Negative tension strings [Archive]

Lubos Motl

Nov15-04, 05:39 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi Jack!\n\nLet\'s hope that Xi\'s harder question about algebraic geometry will be\nanswered, too (although I doubt it). What I write below can have errors in\nit, and you should correct me if you see them!\n\nThe stringy path integral is a path integral over the worldsheet embedded\ninto spacetime with some fixed, predetermined background which is the\npoint around which we expand.\n\nIn Minkowski space, the typical contributions to the path integral are\ntrajectories that are non-differentiable almost everywhere. It\'s always\nthe case for path integrals. The classical intuition that we only sum over\nworldsheets that have the right signature is naive.\n\nEven in ordinary quantum mechanics, the typical path in the path integral\nis non-differentiable almost everywhere. In this case, it\'s superluminal\nalmost everywhere.\n\nThe path integrals are more controllable in Euclidean spacetime because\nthe exponential oscillation of exp(i.S/hbar) is replaced by exp(-S/hbar)\nwhich is exponentially damped. This is the actual path integral that is\neasier to define mathematically, but it\'s true that a typical\nconfiguration that contributes looks like Brownian motion, roughly\nspeaking.\n\nIn string theory, we usually integrate over all embeddings of the\nEuclidean worldsheet into the Euclidean spacetime. There is an independent\nauxilliary metric on the worldsheet. The equations of motion guarantee\nthat this auxilliary metric is proportional to the induced metric. But the\npath integral also covers the configurations that don\'t solve the\nequations of motion, of course.\n\n> I have a (hopefully) basic question. Consider open strings with\n> Chan-Paton charges at there ends in an electric background. The\n> effective tension of the strings will be reduced by the background\n> field. There is a maximum electric field above which strings would...\n\nRight.\n\n> appear to have negative tension and the Dirac-Born-Infeld action\n> becomes ill-defined. Clearly such states are unphysical. In general, ...\n\nOK, now the answer (hopefully):\n\nYou must define the rules of your path integral - and the background -\n*before* you start the perturbative calculations. If the classical\nelectric field in your classical backgrounds is such that the electric\ndipole energy is smaller than the standard stringy tension, then there is\nnothing wrong with the path integral. It is qualitatively the same path\nintegral like in flat space, and you must integrate over all\nconfigurations. You will never get configurations where negative-energy\nstrings exist for a long time and everything is fine.\n\nIf your classical background, around which you expand, has a too high\nelectric field, then your perturbative expansion will break down. But it\'s\nnot because the particular terms in your path integral in the expansion\nare wrong: it\'s because the classical background itself is "wrong".\n\n> the effective string tension will depend on the particular embedding.\n> My question is whether one should include such negative tension\n> strings in a path integral formulation. That is, when I sum over all\n> embeddings of the world sheet, do I have to include embeddings that\n> would correspond to strings of negative tension?\n\nYou must always include *all* configurations. Indeed, if you start with an\n"overcritical" electric field, then including these strings that feel this\nbig electric field will lead to problems. But these are *real* physical\nproblems with the *background*, and the particular contributions to\nparticular amplitudes are just *evidence* that there are problems. But the\nproblems are not the worldsheets contributing to the path integral: the\nproblem is the background itself.\n\n> If I think, for example, about a point particle, the Feynman path\n> integral includes time-like trajectories and any conceivable\n> trajectory needs to be summed over (weighted by its exponentiated\n> action.) Simply by analogy, it is not clear to me how this concept\n> extends to string theory.\n\nIt extends to any theory that you want to describe by a path integral. It\nis the very nature of Feynman\'s path integrals that you integrate over all\nconfigurations, whether or not they look "natural" to you as classical\nconfigurations. In the classical limit, the path integral is dominated by\nthe classical trajectory, solving the classical equations of motion, and\nits neighborhood. But it by no means implies that the configurations that\nlook "wrong" should not be counted.\n\n> Of course, world-sheets with time-like\n> directions are included. But the analog of string tension is the point\n> particle\'s mass and it simply does not depend on the background in the\n> same way the string tension does. Any thoughts?\n\nThe tension that enters the Polyakov action - and the Polyakov action is\nthe counterpart of the action for pointlike particles - is a constant\n(1/2.pi.alpha\'), too. The stringy action also has "dipole" contributions\nfrom the electric field (and B-field) which does not really exist for the\npointlike particles, and by combining these two you get the "effective\ntension". In all cases, you must path-integrate over all configurations.\nThe path integral will show you that the classical backgrounds with\novercritical electric fields are unstable.\n\n> It seems to be impossible to get a consistent theory if\n> negative-tension strings can contribute to the path integral.\n\nThere is no problem like that if you expand around the right backgrounds,\nand if you start with a wrong background, it\'s the problem (instability)\nof the background, not a problem of the stringy perturbative expansions.\n\n> But if they have to be excluded, how do we do that?\n\nYou\'re never allowed to exclude configurations from your path integral\nunless there is a topological discrepancy that eliminates them.\n\nIf you start with an overcritical electric field background, someone might\nwant to exclude the "wrong" worldsheets, so that all the physical strings\nthat he sees in his worldsheet sort of respect the positivity of the total\ntension (for example, they never direct their dipoles so that their\ntension is negative). But such an approach is like closing your eyes on\n9/11 while you\'re at the 90th floor of the first tower. By closing your\neyes (or artificially removing the configurations that *prove*\ninstability) you don\'t remove the instability. You would just fool\nyourself.\n\nThe background is *really* unstable - it\'s inconsistent, if you wish to\nuse this general label.\n\n> What is the justification?\n\nOnce again, there cannot ever be any justification that genetically\nremoves unwanted configurations. You know, people in loop quantum gravity\nare often ready to do this black magic. Once their analysis becomes\nsufficiently sharp so that they see that their path integral cannot lead\nto nearly flat space at long distances (which really don\'t exist at all in\nthe spin foam models), they try to eliminate the "highly unwanted" (very\nfar from flat space) configurations from their spin foam path integral.\nThat\'s equivalent to setting the action for these "unwanted\nconfigurations" to i.infinity. It\'s not just impossible to justify it, but\nit\'s possible to show that such artificial truncations violate unitarity.\n\nSo if you insist that a "consistent theory" must be stable and the vacuum\nmust be the minimal energy state, then the configurations with\novercritical electric fields are inconsistent (because you can *save*\nenergy by creating new open strings in these backgrounds). Using the same\nrigid rules of path integrals, you can see that spin foams are equally\ninconsistent, too.\n\n> Can this be derived from first principles? A condition to exclude\n> certain embeddings looks like something horribly non-local and does\n> not seem to respect any of the important symmetries...\n\nThis intuition of yours is absolutely correct. It\'s just not allowed to\nartificially omit some configurations using a criterion that counts the\n"immediate distance between the endpoints of an open string", for example\n- such truncations would indeed be horribly nonlocal and they would\nviolate the Weyl x diff symmetry on the worldsheet, which would\nconsequently destroy the effective spacetime gauge symmetries etc. The\nnonlocality is an indication that such a procedure would be wrong, but it\nis really the violation of Weyl symmetries that brings the inconsistency.\n\n(Of course, every such nonlocal censorship of the path integral breaks the\nWeyl x diff symmetries.)\n\nIt\'s just not possible to truncate the configurations in this way, and the\nconfigurations with overcritical electric fields are inconsistent indeed.\nMore precisely, they\'re unstable. You know, if you start with a huge\nelectric field, the process of "spontaneous creation of open strings from\nnothing" will really take place, and the electric field from these new\nopen strings will eventually reduce the total electric field below the\ncritical bound. At any rate, it\'s not a too controllable approach to\nexpand the theory around the overcritical strange classical\nconfigurations, and the "strange" worldsheets *prove* that the\n*background* is a wrong starting point. Because they *prove* something\ntrue about physics, we must be grateful to these worldsheets for telling\nus the truth - instead of censoring them our from the path integral. ;-)\n\nThese overcritical backgrounds are analogous to backgrounds with tachyons.\n\nDisagreement welcome.\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>Hi Jack!

Let's hope that \Xi's harder question about algebraic geometry will be
answered, too (although I doubt it). What I write below can have errors in
it, and you should correct me if you see them!

The stringy path integral is a path integral over the worldsheet embedded
into spacetime with some fixed, predetermined background which is the
point around which we expand.

In Minkowski space, the typical contributions to the path integral are
trajectories that are non-differentiable almost everywhere. It's always
the case for path integrals. The classical intuition that we only sum over
worldsheets that have the right signature is naive.

Even in ordinary quantum mechanics, the typical path in the path integral
is non-differentiable almost everywhere. In this case, it's superluminal
almost everywhere.

The path integrals are more controllable in Euclidean spacetime because
the exponential oscillation of \exp(i.S/\hbar) is replaced by \exp(-S/\hbar)
which is exponentially damped. This is the actual path integral that is
easier to define mathematically, but it's true that a typical
configuration that contributes looks like Brownian motion, roughly
speaking.

In string theory, we usually integrate over all embeddings of the
Euclidean worldsheet into the Euclidean spacetime. There is an independent
auxilliary metric on the worldsheet. The equations of motion guarantee
that this auxilliary metric is proportional to the induced metric. But the
path integral also covers the configurations that don't solve the
equations of motion, of course.

> I have a (hopefully) basic question. Consider open strings with
> Chan-Paton charges at there ends in an electric background. The
> effective tension of the strings will be reduced by the background
> field. There is a maximum electric field above which strings would...

Right.

> appear to have negative tension and the Dirac-Born-Infeld action
> becomes ill-defined. Clearly such states are unphysical. In general, ...

OK, now the answer (hopefully):

You must define the rules of your path integral - and the background -
*before* you start the perturbative calculations. If the classical
electric field in your classical backgrounds is such that the electric
dipole energy is smaller than the standard stringy tension, then there is
nothing wrong with the path integral. It is qualitatively the same path
integral like in flat space, and you must integrate over all
configurations. You will never get configurations where negative-energy
strings exist for a long time and everything is fine.

If your classical background, around which you expand, has a too high
electric field, then your perturbative expansion will break down. But it's
not because the particular terms in your path integral in the expansion
are wrong: it's because the classical background itself is "wrong".

> the effective string tension will depend on the particular embedding.
> My question is whether one should include such negative tension
> strings in a path integral formulation. That is, when I sum over all
> embeddings of the world sheet, do I have to include embeddings that
> would correspond to strings of negative tension?

You must always include *all* configurations. Indeed, if you start with an
"overcritical" electric field, then including these strings that feel this
big electric field will lead to problems. But these are *real* physical
problems with the *background*, and the particular contributions to
particular amplitudes are just *evidence* that there are problems. But the
problems are not the worldsheets contributing to the path integral: the
problem is the background itself.

> If I think, for example, about a point particle, the Feynman path
> integral includes time-like trajectories and any conceivable
> trajectory needs to be summed over (weighted by its exponentiated
> action.) Simply by analogy, it is not clear to me how this concept
> extends to string theory.

It extends to any theory that you want to describe by a path integral. It
is the very nature of Feynman's path integrals that you integrate over all
configurations, whether or not they look "natural" to you as classical
configurations. In the classical limit, the path integral is dominated by
the classical trajectory, solving the classical equations of motion, and
its neighborhood. But it by no means implies that the configurations that
look "wrong" should not be counted.

> Of course, world-sheets with time-like
> directions are included. But the analog of string tension is the point
> particle's mass and it simply does not depend on the background in the
> same way the string tension does. Any thoughts?

The tension that enters the Polyakov action - and the Polyakov action is
the counterpart of the action for pointlike particles - is a constant
(1/2.\pi.\alpha'), too. The stringy action also has "dipole" contributions
from the electric field (and B-field) which does not really exist for the
pointlike particles, and by combining these two you get the "effective
tension". In all cases, you must path-integrate over all configurations.
The path integral will show you that the classical backgrounds with
overcritical electric fields are unstable.

> It seems to be impossible to get a consistent theory if
> negative-tension strings can contribute to the path integral.

There is no problem like that if you expand around the right backgrounds,
and if you start with a wrong background, it's the problem (instability)
of the background, not a problem of the stringy perturbative expansions.

> But if they have to be excluded, how do we do that?

You're never allowed to exclude configurations from your path integral
unless there is a topological discrepancy that eliminates them.

If you start with an overcritical electric field background, someone might
want to exclude the "wrong" worldsheets, so that all the physical strings
that he sees in his worldsheet sort of respect the positivity of the total
tension (for example, they never direct their dipoles so that their
tension is negative). But such an approach is like closing your eyes on
9/11 while you're at the 90th floor of the first tower. By closing your
eyes (or artificially removing the configurations that *prove*
instability) you don't remove the instability. You would just fool
yourself.

The background is *really* unstable - it's inconsistent, if you wish to
use this general label.

> What is the justification?

Once again, there cannot ever be any justification that genetically
removes unwanted configurations. You know, people in loop quantum gravity
are often ready to do this black magic. Once their analysis becomes
sufficiently sharp so that they see that their path integral cannot lead
to nearly flat space at long distances (which really don't exist at all in
the spin foam models), they try to eliminate the "highly unwanted" (very
far from flat space) configurations from their spin foam path integral.
That's equivalent to setting the action for these "unwanted
configurations" to i.infinity. It's not just impossible to justify it, but
it's possible to show that such artificial truncations violate unitarity.

So if you insist that a "consistent theory" must be stable and the vacuum
must be the minimal energy state, then the configurations with
overcritical electric fields are inconsistent (because you can *save*
energy by creating new open strings in these backgrounds). Using the same
rigid rules of path integrals, you can see that spin foams are equally
inconsistent, too.

> Can this be derived from first principles? A condition to exclude
> certain embeddings looks like something horribly non-local and does
> not seem to respect any of the important symmetries...

This intuition of yours is absolutely correct. It's just not allowed to
artificially omit some configurations using a criterion that counts the
"immediate distance between the endpoints of an open string", for example
- such truncations would indeed be horribly nonlocal and they would
violate the Weyl x diff symmetry on the worldsheet, which would
consequently destroy the effective spacetime gauge symmetries etc. The
nonlocality is an indication that such a procedure would be wrong, but it
is really the violation of Weyl symmetries that brings the inconsistency.

(Of course, every such nonlocal censorship of the path integral breaks the
Weyl x diff symmetries.)

It's just not possible to truncate the configurations in this way, and the
configurations with overcritical electric fields are inconsistent indeed.
More precisely, they're unstable. You know, if you start with a huge
electric field, the process of "spontaneous creation of open strings from
nothing" will really take place, and the electric field from these new
open strings will eventually reduce the total electric field below the
critical bound. At any rate, it's not a too controllable approach to
expand the theory around the overcritical strange classical
configurations, and the "strange" worldsheets *prove* that the
*background* is a wrong starting point. Because they *prove* something
true about physics, we must be grateful to these worldsheets for telling
us the truth - instead of censoring them our from the path integral. ;-)

These overcritical backgrounds are analogous to backgrounds with tachyons.

Disagreement welcome.

All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
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