View Full Version : [SOLVED] Why truncating path integrals violates unitarity
Lubos Motl
Dec16-04, 06:57 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>We just had a pretty interesting discussion with Nima about many things,\nincluding the topic of truncating the path integrals and unitarity -\nsomething that was discussed recently in the thread "Orbifold tachyons\nfrom SUGRA and other papers".\n\nNima always brings interesting insights about all fundamental and most\nother questions. Among other things, he presented a very quantitative\nparticular model showing why you cannot eliminate "unwanted"\nconfigurations from your path integral.\n\nImagine that you study any quantum system, for example quantum mechanics\nof a single particle - and you only want to include the configurations\n(trajectories) with a finite action - more precisely with the action\nsmaller than some pre-determined upper bound S_{max}.\n\nThis is equivalent to inserting the factor\n\ntheta(S_{max} - S)\n\nto the path integral where theta(x) equals 0 for x<0 and 0 for x>0,\nrespectively.\n\nThis theta function can be written as the Fourier transform\n\nint da exp(i.a.(S-S_{max})) / (a + i.epsilon)\n\nup to some signs and normalizations; the integral over "a" goes from\n-infinity to +infinity. OK, so inserting the theta function is equivalent\nto adding an integral over "a", which is a dual variable to the action\n"S". Let\'s put the integral over "a" at the beginning. The original path\nintegral becomes\n\nint da/(a+i.epsilon) exp(-i.a.S_{max})\nint [path integral] exp(i.S.(1+a))\n\nNote that the path integral is much like before, but the action is\nrescaled by the factor of (1+a) - you may view it as a rescaling of\nPlanck\'s constant.\n\nSo the path integral with the requirement that S is bounded is actually a\nkind of "weighted average" over the original path integrals (and\namplitudes) in the original theory, but with all possible values of hbar.\n\nObviously, the original theory is unitary for every specific value of\nhbar, but if you compute any kind of average of many unitary matrices, the\nresult won\'t be unitary: (U1+U2)/2 for unitary U1,U2 is not unitary unless\nU1=U2. Imposing any constraints over your configurations that cannot be\nvisualized as a local constraint of existing degrees of freedom will end\nup with a nonunitary, inconsistent theory.\n\nSimilar conclusion (non-unitarity) works for the Hawking-like theories\nwhere we average the amplitudes over some auxiliary global parameters that\nHawking could obtain from the "baby Universes".\n\nThis is why all the attempts to discretize gravity - not only loop quantum\ngravity - that require one to keep the global topology or similar features\nunchanged lead to non-unitary theories, and their investigation is always\nbased on a rather simple misconception.\n\nOn the other hand, if you don\'t eliminate the configurations with bubbling\ntopology etc., all known discretizations of gravity in d>3 end up with\nsingular gibberish. Combined with the previous paragraphs, all\ndiscretizations of d>3 non-topological gravities similar to those we know\nare guaranteed to remain rubbish.\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>We just had a pretty interesting discussion with Nima about many things,
including the topic of truncating the path integrals and unitarity -
something that was discussed recently in the thread "Orbifold tachyons
from SUGRA and other papers".
Nima always brings interesting insights about all fundamental and most
other questions. Among other things, he presented a very quantitative
particular model showing why you cannot eliminate "unwanted"
configurations from your path integral.
Imagine that you study any quantum system, for example quantum mechanics
of a single particle - and you only want to include the configurations
(trajectories) with a finite action - more precisely with the action
smaller than some pre-determined upper bound S_{max}.
This is equivalent to inserting the factor
\theta(S_{max} - S)
to the path integral where \theta(x) equals for x<0 and for x>0,
respectively.
This \theta function can be written as the Fourier transform
\int da \exp(i.a.(S-S_{max})) / (a + i.\epsilon)
up to some signs and normalizations; the integral over "a" goes from
-infinity to +infinity. OK, so inserting the \theta function is equivalent
to adding an integral over "a", which is a dual variable to the action
"S". Let's put the integral over "a" at the beginning. The original path
integral becomes
\int da/(a+i.\epsilon) \exp(-i.a.S_{max})\int[/itex] [path integral] [itex]\exp(i.S.(1+a))
Note that the path integral is much like before, but the action is
rescaled by the factor of (1+a) - you may view it as a rescaling of
Planck's constant.
So the path integral with the requirement that S is bounded is actually a
kind of "weighted average" over the original path integrals (and
amplitudes) in the original theory, but with all possible values of \hbar.
Obviously, the original theory is unitary for every specific value of
\hbar, but if you compute any kind of average of many unitary matrices, the
result won't be unitary: (U1+U2)/2 for unitary U1,U2 is not unitary unless
U1=U2. Imposing any constraints over your configurations that cannot be
visualized as a local constraint of existing degrees of freedom will end
up with a nonunitary, inconsistent theory.
Similar conclusion (non-unitarity) works for the Hawking-like theories
where we average the amplitudes over some auxiliary global parameters that
Hawking could obtain from the "baby Universes".
This is why all the attempts to discretize gravity - not only loop quantum
gravity - that require one to keep the global topology or similar features
unchanged lead to non-unitary theories, and their investigation is always
based on a rather simple misconception.
On the other hand, if you don't eliminate the configurations with bubbling
topology etc., all known discretizations of gravity in d>3 end up with
singular gibberish. Combined with the previous paragraphs, all
discretizations of d>3 non-topological gravities similar to those we know
are guaranteed to remain rubbish.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Robert C. Helling
Dec17-04, 05:20 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 16 Dec 2004 19:57:06 -0500, Lubos Motl <motl@feynman.harvard.edu> wrote:\n>\n> Obviously, the original theory is unitary for every specific value of\n> hbar, but if you compute any kind of average of many unitary matrices, the\n> result won\'t be unitary: (U1+U2)/2 for unitary U1,U2 is not unitary unless\n> U1=U2. Imposing any constraints over your configurations that cannot be\n> visualized as a local constraint of existing degrees of freedom will end\n> up with a nonunitary, inconsistent theory.\n\nHmm, what about this trivial manipulation\n\nint D phi e^{-S[phi]}\n\n=\n\nint dk int da exp(ik(a-1)) int D phi exp(-a S[phi])\n\nnow it looks as a weighted sum of theories with different hbar\'s but\nof course it is still the original path integral.\n\n>\n> On the other hand, if you don\'t eliminate the configurations with bubbling\n> topology etc., all known discretizations of gravity in d>3 end up with\n> singular gibberish. Combined with the previous paragraphs, all\n> discretizations of d>3 non-topological gravities similar to those we know\n> are guaranteed to remain rubbish.\n\nI still don\'t see why eliminating non-trivial topologies is the same\nas restricting to configurations with bounded action. There are other\nrestrictions of your theory that usually don\'t harm:\n\nE.g. you can restrict the Feynman diagrams to only diagrams that have\nno loops (classical limit) or to planar diagrams (at least for a large\nclass of theories where you can make sense of a large N limit) and in\nboth cases you end up with well behaved theories.\n\nIf you don\'t like restricting Feynman diagrams but insist on\nrestricting configurations, try doing YM but restrict to only\nconfigurations that where the gauge field is diagonal (in your\nfavourite gauge). The collection of Maxwell theories you end up with\nof course is unitary but of course is slightly boring.\n\nSo, your argument that elemination of configurations leads to rubbish\nmight be a bit too quick. You should at least be more specific about\nthe class of eliminations that you consider.\n\nRobert\n\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 Thu, 16 Dec 2004 19:57:06 -0500, Lubos Motl <motl@feynman.harvard.edu> wrote:
>
> Obviously, the original theory is unitary for every specific value of
> \hbar, but if you compute any kind of average of many unitary matrices, the
> result won't be unitary: (U1+U2)/2 for unitary U1,U2 is not unitary unless
> U1=U2. Imposing any constraints over your configurations that cannot be
> visualized as a local constraint of existing degrees of freedom will end
> up with a nonunitary, inconsistent theory.
Hmm, what about this trivial manipulation
\int D \phi e^{-S[\phi]}
=
\int dk \int da \exp(ik(a-1)) \int D \phi \exp(-a S[\phi])
now it looks as a weighted sum of theories with different \hbar's but
of course it is still the original path integral.
>
> On the other hand, if you don't eliminate the configurations with bubbling
> topology etc., all known discretizations of gravity in d>3 end up with
> singular gibberish. Combined with the previous paragraphs, all
> discretizations of d>3 non-topological gravities similar to those we know
> are guaranteed to remain rubbish.
I still don't see why eliminating non-trivial topologies is the same
as restricting to configurations with bounded action. There are other
restrictions of your theory that usually don't harm:
E.g. you can restrict the Feynman diagrams to only diagrams that have
no loops (classical limit) or to planar diagrams (at least for a large
class of theories where you can make sense of a large N limit) and in
both cases you end up with well behaved theories.
If you don't like restricting Feynman diagrams but insist on
restricting configurations, try doing YM but restrict to only
configurations that where the gauge field is diagonal (in your
favourite gauge). The collection of Maxwell theories you end up with
of course is unitary but of course is slightly boring.
So, your argument that elemination of configurations leads to rubbish
might be a bit too quick. You should at least be more specific about
the class of eliminations that you consider.
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: +49 421-200 3574
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
Dec17-04, 08:00 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 17 Dec 2004, Robert C. Helling wrote:\n\n> Hmm, what about this trivial manipulation\n> int D phi e^{-S[phi]}\n> =\n> int dk int da exp(ik(a-1)) int D phi exp(-a S[phi])\n>\n> now it looks as a weighted sum of theories with different hbar\'s but\n> of course it is still the original path integral.\n\nIs not it the same thing as mine, except that you did not define what phi\nis and what you\'re exactly doing with it? ;-)\n\n> I still don\'t see why eliminating non-trivial topologies is the same\n> as restricting to configurations with bounded action.\n\nBecause both of them are global constraints on the configurations that\ncan\'t be described locally. If you really don\'t believe that the\nsituations are completely analogous, we can analyze more complicated\nexamples than the bounded actions - and you will see that they have the\nsame effect. In all cases, the modified path integral can be translated\ninto an integral over many parameters "phi", perhaps infinitely many, and\nthe resulting amplitudes are always some kind of a weighted average.\n\nFor example trying to restrict the velocities by theta(1-xDOT^2) leads\nto an infinite-dimensional integral over some new auxiliary parameters.\n\nThe quantum gravity path integral itself cannot be used as an example,\nhowever, exactly because it is not well-defined even if it is done\ncorrectly - by "correctly" I mean, of course, summing over the quantum\nfoam and bubbling topologies.\n\n> There are other restrictions of your theory that usually don\'t harm:\n\nI will show you what\'s wrong with your examples.\n\n> E.g. you can restrict the Feynman diagrams to only diagrams that have\n> no loops (classical limit) or to planar diagrams (at least for a large\n> class of theories where you can make sense of a large N limit) and in\n> both cases you end up with well behaved theories.\n\nNope. You cannot "set" N to infinity in any particular gauge theory. If\nyou really talk about a gauge theory, N is finite. The nonplanar diagrams\nare subleading for large N, but you cannot remove them if you want to get\na consistent quantum theory. What you can get it a *classical* string\ntheory - a theory of non-interacting strings or a theory of one string -\nfrom the large N limit of gauge theories.\n\nBut this is just a limit of the gauge theory, not a way to define it!\n\nThe large N limit *is* about getting a classical theory of gravity in\nspacetime, because the radius is much bigger than the Planck length. It is\nnot a limit in which you can say much about the Planckian regime in the\nAdS space. Note that when we talk about the quantum gravity path integral,\nwe definitely want to study the Planckian physics, and the analogy of your\noperation removes the interesting physics completely.\n\nYou seem to be confusing the issue how to define the path integral for a\nspecific theory with the issue how to modify and truncate a theory in an\narbitrary way to get another (or limiting) theory.\n\n> If you don\'t like restricting Feynman diagrams but insist on\n> restricting configurations, try doing YM but restrict to only\n> configurations that where the gauge field is diagonal (in your\n> favourite gauge). The collection of Maxwell theories you end up with\n> of course is unitary but of course is slightly boring.\n\nBy "diagonal", do you mean that you replace a U(N) gauge theory by U(1)^N\ngauge theory? What does it have to do with U(N), and why do you relate\nthis truncation to "my favorite gauge"? It has nothing to do with gauges -\nyou just truncated the whole theory. You totally changed it. You\ncompletely modified your field content. I am not getting at all what does\nit have to do with our discussions. The triangulation gravity people may\nmake wrong assumptions that it is OK to remove bubbling topology, but at\nleast they work with a path integral whose local dynamics has the right\noriginal degrees of freedom - the metric - and it\'s only the weight of\ndifferent configurations that they try to modify, which is naturally\ninterpreted as a bizarre change of the action. You don\'t do the same thing\nbecause you altered the degrees of freedom themselves.\n\nUsing the same highly sloppy approach that you exhibited, cannot I say\nthat all theories are different definitions of the same theory?\n\n> So, your argument that elemination of configurations leads to rubbish\n> might be a bit too quick. You should at least be more specific about\n> the class of eliminations that you consider.\n\nI hope that the second posting of mine will satisfy you, and I hope that\nyour reasoning about this won\'t be a bit too slow. ;-) Cheers, Lubos\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 Fri, 17 Dec 2004, Robert C. Helling wrote:
> Hmm, what about this trivial manipulation
> \int D \phi e^{-S[\phi]}
> =
> \int dk \int da \exp(ik(a-1)) \int D \phi \exp(-a S[\phi])
>
> now it looks as a weighted sum of theories with different \hbar's but
> of course it is still the original path integral.
Is not it the same thing as mine, except that you did not define what \phi
is and what you're exactly doing with it? ;-)
> I still don't see why eliminating non-trivial topologies is the same
> as restricting to configurations with bounded action.
Because both of them are global constraints on the configurations that
can't be described locally. If you really don't believe that the
situations are completely analogous, we can analyze more complicated
examples than the bounded actions - and you will see that they have the
same effect. In all cases, the modified path integral can be translated
into an integral over many parameters "\phi", perhaps infinitely many, and
the resulting amplitudes are always some kind of a weighted average.
For example trying to restrict the velocities by \theta(1-xDOT^2) leads
to an infinite-dimensional integral over some new auxiliary parameters.
The quantum gravity path integral itself cannot be used as an example,
however, exactly because it is not well-defined even if it is done
correctly - by "correctly" I mean, of course, summing over the quantum
foam and bubbling topologies.
> There are other restrictions of your theory that usually don't harm:
I will show you what's wrong with your examples.
> E.g. you can restrict the Feynman diagrams to only diagrams that have
> no loops (classical limit) or to planar diagrams (at least for a large
> class of theories where you can make sense of a large N limit) and in
> both cases you end up with well behaved theories.
Nope. You cannot "set" N to infinity in any particular gauge theory. If
you really talk about a gauge theory, N is finite. The nonplanar diagrams
are subleading for large N, but you cannot remove them if you want to get
a consistent quantum theory. What you can get it a *classical* string
theory - a theory of non-interacting strings or a theory of one string -
from the large N limit of gauge theories.
But this is just a limit of the gauge theory, not a way to define it!
The large N limit *is* about getting a classical theory of gravity in
spacetime, because the radius is much bigger than the Planck length. It is
not a limit in which you can say much about the Planckian regime in the
AdS space. Note that when we talk about the quantum gravity path integral,
we definitely want to study the Planckian physics, and the analogy of your
operation removes the interesting physics completely.
You seem to be confusing the issue how to define the path integral for a
specific theory with the issue how to modify and truncate a theory in an
arbitrary way to get another (or limiting) theory.
> If you don't like restricting Feynman diagrams but insist on
> restricting configurations, try doing YM but restrict to only
> configurations that where the gauge field is diagonal (in your
> favourite gauge). The collection of Maxwell theories you end up with
> of course is unitary but of course is slightly boring.
By "diagonal", do you mean that you replace a U(N) gauge theory by U(1)^N
gauge theory? What does it have to do with U(N), and why do you relate
this truncation to "my favorite gauge"? It has nothing to do with gauges -
you just truncated the whole theory. You totally changed it. You
completely modified your field content. I am not getting at all what does
it have to do with our discussions. The triangulation gravity people may
make wrong assumptions that it is OK to remove bubbling topology, but at
least they work with a path integral whose local dynamics has the right
original degrees of freedom - the metric - and it's only the weight of
different configurations that they try to modify, which is naturally
interpreted as a bizarre change of the action. You don't do the same thing
because you altered the degrees of freedom themselves.
Using the same highly sloppy approach that you exhibited, cannot I say
that all theories are different definitions of the same theory?
> So, your argument that elemination of configurations leads to rubbish
> might be a bit too quick. You should at least be more specific about
> the class of eliminations that you consider.
I hope that the second posting of mine will satisfy you, and I hope that
your reasoning about this won't be a bit too slow. ;-) Cheers, Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Volker Braun
Dec17-04, 10:27 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lets put this again in terms of topology change. You are assuming that\nthere is a spacetime of finite action, with initial space slice and a\ntopologically different final space slice. Then it would be very\nunnatural, indeed inconsistent, to disallow the topology change.\n\nBut how do you know that there is such an interpolating spacetime of\nfinite action? Especially if you start and end with non-cobordant spaces.\nThen we just arrive at the old question again, what kind of singularities\nare allowed and how do we extend the action to accommodate them.\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>Lets put this again in terms of topology change. You are assuming that
there is a spacetime of finite action, with initial space slice and a
topologically different final space slice. Then it would be very
unnatural, indeed inconsistent, to disallow the topology change.
But how do you know that there is such an interpolating spacetime of
finite action? Especially if you start and end with non-cobordant spaces.
Then we just arrive at the old question again, what kind of singularities
are allowed and how do we extend the action to accommodate them.
Lubos Motl
Dec17-04, 03:31 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 17 Dec 2004, Volker Braun wrote:\n\n> Lets put this again in terms of topology change. You are assuming that\n> there is a spacetime of finite action, with initial space slice and a\n> topologically different final space slice.\n\nMy goal was just the opposite - my goal was to show that any reasonable\npath integral is always dominated by non-differentiable configurations\nwhose action is infinite (in the Minkowski version) and whose fluctuations\ncan achieve any conceivable process. Any restriction of the path integral\nto finite actions; bounded actions; differentiable configurations;\nconfigurations with any global constraints will lead to non-unitary\ntheories. The path integral can get *localized* to nice smooth\nconfigurations, but such a localization is already a step in the\ncalculation, not a definition of the path integral.\n\n> Then it would be very unnatural, indeed inconsistent, to disallow the\n> topology change.\n\nGood.\n\n> But how do you know that there is such an interpolating spacetime of\n> finite action?\n\nOnce again what I wanted to convey by the example: the finite-action\nconfigurations are always of measure zero in any path integral analogous\nto any theory we know, I think - the action computed for the\nBrownian-motion-like trajectories is always infinite.\n\n> Especially if you start and end with non-cobordant spaces.\n> Then we just arrive at the old question again, what kind of singularities\n> are allowed and how do we extend the action to accommodate them.\n\nI have nothing to say about this.\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 Fri, 17 Dec 2004, Volker Braun wrote:
> Lets put this again in terms of topology change. You are assuming that
> there is a spacetime of finite action, with initial space slice and a
> topologically different final space slice.
My goal was just the opposite - my goal was to show that any reasonable
path integral is always dominated by non-differentiable configurations
whose action is infinite (in the Minkowski version) and whose fluctuations
can achieve any conceivable process. Any restriction of the path integral
to finite actions; bounded actions; differentiable configurations;
configurations with any global constraints will lead to non-unitary
theories. The path integral can get *localized* to nice smooth
configurations, but such a localization is already a step in the
calculation, not a definition of the path integral.
> Then it would be very unnatural, indeed inconsistent, to disallow the
> topology change.
Good.
> But how do you know that there is such an interpolating spacetime of
> finite action?
Once again what I wanted to convey by the example: the finite-action
configurations are always of measure zero in any path integral analogous
to any theory we know, I think - the action computed for the
Brownian-motion-like trajectories is always infinite.
> Especially if you start and end with non-cobordant spaces.
> Then we just arrive at the old question again, what kind of singularities
> are allowed and how do we extend the action to accommodate them.
I have nothing to say about this.
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Please excuse me if this is a duplicate -- sci.physics.strings was\nmisconfigured here, and I don\'t thnk my first post got through.\n\nLubos Motl <motl@feynman.harvard.edu> wrote:\n\n[...]\n> Imagine that you study any quantum system, for example quantum mechanics\n> of a single particle - and you only want to include the configurations\n> (trajectories) with a finite action - more precisely with the action\n> smaller than some pre-determined upper bound S_{max}.\n\nThere is an easy way to see why this is a bad idea, that relates much\nmore directly to unitarity.\n\nConsider a path integral from a to b. (For a particle, a and b are\ninitial and final positions, but more generally, you can think of\nthem as initial and final configurations.) The path integral version\nof "inserting a complete set of states" is to write this as a path\nintegral from a to c and then c to be, and then to integrate over c.\nFor a unitary theory, this must give the same result as the initial\npath integral (with dur attention to gauge-fixing and boundary terms,\nof course). But action is additive, so it\'s perfectly possible to\nhave a path for which the segment from a to c and the segment from\nc to b each have an action less than S_{max}, but the combined path\nhas an action greater than S_{max}.\n\nWhat\'s wrong with your "maximum action" rule is that it doesn\'t respect\ncomposition of paths. You can get lots of other interesting results\nfrom demanding composition -- see, for example, Laidlaw and (Cecile)\nDeWitt, Phys. Rev. D3 (1971) 1375. But if you have a constraint on\npaths that *does* respect composition, it\'s not at all clear that you\nget additional restrictions from unitarity.\n\nIn particular...\n\n> This is why all the attempts to discretize gravity - not only loop quantum\n> gravity - that require one to keep the global topology or similar features\n> unchanged lead to non-unitary theories, and their investigation is always\n> based on a rather simple misconception.\n\nThis does not follow. It is certainly true that you have to be careful\nabout what restrictions on topology you include, but there are restrictions\nthat respect composition of paths, and I see no reason to expect them to\nlead to nonunitarity. In particular, a restriction to manifolds with the\ntopology RxS with a fixed spatial topology S respects composition.\n\nIn fact, there are two easy counterexamples to your claim. One is (2+1)-\ndimensional gravity, in which the restriction to a fixed topology gives a\nsensible, unitary theory, while it is much less clear that a sum over all\ntopologies does (you typically get divergences from zero-modes). The second\nis two-dimensional gravity, in which both the theory with a fixed spatial\ntopology and the theory with varying topology exist, but are different.\n\nSteve Carlip\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>Please excuse me if this is a duplicate -- sci.physics.strings was
misconfigured here, and I don't thnk my first post got through.
Lubos Motl <motl@feynman.harvard.edu> wrote:
[...]
> Imagine that you study any quantum system, for example quantum mechanics
> of a single particle - and you only want to include the configurations
> (trajectories) with a finite action - more precisely with the action
> smaller than some pre-determined upper bound S_{max}.
There is an easy way to see why this is a bad idea, that relates much
more directly to unitarity.
Consider a path integral from a to b. (For a particle, a and b are
initial and final positions, but more generally, you can think of
them as initial and final configurations.) The path integral version
of "inserting a complete set of states" is to write this as a path
integral from a to c and then c to be, and then to integrate over c.
For a unitary theory, this must give the same result as the initial
path integral (with dur attention to gauge-fixing and boundary terms,
of course). But action is additive, so it's perfectly possible to
have a path for which the segment from a to c and the segment from
c to b each have an action less than S_{max}, but the combined path
has an action greater than S_{max}.
What's wrong with your "maximum action" rule is that it doesn't respect
composition of paths. You can get lots of other interesting results
from demanding composition -- see, for example, Laidlaw and (Cecile)
DeWitt, Phys. Rev. D3 (1971) 1375. But if you have a constraint on
paths that *does* respect composition, it's not at all clear that you
get additional restrictions from unitarity.
In particular...
> This is why all the attempts to discretize gravity - not only loop quantum
> gravity - that require one to keep the global topology or similar features
> unchanged lead to non-unitary theories, and their investigation is always
> based on a rather simple misconception.
This does not follow. It is certainly true that you have to be careful
about what restrictions on topology you include, but there are restrictions
that respect composition of paths, and I see no reason to expect them to
lead to nonunitarity. In particular, a restriction to manifolds with the
topology RxS with a fixed spatial topology S respects composition.
In fact, there are two easy counterexamples to your claim. One is (2+1)-
dimensional gravity, in which the restriction to a fixed topology gives a
sensible, unitary theory, while it is much less clear that a sum over all
topologies does (you typically get divergences from zero-modes). The second
is two-dimensional gravity, in which both the theory with a fixed spatial
topology and the theory with varying topology exist, but are different.
Steve Carlip
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