[SOLVED] Orbifold tachyons from SUGRA and other papers

Nov16-04, 10:04 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>http://motls.blogspot.com/2004/11/orbifold-tachyons-from-sugra-and-other.html\n\nLast time when I commented all the articles, we were impressed how\ninteresting and serious they were. Tonight it\'s slightly easier to\ndescribe all the hep-th papers on the web, but some of them still look\ninteresting:\n\n* http://www.arxiv.org/abs/hep-th/0411148 - Supergravity description\nof a family of tachyons on orbifolds\n\nThis paper by Matthew Headrick and Joris Raeymaekers is obviously\ninteresting. Consider the nonsupersymmetric orbifold of type II string\ntheory - Adams-Polchinski-Silverstein (APS) type of orbifold - on C/Z_n\nfor large n. You know that there are many tachyons in the twisted sectors.\nFor large n, it makes sense to T-dualize around the angular direction of\nC/Z_n. You get some SUGRA solution. Well, many people have definitely\nlooked at the orbifold in this way, using T-duality. But Matt and Joris\nfinally consider the obviously interesting limit in which n is sent to\ninfinity, but you keep n times alpha\' fixed. It\'s some kind of zero slope\nlimit, but the "lightest" tachyons (=closest to being massless) whose\nsquared masses are comparable to (-1/alpha\' n) survive this limit because\nthese squared masses are exactly inverse to the quantity that is kept\nfixed.\n\nNote that the tachyons come from twisted sectors. By interpreting the\nangle as a circle, they\'re winding strings. It means that in the T-dual\npicture, they are momentum modes of a field in the dual string theory\nwhich is effectively supergravity. Finally, the present authors calculate\nsome interactions of the momentum modes in supergravity - which exist\noff-shell - and they show that they agree with the couplings of the\ndifferent tachyons calculated from the CFT - which only exist on-shell.\nThat\'s very interesting. The main thing I worry about is that the result\nis perhaps not too unexpected because instead of the orbifold, one might\nwork directly with the "limiting CFT" on the thin cone and its T-dual.\n\n* http://www.arxiv.org/abs/hep-th/0411149 - Ricci solitons by Nitta\n\nThis author constructs some new solutions of the modified Ricci-flatness\nequations, something that is necessary for a CFT to be well-defined. You\nknow that Ricci flatness is the right equation of motion only if you have\nno fluxes and if the dilaton is constant. If the dilaton is not constant,\nthe Ricci tensor is nonzero - Einstein\'s equations get a source. He or she\ndoes not quite want to talk about a non-constant dilaton. Instead, he or\nshe focuses on another generalization of the CFT and Ricci flatness -\nnamely a CFT with an extra complex field that has an "anomalous\ndimension". My understanding is that it\'s just a matter of notation\nwhether you say that a field has an "anomalous dimension", or whether you\nredefine it by a function of the dilaton, and you allow the dilaton to\nvary. If my understanding is correct, Nitta has effectively found new\nsolutions of the combined Einstein\'s equations with some dilaton-gradient\nsource. These solutions have either a U(N) isometry, or an O(N) isometry.\nIt\'s because the coordinates of his or her manifolds are explicitly\nwritten using U(N) or O(N) covariant coordinates, and the metric only\ndepends on the quadratic invariants. Some of these solutions can be\ninterpreted as generalizations or deformations of the Ricci-flat metric of\nthe conifold with an extra linear dilaton - at least that\'s my impression.\n\n* http://www.arxiv.org/abs/hep-th/0411150 - Finiteness of N=4 SYM\n\nWe often say that the maximally supersymmetric Yang-Mills theory in four\ndimensions is "finite" because of powerful supersymmetric cancellations.\nWell, what is exactly is finite? Clearly, there are operators with\nanomalous dimensions that must be regulated and that are cutoff-dependent,\nand so forth. So it\'s not everything that is finite. However, the\neffective action is something that should be finite - it sort of computes\nthe correlators of the elementary fields in which the divergences are\nsupposed to cancel between the bosons and fermions. In this paper, these\ntwo guys try to prove the finiteness of the N=4 effective action in the\nN=1 superspace language. The effective action of N=4 can be written in\nthis form, due to the Slavnov-Taylor identity, as long as we allow the\nsuperfields to be "dressed". They must go through some un-controversial\nsteps to show that the R-symmetry anomaly cancels due to the N=2\nsupersymmetry, and they re-check that the one-loop beta-function vanishes.\nNevertheless, the final result is that once you express the effective\naction of the N=4 gauge theory in terms of the dressed fields, all the\nterms become independent of the UV cutoff, and the effective theory is\ntherefore finite.\n\n* http://www.arxiv.org/abs/hep-th/0411151 - Lorentz violation again\n\nOK, there was a Lorentz violating paper last time, too, and many comments\nmay be repeated. I still don\'t understand the motivation behind these\nmodels. The space of possible non-relativistic non-stringy quantum field\ntheories is huge. I don\'t really feel what constrains it. For relativistic\ntheories, we may label all fields by their dimension - which is their\ndimension with respect to space as well as time. However, for\nnon-relativistic theories, we must introduce separate spatial and temporal\ndimensional analyses, and I don\'t think that we can really distinguish\n"renormalizable" theories from "non-renormalizable" theories. Of course,\nthis can be done if we write down a non-relativistic theory as a\ndeformation of a relativistic theory, and this is what this groups does,\ntoo. Nevertheless the number of new terms, once you allow the Lorentz\nsymmetry to be broken, is again huge. Moreover, I think that the lessons\nof 1905 are serious lessons, and breaking the Lorentz invariance\nexplicitly should also be accompanied by breaking of the rotational\ninvariance, and I see no reasons to do so. Let\'s stop criticism for a\nwhile.\n\nWhat theory do they consider? Take the usual Maxwell action in four\ndimensions, and imagine that you decide to deform it in some way, by\nadding another action. What is the other action you know for a U(1) gauge\nfield? Well, the Chern-Simons action. There is a problem: the usual\nChern-Simons term only exists in three dimensions. However, that\'s not a\nproblem for those who don\'t care about the Lorentz invariance. Just\nmultiply the Chern-Simons 3-form with a general vector, i.e. a 1-form, to\nget a 4-form, and you can integrate the product over the spacetime. The\nvector picks a priviliged direction which is OK with you. Because the\nChern-Simons action has an epsilon in it, you will break not only the\nLorentz symmetry but also the CPT symmetry - which can happen once the\nLorentz symmetry is gone. To make the things even more confusing, add an\nexternal current J that couples to the gauge field via the J.A term.\n\nThat was too natural so far. Let\'s make something more fancy. Reduce this\nfour-dimensional Lorentz invariant theory to three spacetime dimensions\n(dimensional reduction). Moreover, don\'t reduce it along the priviliged\nvector discussed previously, but along a more general vector. In this\ncase, the three-dimensional Lorentz symmetry will still be violated. If\nyou write down some terms, you will discover mass terms of various types.\nJust do it and derive the equations of motion and solve them and draw\nseveral graphs. And don\'t forget to be excited that the results pick a\npriviliged reference frame (even though you know that it was your starting\npoint). It\'s probably a good feature to violate the Lorentz symmetry.\n\nFinally, let me admit that I am totally lost. I have no idea why they\'re\ndoing what they\'re doing, whether it should be a physically realistic\nmodel or a mathematically interesting one: I just don\'t see the meaning of\nit all. This type of activity is what most of us would be doing today if\nwe had no string theory. Combining random terms that apparently follow no\ndeeper or organizational principles - terms extracted from an infinite\nchaotic ocean of arbitrary terms and their combinations - terms that are\nmuch more ugly and unjustified than the theories that are known to work.\nSorry for being so skeptical; I might simply be dumb.\n\n* http://www.arxiv.org/abs/hep-th/0411152 - Triangulated gravity\n\nThese colleagues first repeat a lot of the commercials about "Causal\nDynamical Triangulations" that they\'ve already written in many previous\npapers. The starting points are very obvious and sort of naive: try to\ndefine the path integral of quantum gravity in a discretized form. (It\'s\nlike spin foams in loop quantum gravity, but you don\'t necessarily require\nthat the details will agree.) OK, so how can you discretize a geometry?\nYou triangulate it into simplices, and you imagine that every simplex has\na region of flat Minkowski spacetime in it.\n\n(That\'s not like loop quantum gravity - the latter assumes that there is\nno geometry "inside" the spin foam simplices - the geometry is\nconcentrated at the singular points and edges of the spin foam.)\n\nThen you write down the Einstein-Hilbert action many times and you\nemphasize that it is discretized. There are many other differences from\nloop quantum gravity: while the minimal positive distance in loop quantum\ngravity is sort of Planckian, in the present case they want to send the\nsize of the simplices to zero and the regulator should be unphysical. Of\ncourse that if you do it, you formally get quantized general relativity\nwith all of its problems: as soon as the resolution becomes strongly\nsubPlanckian, the fluctuation of the metric tensor becomes large. The path\nintegral will be dominated by heavily fluctuating configurations where the\ntopology changes a lot and where the causal relations are totally obscured\n- and the results of these path integrals will be non-renormalizably\ndivergent - at least if you expand them perturbatively. But this is simply\nwhat a correct, authentic quantization of pure gravity gives you.\n\nThese authors are doing something different in one essential aspect. They\ndon\'t want to sum over all configurations, all metrics - the objects that\nyou encounter in the foamy GR path integral above. They don\'t do it\nbecause they sort of know that pure GR at subPlanckian distances is\nrubbish. Instead, they truncate the path integral to contain "nice and\nsmooth" configurations only. The allowed configurations they include must\nbe not only nice, but they must have the trivial causal diagram as well as\na fixed topology - namely S3 x R in their main example. Well, if you\nrestrict your path integral to configurations that look nice, it\'s not\nsurprising that your final pictures will look nice and similar to flat\nspace, too. But it by no means implies that you have found a physical\ntheory.\n\nAny path integral that more or less works simply must be dominated by\nconfigurations that are non-differentiable almost everywhere, by the very\nnature of functional integration and by the uncertainty principle. One can\noften show that the path integral localizes, but that\'s just a result of\ntheorems and calculations. One cannot define the path integral to include\nsmooth and causal histories only. Such a definition simply violates the\nuncertainty principle as well as locality, if you make some global\nconstraints on the way how your 3-geometry can look like. Consequently, it\nalso violates general covariance, and you won\'t decouple the unphysical\npolarizations. If you also make global constraints about the allowed\nshapes as functions of time that cannot be derived from local constraints,\nyou will also violate unitarity.\n\nAs far as I know, none of these "discrete gravity" people ever asked the\nquestion whether these theories are physical, unitary, and so forth. In\nfact, the subset of the "discrete gravity" people called the "loop quantum\ngravity" people declares quite openly that they don\'t care about unitarity\nat all. Unitarity is actually one of their enemies and they claim that it\nfollows from time-translation symmetry, which is of course a\nmisunderstanding of total basics of physics: unitarity is about\nhermiticity of the Hamiltonian, if one exists, while time-translation\nsymmetry is about its time-independence. Unitarity is one of the concepts\nthat must be destroyed by the revolution in physics that they\'ve been\nplanning for quite some time. ;-) They\'re just not getting that unitarity\nis about the probabilities being non-negative numbers that sum up to one,\nand that this rule must work in any context in quantum physics. I am\nafraid that the rest of the "discrete gravity" people does not bother to\ncheck these elementary physics questions either. That may be a good point\nto stop criticism because everyone knows it anyway that I don\'t believe\nthat this line of research will lead to any usable new physics because it\nsimply neglects some totally essential features of quantum physics.\n\nAt any rate, they show that these strange rules of the game admit some\nbig-bang big-crunch cosmological solution described by some collective\ncoordinates (a nice picture animates in front of your eyes), and they\nconstruct or propose a wave function of the Universe that depends on the\nobservable representing the "3-volume of the Universe".\n\nposted by Lumo at 8:17 PM\n______________________________________________ ________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>http://motls.blogspot.com/2004/11/or...and-other.html

Last time when I commented all the articles, we were impressed how
interesting and serious they were. Tonight it's slightly easier to
describe all the hep-th papers on the web, but some of them still look
interesting:

* http://www.arxiv.org/abs/http://www....hep-th/0411148 - Supergravity description
of a family of tachyons on orbifolds

This paper by Matthew Headrick and Joris Raeymaekers is obviously
interesting. Consider the nonsupersymmetric orbifold of type II string
theory - Adams-Polchinski-Silverstein (APS) type of orbifold - on

for large n. You know that there are many tachyons in the twisted sectors.
For large n, it makes sense to T-dualize around the angular direction of

. You get some SUGRA solution. Well, many people have definitely
looked at the orbifold in this way, using T-duality. But Matt and Joris
finally consider the obviously interesting limit in which n is sent to
infinity, but you keep n times $LaTeX Code: \\alphasingle-quote$ fixed. It's some kind of zero slope
limit, but the "lightest" tachyons (=closest to being massless) whose
squared masses are comparable to $LaTeX Code: (-1/\\alphasingle-quote n)$ survive this limit because
these squared masses are exactly inverse to the quantity that is kept
fixed.

Note that the tachyons come from twisted sectors. By interpreting the
angle as a circle, they're winding strings. It means that in the T-dual
picture, they are momentum modes of a field in the dual string theory
which is effectively supergravity. Finally, the present authors calculate
some interactions of the momentum modes in supergravity - which exist
off-shell - and they show that they agree with the couplings of the
different tachyons calculated from the CFT - which only exist on-shell.
That's very interesting. The main thing I worry about is that the result
is perhaps not too unexpected because instead of the orbifold, one might
work directly with the "limiting CFT" on the thin cone and its T-dual.

* http://www.arxiv.org/abs/http://www....hep-th/0411149 - Ricci solitons by Nitta

This author constructs some new solutions of the modified Ricci-flatness
equations, something that is necessary for a CFT to be well-defined. You
know that Ricci flatness is the right equation of motion only if you have
no fluxes and if the dilaton is constant. If the dilaton is not constant,
the Ricci tensor is nonzero - Einstein's equations get a source. He or she
does not quite want to talk about a non-constant dilaton. Instead, he or
she focuses on another generalization of the CFT and Ricci flatness -
namely a CFT with an extra complex field that has an "anomalous
dimension". My understanding is that it's just a matter of notation
whether you say that a field has an "anomalous dimension", or whether you
redefine it by a function of the dilaton, and you allow the dilaton to
vary. If my understanding is correct, Nitta has effectively found new
solutions of the combined Einstein's equations with some dilaton-gradient
source. These solutions have either a U(N) isometry, or an O(N) isometry.
It's because the coordinates of his or her manifolds are explicitly
written using U(N) or O(N) covariant coordinates, and the metric only
depends on the quadratic invariants. Some of these solutions can be
interpreted as generalizations or deformations of the Ricci-flat metric of
the conifold with an extra linear dilaton

least that's my impression.

* http://www.arxiv.org/abs/http://www....hep-th/0411150 - Finiteness of N=4 SYM

We often say that the maximally supersymmetric Yang-Mills theory in four
dimensions is "finite" because of powerful supersymmetric cancellations.
Well, what is exactly is finite? Clearly, there are operators with
anomalous dimensions that must be regulated and that are cutoff-dependent,
and so forth. So it's not everything that is finite. However, the
effective action is something that should be finite

sort of computes
the correlators of the elementary fields in which the divergences are
supposed to cancel between the bosons and fermions. In this paper, these
two guys try to prove the finiteness of the N=4 effective action in the
N=1 superspace language. The effective action of N=4 can be written in
this form, due to the Slavnov-Taylor identity, as long as we allow the
superfields to be "dressed". They must go through some un-controversial
steps to show that the R-symmetry anomaly cancels due to the N=2
supersymmetry, and they re-check that the one-loop $LaTeX Code: \\beta-function$ vanishes.
Nevertheless, the final result is that once you express the effective
action of the N=4 gauge theory in terms of the dressed fields, all the
terms become independent of the UV cutoff, and the effective theory is
therefore finite.

* http://www.arxiv.org/abs/http://www....hep-th/0411151 - Lorentz violation again

OK, there was a Lorentz violating paper last time, too, and many comments
may be repeated. I still don't understand the motivation behind these
models. The space of possible non-relativistic non-stringy quantum field
theories is huge. I don't really feel what constrains it. For relativistic
theories, we may label all fields by their dimension - which is their
dimension with respect to space as well as time. However, for
non-relativistic theories, we must introduce separate spatial and temporal
dimensional analyses, and I don't think that we can really distinguish
"renormalizable" theories from "non-renormalizable" theories. Of course,
this can be done if we write down a non-relativistic theory as a
deformation of a relativistic theory, and this is what this groups does,
too. Nevertheless the number of new terms, once you allow the Lorentz
symmetry to be broken, is again huge. Moreover, I think that the lessons
of 1905 are serious lessons, and breaking the Lorentz invariance
explicitly should also be accompanied by breaking of the rotational
invariance, and I see no reasons to do so. Let's stop criticism for a
while.

What theory do they consider? Take the usual Maxwell action in four
dimensions, and imagine that you decide to deform it in some way, by
adding another action. What is the other action you know for a U(1) gauge
field? Well, the Chern-Simons action. There is a problem: the usual
Chern-Simons term only exists in three dimensions. However, that's not a
problem for those who don't care about the Lorentz invariance. Just
multiply the Chern-Simons 3-form with a general vector, i.e. a 1-form, to
get a 4-form, and you can integrate the product over the spacetime. The
vector picks a priviliged direction which is OK with you. Because the
Chern-Simons action has an $LaTeX Code: \\epsilon$ in it, you will break not only the
Lorentz symmetry but also the CPT symmetry - which can happen once the
Lorentz symmetry is gone. To make the things even more confusing, add an
external current J that couples to the gauge field via the J.A term.

That was too natural so far. Let's make something more fancy. Reduce this
four-dimensional Lorentz invariant theory to three spacetime dimensions
(dimensional reduction). Moreover, don't reduce it along the priviliged
vector discussed previously, but along a more general vector. In this
case, the three-dimensional Lorentz symmetry will still be violated. If
you write down some terms, you will discover mass terms of various types.
Just do it and derive the equations of motion and solve them and draw
several graphs. And don't forget to be excited that the results pick a
priviliged reference frame (even though you know that it was your starting
point). It's probably a good feature to violate the Lorentz symmetry.

Finally, let me admit that I am totally lost. I have no idea why they're
doing what they're doing, whether it should be a physically realistic
model or a mathematically interesting one: I just don't see the meaning of
it all. This type of activity is what most of us would be doing today if
we had no string theory. Combining random terms that apparently follow no
deeper or organizational principles - terms extracted from an infinite
chaotic ocean of arbitrary terms and their combinations - terms that are
much more ugly and unjustified than the theories that are known to work.
Sorry for being so skeptical; I might simply be dumb.

* http://www.arxiv.org/abs/http://www....hep-th/0411152 - Triangulated gravity

These colleagues first repeat a lot of the commercials about "Causal
Dynamical Triangulations" that they've already written in many previous
papers. The starting points are very obvious and sort of naive: try to
define the path integral of quantum gravity in a discretized form. (It's
like spin foams in loop quantum gravity, but you don't necessarily require
that the details will agree.) OK, so how can you discretize a geometry?
You triangulate it into simplices, and you imagine that every simplex has
a region of flat Minkowski spacetime in it.

(That's not like loop quantum gravity - the latter assumes that there is
no geometry "inside" the spin foam simplices - the geometry is
concentrated at the singular points and edges of the spin foam.)

Then you write down the Einstein-Hilbert action many times and you
emphasize that it is discretized. There are many other differences from
loop quantum gravity: while the minimal positive distance in loop quantum
gravity is sort of Planckian, in the present case they want to send the
size of the simplices to zero and the regulator should be unphysical. Of
course that if you do it, you formally get quantized general relativity
with all of its problems: as soon as the resolution becomes strongly
subPlanckian, the fluctuation of the metric tensor becomes large. The path
integral will be dominated by heavily fluctuating configurations where the
topology changes a lot and where the causal relations are totally obscured
- and the results of these path integrals will be non-renormalizably
divergent

least if you expand them perturbatively. But this is simply
what a correct, authentic quantization of pure gravity gives you.

These authors are doing something different in one essential aspect. They
don't want to sum over all configurations, all metrics - the objects that
you encounter in the foamy GR path integral above. They don't do it
because they sort of know that pure GR at subPlanckian distances is
rubbish. Instead, they truncate the path integral to contain "nice and
smooth" configurations only. The allowed configurations they include must
be not only nice, but they must have the trivial causal diagram as well as
a fixed topology - namely S3 x R in their main example. Well, if you
restrict your path integral to configurations that look nice, it's not
surprising that your final pictures will look nice and similar to flat
space, too. But it by no means implies that you have found a physical
theory.

Any path integral that more or less works simply must be dominated by
configurations that are non-differentiable almost everywhere, by the very
nature of functional integration and by the uncertainty principle. One can
often show that the path integral localizes, but that's just a result of
theorems and calculations. One cannot define the path integral to include
smooth and causal histories only. Such a definition simply violates the
uncertainty principle as well as locality, if you make some global
constraints on the way how your 3-geometry can look like. Consequently, it
also violates general covariance, and you won't decouple the unphysical
polarizations. If you also make global constraints about the allowed
shapes as functions of time that cannot be derived from local constraints,
you will also violate unitarity.

As far as I know, none of these "discrete gravity" people ever asked the
question whether these theories are physical, unitary, and so forth. In
fact, the subset of the "discrete gravity" people called the "loop quantum
gravity" people declares quite openly that they don't care about unitarity
at all. Unitarity is actually one of their enemies and they claim that it
follows from time-translation symmetry, which is of course a
misunderstanding of total basics of physics: unitarity is about
hermiticity of the Hamiltonian, if one exists, while time-translation
symmetry is about its time-independence. Unitarity is one of the concepts
that must be destroyed by the revolution in physics that they've been
planning for quite some time. ;-) They're just not getting that unitarity
is about the probabilities being non-negative numbers that sum up to one,
and that this rule must work in any context in quantum physics. I am
afraid that the rest of the "discrete gravity" people does not bother to
check these elementary physics questions either. That may be a good point
to stop criticism because everyone knows it anyway that I don't believe
that this line of research will lead to any usable new physics because it
simply neglects some totally essential features of quantum physics.

At any rate, they show that these strange rules of the game admit some
big-bang big-crunch cosmological solution described by some collective
coordinates (a nice picture animates in front of your eyes), and they
construct or propose a wave function of the Universe that depends on the
observable representing the "3-volume of the Universe".

posted by Lumo at 8:17 PM
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax:

Web: http://lumo.matfyz.cz/
eFax:

work:

home:

(call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Nov17-04, 08:09 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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, 16 Nov 2004 22:04:30 -0500, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> * http://www.arxiv.org/abs/hep-th/0411152 - Triangulated gravity\n>\n> These colleagues first repeat a lot of the commercials about "Causal\n> Dynamical Triangulations" that they\'ve already written in many previous\n> papers.\n\nOnce again, Lubos is much faster than me and I make my comments\nwithout having ready anything of the papers than the abstract. And I\nagree, when I saw this paper on the arxive, my reaction was "another\none of those. how for into the abstract do I have to read to find the\nnew stuff?" as was yours.\n\nHowever, again once again, I am a bit less critical than you are. OK,\nit seems they beat the publicity drum a lot but I think this is fair\nif you are a small group that wants to be noticed in the stringy\natmosphere of hep-th. And I should mention that Jan Ambjorn has worked\non many different things including matrix models (the old ones),\nlattice theories and string theory.\n\nSo let me try to say a couple of words in defence of their approach:\nThis stuff obviously has its background in the matrix model\nliterature and the realization of 2d gravity in terms of dynamical\ntriangulations (dual to matrices) was one of the successes of the\n80s. But you are right, the Euclidean path integral is not only\ndominated by but also seems to localise in non-smooth geometries.\nSo they try to cure this problem by changing the rules of their path\nintegral.\n\n[Moderator\'s note: Well, I understand. That\'s what I criticize.\nEvery path integral in a quantum theory is dominated by\nnon-differentiable configurations because this is necessary\nfor the uncertainty principle. A classical configuration has\nsharp, well-defined values of the fields like X(t) or PHI(x,t)\nor g_{12}(x,t), and by the uncertainty principle, the uncertainty\nof the canonical momentum must therefore be infinite, which is\nreflected in the path integral by the fact that the |derivative|\nof the field is typically infinite, i.e. the non-differentiable\nconfigurations dominate. Do you agree that you could not get\nquantum mechanics if your path integral only summed over\ndifferentiable paths? If you succeeded to define this "truncated"\nintegral in quantum mechanics, it would violate unitarity\nand the rule U(t1,t2)U(t2,t3)=U(t1,t3) because the different\nintervals would disagree "how much differentiable" the functions\nmust be. LM]\n\nIt is probably fair to divide geometry into different levels of\nstructure. One possible distinction is\n\n0) differentiable structure\n1) topology\n2) causal structure\n3) conformal structure\n4) metric structure\n\nIt is up to discussion at which of these levels you start varying in\nyour path integral and which parts you keep fixed.\n\n[Moderator\'s note: It is fair to divide geometries, but it is never\nfair to "cut" some configurations from a path integral, I think.\nWe\'ve had a recent debate on sci.physics.strings about the overcritical\nelectric field which was exactly about this issue - did you agree with\nour conclusion that you can\'t ever omit "unwanted" configurations? LM]\n\nI guess, Lubos wants to vary 1-4 while keeping 0 fixed, Ambjorn and\nfriends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in\nwhich he tried to convince me that one should vary all 0--4.\n\n[Moderator\'s note: I agree with him, you must vary everything.\nThe differential structure only exists "classically" and it is a\nconsequence of dynamics - the appearance of the derivative terms\nin the path integral. But the path integral over sums over all\nconfigurations of the given fields. LM]\n\nNobody has done a really convincing \'sum over geometries\' yet, so I\nthink it should be allowed to try all these approaches.\n\n[Moderator\'s note: Nobody has found a really convincing luminiferous\naether theory, so should all of us divide to different approaches how to\nconstruct aether? Actually I think that these two questions are more\nsimilar, even in details, than you might think. ;-) LM]\n\nWhat Ambjorn etal find is that again in 2d you can solve this model\nexactly (ie compute the partition function with sources) and it agrees\nwith expectations (whatever those would be). Second the typical\nconfigurations look much smoother (something they haven\'t put in, they\nonly demand causality and global topology) than in the Euclidean case.\n\n[Moderator\'s note: I am not getting this point at all. What\'s exactly\nthe difference between the input and output? Typical configurations\nin the gravity path integral have strongly oscillating topology, both\nin the Minkowski and the Euclidean case, and in the Minkowski case,\nthey have also a highly nontrivial and chaotic causal diagram.\nIf you unphysically cut the "ugly" configurations, of course, you will\nend up with the "nice" ones, and because you made more constraints\nabout the allowed configurations in the Minkowski case, you will\nget even nicer configurations than in the Euclidean space at the end. But\nthat\'s not a result, that\'s your assumption. And it\'s an assumption\nthat contradicts quantum mechanics. LM]\n\nOf course, in 2D gravity is not typical, all the dynamics is in the\ncosmological constant and its conjugate variable, the volume,\nrespectively. And in higher dimensions it is not possible to solve the\nproblem analytically, you can only run in on your computer.\n\n[Moderator\'s note: Right, 2D and 3D gravity don\'t really have gravitons\nas local degrees of freedom. All of us know how to compute 2D gravity\nas a path integral over "nice topologies" of two-dimensional spacetime:\nit\'s called the stringy worldsheet. But the conformal structure on\nthe worldsheet is only "nice" because *any* configuration in 2D\ncan be mapped to the "standard ones" by diff x Weyl transformations.\nAnalogous things hold for 2D string theory - one really wants to\ncalculate the path integral over the scalar fields in spacetime\nand their effects. Moreover, my arguments above that talk about the\nuncertainty principle for g_{12} and its time derivative can break down\nin d<4 because there is no such a physical degree of freedom. LM]\n\nAnother success that they claim is that they break the \'c=1 barrier\'. OK,\nI have no idea what that really is because I try to stay away from all\nthis old matrix model technology but Matthias Staudacher, who was around\nin those days, says this is quite non-trivial: In these models, you do not\nhave to restrict yourself to pure gravity, you can couple matter to it:\nFor example you can add an Ising spin degree of freedom to all your\ntriangles and sum over it as well.\n\nAs, you say, in all these models you have to take the continuum limit\nand then you get a conformal field theory. In the old days, it was\nobserved that whatever you did matter wise or matrix wise, you could\nonly get models with central charge <1. But with causal triangulations\ncoupled to matter you can break this barrier.\n\nFinally these people claim their model has a well behaved continuum\nlimit and I see no reason to doubt it.\n\n[Moderator\'s note: in the case of the present paper, I don\'t have\ndifficulties with the word "continuum limit" but rather with the word\n"model". You can define some set of rules that gives you a *classical*\ntheory in some limit, but it by no means implies that your rules,\nbefore you take the limit, define a meaningful quantum theory, does it?\nFor example, you should always ask whether your rules can lead to a\nunitary S-matrix, which path integrals should, and the answer will\nbe NO in the 4D case, I think. LM]\n\nBut in the end this is only gravity if you end up in the correct\nuniversality class. That is, all your weird rules you make up to construct\nyour discretised space-times correspond only to irrelevant operators that\ngo away in this limit. And to show this is of course the hard part.\n\n[Modeator\'s note: That may be a different way to say the same thing.\nYou\'re simply not sure whether the "restricted path integral" has\nanything whatsoever in common with the real path integral. LM]\n\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling School of Science and Engineering\nInternational University Bremen\nprint "Just another Phone: +49 421-200 3574\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 16 Nov 2004 22:04:30

Lubos Motl <motl@feynman.harvard.edu> wrote:

> * http://www.arxiv.org/abs/http://www....hep-th/0411152 - Triangulated gravity
>
> These colleagues first repeat a lot of the commercials about "Causal
> Dynamical Triangulations" that they've already written in many previous
> papers.

Once again, Lubos is much faster than me and I make my comments
without having ready anything of the papers than the abstract. And I
agree, when I saw this paper on the arxive, my reaction was "another
one of those. how for into the abstract do I have to read to find the
new stuff?" as was yours.

However, again once again, I am a bit less critical than you are. OK,
it seems they beat the publicity drum a lot but I think this is fair
if you are a small group that wants to be noticed in the stringy
atmosphere of hep-th. And I should mention that Jan Ambjorn has worked
on many different things including matrix models (the old ones),
lattice theories and string theory.

So let me try to say a couple of words in defence of their approach:
This stuff obviously has its background in the matrix model
literature and the realization of 2d gravity in terms of dynamical
triangulations (dual to matrices) was one of the successes of the
80s. But you are right, the Euclidean path integral is not only
dominated by but also seems to localise in non-smooth geometries.
So they try to cure this problem by changing the rules of their path
integral.

[Moderator's note: Well, I understand. That's what I criticize.
Every path integral in a quantum theory is dominated by
non-differentiable configurations because this is necessary
for the uncertainty principle. A classical configuration has
sharp, well-defined values of the fields like X(t) or $LaTeX Code: \\PHI(x,t)$
or $LaTeX Code: g_{12}(x,t),$ and by the uncertainty principle, the uncertainty
of the canonical momentum must therefore be infinite, which is
reflected in the path integral by the fact that the |derivative|
of the field is typically infinite, i.e. the non-differentiable
configurations dominate. Do you agree that you could not get
quantum mechanics if your path integral only summed over
differentiable paths? If you succeeded to define this "truncated"
integral in quantum mechanics, it would violate unitarity
and the rule

because the different
intervals would disagree "how much differentiable" the functions
must be. LM]

It is probably fair to divide geometry into different levels of
structure. One possible distinction is

0) differentiable structure
1) topology
2) causal structure
3) conformal structure
4) metric structure

It is up to discussion at which of these levels you start varying in
your path integral and which parts you keep fixed.

[Moderator's note: It is fair to divide geometries, but it is never
fair to "cut" some configurations from a path integral, I think.
We've had a recent debate on sci.physics.strings about the overcritical
electric field which was exactly about this issue - did you agree with
our conclusion that you can't ever omit "unwanted" configurations? LM]

I guess, Lubos wants to vary 1-4 while keeping fixed, Ambjorn and
friends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in
which he tried to convince me that one should vary all 0--4.

[Moderator's note: I agree with him, you must vary everything.
The differential structure only exists "classically" and it is a
consequence of dynamics - the appearance of the derivative terms
in the path integral. But the path integral over sums over all
configurations of the given fields. LM]

Nobody has done a really convincing 'sum over geometries' yet, so I
think it should be allowed to try all these approaches.

[Moderator's note: Nobody has found a really convincing luminiferous
aether theory, so should all of us divide to different approaches how to
construct aether? Actually I think that these two questions are more
similar, even in details, than you might think. ;-) LM]

What Ambjorn etal find is that again in 2d you can solve this model
exactly (ie compute the partition function with sources) and it agrees
with expectations (whatever those would be). Second the typical
configurations look much smoother (something they haven't put in, they
only demand causality and global topology) than in the Euclidean case.

[Moderator's note: I am not getting this point at all. What's exactly
the difference between the input and output? Typical configurations
in the gravity path integral have strongly oscillating topology, both
in the Minkowski and the Euclidean case, and in the Minkowski case,
they have also a highly nontrivial and chaotic causal diagram.
If you unphysically cut the "ugly" configurations, of course, you will
end up with the "nice" ones, and because you made more constraints
about the allowed configurations in the Minkowski case, you will
get even nicer configurations than in the Euclidean space at the end. But
that's not a result, that's your assumption. And it's an assumption
that contradicts quantum mechanics. LM]

Of course, in 2D gravity is not typical, all the dynamics is in the
cosmological constant and its conjugate variable, the volume,
respectively. And in higher dimensions it is not possible to solve the
problem analytically, you can only run in on your computer.

[Moderator's note: Right, 2D and 3D gravity don't really have gravitons
as local degrees of freedom. All of us know how to compute 2D gravity
as a path integral over "nice topologies" of two-dimensional spacetime:
it's called the stringy worldsheet. But the conformal structure on
the worldsheet is only "nice" because *any* configuration in 2D
can be mapped to the "standard ones" by diff x Weyl transformations.
Analogous things hold for 2D string theory - one really wants to
calculate the path integral over the scalar fields in spacetime
and their effects. Moreover, my arguments above that talk about the
uncertainty principle for $LaTeX Code: g_{12}$ and its time derivative can break down
in d<4 because there is no such a physical degree of freedom. LM]

Another success that they claim is that they break the

barrier'. OK,
I have no idea what that really is because I try to stay away from all
this old matrix model technology but Matthias Staudacher, who was around
in those days, says this is quite non-trivial: In these models, you do not
have to restrict yourself to pure gravity, you can couple matter to it:
For example you can add an Ising spin degree of freedom to all your
triangles and sum over it as well.

As, you say, in all these models you have to take the continuum limit
and then you get a conformal field theory. In the old days, it was
observed that whatever you did matter wise or matrix wise, you could
only get models with central charge <1. But with causal triangulations
coupled to matter you can break this barrier.

Finally these people claim their model has a well behaved continuum
limit and I see no reason to doubt it.

[Moderator's note: in the case of the present paper, I don't have
difficulties with the word "continuum limit" but rather with the word
"model". You can define some set of rules that gives you a *classical*
theory in some limit, but it by no means implies that your rules,
before you take the limit, define a meaningful quantum theory, does it?
For example, you should always ask whether your rules can lead to a
unitary S-matrix, which path integrals should, and the answer will
be NO in the 4D case, I think. LM]

But in the end this is only gravity if you end up in the correct
universality class. That is, all your weird rules you make up to construct
your discretised space-times correspond only to irrelevant operators that
go away in this limit. And to show this is of course the hard part.

[Modeator's note: That may be a different way to say the same thing.
You're simply not sure whether the "restricted path integral" has
anything whatsoever in common with the real path integral. LM]

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling School of Science and Engineering
International University Bremen
print "Just another Phone:

3574
stupid $LaTeX Code: .sig\\n";$ http://www.aei-potsdam.mpg.de/~helling

Nov17-04, 09:16 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>"Robert C. Helling" <robert@helling-dell600.iuhb02.iu-bremen.de> schrieb im\nNewsbeitrag news:slrncpm6ba.91c.robert-100000@localhost.localdomain...\n\n> It is probably fair to divide geometry into different levels of\n> structure. One possible distinction is\n>\n> 0) differentiable structure\n> 1) topology\n> 2) causal structure\n> 3) conformal structure\n> 4) metric structure\n>\n> It is up to discussion at which of these levels you start varying in\n> your path integral and which parts you keep fixed.\n> I guess, Lubos wants to vary 1-4 while keeping 0 fixed, Ambjorn and\n> friends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in\n> which he tried to convince me that one should vary all 0--4.\n\nWhat can be said about varying differentiable structures in a physical\ntheory? As far as I know there are not many examples where several smooth\nstructures on a given manifold are known. First and famous is the exotic\nS^7. Then Donaldson showed that there are exotic R^4s, but these are only\nimplicitly known to exist.\n\nWhat would a field theory on an exotic R^4 look like?\n\n\n(Concerning 3)+4): Doesn\'t causal structure plus conformal structure imply a\nmetric structure?)\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>"Robert C. Helling" <robert@helling-dell600.iuhb02.iu-bremen.de> schrieb im
Newsbeitrag news:slrncpm6ba.91c.robert-100000@lo...localdomain...

> It is probably fair to divide geometry into different levels of
> structure. One possible distinction is
>
> 0) differentiable structure
> 1) topology
> 2) causal structure
> 3) conformal structure
> 4) metric structure
>
> It is up to discussion at which of these levels you start varying in
> your path integral and which parts you keep fixed.
> I guess, Lubos wants to vary 1-4 while keeping fixed, Ambjorn and
> friends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in
> which he tried to convince me that one should vary all 0--4.

What can be said about varying differentiable structures in a physical
theory? As far as I know there are not many examples where several smooth
structures on a given manifold are known. First and famous is the exotic

. Then Donaldson showed that there are exotic $LaTeX Code: R^{4s},$ but these are only
implicitly known to exist.

What would a field theory on an exotic

look like?

(Concerning

Doesn't causal structure plus conformal structure imply a
metric structure?)

Nov17-04, 10:15 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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 Wed, 17 Nov 2004, Urs Schreiber wrote:\n\n> What can be said about varying differentiable structures in a physical\n> theory? As far as I know there are not many examples where several smooth\n> structures on a given manifold are known. First and famous is the exotic\n> S^7. Then Donaldson showed that there are exotic R^4s, but these are only\n> implicitly known to exist.\n\nThese are very interesting things that a priori only look like some picky\nmathematical curiosities, but they potentially can even have physical\nconsequences. However, in the path integral, it seems reasonable to guess\nthat we don\'t really want to define any particular differentiable\nstructure. And even if we do define it, we should probably sum over all of\nthem.\n\nThis is my favorite numerology: M-theory has 11 dimensions, so let\'s look\nat exotic 11-spheres. There are 992 of them, which is twice as many as the\ndimension of the gauge groups in 10 dimensions (namely 496, of SO(32) or\nE8 x E8), and it is one half of the year in which Green and Schwarz\ncalculated that the dimension 496 was necessary (namely 1984). ;-)\n\n> What would a field theory on an exotic R^4 look like?\n\nIs it really a well-posed question? Does the usual path integral focuses\non the "normal" R^4 only, or does it sum over the structures?\n\n> (Concerning 3)+4): Doesn\'t causal structure plus conformal structure imply a\n> metric structure?)\n\nSorry, but locally, causal structure and conformal structure of a manifold\nwith Minkowski signature is the same thing - they determine the metric up\nto a local Weyl scaling, don\'t they? If you say what is the infinitesimal\nfuture light cone of a given point, you\'ve defined where ds^2 vanishes,\nwhich means that assuming that ds^2 has the usual quadratic form, you\'ve\ndefined ds^2 up to a scaling.\n\nThis is why the Penrose diagrams, called causal diagrams, are also said to\nencode the conformal structure of spacetime, is not it?\n_____________________________________________ _________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 17 Nov 2004, Urs Schreiber wrote:

> What can be said about varying differentiable structures in a physical
> theory? As far as I know there are not many examples where several smooth
> structures on a given manifold are known. First and famous is the exotic
>

. Then Donaldson showed that there are exotic $LaTeX Code: R^{4s},$ but these are only
> implicitly known to exist.

These are very interesting things that a priori only look like some picky
mathematical curiosities, but they potentially can even have physical
consequences. However, in the path integral, it seems reasonable to guess
that we don't really want to define any particular differentiable
structure. And even if we do define it, we should probably sum over all of
them.

This is my favorite numerology: M-theory has 11 dimensions, so let's look
at exotic 11-spheres. There are 992 of them, which is twice as many as the
dimension of the gauge groups in 10 dimensions (namely 496, of SO(32) or
E8 x E8), and it is one half of the year in which Green and Schwarz
calculated that the dimension 496 was necessary (namely 1984). ;-)

> What would a field theory on an exotic

look like?

Is it really a well-posed question? Does the usual path integral focuses
on the "normal

only, or does it sum over the structures?

> (Concerning

Doesn't causal structure plus conformal structure imply a
> metric structure?)

Sorry, but locally, causal structure and conformal structure of a manifold
with Minkowski signature is the same thing - they determine the metric up
to a local Weyl scaling, don't they? If you say what is the infinitesimal
future light cone of a given point, you've defined where

vanishes,
which means that assuming that

has the usual quadratic form, you've
defined

to a scaling.

This is why the Penrose diagrams, called causal diagrams, are also said to
encode the conformal structure of spacetime, is not it?
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax:

Web: http://lumo.matfyz.cz/
eFax:

work:

home:

(call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Nov17-04, 10:48 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0411171008100.2231 2-100000@feynman.harvard.edu...\n> On Wed, 17 Nov 2004, Urs Schreiber wrote:\n>> What would a field theory on an exotic R^4 look like?\n>\n> Is it really a well-posed question? Does the usual path integral focuses\n> on the "normal" R^4 only, or does it sum over the structures?\n\nYou can imagine a path integral for gravity to be over all these structure.\nI was here thinking of ordinary non-gravitational field theory on a fixed\nexotic R^4. Just to get a feeling for how exotic exotic is.\n\n\n>> (Concerning 3)+4): Doesn\'t causal structure plus conformal structure\n>> imply a\n>> metric structure?)\n>\n> Sorry, but locally, causal structure and conformal structure of a manifold\n> with Minkowski signature is the same thing\n\nRight, I was confused. I thought Robert meant the specification of a volume\nelement at every point by "conformal structure".\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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0411171008100.223...harvard.edu...
> On Wed, 17 Nov 2004, Urs Schreiber wrote:
>> What would a field theory on an exotic

look like?
>
> Is it really a well-posed question? Does the usual path integral focuses
> on the "normal

only, or does it sum over the structures?

You can imagine a path integral for gravity to be over all these structure.
I was here thinking of ordinary non-gravitational field theory on a fixed
exotic

. Just to get a feeling for how exotic exotic is.

>> (Concerning

Doesn't causal structure plus conformal structure
>> imply a
>> metric structure?)
>
> Sorry, but locally, causal structure and conformal structure of a manifold
> with Minkowski signature is the same thing

Right, I was confused. I thought Robert meant the specification of a volume
element at every point by "conformal structure".

Nov17-04, 12:07 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>> What can be said about varying differentiable structures in a physical\n> theory? As far as I know there are not many examples where several smooth\n> structures on a given manifold are known. First and famous is the exotic\n> S^7. Then Donaldson showed that there are exotic R^4s, but these are only\n> implicitly known to exist.\n\nI think there are plenty of examples of topological four-manifolds that\nadmit different smooth structures. The quitic surface for example. It\'s\nreally in some sense a generic phenomenon, since two four-manifolds are\nalways homeomorphic if they have the same intersection form, but it\'s much\nharder to show that they are diffeomorphic. There are also uncountably\nmany exotic R^4\'s. I think the construction of them using Casson handles\nare actually quite explicit, although the construction is so complicated\nthat it\'s truly difficult to imagine how it could ever show up in physics.\n\n> What would a field theory on an exotic R^4 look like?\n>\n> (Concerning 3)+4): Doesn\'t causal structure plus conformal structure imply a\n> metric structure?)\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>> What can be said about varying differentiable structures in a physical
> theory? As far as I know there are not many examples where several smooth
> structures on a given manifold are known. First and famous is the exotic
>

. Then Donaldson showed that there are exotic $LaTeX Code: R^{4s},$ but these are only
> implicitly known to exist.

I think there are plenty of examples of topological four-manifolds that
admit different smooth structures. The quitic surface for example. It's
really in some sense a generic phenomenon, since two four-manifolds are
always homeomorphic if they have the same intersection form, but it's much
harder to show that they are diffeomorphic. There are also uncountably
many exotic

. I think the construction of them using Casson handles
are actually quite explicit, although the construction is so complicated
that it's truly difficult to imagine how it could ever show up in physics.

> What would a field theory on an exotic

look like?
>
> (Concerning

Doesn't causal structure plus conformal structure imply a
> metric structure?)

Nov17-04, 12:10 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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 Wed, 17 Nov 2004 08:09:21 -0500, Robert C. Helling wrote:\n\n> 0) differentiable structure\n> 1) topology\n> 2) causal structure\n> 3) conformal structure\n> 4) metric structure\n\nLet me reorder that...\n\n-1) sets\n0) topology\n1) differentiable structure\n2) causal structure + conformal structure (lets not get into details here)\n4) metric structure\n\n> [Moderator\'s note: It is fair to divide geometries, but it is never\n> fair to "cut" some configurations from a path integral, I think.\n\nI disagree. To properly write down the path integral, the first thing\nyou must define what you are integrating over.\n\nLets take some old QFT (as opposed to some fancy theory of gravity),\nand "fields" just some sort of function. You should really write down\nfirst which functions you want to admit, probably all smooth ones plus\nsome more. Probably differentiable functions with suitable decay at\ninfinity, and then the closure of that space (and extend the derivative\noperators to this Hilbert space).\n\nOf course, a good physicist never does that and hopes that the details of\nSobolev spaces enter the final answer just like the choice of regulator,\nthat is not at all.\n\nSo back to gravity, I think the LModerator agrees that we should not\nintegrate over all -1..4, since then we immediately have nasty\nnon-separable Hilbert spaces. We\'ll have to extract something separable\nin-between that allows a reasonably well-behaved extension of the action.\nOf course, If I knew the specifics I\'d be writing the paper right now (and\nthen cross-list to hep/th from the functional analysis archive).\n\n-Volker\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 17 Nov 2004 08:09:21

Robert C. Helling wrote:

> 0) differentiable structure
> 1) topology
> 2) causal structure
> 3) conformal structure
> 4) metric structure

Let me reorder that...

-1) sets
0) topology
1) differentiable structure
2) causal structure + conformal structure (lets not get into details here)
4) metric structure

> [Moderator's note: It is fair to divide geometries, but it is never
> fair to "cut" some configurations from a path integral, I think.

I disagree. To properly write down the path integral, the first thing
you must define what you are integrating over.

Lets take some old QFT (as opposed to some fancy theory of gravity),
and "fields" just some sort of function. You should really write down
first which functions you want to admit, probably all smooth ones plus
some more. Probably differentiable functions with suitable decay at
infinity, and then the closure of that space (and extend the derivative
operators to this Hilbert space).

Of course, a good physicist never does that and hopes that the details of
Sobolev spaces enter the final answer just like the choice of regulator,
that is not at all.

So back to gravity, I think the LModerator agrees that we should not
integrate over all -1..4, since then we immediately have nasty
non-separable Hilbert spaces. We'll have to extract something separable
in-between that allows a reasonably well-behaved extension of the action.
Of course, If I knew the specifics I'd be writing the paper right now (and
then cross-list to

from the functional analysis archive).

-Volker

Nov17-04, 12:34 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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 Wed, 17 Nov 2004, Volker Braun wrote:\n\n> I disagree. To properly write down the path integral, the first thing\n> you must define what you are integrating over.\n\nI agree with this sentence, but it seems to support my point. Defining\nwhat you\'re integrating over means choosing your fields or other degrees\nof freedom and their range, and you\'re always integrating over all of\nthem. There can be constraints such as "you only integrate over manifolds\nwith a spin structure". Such constraints look global in character, but\nthey\'re always about our ability to define some fields - not necessarily\nfields with local dynamics - at every point. In other words, there cannot\nbe any nontrivial global constraints. Once you define your fields and the\nset in which they take their values, you must always integrate over all of\ntheir values, I think. Making some additional constraints would break\nlocality and unitarity.\n\nBy the way, omitting some configurations is equivalent to redefining their\naction to infinity, which is usually a very strange attempt to modify your\ndynamics.\n\nIncidentally, Andy Neitzke tells me that :\n\n1. Michael Green claimed that there was really a relation of the number\n496 with the number of spheres, or something like that\n\n2. There has been a paper by Edward Witten relating these exotic\ndifferential structures and global gravitational anomalies, but not\nnecessarily involving the interesting numerology with 992\n\nWe would be very interested if someone could say something about it.\n\n> Lets take some old QFT (as opposed to some fancy theory of gravity),\n> and "fields" just some sort of function. You should really write down\n> first which functions you want to admit, probably all smooth ones plus\n> some more. Probably differentiable functions with suitable decay at\n> infinity, and then the closure of that space (and extend the derivative\n> operators to this Hilbert space).\n\nDo I understand you well? The path integral in a quantum field theory is\ndominated by functions that don\'t satisfy any of your criteria. The path\nintegral may be *localized* on smooth functions, if you can prove it, but\nit is *defined* in such a way that most of the contributions come from\nfunctions that don\'t have derivative almost anywhere.\n\nPath integrals are very subtle things. Mathematicians have problems to\ndefine them rigorously. Nevertheless they mathematically work for physical\npurposes. But they only work once you carefully follow all the rules - and\none of the rules definitely is that one must integrate over all\nconfigurations of the fields that he starts with. Messing up with this\nrule ends up with a nonsensical theory, certainly in a generic case.\n\nIt\'s a sort of miracle that one can define meaningful functional integrals\nin quantum theories at all, but this procedure is extremely sensitive.\n\n> Of course, a good physicist never does that and hopes that the details of\n> Sobolev spaces enter the final answer just like the choice of regulator,\n> that is not at all.\n\nDo I understand you well that you say that it is possible to define\nphysically meaningful functional integrals on Sobolev spaces with positive\n"p" i.e. only on spaces of functions whose first "p" derivatives are L2\nintegrable? Wow. ;-)\n\nYou can never restrict your path integral to such Sobolev spaces simply\nbecause the dominant contributions to any meaningful path integral with\nlocal physical degrees of freedom comes from non-differentiable\nconfigurations whose derivatives are definitely not L2-integrable. Quantum\nmechanics is jittery. Imposing some very strong differentiability criteria\non the configurations in your path integral is a road to hell.\n\n> So back to gravity, I think the LModerator agrees that we should not\n> integrate over all -1..4, since then we immediately have nasty\n> non-separable Hilbert spaces.\n\nWhat does it exactly have to do with Hilbert spaces? You must integrate\nover -1..4 because this is how Feynman\'s rules work - integrate over\neverything. What is exactly a Hilbert space is not so transparent in the\npath integral formalism.\n\nBut if you\'re able to derive that the standard Feynman rules of path\nintegrals end up with a non-separable Hilbert space (or another problem),\nthen the theory simply has a non-separable Hilbert space (or the other\nproblem) and it is a physically uninteresting theory. You will not be\nable to revive such a theory.\n\n> We\'ll have to extract something separable in-between that allows a\n> reasonably well-behaved extension of the action.\n\nNope. The reason why the path integral seems to behave badly for pure\ngeneral relativity in d>3 is that pure general relativity in d>3 is a bad\nquantum theory. The right way to fix it is to switch to a correct theory -\nadd new dynamics, new fields, new interactions (such as those derivable\nfrom string theory). The wrong way to fix it is to try to modify the rules\nof quantum mechanics - such as the rule that Feynman\'s path integral is an\nintegral over all configurations of your fields.\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)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\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>On Wed, 17 Nov 2004, Volker Braun wrote:

> I disagree. To properly write down the path integral, the first thing
> you must define what you are integrating over.

I agree with this sentence, but it seems to support my point. Defining
what you're integrating over means choosing your fields or other degrees
of freedom and their range, and you're always integrating over all of
them. There can be constraints such as "you only integrate over manifolds
with a spin structure". Such constraints look global in character, but
they're always about our ability to define some fields - not necessarily
fields with local dynamics

every point. In other words, there cannot
be any nontrivial global constraints. Once you define your fields and the
set in which they take their values, you must always integrate over all of
their values, I think. Making some additional constraints would break
locality and unitarity.

By the way, omitting some configurations is equivalent to redefining their
action to infinity, which is usually a very strange attempt to modify your
dynamics.

Incidentally, Andy Neitzke tells me that :

1. Michael Green claimed that there was really a relation of the number
496 with the number of spheres, or something like that

2. There has been a paper by Edward Witten relating these exotic
differential structures and global gravitational anomalies, but not
necessarily involving the interesting numerology with 992

We would be very interested if someone could say something about it.

> Lets take some old QFT (as opposed to some fancy theory of gravity),
> and "fields" just some sort of function. You should really write down
> first which functions you want to admit, probably all smooth ones plus
> some more. Probably differentiable functions with suitable decay at
> infinity, and then the closure of that space (and extend the derivative
> operators to this Hilbert space).

Do I understand you well? The path integral in a quantum field theory is
dominated by functions that don't satisfy any of your criteria. The path
integral may be *localized* on smooth functions, if you can prove it, but
it is *defined* in such a way that most of the contributions come from
functions that don't have derivative almost anywhere.

Path integrals are very subtle things. Mathematicians have problems to
define them rigorously. Nevertheless they mathematically work for physical
purposes. But they only work once you carefully follow all the rules - and
one of the rules definitely is that one must integrate over all
configurations of the fields that he starts with. Messing up with this
rule ends up with a nonsensical theory, certainly in a generic case.

It's a sort of miracle that one can define meaningful functional integrals
in quantum theories at all, but this procedure is extremely sensitive.

> Of course, a good physicist never does that and hopes that the details of
> Sobolev spaces enter the final answer just like the choice of regulator,
> that is not at all.

Do I understand you well that you say that it is possible to define
physically meaningful functional integrals on Sobolev spaces with positive
"p" i.e. only on spaces of functions whose first "p" derivatives are L2
integrable? Wow. ;-)

You can never restrict your path integral to such Sobolev spaces simply
because the dominant contributions to any meaningful path integral with
local physical degrees of freedom comes from non-differentiable
configurations whose derivatives are definitely not L2-integrable. Quantum
mechanics is jittery. Imposing some very strong differentiability criteria
on the configurations in your path integral is a road to hell.

> So back to gravity, I think the LModerator agrees that we should not
> integrate over all -1..4, since then we immediately have nasty
> non-separable Hilbert spaces.

What does it exactly have to do with Hilbert spaces? You must integrate
over -1..4 because this is how Feynman's rules work - integrate over
everything. What is exactly a Hilbert space is not so transparent in the
path integral formalism.

But if you're able to derive that the standard Feynman rules of path
integrals end up with a non-separable Hilbert space (or another problem),
then the theory simply has a non-separable Hilbert space (or the other
problem) and it is a physically uninteresting theory. You will not be
able to revive such a theory.

> We'll have to extract something separable in-between that allows a
> reasonably well-behaved extension of the action.

Nope. The reason why the path integral seems to behave badly for pure
general relativity in d>3 is that pure general relativity in d>3 is a bad
quantum theory. The right way to fix it is to switch to a correct theory -
add new dynamics, new fields, new interactions (such as those derivable
from string theory). The wrong way to fix it is to try to modify the rules
of quantum mechanics - such as the rule that Feynman's path integral is an
integral over all configurations of your fields.

All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax:

Web: http://lumo.matfyz.cz/
eFax:

work:

home:

(call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Nov17-04, 01:08 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>In article <pan.2004.11.17.16.25.54.200129-100000@physik.hu-berlin.de>,\nVolker Braun <volker.braun@physik.hu-berlin.de> wrote:\n\n> I disagree. To properly write down the path integral, the first thing\n> you must define what you are integrating over.\n>\n> Lets take some old QFT (as opposed to some fancy theory of gravity),\n> and "fields" just some sort of function. You should really write down\n> first which functions you want to admit, probably all smooth ones plus\n> some more. Probably differentiable functions with suitable decay at\n> infinity, and then the closure of that space (and extend the derivative\n> operators to this Hilbert space).\n\nThere\'s a real sense in that the only relevant contributions to the path\nintegral come from nondifferentiable functions. See Appendix 3 of\nColeman\'s "The sses of instantons" in Aspects of Symmetry.\n\nAaron\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>In article <pan.2004.11.17.16.25.54.200129-1000....hu-berlin.de>,
Volker Braun <volker.braun@physik.hu-berlin.de> wrote:

> I disagree. To properly write down the path integral, the first thing
> you must define what you are integrating over.
>
> Lets take some old QFT (as opposed to some fancy theory of gravity),
> and "fields" just some sort of function. You should really write down
> first which functions you want to admit, probably all smooth ones plus
> some more. Probably differentiable functions with suitable decay at
> infinity, and then the closure of that space (and extend the derivative
> operators to this Hilbert space).

There's a real sense in that the only relevant contributions to the path
integral come from nondifferentiable functions. See Appendix 3 of
Coleman's "The sses of instantons" in Aspects of Symmetry.

Aaron

Nov17-04, 02:25 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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 Wed, 17 Nov 2004 12:34:38 -0500, Lubos Motl wrote:\n\n> [...] Such constraints look global in character,\n> but they\'re always about our ability to define some fields - not\n> necessarily fields with local dynamics - at every point. In other words,\n> there cannot be any nontrivial global constraints.\n\nI fail to see how you distinguish constraints that "look" global from\nthose which "are" global. On some level, they are only distinguished by\nyour aesthetic preferences. Of course, if anything breaks locality or\nunitarity, it is of course bad.\n\n> Incidentally, Andy Neitzke tells me that : [...]\n> 2. There has been a paper by Edward Witten relating these exotic\n> differential structures and global gravitational anomalies, but not\n> necessarily involving the interesting numerology with 992\n> We would be very interested if someone could say something about it.\n\nFor the record, I\'m pretty sure you refer to:\n\nWitten, Edward(1-PRIN-H)\nGlobal gravitational anomalies.\nComm. Math. Phys. 100 (1985), no. 2, 197--229.\n\n>> Lets take some old QFT (as opposed to some fancy theory of gravity), and\n>> "fields" just some sort of function. You should really write down first\n>> which functions you want to admit, probably all smooth ones plus some\n>> more. Probably differentiable functions with suitable decay at infinity,\n>> and then the closure of that space (and extend the derivative operators\n>> to this Hilbert space).\n>\n> Do I understand you well? The path integral in a quantum field theory is\n> dominated by functions that don\'t satisfy any of your criteria. The path\n> integral may be *localized* on smooth functions, if you can prove it, but\n> it is *defined* in such a way that most of the contributions come from\n> functions that don\'t have derivative almost anywhere.\n\nYes, I agree completely. Those things crept in where I said "closure".\nSorry, should have been more explicit.\n\n>> So back to gravity, I think the LModerator agrees that we should not\n>> integrate over all -1..4, since then we immediately have nasty\n>> non-separable Hilbert spaces.\n>\n> What does it exactly have to do with Hilbert spaces? You must integrate\n> over -1..4 because this is how Feynman\'s rules work - integrate over\n> everything. What is exactly a Hilbert space is not so transparent in the\n> path integral formalism.\n\nRemember that point number -1) were just point sets, without any topology.\nThere is no way you can get a separable Hilbert space out of that.\n\n>> We\'ll have to extract something separable in-between that allows a\n>> reasonably well-behaved extension of the action.\n>\n> Nope. The reason why the path integral seems to behave badly for pure\n> general relativity in d>3 is that pure general relativity in d>3 is a bad\n> quantum theory. The right way to fix it is to switch to a correct theory -\n> add new dynamics, new fields, new interactions (such as those derivable\n> from string theory).\n\nI agree that string theory is the best bet here. In fact, I was thinking\nof string theory as the "something in-between". Somewhere between point\nsets and complete geometry, we can try to study spaces by their loop\nspace. Well ok, that is just words, but this is string theory to me.\n\nBest,\nVolker\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 17 Nov 2004 12:34:38

Lubos Motl wrote:

> [...] Such constraints look global in character,
> but they're always about our ability to define some fields - not
> necessarily fields with local dynamics

every point. In other words,
> there cannot be any nontrivial global constraints.

I fail to see how you distinguish constraints that "look" global from
those which "are" global. On some level, they are only distinguished by
your aesthetic preferences. Of course, if anything breaks locality or
unitarity, it is of course bad.

> Incidentally, Andy Neitzke tells me that : [...]
> 2. There has been a paper by Edward Witten relating these exotic
> differential structures and global gravitational anomalies, but not
> necessarily involving the interesting numerology with 992
> We would be very interested if someone could say something about it.

For the record, I'm pretty sure you refer to:

Witten, Edward(1-PRIN-H)
Global gravitational anomalies.
Comm. Math. Phys. 100 (1985), no. 2, 197--229.

>> Lets take some old QFT (as opposed to some fancy theory of gravity), and
>> "fields" just some sort of function. You should really write down first
>> which functions you want to admit, probably all smooth ones plus some
>> more. Probably differentiable functions with suitable decay at infinity,
>> and then the closure of that space (and extend the derivative operators
>> to this Hilbert space).
>
> Do I understand you well? The path integral in a quantum field theory is
> dominated by functions that don't satisfy any of your criteria. The path
> integral may be *localized* on smooth functions, if you can prove it, but
> it is *defined* in such a way that most of the contributions come from
> functions that don't have derivative almost anywhere.

Yes, I agree completely. Those things crept in where I said "closure".
Sorry, should have been more explicit.

>> So back to gravity, I think the LModerator agrees that we should not
>> integrate over all -1..4, since then we immediately have nasty
>> non-separable Hilbert spaces.
>
> What does it exactly have to do with Hilbert spaces? You must integrate
> over -1..4 because this is how Feynman's rules work - integrate over
> everything. What is exactly a Hilbert space is not so transparent in the
> path integral formalism.

Remember that point number -1) were just point sets, without any topology.
There is no way you can get a separable Hilbert space out of that.

>> We'll have to extract something separable in-between that allows a
>> reasonably well-behaved extension of the action.
>
> Nope. The reason why the path integral seems to behave badly for pure
> general relativity in d>3 is that pure general relativity in d>3 is a bad
> quantum theory. The right way to fix it is to switch to a correct theory -
> add new dynamics, new fields, new interactions (such as those derivable
> from string theory).

I agree that string theory is the best bet here. In fact, I was thinking
of string theory as the "something in-between". Somewhere between point
sets and complete geometry, we can try to study spaces by their loop
space. Well ok, that is just words, but this is string theory to me.

Best,
Volker

Nov17-04, 06:58 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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 Wed, 17 Nov 2004, Volker Braun wrote:\n\n> I fail to see how you distinguish constraints that "look" global from\n> those which "are" global. On some level, they are only distinguished by\n> your aesthetic preferences. Of course, if anything breaks locality or\n> unitarity, it is of course bad.\n\nRight, the criterion is, of course, whether or not you really violate\nunitarity or (macroscopically) locality. What I proposed was a description\nof the "allowed global" constraints that don\'t violate locality - and I\nsaid that they "look global". For example, it is OK to require that the\nmanifolds we integrate over have a spin structure - because it is about\nthe pointwise existence of spinorial degrees of freedom - but it is not\ncorrect to require that all noncontractible circles in your geometry must\nbe longer than 5 meters - because such a constraint would be "really"\nglobal - it would be correlating physics in different places by a nonlocal\nlink - and it would violate locality. I might be using inappropriate\nwords, but you may understand me anyway.\n\n> Witten, Edward(1-PRIN-H)\n> Global gravitational anomalies.\n> Comm. Math. Phys. 100 (1985), no. 2, 197--229.\n\nThanks! We will have a look.\n\n> Yes, I agree completely. Those things crept in where I said "closure".\n> Sorry, should have been more explicit.\n\nGood.\n\n> Remember that point number -1) were just point sets, without any topology.\n> There is no way you can get a separable Hilbert space out of that.\n\nI don\'t really know what it means to integrate over "just" sets, but if\n"just" sets were defining the configurations in whatever theory you would\nstudy, you would have to sum/integrate over them. If you can\'t get a\nreasonable Hilbert space in this way, then it means that your degrees of\nfreedom should never be described just by "sets". It does not mean that\nyou should change the rules of quantum mechanics.\n\nOne may replace the word "set" in the previous paragraph by "derived\ncategories" or something else to get a more meaningful statement. ;-)\n\n> I agree that string theory is the best bet here. In fact, I was thinking\n> of string theory as the "something in-between". Somewhere between point\n> sets and complete geometry, we can try to study spaces by their loop\n> space. Well ok, that is just words, but this is string theory to me.\n\nWell, you have some particular sets, conformal structure and perhaps even\nthe metric tensor on the space of possible feelings that one can get by\nlooking on a theory. ;-) This metric may be different from others\'. In my\nopinion, it is still more realistic to imagine string theory as being\ndescribed by an even bigger path integral than the quantum gravity\nintegrals with fluctuating topology - string field theory (closed, in this\ncase) pictures physics as the path integral over all possible geometries\nbut also all possible configurations of infinitely many other fields, and\nso on. ;-) Of course that this string field theory impression\noverestimates the number of degrees of freedom in quantum gravity -\nholography tells us that the path integral should be equivalent to a path\nintegral of a (d-1)-dimensional non-gravitational theory - which is what\nworks in AdS/CFT. These are subtle things, but whatever description one\nchooses, I think that he must simply integrate over all configurations\nwith the pre-determined degrees of freedom.\n________________________________________ ______________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 17 Nov 2004, Volker Braun wrote:

> I fail to see how you distinguish constraints that "look" global from
> those which "are" global. On some level, they are only distinguished by
> your aesthetic preferences. Of course, if anything breaks locality or
> unitarity, it is of course bad.

Right, the criterion is, of course, whether or not you really violate
unitarity or (macroscopically) locality. What I proposed was a description
of the "allowed global" constraints that don't violate locality - and I
said that they "look global". For example, it is OK to require that the
manifolds we integrate over have a spin structure - because it is about
the pointwise existence of spinorial degrees of freedom - but it is not
correct to require that all noncontractible circles in your geometry must
be longer than 5 meters - because such a constraint would be "really"
global

would be correlating physics in different places by a nonlocal
link - and it would violate locality. I might be using inappropriate
words, but you may understand me anyway.

> Witten, Edward(1-PRIN-H)
> Global gravitational anomalies.
> Comm. Math. Phys. 100 (1985), no. 2, 197--229.

Thanks! We will have a look.

> Yes, I agree completely. Those things crept in where I said "closure".
> Sorry, should have been more explicit.

Good.

> Remember that point number -1) were just point sets, without any topology.
> There is no way you can get a separable Hilbert space out of that.

I don't really know what it means to integrate over "just" sets, but if
"just" sets were defining the configurations in whatever theory you would
study, you would have to sum/integrate over them. If you can't get a
reasonable Hilbert space in this way, then it means that your degrees of
freedom should never be described just by "sets". It does not mean that
you should change the rules of quantum mechanics.

One may replace the word "set" in the previous paragraph by "derived
categories" or something else to get a more meaningful statement. ;-)

> I agree that string theory is the best bet here. In fact, I was thinking
> of string theory as the "something in-between". Somewhere between point
> sets and complete geometry, we can try to study spaces by their loop
> space. Well ok, that is just words, but this is string theory to me.

Well, you have some particular sets, conformal structure and perhaps even
the metric tensor on the space of possible feelings that one can get by
looking on a theory. ;-) This metric may be different from others'. In my
opinion, it is still more realistic to imagine string theory as being
described by an even bigger path integral than the quantum gravity
integrals with fluctuating topology - string field theory (closed, in this
case) pictures physics as the path integral over all possible geometries
but also all possible configurations of infinitely many other fields, and
so on. ;-) Of course that this string field theory impression
overestimates the number of degrees of freedom in quantum gravity -
holography tells us that the path integral should be equivalent to a path
integral of a (d-1)-dimensional non-gravitational theory - which is what
works in

. These are subtle things, but whatever description one
chooses, I think that he must simply integrate over all configurations
with the pre-determined degrees of freedom.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax:

Web: http://lumo.matfyz.cz/
eFax:

work:

home:

(call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Nov17-04, 09:24 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>In article <Pine.LNX.4.31.0411171210200.22715-100000@feynman.harvard.edu>,\nLubos Motl <motl@feynman.harvard.edu> wrote:\n\n> Incidentally, Andy Neitzke tells me that :\n>\n> 1. Michael Green claimed that there was really a relation of the number\n> 496 with the number of spheres, or something like that\n\nGlobal gravitational anomalies have to do with pi_0(Diff^+(M)).\n\nSimilarly, you can construct exotic spheres by a clutching-type\nconstruction. In particular, the equator of a S^n is an S^{n-1}. If\npi_0(Diff^+(S^{n-1})) =/= 0, glue the two hemispheres together with a\ndiffeomorphism that isn\'t in the identity component.\n\nThat\'s basically the connection.\n\nAaron\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>In article <Pine.LNX.4.31.0411171210200.22715-1...an.harvard.edu>,
Lubos Motl <motl@feynman.harvard.edu> wrote:

> Incidentally, Andy Neitzke tells me that :
>
> 1. Michael Green claimed that there was really a relation of the number
> 496 with the number of spheres, or something like that

Global gravitational anomalies have to do with $LaTeX Code: \\pi_0(Diff^+(M))$ .

Similarly, you can construct exotic spheres by a clutching-type
construction. In particular, the equator of

is an $LaTeX Code: S^{n-1}$ . If
$LaTeX Code: \\pi_0(Diff^+(S^{n-1})) =/= 0,$ glue the two hemispheres together with a
diffeomorphism that isn't in the identity component.

That's basically the connection.

Aaron

Nov19-04, 01:30 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>"Xi Yin" <xiyin@fas.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.58.0411171115580.2412 1-100000@ls02.fas.harvard.edu...\n\n> I think there are plenty of examples of topological four-manifolds that\n> admit different smooth structures. The quitic surface for example. It\'s\n> really in some sense a generic phenomenon, since two four-manifolds are\n> always homeomorphic if they have the same intersection form, but it\'s much\n> harder to show that they are diffeomorphic. There are also uncountably\n> many exotic R^4\'s. I think the construction of them using Casson handles\n> are actually quite explicit,\n\n\nCan you sketch how that works?\n\n\n> although the construction is so complicated\n> that it\'s truly difficult to imagine how it could ever show up in physics.\n\n\nYes, but Lubos was imagining having a path integral where all of them are\nsummed over, which would make them "show up" all over the place. To get a\nfeeling for what this would mean I was wondering about the following: Break\nthat hypothetical path integral up into the sum/integration over the smooth\nstructures and the integration over the rest. So for each given smooth\nstructure the rest is an "ordinary" quantum (gravitational) field theory,\nalbeit on an exotic space. What would ordinary QFT on a fixed exotic smooth\nstructure be like?\n\nFor instance: Can we say anything about the spectrum of the Laplace operator\non an exotic sphere?\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>"\Xi Yin" <xiyin@fas.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.58.0411171115580.241...harvard.edu...

> I think there are plenty of examples of topological four-manifolds that
> admit different smooth structures. The quitic surface for example. It's
> really in some sense a generic phenomenon, since two four-manifolds are
> always homeomorphic if they have the same intersection form, but it's much
> harder to show that they are diffeomorphic. There are also uncountably
> many exotic

. I think the construction of them using Casson handles
> are actually quite explicit,

Can you sketch how that works?

> although the construction is so complicated
> that it's truly difficult to imagine how it could ever show up in physics.

Yes, but Lubos was imagining having a path integral where all of them are
summed over, which would make them "show up" all over the place. To get a
feeling for what this would mean I was wondering about the following: Break
that hypothetical path integral up into the sum/integration over the smooth
structures and the integration over the rest. So for each given smooth
structure the rest is an "ordinary" quantum (gravitational) field theory,
albeit on an exotic space. What would ordinary QFT on a fixed exotic smooth
structure be like?

For instance: Can we say anything about the spectrum of the Laplace operator
on an exotic sphere?

Nov20-04, 04:29 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>Xi Yin wrote:\n\n> > I think there are plenty of examples of topological four-manifolds that\n> > admit different smooth structures. The quitic surface for example. It\'s\n> > really in some sense a generic phenomenon, since two four-manifolds are\n> > always homeomorphic if they have the same intersection form, but it\'s much\n> > harder to show that they are diffeomorphic. There are also uncountably\n> > many exotic R^4\'s. I think the construction of them using Casson handles\n> > are actually quite explicit,\n\nUrs Schreiber wrote:\n\n> Can you sketch how that works?\n\nYou mean the exotic R^4?\n\nFirst one needs to construct a Casson handle. The standard Casson handle\nis basicly an open 2-handle, namely D^2 x int(D^2), but you cut out a\npiece which is obtained much like the Whitehead continuum. You can find a\nmap g: D^2xD^2 -> D2xD2 such that its restriction to D2xS1 -> D2xS1 is\nthe map f that takes a solid torus to a thicken link inside this solid\ntorus. By repeated acting g on D2xD2 and take the intersection of all of\nthe images, you get the piece that you need to cut out to get the Casson\nhandle. There are many generalizations of this procedure, which gives\nuncountably many Casson handles.\n\nNow it is a hard result of Freedman that the Casson handle is actually\nhomeomorphic to the standard open 2-handle. I don\'t know the proof of\nit. It\'s the key step in Freedman\'s classification of topological\n4-manifolds. But if you think about it, the Casson handle is very exotic\nand it doesn\'t look like the ordinary 2-handle at all.\n\nLet\'s call the Casson handle CH, and the ordinary open 2-handle H. Knowing\nthat CH is homeomorphic to H, one knows that there is a TOPOLOGICALLY\nembedded disk. Ordinarily one can take a 4-ball union with two open\n2-handles and get S^2xS^2 with a 4-ball deleted - much like that you can\ntake a disk plus two strips and get a torus with a hole. Now instead of\nusing the ordinary handle you can use the Casson handle CH. So you get\nsomething homeomorphic to S^2xS^2 with a 4-ball cutout, and there are two\ntopologically embedded disks in the two CH\'s, which can be extended to\ngive a topologically embedded S^2vS^2 (wedge of two spheres). You can\ninclude the missing 4-ball and get S^2xS^2, with a topologically embedded\nS^2vS^2 (but embedded in a very exotic way).\n\nFinally you take the S^2xS^2, and cut out the topologically S^2vS^2, you\nget the exotic R^4. Let\'s call it X. It is clear that X is homeomorphic to\nR^4 (remember that to prove this we need the result of Freedman on CH).\n\nLet\'s pretend that X is diffeomorphic to R^4, and try to get\ncontradiction. If X is diffeomorphic to R^4, for any compact subset K in\nX, one can find a smoothly embedded 3-sphere that separates X into two\nparts A and B, such that A is compact and contains K. For example, we can\ntake K to be the missing 4-ball we added to get S^2xS^2 in constructing X.\nSo there is a compact manifold A with boundary S^3 that contains K, and\nanother compact manifold B with boundary S^3 that contains the\ntopologically embedded wedge of two S^2vS^2 which we cut out from S^2xS^2\nto get X. Notice that B has the same homotopy type as S^2xS^2.\n\nPreviously we used the open 4-ball with two Casson handles attached. Let\'s\ncall it CW. So B is contained in CW, with its boundary S^3 smoothly\nembedded.\n\nIt is a result of Casson and Freedman that you can embed CW topologically\nin any 4-manifold. In particular, you can embed 3 copies of CW\'s in a K3\nsurface. There is a B in each embedded CW, whose boundary S^3 is smoothly\nembedded. You can then cut out all the B\'s and replace them with 4-balls.\nThe leftover of the K3 surface after this surgery would have intersection\nform -E8-E8, which is impossible for any smooth four-manifold by the\ntheorem of Donaldson. This proves contradiction.\n\nThere are other examples of topological four-manifolds with more than one\nsmooth structures that are much easier to describe. For example, the\nquintic surface in CP^3 is homeomorphic but not diffeomorphic to the\nconnected sum of 9 copies of CP^2 and 44 copies of CP^2 bar (meaning CP^2\nwith opposite orientation). They are not diffeomorphic because they have\ndifferent Seiberg-Witten invariants.\n\n> > Xi Yin: ... although the construction is so complicated\n> > that it\'s truly difficult to imagine how it could ever show up in physics.\n>\n> Urs: Yes, but Lubos was imagining having a path integral where all of them are\n> summed over, which would make them "show up" all over the place. To get a\n> feeling for what this would mean I was wondering about the following: Break\n> that hypothetical path integral up into the sum/integration over the smooth\n> structures and the integration over the rest. So for each given smooth\n> structure the rest is an "ordinary" quantum (gravitational) field theory,\n> albeit on an exotic space. What would ordinary QFT on a fixed exotic smooth\n> structure be like?\n>\n> For instance: Can we say anything about the spectrum of the Laplace operator\n> on an exotic sphere?\n\nThe exotic spheres are not that exotic by themselves. For example, the\nexotic S^7 can be obtained from R^4 bundles over S^3. I don\'t know any\nresults concerning the spectrum of the Laplacian on these spaces but it\'s\nnot any more difficult than on other generic smooth 7-manifolds.\n\n-Xi\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>Xi Yin wrote:

> > I think there are plenty of examples of topological four-manifolds that
> > admit different smooth structures. The quitic surface for example. It's
> > really in some sense a generic phenomenon, since two four-manifolds are
> > always homeomorphic if they have the same intersection form, but it's much
> > harder to show that they are diffeomorphic. There are also uncountably
> > many exotic

. I think the construction of them using Casson handles
> > are actually quite explicit,

Urs Schreiber wrote:

> Can you sketch how that works?

You mean the exotic

?

First one needs to construct a Casson handle. The standard Casson handle
is basicly an open 2-handle, namely $LaTeX Code: D^2 x \\int(D^2),$ but you cut out a
piece which is obtained much like the Whitehead continuum. You can find a
map $LaTeX Code: g: D^{2xD}^2 ->$ D2xD2 such that its restriction to D2xS1 -> D2xS1 is
the map f that takes a solid torus to a thicken link inside this solid
torus. By repeated acting g on D2xD2 and take the intersection of all of
the images, you get the piece that you need to cut out to get the Casson
handle. There are many generalizations of this procedure, which gives
uncountably many Casson handles.

Now it is a hard result of Freedman that the Casson handle is actually
homeomorphic to the standard open 2-handle. I don't know the proof of
it. It's the key step in Freedman's classification of topological
4-manifolds. But if you think about it, the Casson handle is very exotic
and it doesn't look like the ordinary 2-handle at all.

Let's call the Casson handle CH, and the ordinary open 2-handle H. Knowing
that CH is homeomorphic to H, one knows that there is a TOPOLOGICALLY
embedded disk. Ordinarily one can take a 4-ball union with two open
2-handles and get $LaTeX Code: S^{2xS}^2$ with a 4-ball deleted - much like that you can
take a disk plus two strips and get a torus with a hole. Now instead of
using the ordinary handle you can use the Casson handle CH. So you get
something homeomorphic to $LaTeX Code: S^{2xS}^2$ with a 4-ball cutout, and there are two
topologically embedded disks in the two CH's, which can be extended to
give a topologically embedded $LaTeX Code: S^{2vS}^2$ (wedge of two spheres). You can
include the missing 4-ball and get $LaTeX Code: S^{2xS}^2,$ with a topologically embedded
$LaTeX Code: S^{2vS}^2$ (but embedded in a very exotic way).

Finally you take the $LaTeX Code: S^{2xS}^2,$ and cut out the topologically $LaTeX Code: S^{2vS}^2,$ you
get the exotic

. Let's call it X

is clear that X is homeomorphic to

(remember that to prove this we need the result of Freedman on CH).

Let's pretend that X is diffeomorphic

and try to get
contradiction. If X is diffeomorphic to

for any compact subset K in
X, one can find a smoothly embedded 3-sphere that separates X into two
parts A and B, such that A is compact and contains K. For example, we can
take K to be the missing 4-ball we added to get $LaTeX Code: S^{2xS}^2$ in constructing X.
So there is a compact manifold A with boundary

that contains K, and
another compact manifold B with boundary

that contains the
topologically embedded wedge of two $LaTeX Code: S^{2vS}^2$ which we cut out from $LaTeX Code: S^{2xS}^2$
to get X. Notice that B has the same homotopy type as $LaTeX Code: S^{2xS}^2$ .

Previously we used the open 4-ball with two Casson handles attached. Let's
call it CW. So B is contained in CW, with its boundary

smoothly
embedded.

It is a result of Casson and Freedman that you can embed CW topologically
in any 4-manifold. In particular, you can embed 3 copies of CW's in a K3
surface. There is a B in each embedded CW, whose boundary

is smoothly
embedded. You can then cut out all the B's and replace them with 4-balls.
The leftover of the K3 surface after this surgery would have intersection
form

which is impossible for any smooth four-manifold by the
theorem of Donaldson. This proves contradiction.

There are other examples of topological four-manifolds with more than one
smooth structures that are much easier to describe. For example, the
quintic surface in

is homeomorphic but not diffeomorphic to the
connected sum of 9 copies of

and 44 copies of

bar (meaning

with opposite orientation). They are not diffeomorphic because they have
different Seiberg-Witten invariants.

> > \Xi Yin: ... although the construction is so complicated
> > that it's truly difficult to imagine how it could ever show up in physics.
>
> Urs: Yes, but Lubos was imagining having a path integral where all of them are
> summed over, which would make them "show up" all over the place. To get a
> feeling for what this would mean I was wondering about the following: Break
> that hypothetical path integral up into the sum/integration over the smooth
> structures and the integration over the rest. So for each given smooth
> structure the rest is an "ordinary" quantum (gravitational) field theory,
> albeit on an exotic space. What would ordinary QFT on a fixed exotic smooth
> structure be like?
>
> For instance: Can we say anything about the spectrum of the Laplace operator
> on an exotic sphere?

The exotic spheres are not that exotic by themselves. For example, the
exotic

can be obtained from

bundles over

. I don't know any
results concerning the spectrum of the Laplacian on these spaces but it's
not any more difficult than on other generic smooth 7-manifolds.

$LaTeX Code: -\\Xi$

Nov20-04, 04:42 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>These Casson handles are very interesting objects, and there are some\npotentially interesting, but probably not quite correct, papers about it,\nlike the following:\n\nhttp://www.arxiv.org/abs/quant-ph/0303089\n\nIn this paper, Jerzy Krol uses the exotic "small" R^4 - one with the\nCasson Handle - to break supersymmetry in AdS/CFT. ;-) If you extract\nsomething out of this paper, it may be interesting if you describe your\nfindings.\n\nI\'m now gonna spend some time with Casson handles and Whitehead continua. ;-)\n_______________________________________________ _______________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\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>These Casson handles are very interesting objects, and there are some
potentially interesting, but probably not quite correct, papers about it,
like the following:

http://www.arxiv.org/abs/http://www....ant-ph/0303089

In this paper, Jerzy Krol uses the exotic "small

one with the
Casson Handle - to break supersymmetry in

. ;-) If you extract
something out of this paper, it may be interesting if you describe your
findings.

I'm now gonna spend some time with Casson handles and Whitehead continua. ;-)
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax:

Web: http://lumo.matfyz.cz/
eFax:

work:

home:

(call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Nov21-04, 07:39 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>Xi Yin wrote:\n\n> I think there are plenty of examples of topological four-manifolds\n> that admit different smooth structures. The quintic surface for\n> example. It\'s really in some sense a generic phenomenon, since two\n> four-manifolds are always homeomorphic if they have the same\n> intersection form, but it\'s much harder to show that they are\n> diffeomorphic.\n\nYou mean *simply connected* (closed smoothable oriented) four-manifolds,\nof course.\n\n-- Marc Nardmann\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>Xi Yin wrote:

> I think there are plenty of examples of topological four-manifolds
> that admit different smooth structures. The quintic surface for
> example. It's really in some sense a generic phenomenon, since two
> four-manifolds are always homeomorphic if they have the same
> intersection form, but it's much harder to show that they are
> diffeomorphic.

You mean *simply connected* (closed smoothable oriented) four-manifolds,
of course.

-- Marc Nardmann