in the class which reproduces the (current or\nhypothetical future) observations. It does not matter whether the\nvacua which do it are "complicated" or whether we dislike them for\nsome other subjective reason. Such falsifiability was expected in\nprevious discussions in which people assumed the existence of millions\nor billions of vacua. A major point of my work is that even numbers\nwhich seem large, like 10^300, are not necessarily too large from this\npoint of view.\n\nThe reason 10^300 should not be approximated by infinity, while\nperhaps 10^1000 could be, is just the amount of data at hand and the\nstructure of the problem. As discussed in 0303194, we measure many\nparameters of the SM to some precision, and the cosmological constant\nto fantastic accuracy, and it is the problem of matching this data\nwhich leads to numbers like 10^240.\n\nAt present it is not at all clear what the actual number and\ndistribution of vacua will turn out to be. Indeed the simplest\nconjecture (which I raised in 2001 at JHS60) is that the number is\ninfinite -- why not. I worked for over a year around 2002 looking for\nevidence for infinite series of vacua which\nmight roughly match observations (infinite series of CY\'s, of vector\nbundles, of brane configurations etc.) and my tentative conclusion is\nthat there are not; there are many infinite series (arbitrary flux in\nAdS_5 \\times S^5 being the simplest illustration) which however do not\nmatch observation because they have towers of light states coming\ndown. An infinite series of CY_3\'s might not have this property, but\nthe finitude of CY_3\'s is a famous conjecture at this point --\nthis is not to say we know it is true, just that many people have\nthought about it and it is consistent with everything else we know.\nAnyways, more people should be working on trying to find infinite\nseries of potentially realistic vacua.\n\nAlternatively, it might turn out in the end that the number is more\nlike 10^1000. Suppose we prove to our own satisfaction that there are\n10^1000 relevant vacua, which are uniformly enough distributed that\nstring theory can reproduce a huge variety of extensions of the\nStandard Model. Now I would consider this a huge victory for string\ntheorists, in that we answered a primary question, even if from some\npoint of view the answer was negative. We would have a theory which\ncould fit the data, and might lead to interesting new predictions in\nyet unrealized situations.\n\nBut one could also go on, and try to propose a principle to narrow\ndown the class of vacua, call it principle X, and regain predictivity.\nOne could go on to propose arguments justifying it, say that this\nclass of vacua comes out of the "preferred initial conditions." But\nthis is not absolutely necessary -- if a principle cuts down the\nnumbers of vacua sufficiently, one could assume it and try to make\npredictions along the lines I am discussing, and the principle might\njustify itself by its predictions. What is necessary is that the\nprinciple be precise and pick out a precise class of vacua. If you\nthink about this, since the principle will pick a subclass out of the\npreexisting set of possible vacua, using it will still require the\ndistribution information which we are studying now. Anyways, I think\nit is interesting to formulate such candidate vacuum selection\nprinciples.\n\nNow, predictions under assumptions such as principle X, cannot\nliterally falsify the theory, they can only falsify the combination of\ntheory + assumptions. Suppose no vacuum satisfying X reproduces the\nobservations, while some other vacuum Y not satisfying X, say it\nrequires special tuning of the initial conditions, has parameters\nwhich are not of order 1, etc., actually does. Will you go on to tell\nme that string theory is wrong, that vacuum Y is no good? You better\nhave a pretty convincing principle, more so than your (##) I think.\n\nI think it would be valuable for you to propose any precise version of\nyour (##). Whether (##) might also deserve to be called "naturalness"\nI think depends on what assumptions it is believed to follow from\n(note that there are many variations on the traditional definition of\nnaturalness as well, with different names). But it is not worth\ndiscussing without a precise definition. Is compactification on a CY\nwith hundreds of cycles "simpler" or "more complicated" than one with\na few cycles? Why? As for fluxes, the explicit discrete parameters\ndepend on a choice of basis, ambiguous up to Sp(b_3,Z)\ntransformations. How do I decide if they are "order one"? And so\non...\n\nPlease think about this as while I agree with some of your comments,\nothers seem basically wrong to me. For example, the comment early on\nthat "According to (**), the more ambiguous and unpredictive something\nis, the better." This is totally backwards as the most interesting\ncase is of course when the distribution of some observable is highly\npeaked. So, if it were to turn out that the distribution of the rank\nN of the gauge group among vacua was highly peaked at 4, that would be\nquite interesting, and many would consider it a candidate explanation\nfor the observed rank of the Standard Model gauge group. According to\n(**), the property N=4 would be highly natural. What does (##) say\nabout it?\n\nBest, Mike\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,
Thanks for the quick reply. I appreciate it, coming from someone who
has given real thought to the subject, as is evident in your paper
0007206.
Your message deserves a longer reply than I have time for right now,
but let me encourage anyone reading this to look at 0409207 and my
other papers, especially the first two and last sections of 0303194,
for a real explanation of my point of view. But here is a brief
response to your comments.
First, it is very easy to get hung up in this context on definitions,
philosophical discussions of what it means to "explain" something,
etc. The best way to avoid this is to try to work towards some
objective claim. I have been trying to see if string theory can be
falsified, in the sense that we could show that some phenomenon which
might be observed could NOT be reproduced from string theory. Perhaps
the best candidate for this I know of is time variation
of the fine structure constant, along lines argued in my
http://www.arxiv.org/abs/hep-ph/0112059 with Banks and Dine. But it is interesting to consider
other phenomena, even the standard predictions of low energy
supersymmetry, in this light.
There are various motivations for the definition (**) of "stringy
naturalness," as I explain in 0409207 and elsewhere. The main reason
for the name is just that it substitutes for and in some cases gives
different predictions from the traditional definitions of naturalness.
Another, as I discuss further below, is that it is "natural"
information about the set of possibilities which can come out of
string theory, which is an important input into almost any candidate
vacuum selection principle.
But perhaps the strongest motivation for the definition (**), as I
explain in 0409207 and elsewhere, is that under certain assumptions --
mostly, that the number of vacua that we believe are candidates is in
this 10^100-10^300 range, (**) could lead to believable arguments that
certain possible physics could NOT be obtained from string/M theory,
because there are just not enough vacua of that type to tune the
cosmological constant and get the other parameters right.
I believe that to falsify string theory, one really needs to argue
that there are NO consistent vacua (coming out of consistent
cosmologies) in the class which reproduces the (current or
hypothetical future) observations. It does not matter whether the
vacua which do it are "complicated" or whether we dislike them for
some other subjective reason. Such falsifiability was expected in
previous discussions in which people assumed the existence of millions
or billions of vacua. A major point of my work is that even numbers
which seem large, like 10^300, are not necessarily too large from this
point of view.
The reason 10^300 should not be approximated by infinity, while
perhaps 10^1000 could be, is just the amount of data at hand and the
structure of the problem. As discussed in 0303194, we measure many
parameters of the SM to some precision, and the cosmological constant
to fantastic accuracy, and it is the problem of matching this data
which leads to numbers like 10^240.
At present it is not at all clear what the actual number and
distribution of vacua will turn out to be. Indeed the simplest
conjecture (which I raised in 2001 at JHS60) is that the number is
infinite -- why not. I worked for over a year around 2002 looking for
evidence for infinite series of vacua which
might roughly match observations (infinite series of CY's, of vector
bundles, of brane configurations etc.) and my tentative conclusion is
that there are not; there are many infinite series (arbitrary flux in
AdS_5 \times S^5 being the simplest illustration) which however do not
match observation because they have towers of light states coming
down. An infinite series of CY_3's might not have this property, but
the finitude of CY_3's is a famous conjecture at this point --
this is not to say we know it is true, just that many people have
thought about it and it is consistent with everything else we know.
Anyways, more people should be working on trying to find infinite
series of potentially realistic vacua.
Alternatively, it might turn out in the end that the number is more
like 10^1000. Suppose we prove to our own satisfaction that there are
10^1000 relevant vacua, which are uniformly enough distributed that
string theory can reproduce a huge variety of extensions of the
Standard Model. Now I would consider this a huge victory for string
theorists, in that we answered a primary question, even if from some
point of view the answer was negative. We would have a theory which
could fit the data, and might lead to interesting new predictions in
yet unrealized situations.
But one could also go on, and try to propose a principle to narrow
down the class of vacua, call it principle X, and regain predictivity.
One could go on to propose arguments justifying it, say that this
class of vacua comes out of the "preferred initial conditions." But
this is not absolutely necessary -- if a principle cuts down the
numbers of vacua sufficiently, one could assume it and try to make
predictions along the lines I am discussing, and the principle might
justify itself by its predictions. What is necessary is that the
principle be precise and pick out a precise class of vacua. If you
think about this, since the principle will pick a subclass out of the
preexisting set of possible vacua, using it will still require the
distribution information which we are studying now. Anyways, I think
it is interesting to formulate such candidate vacuum selection
principles.
Now, predictions under assumptions such as principle X, cannot
literally falsify the theory, they can only falsify the combination of
theory + assumptions. Suppose no vacuum satisfying X reproduces the
observations, while some other vacuum Y not satisfying X, say it
requires special tuning of the initial conditions, has parameters
which are not of order 1, etc., actually does. Will you go on to tell
me that string theory is wrong, that vacuum Y is no good? You better
have a pretty convincing principle, more so than your (##) I think.
I think it would be valuable for you to propose any precise version of
your (##). Whether (##) might also deserve to be called "naturalness"
I think depends on what assumptions it is believed to follow from
(note that there are many variations on the traditional definition of
naturalness as well, with different names). But it is not worth
discussing without a precise definition. Is compactification on a CY
with hundreds of cycles "simpler" or "more complicated" than one with
a few cycles? Why? As for fluxes, the explicit discrete parameters
depend on a choice of basis, ambiguous up to Sp(b_3,Z)
transformations. How do I decide if they are "order one"? And so
on...
Please think about this as while I agree with some of your comments,
others seem basically wrong to me. For example, the comment early on
that "According to (**), the more ambiguous and unpredictive something
is, the better." This is totally backwards as the most interesting
case is of course when the distribution of some observable is highly
peaked. So, if it were to turn out that the distribution of the rank
N of the gauge group among vacua was highly peaked at 4, that would be
quite interesting, and many would consider it a candidate explanation
for the observed rank of the Standard Model gauge group. According to
(**), the property N=4 would be highly natural. What does (##) say
about it?
Best, Mike
Lubos Motl
Oct8-04, 12:56 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 Mike,\n\nthank you very much for your time and extensive reply. I think that your\npoints became more comprehensible, and many of them have my sympathies.\nIt\'s clear that your time is valuable, so don\'t hurry up with another\nanswer if any.\n\n> But perhaps the strongest motivation for the definition (**), as I\n> explain in 0409207 and elsewhere, is that under certain assumptions --\n> mostly, that the number of vacua that we believe are candidates is in\n> this 10^100-10^300 range, (**) could lead to believable arguments that\n> certain possible physics could NOT be obtained from string/M theory,\n> because there are just not enough vacua of that type to tune the\n> cosmological constant and get the other parameters right.\n\nFor example, I totally appreciate this point, and use it all the time in\nthe discussions. For example, 10^{300} may seem like a lot, but if one\nassumes a more or less uniform Lambda, getting it more or less right\nreduces the number to 10^{200}. Most of them do not have a correct\nStandard Model, and so on. So if one ends up with 10^{100} Standard Model\nvacua in string theory, it may not be enough to get the parameters right:\nchoosing one vacuum among 10^{100} is like choosing 100 decimal digits of\ninformation (input), which means that the 30 parameters of the Standard\nModel (neutrino masses counted) can only be adjusted roughly to 3\ndecimal digits per each parameter.\n\nOur experimental precision may easily become better. So if we were really\nable to calculate the stringy vacua properly and accurately enough - and\nall of them ;-), it seems pretty clear to me that we could either falsify\nstring theory (or at least these checked classes of vacua), or find the\ncorrect vacuum that would have to convince everyone.\n\nWe\'re not there yet.\n\n> It does not matter whether the vacua which do it are "complicated" or\n> whether we dislike them for some other subjective reason.\n\nIt would not matter assuming that you could really extract all the\nnecessary information with a high enough precision - and verify that a\ngiven vacuum (say, a generic vacuum from a large class) agrees with\nreality exactly, giving you a plenty of nontrivial (and successfully\ntested) digits to check beyond those that you inserted as input. In that\ncase, I think that no rational person would have serious doubts that this\nvacuum is correct, even if it looks slightly contrived, and the selection\nmechanism, cosmology, and everything else would have to be adjusted in\nsuch a way that it agrees with this obviously correct vacuum.\n\nI hope this answer makes you a bit happier, and I also hope that you would\nagree that the same approval would apply to a theory from a very small\nclass, once it\'s successfully tested.\n\nBut in my opinion, if we only consider some sort of approximate,\nqualitative agreement, it is less convincing to explain some class of\nphenomena with a more arbitrary theory (a theory with many choices/vacua,\nalthough they are just discrete) than with a more rigid theory (by\n"theory" I can mean a particular class of string vacua that do *not* come\nin too many flavors).\n\n> The reason 10^300 should not be approximated by infinity, while\n> perhaps 10^1000 could be, is just the amount of data at hand and the\n> structure of the problem. As discussed in 0303194, we measure many\n> parameters of the SM to some precision, and the cosmological constant\n> to fantastic accuracy, and it is the problem of matching this data\n> which leads to numbers like 10^240.\n\nI agree that 10^{300} might not be enough at the end ;-), but we are not\nat the end yet. Your texts simply lead me to the feeling that you believe\nthat if we find a proof that the number of vacua in some "class" XY -\nwhatever grouping to the classes one chooses - is much bigger than the\nnumber of vacua in another class VW, then we should focus most of our\nresearch energy on XY.\n\nThis is the point that I can\'t agree with, because I think that a small,\nunderpopulated class of vacua has roughly the same chance to be relevant\nfor reality as an overpopulated class of vacua - and the explanation of\nreality with the "small" class would even be more satisfactory and\nimpressive because more detailed facts about the structure of the Universe\nwould be explained. One of your assumptions seems to be that the final\nexplanation of reality *cannot* be that impressive and we must live in a\ngeneric vacuum of some type where very many things remain unexplained (a\nresult of evironmental accidents) which I view to be the *unscientific*\npart of the anthropic reasoning.\n\nIn other words, I think that the amount of research energy and optimism\nthat we focus on a given class of vacua XY should be proportional to the\namount of interesting ideas (both mathematically, as well as the ideas\nthat seem to be relevant for reality) in this class, which is a very\ndifferent quantity than the number of discrete vacua in this class.\n\nLet me interpret your approach - as I see it - in a slightly different\nway. You seem to be emphasizing the populated classes of vacua, and the\n"generic" vacua in this counting. And the reason seems to be that these\nare the vacua where you have the greatest probability to find one that\nagrees with reality pretty well assuming that all parameters predicted by\nthe vacua in this class are distributed sort of randomly.\n\nBut that does not seem to be the purpose of physics because we don\'t want\nto find a vacuum that just "looks", within the experimental errors,\ncompatible with reality, do we? This would be a pure description, not a\nquest for a predictive correct theory. We want to find the *exact* vacuum,\none that is able to predict the numbers about the *future* experiments\nwith an arbitrary, ever increasing accuracy, and I just see no reason why\nthe *exact* vacuum describing the Universe around should be "generic" in\nthe counting with the uniform measure.\n\nThis statement about "genericity" of the vacuum around us could only\nresult from some "metathermodynamic" ensemble of Universes that, for a\nreason totally mysterious to me, assigns the same (or roughly the same)\nprobability for any vacuum. Do you think that such a mechanism exists\nanywhere (in early cosmology? Cosmology of the multiverses?).\n\nIf there are two very different classes of vacua with very different\nnumbers of members, and they seem to have the same amount of interesting\nideas, and they seem roughly equally realistic, then I would only agree\nthat the "large" class will have a bigger chance to shoot near the real\nUniverse assuming that the theory as wrong and the agreements are pure\ncoincidences. If we want to maximize our chances to pretend the agreement\nand show that the theory looks rather correct (even though it is not\nexactly correct), then the large class is a better choice.\n\nOn the other hand, if the theory is valid and one of the vacua is exactly\ncorrect, the statements that it must come from the smaller (or larger)\ngroup seem equally reasonable to me, and the answer "smaller" is certainly\nmore appealing, predictive, and satisfactory. If we want to find a\n*completely* correct theory that can give us arbitrarily many arbitrarily\naccurate new predictions, then the correct predictions from such a theory\ncannot be guessed by "coincidence" (the probability to get everything\nexactly right is zero, much less than 10^{-1000}), and the large number of\nvacua in a class simply does not help. What helps are well-motivated\nideas.\n\nI think that the field theory counterpart of your approach would be the\nfollowing: imagine that we don\'t know the Standard Model (nor string\ntheory) yet, and we are looking for a QFT that describes the phenomena up\nto 200 GeV properly within the class of renormalizable 4D quantum field\ntheories. I think that a reasoning similar to yours would lead us to the\ntheories with very many (infinitely many?) fields and adjustable\nparameters, because the "number" of these theories - understood as the\nspace of parameters (coupling constants) - is peaked for these convoluted\ntheories. In this field theory context I am sure that you will agree that\nthis specific implementation of your rule is not only un-predictive, but\nit is also incorrect because the correct theory, namely the Standard\nModel, only has these less than 30 parameters as we already know.\n\nIn my opinion, the only relevant difference between the quantum field\ntheory example above and the case of string theory landscape is that they\norganize the parameters differently. It remains unreasonable to focus on\nthe "huge classes" of vacua - whose predictive power is inevitably limited\nby the large input. The only way how the string theory situation could be\nvery different is the case when we actually use the fact that these vacua\nare dynamically connected within a single theory - but this fact can only\nbe used once we compute the actual dynamics (tunneling between the\ndifferent Universes right after the Big Bang); but if we calculate it this\nway, I am pretty sure that the result will be extremely far from your\nuniform probability measure on the vacua.\n\n> might roughly match observations (infinite series of CY\'s, of vector\n> bundles, of brane configurations etc.) and my tentative conclusion is\n> that there are not; there are many infinite series (arbitrary flux in\n> AdS_5 \\times S^5 being the simplest illustration) which however do not\n> match observation because they have towers of light states coming\n> down. An infinite series of CY_3\'s might not have this property, but\n> the finitude of CY_3\'s is a famous conjecture at this point --\n> this is not to say we know it is true, just that many people have\n> thought about it and it is consistent with everything else we know.\n> Anyways, more people should be working on trying to find infinite\n> series of potentially realistic vacua.\n\nRight. As far as I understand, it\'s proved (according to what you say)\nthat a finite number of CY_3 topologies implies a finite number of\ncorresponding type II (and heterotic?) geometric vacua - i.e. the fluxes\nand bundles can\'t multiply it by infinity. Is that correct? Cannot you\nstill have an infinite number of realistic nongeometric vacua?\n\n> Alternatively, it might turn out in the end that the number is more\n> like 10^1000. Suppose we prove to our own satisfaction that there are\n> 10^1000 relevant vacua, which are uniformly enough distributed that\n> string theory can reproduce a huge variety of extensions of the\n> Standard Model. Now I would consider this a huge victory for string...\n\nWell, yes, we clearly differ on this point. I don\'t see any victory about\nthis situation; in fact, we are not terribly far from this situation, and\nthe current situation does not seem as a final victory to me (or most\nothers). ;-)\n\n> theorists, in that we answered a primary question, even if from some\n> point of view the answer was negative. We would have a theory which\n> could fit the data, and might lead to interesting new predictions in\n> yet unrealized situations.\n\nI don\'t understand how can one make predictions if virtually all - or at\nleast extremely many possibilities - exist for the new physics. Do you\nmean quantitative predictions, or some qualitative predictions?\n\n> Now, predictions under assumptions such as principle X, cannot\n> literally falsify the theory, they can only falsify the combination of\n> theory + assumptions. Suppose no vacuum satisfying X reproduces the\n> observations, while some other vacuum Y not satisfying X, say it\n> requires special tuning of the initial conditions, has parameters\n> which are not of order 1, etc., actually does. Will you go on to tell\n> me that string theory is wrong, that vacuum Y is no good?\n\nAs I\'ve said, I will only approve Y as the correct vacuum once it\'s able\nto successfully predict - in agreement with observations - many more\npieces of information (for example, couplings in the Standard Model with\nvery good accuracy) than the information input that you inserted (in the\nform of the discrete choice of Y within its class). As long as you only\ntalk about the potential to agree with reality exactly enough, I will\ninsist that we don\'t know whether Y is quite good, and the large number of\nrepresentatives similar to Y does not make the probability that Y is\n*exactly* correct any more likely.\n\n> I think it would be valuable for you to propose any precise version of\n> your (##).\n\nRight, I would need a cosmological mechanism that decides what is the\n"thermal weight" of the vacuum in some big-bang ensemble of all Universes,\nwhatever it means. This leads us to the problems of quantum cosmology. But\nin a sense, the goal is that one starts with a Planckian seed which may be\nuniversal in some sense - or whose details do not matter - and evolve it\naccording some rules that can still tell us what is the probability for\ndimensions to grow into some given shapes/vacua of string theory. I don\'t\nknow how to calculate this early cosmology or whatever it is, but it\nshould be done at one moment, and the result will be almost definitely\ndifferent from the uniform distribution of the probabilities per vacuum.\n\nThe uniform distribution is also not what one gets from the anthropic\nreasoning - where every vacuum is weighted by the number of people that\nwill occur in the Universe.\n\n> a few cycles? Why? As for fluxes, the explicit discrete parameters\n> depend on a choice of basis, ambiguous up to Sp(b_3,Z)\n> transformations. How do I decide if they are "order one"? And so\n> on...\n\nWell, I agree. The definition - or derivation - which vacua are "simpler"\nin this class would have to be more specific, and it is conceivable that\nthe "simple" flux vacua would have the property that there exists a basis\n(or a Sp(b_3,Z) transformation of the original basis) in which most fluxes\nare zero, for example. I don\'t know. It is also conceivable that the large\ndegeneracy of these vacua will be viewed, at the end, as a reason why this\nwhole class is not interesting.\n\n> Please think about this as while I agree with some of your comments,\n> others seem basically wrong to me. For example, the comment early on\n> that "According to (**), the more ambiguous and unpredictive something\n> is, the better." This is totally backwards as the most interesting\n> case is of course when the distribution of some observable is highly\n> peaked. So, if it were to turn out that the distribution of the rank\n> N of the gauge group among vacua was highly peaked at 4, that would be\n> quite interesting, and many would consider it a candidate explanation\n\nIt would just not be too interesting to people with similar feelings as I\nbecause the understanding that the gauge group\'s rank - one number - is\ngenerically equal to four is very far from being able to predict the\nresults of billions of experiments that we can make - simply because the\nrank is a very tiny part of the total information about the model. The\nconcept of the heterotic strings on Calabi-Yau spaces naturally predicts\nreasonable representations for the fermions, which would still seem as a\nmuch bigger success than some calculation leading to "average rank equals\nfour".\n\nMoreover, the rank is certainly not peaked around four in the class of\n*all* vacua - your/Juan\'s infinite class "AdS5 x S5 with flux N" has\ndifferent ranks, for example. Therefore, you only want to look for "ranks\npeaked around four" within a subclass of vacua that match the reality in\nother aspects. In this sense, you are looking for realistic correlations\nbetween different realistic features of the stringy vacua - and you want\nto show them as successes. Well, I don\'t think that such a statistical\ncorrelation is an impressive success, especially because between a large\nnumber of variables you can always find a pair of variables that are\ncorrelated the way you want.\n\nKumar and Wells, for example, seem to disagree with you, too (and agree\nwith me). Their motivation is, on the contrary, to look for properties of\nthe stringy vacua that are *not* peaked around the right values, because\nthese properties are the best guides to reduce the set of candidate vacua\n- they give us the maximal possible information. This is a possible path\nto progress which is better than to look for things that already seem\npretty OK generically. Do you agree?\n\nConsider three classes of the vacua as an example (the properties\ndescribed below are simplified, and many other properties are not listed;\nit is a thought experiment).\n\n1. The classical 1980s vacua - heterotic strings on Calabi-Yaus.\nThey naturally lead to reasonable gauge groups and matter\nspectrum\n\n2. The intersecting brane models. They can give us realistic\nSM-like models, and they naturally lead to the exponential\nhierarchy of fermionic masses\n\n3. The type IIB flux vacua. They can lead to many different\nUniverses including those SM-like ones, and their main virtue (?)\nis that the number of these vacua is largest, around 10^{1000}\n\nWhich class would we prefer as a candidate for reality? I would definitely\nprefer 1,2 because of their specific physical properties listed above,\nwhile the "virtue" of 3 does not impress me at all. Do you differ?\n\n> for the observed rank of the Standard Model gauge group. According to\n> (**), the property N=4 would be highly natural. What does (##) say\n> about it?\n\nI completely agree. 11-dimensional M-theory or 10-dimensional string\ntheory *are* natural (and simple) vacua, and we probably *must* use some\nanthropic argument to explain "why" we don\'t live in a higher-dimensional\nworld or a world vith a higher supersymmetry. The compactifications are OK\nand one can add ingredients, but the more ingredients (that carry a lot of\ndiscrete information/input) we add, the less rigid and convincing the\nstructure is. I am not questioning that string theory allows us to include\nthe fluxes and branes consistently, but I am questioning the idea that a\nlarge number of fluxes and wrapped branes on a complicated internal\nmanifold is what can be naturally expected to arise from early cosmology\n(or from another rational explanation of the structure of the extra\ndimensions).\n\nWell, the more ingredients and fluxes we add, the easier it is to agree\nwith all known data within some accuracy - but this is a very different\nvalue from having a correct predictive theory, I think.\n\nCould I ask you a specific question once more? If you or someone else\nproved with certainty that the number of vacua whose best description is\nM-theory on G_2 holonomy manifolds is 10^{150} times smaller than the\nnumber of type IIB flux vacua, would you recommend all the researchers on\nG_2 holonomy manifolds (perhaps, except for 10^{-148} percent of them,\nwhich is much less than one electron) to switch to type IIB because of\nthis counting?\n\nOf course, my answer would definitely be "No" - maybe even "on the\ncontrary".\n\nThanks again, and all 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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Dear Mike,
thank you very much for your time and extensive reply. I think that your
points became more comprehensible, and many of them have my sympathies.
It's clear that your time is valuable, so don't hurry up with another
answer if any.
> But perhaps the strongest motivation for the definition (**), as I
> explain in 0409207 and elsewhere, is that under certain assumptions --
> mostly, that the number of vacua that we believe are candidates is in
> this 10^100-10^300 range, (**) could lead to believable arguments that
> certain possible physics could NOT be obtained from string/M theory,
> because there are just not enough vacua of that type to tune the
> cosmological constant and get the other parameters right.
For example, I totally appreciate this point, and use it all the time in
the discussions. For example, 10^{300} may seem like a lot, but if one
assumes a more or less uniform \Lambda, getting it more or less right
reduces the number to 10^{200}. Most of them do not have a correct
Standard Model, and so on. So if one ends up with 10^{100} Standard Model
vacua in string theory, it may not be enough to get the parameters right:
choosing one vacuum among 10^{100} is like choosing 100 decimal digits of
information (input), which means that the 30 parameters of the Standard
Model (neutrino masses counted) can only be adjusted roughly to 3
decimal digits per each parameter.
Our experimental precision may easily become better. So if we were really
able to calculate the stringy vacua properly and accurately enough - and
all of them ;-), it seems pretty clear to me that we could either falsify
string theory (or at least these checked classes of vacua), or find the
correct vacuum that would have to convince everyone.
We're not there yet.
> It does not matter whether the vacua which do it are "complicated" or
> whether we dislike them for some other subjective reason.
It would not matter assuming that you could really extract all the
necessary information with a high enough precision - and verify that a
given vacuum (say, a generic vacuum from a large class) agrees with
reality exactly, giving you a plenty of nontrivial (and successfully
tested) digits to check beyond those that you inserted as input. In that
case, I think that no rational person would have serious doubts that this
vacuum is correct, even if it looks slightly contrived, and the selection
mechanism, cosmology, and everything else would have to be adjusted in
such a way that it agrees with this obviously correct vacuum.
I hope this answer makes you a bit happier, and I also hope that you would
agree that the same approval would apply to a theory from a very small
class, once it's successfully tested.
But in my opinion, if we only consider some sort of approximate,
qualitative agreement, it is less convincing to explain some class of
phenomena with a more arbitrary theory (a theory with many choices/vacua,
although they are just discrete) than with a more rigid theory (by
"theory" I can mean a particular class of string vacua that do *not* come
in too many flavors).
> The reason 10^300 should not be approximated by infinity, while
> perhaps 10^1000 could be, is just the amount of data at hand and the
> structure of the problem. As discussed in 0303194, we measure many
> parameters of the SM to some precision, and the cosmological constant
> to fantastic accuracy, and it is the problem of matching this data
> which leads to numbers like 10^240.
I agree that 10^{300} might not be enough at the end ;-), but we are not
at the end yet. Your texts simply lead me to the feeling that you believe
that if we find a proof that the number of vacua in some "class" XY -
whatever grouping to the classes one chooses - is much bigger than the
number of vacua in another class VW, then we should focus most of our
research energy on XY.
This is the point that I can't agree with, because I think that a small,
underpopulated class of vacua has roughly the same chance to be relevant
for reality as an overpopulated class of vacua - and the explanation of
reality with the "small" class would even be more satisfactory and
impressive because more detailed facts about the structure of the Universe
would be explained. One of your assumptions seems to be that the final
explanation of reality *cannot* be that impressive and we must live in a
generic vacuum of some type where very many things remain unexplained (a
result of evironmental accidents) which I view to be the *unscientific*
part of the anthropic reasoning.
In other words, I think that the amount of research energy and optimism
that we focus on a given class of vacua XY should be proportional to the
amount of interesting ideas (both mathematically, as well as the ideas
that seem to be relevant for reality) in this class, which is a very
different quantity than the number of discrete vacua in this class.
Let me interpret your approach - as I see it - in a slightly different
way. You seem to be emphasizing the populated classes of vacua, and the
"generic" vacua in this counting. And the reason seems to be that these
are the vacua where you have the greatest probability to find one that
agrees with reality pretty well assuming that all parameters predicted by
the vacua in this class are distributed sort of randomly.
But that does not seem to be the purpose of physics because we don't want
to find a vacuum that just "looks", within the experimental errors,
compatible with reality, do we? This would be a pure description, not a
quest for a predictive correct theory. We want to find the *exact* vacuum,
one that is able to predict the numbers about the *future* experiments
with an arbitrary, ever increasing accuracy, and I just see no reason why
the *exact* vacuum describing the Universe around should be "generic" in
the counting with the uniform measure.
This statement about "genericity" of the vacuum around us could only
result from some "metathermodynamic" ensemble of Universes that, for a
reason totally mysterious to me, assigns the same (or roughly the same)
probability for any vacuum. Do you think that such a mechanism exists
anywhere (in early cosmology? Cosmology of the multiverses?).
If there are two very different classes of vacua with very different
numbers of members, and they seem to have the same amount of interesting
ideas, and they seem roughly equally realistic, then I would only agree
that the "large" class will have a bigger chance to shoot near the real
Universe assuming that the theory as wrong and the agreements are pure
coincidences. If we want to maximize our chances to pretend the agreement
and show that the theory looks rather correct (even though it is not
exactly correct), then the large class is a better choice.
On the other hand, if the theory is valid and one of the vacua is exactly
correct, the statements that it must come from the smaller (or larger)
group seem equally reasonable to me, and the answer "smaller" is certainly
more appealing, predictive, and satisfactory. If we want to find a
*completely* correct theory that can give us arbitrarily many arbitrarily
accurate new predictions, then the correct predictions from such a theory
cannot be guessed by "coincidence" (the probability to get everything
exactly right is zero, much less than 10^{-1000}), and the large number of
vacua in a class simply does not help. What helps are well-motivated
ideas.
I think that the field theory counterpart of your approach would be the
following: imagine that we don't know the Standard Model (nor string
theory) yet, and we are looking for a QFT that describes the phenomena up
to 200 GeV properly within the class of renormalizable 4D quantum field
theories. I think that a reasoning similar to yours would lead us to the
theories with very many (infinitely many?) fields and adjustable
parameters, because the "number" of these theories - understood as the
space of parameters (coupling constants) - is peaked for these convoluted
theories. In this field theory context I am sure that you will agree that
this specific implementation of your rule is not only un-predictive, but
it is also incorrect because the correct theory, namely the Standard
Model, only has these less than 30 parameters as we already know.
In my opinion, the only relevant difference between the quantum field
theory example above and the case of string theory landscape is that they
organize the parameters differently. It remains unreasonable to focus on
the "huge classes" of vacua - whose predictive power is inevitably limited
by the large input. The only way how the string theory situation could be
very different is the case when we actually use the fact that these vacua
are dynamically connected within a single theory - but this fact can only
be used once we compute the actual dynamics (tunneling between the
different Universes right after the Big Bang); but if we calculate it this
way, I am pretty sure that the result will be extremely far from your
uniform probability measure on the vacua.
> might roughly match observations (infinite series of CY's, of vector
> bundles, of brane configurations etc.) and my tentative conclusion is
> that there are not; there are many infinite series (arbitrary flux in
> AdS_5 \times S^5 being the simplest illustration) which however do not
> match observation because they have towers of light states coming
> down. An infinite series of CY_3's might not have this property, but
> the finitude of CY_3's is a famous conjecture at this point --
> this is not to say we know it is true, just that many people have
> thought about it and it is consistent with everything else we know.
> Anyways, more people should be working on trying to find infinite
> series of potentially realistic vacua.
Right. As far as I understand, it's proved (according to what you say)
that a finite number of CY_3 topologies implies a finite number of
corresponding type II (and heterotic?) geometric vacua - i.e. the fluxes
and bundles can't multiply it by infinity. Is that correct? Cannot you
still have an infinite number of realistic nongeometric vacua?
> Alternatively, it might turn out in the end that the number is more
> like 10^1000. Suppose we prove to our own satisfaction that there are
> 10^1000 relevant vacua, which are uniformly enough distributed that
> string theory can reproduce a huge variety of extensions of the
> Standard Model. Now I would consider this a huge victory for string...
Well, yes, we clearly differ on this point. I don't see any victory about
this situation; in fact, we are not terribly far from this situation, and
the current situation does not seem as a final victory to me (or most
others). ;-)
> theorists, in that we answered a primary question, even if from some
> point of view the answer was negative. We would have a theory which
> could fit the data, and might lead to interesting new predictions in
> yet unrealized situations.
I don't understand how can one make predictions if virtually all - or at
least extremely many possibilities - exist for the new physics. Do you
mean quantitative predictions, or some qualitative predictions?
> Now, predictions under assumptions such as principle X, cannot
> literally falsify the theory, they can only falsify the combination of
> theory + assumptions. Suppose no vacuum satisfying X reproduces the
> observations, while some other vacuum Y not satisfying X, say it
> requires special tuning of the initial conditions, has parameters
> which are not of order 1, etc., actually does. Will you go on to tell
> me that string theory is wrong, that vacuum Y is no good?
As I've said, I will only approve Y as the correct vacuum once it's able
to successfully predict - in agreement with observations - many more
pieces of information (for example, couplings in the Standard Model with
very good accuracy) than the information input that you inserted (in the
form of the discrete choice of Y within its class). As long as you only
talk about the potential to agree with reality exactly enough, I will
insist that we don't know whether Y is quite good, and the large number of
representatives similar to Y does not make the probability that Y is
*exactly* correct any more likely.
> I think it would be valuable for you to propose any precise version of
> your (##).
Right, I would need a cosmological mechanism that decides what is the
"thermal weight" of the vacuum in some big-bang ensemble of all Universes,
whatever it means. This leads us to the problems of quantum cosmology. But
in a sense, the goal is that one starts with a Planckian seed which may be
universal in some sense - or whose details do not matter - and evolve it
according some rules that can still tell us what is the probability for
dimensions to grow into some given shapes/vacua of string theory. I don't
know how to calculate this early cosmology or whatever it is, but it
should be done at one moment, and the result will be almost definitely
different from the uniform distribution of the probabilities per vacuum.
The uniform distribution is also not what one gets from the anthropic
reasoning - where every vacuum is weighted by the number of people that
will occur in the Universe.
> a few cycles? Why? As for fluxes, the explicit discrete parameters
> depend on a choice of basis, ambiguous up to Sp(b_3,Z)
> transformations. How do I decide if they are "order one"? And so
> on...
Well, I agree. The definition - or derivation - which vacua are "simpler"
in this class would have to be more specific, and it is conceivable that
the "simple" flux vacua would have the property that there exists a basis
(or a Sp(b_3,Z) transformation of the original basis) in which most fluxes
are zero, for example. I don't know. It is also conceivable that the large
degeneracy of these vacua will be viewed, at the end, as a reason why this
whole class is not interesting.
> Please think about this as while I agree with some of your comments,
> others seem basically wrong to me. For example, the comment early on
> that "According to (**), the more ambiguous and unpredictive something
> is, the better." This is totally backwards as the most interesting
> case is of course when the distribution of some observable is highly
> peaked. So, if it were to turn out that the distribution of the rank
> N of the gauge group among vacua was highly peaked at 4, that would be
> quite interesting, and many would consider it a candidate explanation
It would just not be too interesting to people with similar feelings as I
because the understanding that the gauge group's rank - one number - is
generically equal to four is very far from being able to predict the
results of billions of experiments that we can make - simply because the
rank is a very tiny part of the total information about the model. The
concept of the heterotic strings on Calabi-Yau spaces naturally predicts
reasonable representations for the fermions, which would still seem as a
much bigger success than some calculation leading to "average rank equals
four".
Moreover, the rank is certainly not peaked around four in the class of
*all* vacua - your/Juan's infinite class "AdS5 x S5 with flux N" has
different ranks, for example. Therefore, you only want to look for "ranks
peaked around four" within a subclass of vacua that match the reality in
other aspects. In this sense, you are looking for realistic correlations
between different realistic features of the stringy vacua - and you want
to show them as successes. Well, I don't think that such a statistical
correlation is an impressive success, especially because between a large
number of variables you can always find a pair of variables that are
correlated the way you want.
Kumar and Wells, for example, seem to disagree with you, too (and agree
with me). Their motivation is, on the contrary, to look for properties of
the stringy vacua that are *not* peaked around the right values, because
these properties are the best guides to reduce the set of candidate vacua
- they give us the maximal possible information. This is a possible path
to progress which is better than to look for things that already seem
pretty OK generically. Do you agree?
Consider three classes of the vacua as an example (the properties
described below are simplified, and many other properties are not listed;
it is a thought experiment).
1. The classical 1980s vacua - heterotic strings on Calabi-Yaus.
They naturally lead to reasonable gauge groups and matter
spectrum
2. The intersecting brane models. They can give us realistic
SM-like models, and they naturally lead to the exponential
hierarchy of fermionic masses
3. The type IIB flux vacua. They can lead to many different
Universes including those SM-like ones, and their main virtue (?)
is that the number of these vacua is largest, around 10^{1000}
Which class would we prefer as a candidate for reality? I would definitely
prefer 1,2 because of their specific physical properties listed above,
while the "virtue" of 3 does not impress me at all. Do you differ?
> for the observed rank of the Standard Model gauge group. According to
> (**), the property N=4 would be highly natural. What does (##) say
> about it?
I completely agree. 11-dimensional M-theory or 10-dimensional string
theory *are* natural (and simple) vacua, and we probably *must* use some
anthropic argument to explain "why" we don't live in a higher-dimensional
world or a world vith a higher supersymmetry. The compactifications are OK
and one can add ingredients, but the more ingredients (that carry a lot of
discrete information/input) we add, the less rigid and convincing the
structure is. I am not questioning that string theory allows us to include
the fluxes and branes consistently, but I am questioning the idea that a
large number of fluxes and wrapped branes on a complicated internal
manifold is what can be naturally expected to arise from early cosmology
(or from another rational explanation of the structure of the extra
dimensions).
Well, the more ingredients and fluxes we add, the easier it is to agree
with all known data within some accuracy - but this is a very different
value from having a correct predictive theory, I think.
Could I ask you a specific question once more? If you or someone else
proved with certainty that the number of vacua whose best description is
M-theory on G_2 holonomy manifolds is 10^{150} times smaller than the
number of type IIB flux vacua, would you recommend all the researchers on
G_2 holonomy manifolds (perhaps, except for 10^{-148} percent of them,
which is much less than one electron) to switch to type IIB because of
this counting?
Of course, my answer would definitely be "No" - maybe even "on the
contrary".
Thanks again, and all the best
Lubos
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