Background Independence
d70yxj
As far as I can tell, the usual statement of background independence in
string theory is that even though one chooses a specific metric (and
other target space fields) in the string theory path integral, a
particular choice of background field is exactly equivalent to any
other choice - so long as one inserts appropriate vertex operators in
the path integral. For example, the path integral with a curved metric
is the same as the path integral with a flat target space AND a certain
coherent state of gravitons inserted.
In this sense, two different background fields do not give two
different theories, but can be thought of as two different states of
the same string theory. Hopefully this is OK so far, and agrees with
the usual interpretations.
If so, my question is as follows; suppose I consider a target space
manifold with, say, a flat metric for simplicity, but with non-trivial
homology groups. Can I think of the string theory path integral with
this target space as just being the flat-space R^10 path integral but
with an insertion of some appropriate string vertex operators?
In other words, is there an obvious way to think of two
topologically-different target manifolds as corresponding to different
states of the same string theory, just as there is for two different
choices of background fields?
Sorry, I possibly haven't said this too clearly, but if anyone can put
me straight it's just something that's been bugging me....
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Urs Schreiber
news:d70yxj.1aq...@physicsforums.com...
> If so, my question is as follows; suppose I consider a target space
> manifold with, say, a flat metric for simplicity, but with non-trivial
> homology groups. Can I think of the string theory path integral with
> this target space as just being the flat-space R^10 path integral but
> with an insertion of some appropriate string vertex operators?
No, if the metric is flat, no vertex operators are inserted.
> In other words, is there an obvious way to think of two
> topologically-different target manifolds as corresponding to different
> states of the same string theory, just as there is for two different
> choices of background fields?
I believe this is best thought of in terms of covering spaces. work in the
covering space (e.g. flat space for toroidal compactifications) and take
care of the correct identifications.
d70yxj
> > > In other words, is there an obvious way to think of two
> > > topologically-different target manifolds as corresponding to different
> > > states of the same string theory, just as there is for two different
> > > choices of background fields?[/color]
>
> > I believe this is best thought of in terms of covering spaces. work in the
> > covering space (e.g. flat space for toroidal compactifications) and take
> > care of the correct identifications.
So I do the path integral with a cover of my original manifold, and
then do something (what?) at the end to get to the correct answer? This
doesn't seem as obviously `stringy' as the way different background
fields can be seen as string states sitting on a flat background.
Also, a covering space for a given manifold needn't be topologically
trivial anyway. So it's not obvious to me that I can start with any
target manifold and relate it back to R^10 using what you say.
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Rufus Anton
> In this sense, two different background fields do not give two
> different theories, but can be thought of as two different states of
> the same string theory. Hopefully this is OK so far, and agrees with
> the usual interpretations.
It think that's fine so far.
> If so, my question is as follows; suppose I consider a target space
> manifold with, say, a flat metric for simplicity, but with non-trivial
> homology groups. Can I think of the string theory path integral with
> this target space as just being the flat-space R^10 path integral but
> with an insertion of some appropriate string vertex operators?
>
> In other words, is there an obvious way to think of two
> topologically-different target manifolds as corresponding to different
> states of the same string theory, just as there is for two different
> choices of background fields?
Well, let's back off for a second. The Polyakov path integral
prescription tells you to gauge fix as follows: Divide the space of
all metrics that needs to be summed over in the path integral into
equivalence classes, then (arbitrarily) choose a fiducial metric in
each equivalence class and sum over those fiducial metrics instead.
This procedure mods out the diff x Weyl gauge group volumes (at the
expense of introducing ghosts).
Now, if you need to include a topologically non-trivial sector into
your path integral, this will necessarily involve an equivalence class
different from the one that contains the topologically trivial R^10.
Hence you will have to choose a different fiducial metric in that
sector.
Within each equivalence class different choices of the fiducial metric
correspond to different insertions of vertex operators. But you can
never get from one equivalence class to another by any insertion of
vertex operators. This should be clear as the vertex operators are
local insertions (on the worldsheet) and as such cannot account for a
global change of topology. The trick of thinking of a curved target
space metric in terms of a coherent states of gravitons is cute, but
it should be pointed out that it only works as long as the *global*
topology of the curved metric is that of the flat R^10, that is, the
homology groups must be the same.
The statement of background independence is preserved as follows.
Before gauge fixing nothing depends on a specific metric because you
sum over all of them. (An integral is always independent of the
integration variable...) After gauge fixing you have a sum over
fiducial metrics, so superficially you might worry that this seems to
depend on the choice of fiducial metrics you had to make. But, by
construction, the coice of fiducial metrics is irrelevant. The sum
over them is equivalent to the initial sum over everything and hence
background independent.
Hope that helps.
Best,
Rufus
[Moderator's note: have not you confused the worldsheet metric and the
target manifold's metric, Rufus? In string theory, the target space
metric is fixed at the beginning, and one calculates the S-matrix on
this particular background. Physics at other backgrounds of the same
topology can be obtained by inserting vertex operators to the action,
by deforming the worldsheet action with perturbative string states.
It's because an infinitesimal change of geometry corresponds to a
condensate of closed strings. However, if you want to obtain a manifold
with completely different homology groups, you must make a true
topology change transition, and switching from one topology to another
topology (branch) is represented by a condensation of non-perturbative
states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe
for an elementary introduction). Massless D3-branes are not really
local vertex operators on the worldsheet; in some sense, they can be
described by nonlocal vertex operators that add a boundary to the
worldsheet. At any rate, the stringy perturbative expansion breaks
down once you change the geometry in such a way that some wrapped
D3-branes become massless. For a more technical description of the
conifold transition, see e.g. http://arxiv.org/abs/hep-th/9504145 LM]
d70yxj
the
> target manifold's metric, Rufus? In string theory, the target space
> metric is fixed at the beginning, and one calculates the S-matrix on
> this particular background. Physics at other backgrounds of the same
> topology can be obtained by inserting vertex operators to the
action,
> by deforming the worldsheet action with perturbative string states.
> It's because an infinitesimal change of geometry corresponds to a
> condensate of closed strings.
> However, if you want to obtain a manifold
> with completely different homology groups, you must make a true
> topology change transition, and switching from one topology to
another
> topology (branch) is represented by a condensation of
non-perturbative
> states,
Thanks moderator, I think this is exactly the question I was asking.
> in some sense, they can be
> described by nonlocal vertex operators that add a boundary to the
> worldsheet.
Is there a good reference to see it in this way? The Greene et al paper
doesn't seem to talk about it in quite this language.
> At any rate, the stringy perturbative expansion breaks
> down once you change the geometry in such a way that some wrapped
> D3-branes become massless.
So I specify infinite string coupling in order that the tension of the
D3-brane goes to zero, is that right? Maybe I've misinterpreted that,
but it somehow seems a bit odd; perturbative string theory on a target
space with non-trivial homology groups can only be thought of as an
infinite coupling string theory on flat space (with D3-branes added).
Whereas perturbative string theory in manifold with just different
geometry can always be thought of as perturbative string theory on flat
space (with gravitons). Is that odd? Maybe it's what I should expect,
I'm not sure. Thanks for the help, anyway.
I have a slightly different question, partly related to this, and also
to the recent Hawking material. In his talk here he emphasised that in
AdS/CFT one should sum over all supergravity configurations compatible
with the appropriate boundary conditions. (I guess this is also
emphasised in the Maldacena paper he references, and in 9803131 too).
So at the level of partition functions you have
Z_cft = Z_s where Z_s is obtained by integrating or summing the sugra
action over all appropriate metrics. In particular, in those above
examples the interesting thing is that both AdS and AdS-Schwartzschild
need to be included when comparing bulk results for correlation
functions etc to results in the CFT.
So what happens when I move from just supergravity in the bulk, to
string theory? Do I also need to sum over all appropriate target space
metrics, or can I just choose one and string theory somehow does the
rest for me?
(And is AdS-Schwartschild target space related to an AdS target space
but with some kind of condensation of non-perturbative states?)
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arivero
> Dear Forum,
>
> As far as I can tell, the usual statement of background independence
> in string theory is that even though one chooses a specific metric
> (and other target space fields) in the string theory path integral, a
> particular choice of background field is exactly equivalent to any
At the epoch of the T-duality revolution (about 1995, I mean) , there
was around another concept of background independence; some lecturers
presented the target space as a temporary concept, to be substituted in
the future by some other calculation technique where only "spaces of
strings" were to be consirered, and "spaces of points" were to be not
needed anymore. I have not idea is such conception has been preserved
in the publications.
Univ Zaragoza, PhD Science------------------------------------------------------------------------
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Rufus Anton
> Moderator's note: have not you confused the worldsheet metric and the
> target manifold's metric, Rufus? In string theory, the target space
> metric is fixed at the beginning, and one calculates the S-matrix on
> this particular background.
Thanks for your remarks, Lubos. Apparently I didn't make this very
clear, but the point is that the set of equivalence classes of
*worldsheet metrics* depends in an essential way on the homology
groups of the *target space*. For every nontrivial cycle on the target
space there will be a set of equivalence classes of worldsheet metrics
coming from world sheets that wrap this cycle. Toroidal
compactification is possibly the simplest example for this: The
partition function breaks into a sum of topologically distinct sectors
labeled by winding numbers.
[Moderator's note: Yes, these configurations are called the worldsheet
instantons. LM]
> Physics at other backgrounds of the same
> topology can be obtained by inserting vertex operators to the action,
> by deforming the worldsheet action with perturbative string states.
> It's because an infinitesimal change of geometry corresponds to a
> condensate of closed strings. However, if you want to obtain a manifold
> with completely different homology groups, you must make a true
> topology change transition, and switching from one topology to another
> topology (branch) is represented by a condensation of non-perturbative
> states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe
> for an elementary introduction). Massless D3-branes are not really
> local vertex operators on the worldsheet; in some sense, they can be
> described by nonlocal vertex operators that add a boundary to the
> worldsheet. At any rate, the stringy perturbative expansion breaks
> down once you change the geometry in such a way that some wrapped
> D3-branes become massless. For a more technical description of the
> conifold transition, see e.g. http://arxiv.org/abs/hep-th/9504145 LM
Why do you think this is in conflict with what I said? If you insert
something nonlocal into the path integral, so that you add another
boundary to the worldsheet as you say, then you end up in a different
equivalence class within the space of worldsheet metrics. But if you
started with a *complete* set of such classes, as I suggested, it was
already there to begin with. This is the very definition of the
(first-quantized) approach to string theory as we understand it and it
contains both perturbative and non-perturbative states. So all you do
is pretending that you can transcend the classification of metrics
into equivalence classes by allowing for non-local vertex operator
insertions.
[Moderator's note: I just don't quite understand the role that you want to
assign to worldsheet instantons in answering the question. No doubt,
worldsheet instantons DO play a very important role in topology change,
as Witten showed in the example of the flop transition, see
http://arxiv.org/abs/hep-th/9301042 - but the role seems to be different
than your comments. The worldsheet instantons contribute to various
observables such as the particle masses (or Yukawa couplings) and
guarantee that the total value is continuous throughout the transition,
even though the contribution of the classical Yukawa couplings has
a discontinuity. Nevertheless, different target space manifolds have
different spectrum of possible worldsheet instantons - they have
different second homology - and therefore the division of the worldsheet
configurations into topological classes (the division into worldsheet
instantons) DOES depend on the target space manifold's topology. You
seem to propose a way how to mask this difference and use a unified
treatment, but I just don't understand how it works and whether there
is anything physical about such a description. LM]
d70yxj
classes of worldsheet metrics, one is actually summing over all
possible topologies and geometries of the target space, almost without
realizing it?
I don't think I've seen this expressed in quite this way in the usual
sources, but perhaps I've missed something. Is there a good reference?
[Moderator's note: This posting has been delayed because of communication
problems with physicsforums.com. Summing over different worldsheets
means summing over different embeddings of the worldsheet into
a fixed, pre-determined spacetime background. Physically, such a
calculation can be equivalent to changing the background in various
contexts, and one can show in this formalism that the target space
geometry including its topology is dynamical - but nevertheless
the calculation in string theory requires a fixed spacetime
background and fixed topology to start with. Rufus (and others),
please feel more than free to offer a better answer. LM]
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Rufus Anton
> Rufus, do you mean that by summing correctly over all equivalence
> classes of worldsheet metrics, one is actually summing over all
> possible topologies and geometries of the target space, almost without
> realizing it?
Let me first say that what I am saying is not half as outlandish as
you apparently think and it certainly is nothing new. It may just be a
view point that is slightly different from the standard lore.
D-branes were first discovered as soliton solutions of the low-energy
effective equations of motion of (super-) string theory. Polchinski's
accomplishment (in '96) was to show that these solitons of the
effective theory correspond to nonperturbative microscopic states in
the full-fledged string theory. Nevertheless, Polchinski's treatment
still pertains to a description in terms of classical target space
properties. While such a description provides intuitive pictures of
the low-energy description and is useful for a number of important
issues (including the structure of moduli space), it is not quite
sufficient. After all, the great thing about string theory is that it
is a theory of quantum gravity and, hence, we should not expect to be
able to describe it solely in terms of classical geometry. Maybe in
the future a more natural and/or complete understanding of the quantum
geometry of target space will emerge, but for now we can study the
quantum regime by studying the CFT on the world sheet. The spectrum of
this CFT includes both perturbative and non-perturbative states and
encodes the (quantum) geometry of target space. Surely if you
formulate the CFT on the world-sheet general enough it will have
excitations in its spectrum that correspond to target spaces of
different topolgy in the low-energy effective description.
How does this relate to Lubos' remarks concerning, e.g., Witten's
work? Well, if you insist on describing the target space in classical
terms, some hard work is needed to show that physical properties
remain continous even if the topology changes. The reason why one
would choose the classical description is, of course, that CFTs are
not sufficiently under control to calculate all the things we want to
know. It is simply too hard to solve them.
However, even if we can't calculate many things explicitly, we can
still learn a great deal about the conceptual workings of string
theory. Despite the fact that the most general CFT that is consistent
with all classical string backgrounds hasn't been found yet, it
presumably does exist. The original question in this thread was about
background independence. This question has a particularly simple and
general answer in the abstract CFT language. That's what I tried to
explain using familiar terms.
Best,
Rufus
Urs Schreiber
news:a1c70df8.04090910...@posting.google.com...
> Surely if you
> formulate the CFT on the world-sheet general enough it will have
> excitations in its spectrum that correspond to target spaces of
> different topolgy in the low-energy effective description.
[...]
>Despite the fact that the most general CFT that is consistent
> with all classical string backgrounds hasn't been found yet, it
> presumably does exist.
I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
with the correct central charge) corresponds to precisely one classical
background of string theory. This indeed is precisely the general definition
of "classical background" in string theory, isn't it?
Lubos Motl
> I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
> with the correct central charge) corresponds to precisely one classical
> background of string theory. This indeed is precisely the general definition
> of "classical background" in string theory, isn't it?
My guess is that Rufus might have meant a general (S)CFT with some
parameters and other defining features whose choice correspond to a
selection of a particular classical background - in other words, Rufus
wanted to find a map of the whole perturbative beach of the landscape,
which means the universal description of all backgrounds (at least all
perturbative backgrounds); do I understand you well, Rufus?
______________________________________________________________________________
E-mail: lu...@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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Rufus Anton
Yes, you do. Thanks for clarifying. In the early 90s some people
thought that finding the "master (S)CFT" would be the key to making
progress. But then spacetime rather worldsheet symmetries took over as
the most important tools for understanding more features of string
theory. Maybe the original plan should not be completely forgotten...
Urs Schreiber
news:Pine.LNX.4.31.040909...@feynman.harvard.edu...
> On Thu, 9 Sep 2004, Urs Schreiber wrote:
>
> > I am wondering what you mean by that. I'd say that every CFT (sort for:
SCFT
> > with the correct central charge) corresponds to precisely one classical
> > background of string theory. This indeed is precisely the general
definition
> > of "classical background" in string theory, isn't it?
>
> My guess is that Rufus might have meant a general (S)CFT with some
> parameters and other defining features whose choice correspond to a
> selection of a particular classical background - in other words, Rufus
> wanted to find a map of the whole perturbative beach of the landscape,
> which means the universal description of all backgrounds (at least all
> perturbative backgrounds); do I understand you well, Rufus?
Ok. And background *independence* (which is desireable as opposed to
background *freedom* which may be problematic) means that each such SCFT can
be obtained from any other one by turning on some sort of coherent state in
the latter, roughly.
Robert C. Helling
> I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
> with the correct central charge) corresponds to precisely one classical
> background of string theory. This indeed is precisely the general definition
> of "classical background" in string theory, isn't it?
What exactly do you mean by "classical"? And targets related by
T-duality (or mirror symmetry if you like) have the same CFT but with
two classical backgrounds described by it. Furthermore, I am not sure
that the converse is true: That would mean that each CFT has at least
one point in its moduli space ("large volume") where it is described
by a sigma model.
Robert
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Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Urs Schreiber
Newsbeitrag news:2qdasnFuim...@uni-berlin.de...
> On Thu, 9 Sep 2004 15:22:28 -0400, Urs Schreiber
<Urs.Sc...@uni-essen.de> wrote:
>
> > I am wondering what you mean by that. I'd say that every CFT (sort for:
SCFT
> > with the correct central charge) corresponds to precisely one classical
> > background of string theory. This indeed is precisely the general
definition
> > of "classical background" in string theory, isn't it?
>
> What exactly do you mean by "classical"?
For instance that the (S)CFT is a classical solution of the corresponding
string field theory action.
That's what a background is, isn't it? A classical solution, i.e. a saddle
point of the full string field action that we'd like to compute the path
integral of (if only we could) but which we can only perturb about.
(From the classical solution Phi to the string field theory one gets a
deformed BRST operator Q_Phi which is the BRST operator of the new (S)CFT.)
There is some fine print here, of course, but up to that defining a (S)CFT
means defining the (generalized) classical background that the string whose
worldsheet dynamics is described by that CFT.
I guess the reason why you feel uncomfortable with me saying "classical
background" is that the most general such background is far from being a
"classical spacetime" with smooth space and everything. It may be an exotic
quantum gravitic thingy. But it is still the saddle point solution on which
the perturbative string propagates. Wouldn't you agree?
> And targets related by
> T-duality (or mirror symmetry if you like) have the same CFT but with
> two classical backgrounds described by it.
Ok, right. I should have said that the SCFT describes the background up to
symmetries like gauge symmetries and dualities. But that's more a matter of
language, depending on if you consider two spacetime theories related by
duality to be "different".
> Furthermore, I am not sure
> that the converse is true: That would mean that each CFT has at least
> one point in its moduli space ("large volume") where it is described
> by a sigma model.
Oh, no, that's not what I mean. By "classical background" I don't
necessarily mean one described by a CFT which is a sigma model.
WL
World-sheet concepts like CFT are pretty much useless for describing
non-perturbative backgrounds, such as eg F-theory compactifications
which are non-perturbative backgrounds of the type IIB string
(involving eg mutually non-local 7-branes; I wouldn't know of any
CFT description of this situation). In the space of all consistent
ground states, it seems that only a small subset has a
perturbative decription in terms of weakly coupled worldsheet
theories, like sigma models on gently curved "classical" manifolds.
Such backgrounds are fine as toy models, but if one wants to gain
a better understanding of the whole space of string backgrounds,
one definitely needs to go beyond on-shell worldsheet physics.
[Moderator's note: I agree, of course, but my feeling was that the
discussion was focussed on weakly-coupled perturbative backgrounds
that still do not have to be geometric sigma-models, such as various
Gepner-like models. One can still ask whether they can be always
connected with geometric backgrounds. Various islands, orbifolds
by T-dualities and other CFTs with possibly frozen "size" modulus
suggest that the answer is "No", at least if we want them to connect
on-shell. LM]
jgraber
> Ok. And background *independence* (which is desireable as opposed to
> background *freedom* which may be problematic) means that each such
SCFT can
>be obtained from any other one by turning on some sort of coherent
state in
>the latter, roughly.
I find this very interesting. Could you take a minute to explain why
background independence is desirable, but background freedom is
(possibly) problematic.
Also it would help me understand if you defined the difference, or gave
references to appropriate definitions for each term. Of course, I see
an implied partial definition for background independence, at least in
one particular context, in the post I am responding to. TIA. Jim
Graber
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Urs Schreiber
> Urs Schreiber wrote (in part):
>
> > Ok. And background *independence* (which is desireable as opposed to
> > background *freedom* which may be problematic) means that each such
> SCFT can
> >be obtained from any other one by turning on some sort of coherent
> state in
> >the latter, roughly.
>
> I find this very interesting. Could you take a minute to explain why
> background independence is desirable, but background freedom is
> (possibly) problematic.
> Also it would help me understand if you defined the difference, or gave
> references to appropriate definitions for each term. Of course, I see
> an implied partial definition for background independence, at least in
> one particular context, in the post I am responding to. TIA. Jim
> Graber
First of all it is important to clarify what we mean by "background".
You'll sometimes hear people say things like:
"The big lesson learned grom general relativity is that physics must be
background free."
The next sentence will either advertize LQG as a background free theory
taking this lesson serious or criticize string theory for apparently not
taking this lesson serious.
But a little reflection shows that the meaning of "background" in the
second sentence is not the same as in the first one.
GR is "background free" in the sense that an object in its action which is
not varied (integrated over in the path integral) in other field
theories like pure Yang-Mills for instance, now is varied
(integrated) - namely the metric tensor. Hence GR does not assume a fixed
metric background structure. (It still assumes a fixed topological
manifold, though for instance, which is not "varied".)
But if you look at the effective target space action of string theory
you'll note that it is precisely of the same "background free" form as
the Einstein-Hilbert action (no wonder, because it is just
Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
metric indeed is dynamical, just as in GR. That's pretty obvious.
But now when a theory is quantized there is a new notion of "background"
arising. Quantization means evaluating a path integral and perturbatively
quantizing means approximating the path integral at a saddle point and
then computing corrections to that approximation.
So when we say "quantizing theory X on background Y" we mean "Pick the
saddle point of the path integral given by the classical solution Y of
theory X and then compute quantum fluctuations of it".
This notion of background may seem similar but is completely different
from the one above. Every classical solution of a theory X which is
"background free" in the first sense is a "background" in the second
sense.
And this already tells you why "background freedom" in the first
sense is good (everything should be dynamical), while "background
freedom" in the second sense is bad - it would mean that there are no
classical solutions of your field theory!
String theory is nicely "background free" in the first sense, even more so
than ordinary GR, for instance. Not only is the metric a dynamical
quantity, but even the number of (macroscopic) dimensions, the coupling
constant, and to a large extend the entire field (particle) content of
the theory is not a fixed ingredient of the Lagrangian but is dynamical.
That's the very reason why one can even consider dynamics in the string
theory "landscape". String theory is so immensely "background free" in the
first sense of the word that it is at present very hard to say anything
about which values all these dynamical quantities it contains actually
will obtain after some evolution. The landscape discussion is one of
dealing with a theory which is highly background free, so that you first
have to solve equations of motion to even be able to say something about
the matter content of the theory. This is quite in contrast to other
approaches to quantum gravity, which are often considered to be truly
"background free", where all the matter content, the coupling constants,
the number of dimensions is fixed by hand.
So the problems with answering the "landscape question" in string theory
should be seen in light of the fact that they arise due to a difficult
question which other theories can't even ask.
This directly leads to the next point: String theory is certainly not
"background free" in the second sense. It does have classical solutions!
And that's good. These classical solutions can be used for perturbative
quantization by using them as a "background" about which to start a
perturbative expansion. But it must be realized that this notion of
"background" is one of how to do practical computations, not one of
principle.
In principle you could use other tools to compute the quantum theory, like
for instance computing in non-perturbatively. And there are ways to do
that in string theory, too, but these are as yet not completely general.
It is this non-perturbartive quantization which people really have in mind
when they say that, for instance, "LQG is a background free quantization
of gravity". This just means that, indeed, LQG is an attempt to quantize
gravity in one stroke, without perturbing about classical solutions of it
(="backgrounds" in the second sense).
But currently it seems that LQG, which is certainly "background free" in
the first sense of the word (the metric is dynamical) is maybe even
"background free" in the second sense of the word - which however
would mean that it admits no classical solutions! But that would be very
undesirable, since the world we perceive is so obviously well described by
classical gravity to good approximation, that it is a great challange for
experimentalists to find any quantum fluctuations (of gravity).
(Personally I feel that it is maybe not such a great surprise that the
quantization method used in LQG has problems finding sensible solutions,
since we know that when applied to 1+1 dimensional gravity it does not
reproduce the correct path integral quantization. In 1+1 dimensions there
is a well known theorem that when you solve all the diffeomorphism
constraints the Hamiltonian constraint cannot have any solutions at all.
This seems to be precisely the problem encountered for years in 3+1d LQG,
too.)
Next, what is the difference between "background freedom" and "background
independence". Well, I guess when people use "background freedom" in the
first sense (meaning that the metric is dynamical) it is pretty much
synonymous to "background independence" in this first sense.
But in the second sense of the word "background" (=classical solution
used in quantum perturbation theory) the term "background indepence" is
an important concept. It refers to the question if the results obtained by
perturbing about one background will coincide with the results obtained by
perturbing about another one. This is a nontrivial question and an
important consistency check of perturbative string theory. And indeed, it
can be seen in various nice ways that perturbative string theory, while
requiring the specification of some background (=classical solution) is
independent of this choice. So it is a fixed but arbitrary background (in
the second sense) that is used in perturbative string theory.
And that's a good thing.
P.S.
See the recent discussion on SCFTs and their relation to "backgrounds" to
see more fine print to that last paragraph.
Robert C. Helling
> GR is "background free" in the sense that an object in its action which is
> not varied (integrated over in the path integral) in other field
> theories like pure Yang-Mills for instance, now is varied
> (integrated) - namely the metric tensor. Hence GR does not assume a fixed
> metric background structure. (It still assumes a fixed topological
> manifold, though for instance, which is not "varied".)
Yesterday, my office mate and me spend nearly two hours discussing
this (or a closely related) question, we tried to understand the
statement that GR is diffeomorphism invariant or that diffeomorphisms
are a gauge symmetry in GR. I think we solved it in the end and sorted
out that there are at least two different meanings to these words but
let me get you thinking in this direction: There is an obvious tension
between the following statement that each individually seem to be
true:
1) GR is diffeomorphism invariant.
2) Diffeomorphisms are a gauge symmetry in GR in the sense that before
modding out the description contains redundant degrees of freedom.
3) Gauge symmetries cannot be broken spontaneously (as the degrees of
freedom are not really there). [This might sound surprising but it is
not: In the Higgs effect it is not the local gauge freedom that is
broken it is the global one]
4) The ground state of (quantum) GR should be diffeomorphism invariant
as otherwise diffeomorphisms would be spontaneously broken.
5) Minkowski space is not diffeomorphism invariant.
6) In fact no classical pseudo Riemannian manifold is diffeomorphism
invariant.
This appears to prove that classical manifolds can never be ground
states of quantum gravity (for some definition of "ground" state, not
necessarily referring to some energy).
Once you manage to sort this out for yourselves you learned a great
deal about LQG, I promise you.
> But if you look at the effective target space action of string theory
> you'll note that it is precisely of the same "background free" form as
> the Einstein-Hilbert action (no wonder, because it is just
> Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
> metric indeed is dynamical, just as in GR. That's pretty obvious.
What exactly are you talking about, sugra or string field theory?
Sugra definitelty is a classical theory (and non-renormalizable as a
quantum theory) and my impression from vague looks at the literature
is that string field theory (of Witten type and its possible
generalizations to super and closed) is only used to compute tree
diagrams (although in light cone sft there are loop calculations but
other problems). Is this correct?
My point is, that what you describe is at best theoretical because for
proper quantum (ie loop) processes you have methods available only for
a very limited number of backgrounds (flat space, group manifolds,
orbifolds etc). And you don't really know if it is possible to
generalize these methods that are the only ones so far to reliably
compute scattering data to general backgrounds even in principle. It
seems to me it is not impossible but this is a question of faith. Of
course I have in mind the example of strings in AdS5 which is still a
symmetric space so sufficiently simple but turns out very hard to
treat with world sheet methods.
Now you can come and say that in the past 10 years we have learned so
much about non-perturbative string theory. True. But this is mostly
about states and moduli spaces etc and not yet a full theory that
contains all the dynamics.
Lubos Motl
> 1) GR is diffeomorphism invariant.
Very good.
> 2) Diffeomorphisms are a gauge symmetry in GR in the sense that before
> modding out the description contains redundant degrees of freedom.
Agreed.
> 3) Gauge symmetries cannot be broken spontaneously (as the degrees of
> freedom are not really there). [This might sound surprising but it is
> not: In the Higgs effect it is not the local gauge freedom that is
> broken it is the global one]
First of all - yes, this is a correct interpretation if you do it
carefully. Second: it is not quite fair - the global symmetry is a
subgroup of the local symmetry group, so you can't really break the former
without breaking the latter (at least a little bit). What you probably
wanted to say is that the state must be invariant under "normalizable"
local transformations - those that converge to the identity at infinity
quickly enough. The "asymptotically changing" transformations can be
broken - this is probably what you mean by the "global ones"?.
The electroweak symmetry breaking can be phrased as spontaneous symmetry
breaking. In this ordinary approach, the vacuum is not invariant under the
global transformations of the gauge group (not even under those that only
affect a small piece of space) - which means that it partially breaks the
gauge group (spontaneously), too. But I agree, the more correct way to
describe the situation is to "dress" all operators with a function of the
Higgs, and take the vacuum which is the usual symmetry-breaking vacuum,
but averaged over all gauge copies of itself.
The situation in GR is analogous. You can take a background metric tensor,
and it breaks most of the coordinate transformations, except for the
isometries of the background. If you were picky, you could also work with
a general state invariant under *all* coordinate transformations, and
dress all operators with some (very nonlocal) functions of the metric.
This would completely obscure the geometric interpretation. I don't know
how exactly this procedure should be done, and I am not sure whether it is
too important, interesting, or useful.
I think that this way of thinking - with the completely non-geometric
mess forming your Hilbert space - is the first thing that one must avoid
if she wants to escape from the trap of pseudo-physics where everything is
obscure. I don't know if you agree, but the standard particle physics
treatment - the expansion around a chosen value of the vacuum expectation
value (of the Higgs, in the electroweak case, or the metric, in the GR
case) is a perfectly legitimate approach that preserves the general
diffeomorphism symmetry - it just reinterprets them in terms of new
formulae. It is the way to go. It may be derivable from something deeper
and more obscure, but this derived formalism (expansions around a
background) will remain correct.
> 4) The ground state of (quantum) GR should be diffeomorphism invariant
> as otherwise diffeomorphisms would be spontaneously broken.
First of all, the very notion of "background independence" or the rules of
GR are meant to emphasize the fact that GR does not have a unique ground
state - it has many solutions, certainly if you allow the asymptotic
behavior of the metric tensor to be whatever you want.
In the previous paragraphs I tried to explain that the "non-normalizable"
diffeomorphisms *can* be spontaneously broken, using the first formalism,
and they allow you to fix the asymptotic behavior of the metric tensor at
infinity. You may want to require such a state to be invariant under the
"normalizable" local transformations, which simply means that you would
average it over all coherent states containing the null (pure gauge)
unphysical polarizations of the graviton. We usually don't do it - not
even in gauge theories - and I don't know what you precisely want to gain.
We simply define our physical Hilbert space as excitations of a given
background, and remove the unphysical polarizations corresponding to the
gauge symmetry.
There is nothing wrong to take a geometrically interpretable state which
is not gauge-invariant and work with it, imagining that the allowed state
is the average of your state over the gauge group.
> 5) Minkowski space is not diffeomorphism invariant.
Right. The Minkowski space is Poincare invariant. The Poincare group is a
subgroup of general diffeomorphisms (of any manifold with the trivial
topology); it is the isometry group of the Minkowski space. There is
nothing wrong with it. Minkowski space is an example of a state that
spontaneously breaks most of the general coordinate symmetries. Those
transformations that change the asymptotics are allowed to be broken
spontaneously because they mix different superselection sectors of the
theory - this is why the (ADM) energy can be nonzero even though the
energy is superficially a generator of coordinate transformations that
should annihilate all physical states. It is completely analogous to the
fact that the total electric charge can be nonzero in QED, even though it
naively generates a gauge transformation. But it is a gauge transformation
that changes the asymptotic behavior (of the fields) at infinity, and
therefore it maps the states to states in a different superselection
sector. Therefore it is allowed to break the symmetry under this operation
spontaneously.
> 6) In fact no classical pseudo Riemannian manifold is diffeomorphism
> invariant.
Right.
> This appears to prove that classical manifolds can never be ground
> states of quantum gravity (for some definition of "ground" state, not
> necessarily referring to some energy).
I hope that you agree that if there is any sense in which the manifolds
are "not allowed", it is just an error or a very awkward artifact of a
particular formalism - not anything that should be labeled "a physical
insight". The universe certainly looks much like a pseudo-Riemannian
manifold, and classical GR models the Cosmos in the same way. A theory
that seriously predicts that there is no physics of manifolds implied by
quantum GR is ruled out both theoretically as well as experimentally - it
is just a mere stupidity.
> Once you manage to sort this out for yourselves you learned a great
> deal about LQG, I promise you.
Once again.
If you're seriously studying a theory that does not imply (or even does
not want to imply) physics in spacetime that looks like a
pseudo-Riemannian manifold, not even in the classical limit, then you are
not doing any natural science, because you are contradicting the very
basic property of our Universe - that it *does* behave like a manifold.
All experiments in all sciences that have been done so far support this
claim, and any meaningful theory of physics must be able to reconstruct
it. Moreover, if you derived that your "quantum theory of gravity" cannot
reproduce any physics of its classical counterpart - which certainly
*does* describe the spacetime as a manifold - then you have certainly made
a serious error in your quantization.
If you want to learn something about loop quantum gravity, see
http://en.wikipedia.org/wiki/Loop_gravity
> What exactly are you talking about, sugra or string field theory?
> Sugra definitelty is a classical theory (and non-renormalizable as a
> quantum theory) and my impression from vague looks at the literature
> is that string field theory (of Witten type and its possible
> generalizations to super and closed) is only used to compute tree
> diagrams (although in light cone sft there are loop calculations but
> other problems). Is this correct?
That's definitely not correct. Witten's open string field theory was
constructed to reproduce the whole perturbative expansions, and if you ask
Barton Zwiebach, for example, he will certainly be able to show you the
proof that the Feynman calculations in Witten's cubic SFT cover all
Riemann surfaces with at least one boundary. So SFT has certainly
succeeded to reproduce the whole perturbative quantum theory, otherwise it
would be in a very bad shape. SFT was even meant to be a
non-perturbatively complete formulation. Well, it can construct D-branes
as classical solutions (lumps), but otherwise I think that SFT has failed
in this more ambitious task. But it can certainly give you loop
amplitudes!
The same holds for closed string field theory - the only new major
problem is that the action of closed SFT is itself an infinite expansion
in the coupling constant, and the local symmetries must be dealt with the
BV formalism, the annoying complicated generalization of the BRST methods.
Fields, antifields, ghosts for ghosts etc.
> My point is, that what you describe is at best theoretical because for
> proper quantum (ie loop) processes you have methods available only for
> a very limited number of backgrounds (flat space, group manifolds,
> orbifolds etc). And you don't really know if it is possible to
> generalize these methods that are the only ones so far to reliably
> compute scattering data to general backgrounds even in principle.
I have no idea what you mean. Have you heard the phrase "non-linear sigma
model"? It is a description of perturbative string theory in any
acceptable geometrical background. Are you complaining that the sigma
models do not exist, or do you criticize the fact that a general enough
sigma model does not allow us to write compact formulae for all quantities
- that it is not integrable? Well, most things in the world, including
classical physics of 3 interacting bodies, are not integrable. Or are you
complaining that we do not have nonperturbative definition for general
backgrounds? Yes, I agree - but we do not even have any non-perturbative
definitions for flat space, e.g. type II string theory on T^6 at a general
value of the coupling.
> It seems to me it is not impossible but this is a question of faith.
> Of course I have in mind the example of strings in AdS5 which is still
> a symmetric space so sufficiently simple but turns out very hard to
> treat with world sheet methods.
The fully curved AdS5 is not well-formulated because we don't know how to
treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know
everything (perturbatively). But you seem to have doubts that a nonlinear
sigma model, that can be proved to be a conformal theory, is a correct
theory of physics at the appropriate background. I have no idea how would
you want to justify these doubts.
Best wishes
Lubos
Urs Schreiber
> 6) In fact no classical pseudo Riemannian manifold is diffeomorphism
> invariant.
I don't see what you mean by that. Could you explain?
Say my manifold is flat R^3. To me this statement is quite independent of
any diffeomorpisms R^3 -> R^3. So I don't even know in which sense a
Riemannian manifold could be "diffeomorphism invariant" or not.
> This appears to prove that classical manifolds can never be ground
> states of quantum gravity (for some definition of "ground" state, not
> necessarily referring to some energy).
>
> Once you manage to sort this out for yourselves you learned a great
> deal about LQG, I promise you.
Maybe, but I don't see yet what you are getting at, sorry.
You may have noticed that I have developed a weakness for first thinking
about this stuff in 2-dimensional gravity. So let me suggest we discuss this
point you have in mind (which I don't see yet) for 2d gravity on S^1 x R.
The spatial diffeomorphism constrains are L_n - \bar L_{-n} for all n.
They act, roughly, on the space of fuctionals of maps from the circle into
target space. A state is diffeo invariant, i.e. annihilated by all these
L_n - \bar L_{-n}, precisely if takes the same value on any one of these
maps that it takes on a map that differs from the first only by a
reparameterization.
Such states includes for instance traced Wilson lines along the loop with
respect to some (auxiliary) connection on target space. One can show that we
can deal with the quantum operator ordering issues in these (for instance by
relating them to DDF states).
So such reparameterization invariant states do exist in the quantum theory
of 2d gravity. Could you point out in terms of these what you have in mind
concerning more general quantum gravity?
(BTW, as I have mentioned before, these states in principle cannot be in the
kernel of the Hamiltonian constraint operators.)
> > But if you look at the effective target space action of string theory
> > you'll note that it is precisely of the same "background free" form as
> > the Einstein-Hilbert action (no wonder, because it is just
> > Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
> > metric indeed is dynamical, just as in GR. That's pretty obvious.
>
> What exactly are you talking about, sugra or string field theory?
> Sugra definitelty is a classical theory (and non-renormalizable as a
> quantum theory) and my impression from vague looks at the literature
> is that string field theory (of Witten type and its possible
> generalizations to super and closed) is only used to compute tree
> diagrams (although in light cone sft there are loop calculations but
> other problems). Is this correct?
It may be correct that in practice it is mostly used for tree level
calculations, but as far as I know it is also true that SFT is known to
reproduce all higher loop diagrams.
So I was thinking about string field theory all along, because it makes the
comparison of string theory to other approaches to quantum gravity most
convenient. What we really should be doing is computing the path integral
for the string field action. That statement can be found explicitly for
instance in discussions of AdS-CFT. where the conjecture really is that the
CFT on the boundary is computed by that SFT integral in the bult. Of course
one mostly discusses the limit where the SFT action becomes just the
ordinary sugra action. It's a limit.
> My point is, that what you describe is at best theoretical because for
> proper quantum (ie loop) processes you have methods available only for
> a very limited number of backgrounds (flat space, group manifolds,
> orbifolds etc). And you don't really know if it is possible to
> generalize these methods that are the only ones so far to reliably
> compute scattering data to general backgrounds even in principle.
In principle given any SCFT with c=15 one can compute its correlators on
parameter spaces of arbitrary genus. Are this not the loop processes that
you are talking about?
> Now you can come and say that in the past 10 years we have learned so
> much about non-perturbative string theory. True. But this is mostly
> about states and moduli spaces etc and not yet a full theory that
> contains all the dynamics.
I'd say the most important problem of the known non-perturbative methods is
that they rely on asymptotic boundary conditions, like asymptotic Minkowski
or asymptotic AdS/CFT, which are not quite those that we would like to
consider.
Urs Schreiber
> "Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> schrieb im
> Newsbeitrag news:2qntssF10f4...@uni-berlin.de...
>
> > 6) In fact no classical pseudo Riemannian manifold is diffeomorphism
> > invariant.
>
> I don't see what you mean by that. Could you explain?
I just see Lubos' reply to this and see that I should clarify: Of course
any given diffeomorphism is not an in general an isometry. Lubos is, I
think, referring to isometries, while I was thinking of general diffeos.
Lubos Motl
We are talking about all these concepts simultaneously. Let me say once
again the statements that I find important: the isometry group of a
manifold is the only unbroken part of the diffeomorphism symmetry group;
the rest of the diffeomorphism symmetry is spontaneously broken by the vev
of the metric, i.e. by the geometry.
Someone may argue that local symmetries should not really be broken, and
all physical states should be invariant under all diffeomorphisms. But
this condition can be given up for all diffeomorphisms that are "large",
non-normalizable - those that move the points even at infinity. Such
diffeomorphisms are typically mapping states from a superselection sector
A into states in a different superselection sector B.
In other words, consistency only requires that we demand the physical
states to be annihilated by the generators of the normalizable
diffeomorphisms - those that move the points "in the middle of the
spacetime". Such invariant states are easily obtained from a completely
geometric state, just by demanding that we average it over the gauge
group. Averaging over the "normalizable" part of the gauge group is
equivalent to adding the mixture of coherent states of the pure gauge
unphysically polarized gravitons; averaging over the "non-normalizable"
part of the gauge symmetry combines the state with its copies in other
superselection sectors, and it is totally legitimate to work with a single
representative within a single sector, and we obtain the correct results.
If we recall that the full state may be thought of as a mixture of states
from different sectors, we also realize that the isometries of the
superselection sectors don't have to annihilate the physical states.
The goal of these two messages is to convince the reader that the gauge
symmetries are subtle, there are various types and they can be treated in
many ways - and the idea that a state in quantum gravity must always be a
non-geometrical obscure mixture of everything in the world, because it
must be invariant under all transformations, and that geometrical way of
thinking is impossible if we deal with quantum gravity properly, is naive
and would not lead anywhere. Gauge symmetries are finally not as
fundamental as Robert tries to picture - they represent a nice tool to get
rid of some degrees of freedom, especially some polarizations of vectors
and tensors that would otherwise have a negative norm. The theories with
gauge symmetries can be equivalently rewritten in other formalisms, for
example the BRST formalism or a gauge-fixed form such as the light-cone
gauge. The physics is unchanged, and if Robert were working with the BRST
invariance (or, perhaps, a BV-type formalism, if necessary), he would not
encounter his restrictions and the geometrically clear states would be
clearly possible.
I realize that many people in loop quantum gravity often and explicitly
say that there is no Lorentz or other symmetry in GR, and I just don't
understand what meaningful can they ever mean. GR has a symmetry under
*all* coordinate transformations. For a trivial topology, there are
infinitely many ways to embed the Lorentz group within the group of all
diffeomorphisms, and the theory is invariant under all these
transformations. The symmetry under a diffeomorphism is only broken
spontaneously, by the values of the metric (which we call the "vev"s in
the quantum theory).
But even in this case, if we only consider local physics of classical GR
in a sufficiently small piece of spacetime - and let us pick the locally
inertial frame - then physics of GR always reduces to special relativity,
by the princple of equivalence, and curvature etc. can be neglected. It is
only the combination of the locally valid Lorentz symmetry *with* the
"background independence" - or the ability to curve spacetime - that makes
GR nontrivial and constrained. A theory whose local physics does not
respect the rules of special relativity cannot be called general
relativity (or its ramification).
Note that the Lorentz symmetry enters GR in many different ways. It is an
unbroken symmetry of local physics, but it is also a symmetry applicable
globally in a topologically trivial spacetime - the latter copy of the
Lorentz symmetry is usually spontaneously broken by most backgrounds
(except the Minkowski space).
Robert C. Helling
> First of all - yes, this is a correct interpretation if you do it
> carefully. Second: it is not quite fair - the global symmetry is a
> subgroup of the local symmetry group, so you can't really break the former
> without breaking the latter (at least a little bit). What you probably
> wanted to say is that the state must be invariant under "normalizable"
> local transformations - those that converge to the identity at infinity
> quickly enough. The "asymptotically changing" transformations can be
> broken - this is probably what you mean by the "global ones"?.
Indeed. When I say local gauge transformations I mean those that tend
to the identity at infinity (in some precise sense that you call
"normalizable". The global ones are then in all gauge trafos mod local
gauge trafos and that coset should be a copy of the gauge group (at
least if "infinity" is simple enough).
>> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism invariant.
>
> Right.
Actually, this is where I would probably say no. All this 'confusion'
(which I claim is wide spread in certain circles) comes from the
mixing of two meanings of "diffeomorphism invariant": It always means
"invariant under general changes of coordinates" but the difference
between the two meanings is whether you transform your fields as well:
You can describe Minkowski space in many different coordinate systems,
often for example radial coordinates are quite usefui. But of course
you have to change the metric as well: The metric in radial
coordinates is different from diag(-1,1,1,1). But this change of
coordinates is a diffeomorphism. (There are isometries as well, where
you don't have to pull back the metric but those are only special
diffeos). And it is these changes of coordinates that GR is invarant
under. As is any other theories that I can formulate in terms of
tensors (that I pull back when applying diffeomorphisms). It is just
that in "field theory on flat space" one of these tensors (eta) is not
dynamical and thus background. Even Newtonian gravity is diffeo
invariant (see MTW), there there is background one form dt as well.
What one should really require is that the result of the quantization
procedure does not depend on the choice of coordinates. And therefore
if one uses coordinates at some point in the quantization, one has to
find the unitary operators that do the pull back on the metric and
impose the constraints that under changes of coordinates also the
quantum analogue of the metric is pulled back correctly.
>> This appears to prove that classical manifolds can never be ground
>> states of quantum gravity (for some definition of "ground" state, not
>> necessarily referring to some energy).
Again this mixes the two meanings and would correctly be translated to
"The proper quantum state should have all diffeos as isometries" which
is of course nonsense.
> I hope that you agree that if there is any sense in which the manifolds
> are "not allowed", it is just an error or a very awkward artifact of a
> particular formalism
100% ACK. That was what I wanted to point out.
>> My point is, that what you describe is at best theoretical because for
>> proper quantum (ie loop) processes you have methods available only for
>> a very limited number of backgrounds (flat space, group manifolds,
>> orbifolds etc). And you don't really know if it is possible to
>> generalize these methods that are the only ones so far to reliably
>> compute scattering data to general backgrounds even in principle.
>
> I have no idea what you mean. Have you heard the phrase "non-linear sigma
> model"? It is a description of perturbative string theory in any
> acceptable geometrical background. Are you complaining that the sigma
> models do not exist, or do you criticize the fact that a general enough
> sigma model does not allow us to write compact formulae for all quantities
> - that it is not integrable?
Neither. I complain that for most non-linear sigmar models you don't
know what the asymptotic states are, so you cannot go an compute
scattering amplitudes. You don't even know the particle spectrum.
> The fully curved AdS5 is not well-formulated because we don't know how to
> treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know
> everything (perturbatively). But you seem to have doubts that a nonlinear
> sigma model, that can be proved to be a conformal theory, is a correct
> theory of physics at the appropriate background. I have no idea how would
> you want to justify these doubts.
Correct yes. But it does not tell you much about observed physics.
Urs Schreiber
> What one should really require is that the result of the quantization
> procedure does not depend on the choice of coordinates. And therefore
> if one uses coordinates at some point in the quantization, one has to
> find the unitary operators that do the pull back on the metric and
> impose the constraints that under changes of coordinates also the
> quantum analogue of the metric is pulled back correctly.
But here one has to be very careful what one really means. The standard
quantizations of 2d gravity don't depend on coordinates, yet physical states
are *not* annihilated by the generator L_n - \bar L_{-n} of spatial
coordinate transformations. They cannot. Coordinate invariance is restored
only for the expectation values and amplitudes. The states themselves are
not rep invariant.
If you wish to impose spatial rep invariance for conformal 1+1d gravity on
the cylinder you can do so, but you end up with "boundary states" (off-shell
states) not physical states. They can never be annihilated by any mode of
the Hamiltonian constraint.
I was recently hoping (in vain) that Lee Smolin might reply to that
observation and its obvious implication for non-perturbative quantization of
higher dimensional gravity:
http://golem.ph.utexas.edu/string/archives/000420.html
Lubos Motl
> >> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism invariant.
> >
> > Right.
>
> Actually, this is where I would probably say no. All this 'confusion'
> (which I claim is wide spread in certain circles) comes from the
> mixing of two meanings of "diffeomorphism invariant":
If you now say that the Minkowski space preserves *all* diffeomorphisms,
then you should consider the option that it is you who is very confused.
> It always means "invariant under general changes of coordinates" but
> the difference between the two meanings is whether you transform your
> fields as well:
If you want to do the transformation twice and undo it at the same moment,
then one cannot be surprised that you get the same thing back. You don't
really mean "invariant under gauge transformations". You rather mean
"invariant under gauge transformations up to a gauge transformation".
However, this is a tautology. If you used your definition of "being
invariant under something", then everything would always be invariant
under everything.
Incidentally, you want to mask this trick by making the transformation
active, and undo it in a passive way.
In any theory of fields - that includes GR - we do not really mean "change
of coordinates" but usually the associated transformation of the fields.
Because the diffeomorphisms act equally on all fields, it is sort of
enough to describe the action on the coordinates. But this statement is a
special property of diffeomorphisms, and if we want to use the standard
machinery of field theory rules and methods, we *always* describe any
transformation as a transformation acting on the fields because the fields
are the only dynamical degrees of freedom. The coordinates are not
dynamical degrees of freedom in GR, because GR is a theory of fields, and
it is conceptually misleading to transform them.
For example, a scalar field transforms as
\phi(x^\mu) -> \phi'(x^\mu) := \phi(f^\mu(\{x^\nu\}))
while the non-scalar tensor fields also include one or more copies of the
Jacobi matrix, and so on. This is the only way how to treat
diffeomorphisms uniformly with all other transformations that one can find
in a theory of fields. If you paid attention to supersymmetry, for
example, you would also realize that the SUSic theories without a
superspace formulation simply force you to study the transformation of the
fields, not coordinates, because this is the only way how SUSY is
realized in this context.
> You can describe Minkowski space in many different coordinate systems,
> often for example radial coordinates are quite usefui. But of course
> you have to change the metric as well: The metric in radial
> coordinates is different from diag(-1,1,1,1).
This is why the flat space is *not* invariant under the diffeomorphism
that moves the point (x,y) to (x',y') = (r,\phi) where r\exp(i\phi) =
x+iy. If you transform the fields (!) according to the correct rule, which
is my rule, not yours, you get different fields - a different form of
metric, as you observed, too. The metric tensor describing a manifold is
invariant under the isometries only; others are spontaneously broken. The
word "different" means that the configuration is *not* invariant under the
operation. Do you think that "different" means that it *is* invariant, or
what do you exactly think that the word "invariance" means?
> But this change of coordinates is a diffeomorphism.
Yes, it is a diffeomorphism, but because it does not preserve the
numerical values of the components of the metric, the metric is not
invariant under this diffeomorphism. By the "metric" or the "metric
tensor" we do not mean the "asthetic impression that we get by visualizing
it as a manifold" which is probably what you think it is; instead, we are
doing quantitative science and we mean the collection of 10 functions of 4
coordinates (in the d=4 case). These functions change, and therefore they
(the metric) is *not* invariant under the operation.
> (There are isometries as well, where
> you don't have to pull back the metric but those are only special
> diffeos). And it is these changes of coordinates that GR is invarant
> under.
If you say that the only transformations under which GR is invariant is
your operation in which you transform the coordinates, which is equivalent
to transforming all fields, but then you undo this transformation by
another action on the fields, then you are very confused. GR is diff.
invariant only because it is invariant under the transformation of the
fields only, those described above. The laws (and the action, as the
invariant scalar) of GR is invariant under those; the particular
configurations themselves are not, unless you consider an isometry of the
background.
> As is any other theories that I can formulate in terms of
> tensors (that I pull back when applying diffeomorphisms). It is just
> that in "field theory on flat space" one of these tensors (eta) is not
> dynamical and thus background. Even Newtonian gravity is diffeo
> invariant (see MTW), there there is background one form dt as well.
If you add new fields, you can make virtually any theory be invariant
under virtually any group. The fields that you must add to Newton's theory
to make it generally covariant are non-dynamical and they can easily be
erased, together with these symmetries, and therefore Newton's theory did
not morally satisfy Einstein's requirement of a diff. invariant theory.
When we talk about a gauge symmetry of a theory, we must say in which
degrees of freedom we formulate the theory because different choices can
have different gauge symmetries. Of course, in the case of GR, I meant the
metric tensor (plus matter fields, if you wish) that are functions of the
spacetime coordinates. If we know the degrees of freedom and the action,
we can decide what the symmetry group is.
> What one should really require is that the result of the quantization
> procedure does not depend on the choice of coordinates.
This is not the field theory approach. In field theory, GR is invariant
under the transformations of the fields associated with a transformation
of the coordinates. Once again, the fields - including the metric - are
the only real degrees of freedom, which means that they are the only
objects used in constructing any questions, and the only objects that
should be transformed.
> And therefore if one uses coordinates at some point in the
> quantization,
If we o quantitative physics whose purpose is to be compared with
experiments, we almost always use coordinates and fields that are
functions of them. You might dislike it, but it is the only way to get
quantitative results as opposed to vague hand-waving and unusable abstract
formulae.
> one has to find the unitary operators that do the pull back on the
> metric and impose the constraints that under changes of coordinates
> also the quantum analogue of the metric is pulled back correctly.
What is important is that the operators (the metric, for example) are
*not* invariant under the conjugation by the unitary operator associated
with a generic diffeomorphism. I say "conjugation" because this is how
operators transform under the operation encoded in a unitary operator.
It would be totally wrong to *require* the operators to be strongly
invariant under the conjugation indicated above. For example, for scalar
fields such a requirement would mean that the scalar field is a constant
function of spacetime - a clear nonsense because we shuold not really call
it a field in this case.
Let me tell you a trivial example in gauge theory. The fields are by no
means required to be invariant under gauge transformations, not even the
global ones. If they were, it would mean that all charged fields are
prohibited, for example, because charged fields *do* transform under
U!)_{elmg}, for example. In the same way, I obtained the nonsensical
result above that the fields in GR would have zero momentum (constant
functions of spacetime) if you required *them* to be invariant. If we
require something like that, it is the invariance of the *states* - or,
equivalently, the BRST-invariance of the states.
Also, the "operators" of the metric tensor only exist in the low energy
limit and it is by no means guaranteed that the full theory has these
operators exactly. Well, string theory probably does not allow these
operators to be defined in the short-distance regime, yet it is a
consistent general covariant theory; the unphysical polarizations of the
gravitons decouple, which is the perturbative incarnation of the local
symmetry.
> Neither. I complain that for most non-linear sigmar models you don't
> know what the asymptotic states are, so you cannot go an compute
> scattering amplitudes. You don't even know the particle spectrum.
This is one of the options I gave you. So you *are* complaining that a
generic model is not solvable. Well, even the three-body Newtonian system
is not solvable; if you don't like that most models in this Universe are
not solvable and one cannot calculate the particle spectrum analytically,
you should consider moving into a different Universe.
> > treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know
> > everything (perturbatively). But you seem to have doubts that a nonlinear
> ...
> Correct yes. But it does not tell you much about observed physics.
What does not tell me much about observed physics? The complete solution
of the strings on the pp-wave? It tells me everything I want to know about
observed physics (in that Universe).
Cheers,
Urs Schreiber
> On Wed, 15 Sep 2004, Robert C. Helling wrote:
>
> > >> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism
invariant.
> > >
> > > Right.
> >
> > Actually, this is where I would probably say no. All this 'confusion'
> > (which I claim is wide spread in certain circles) comes from the
> > mixing of two meanings of "diffeomorphism invariant":
>
> If you now say that the Minkowski space preserves *all* diffeomorphisms,
> then you should consider the option that it is you who is very confused.
I think you clarify this further below. When saying "Minkowski space" you
are thinking of a manifold with coordinates such that the metric is in the
form diag(-1,1,...,1) and this is indeed not preserved by all
diffeomorphisms, but only by the isometries (which form the Poincare group
in this case).
When I say "Minkowski space" I usually mean "a flat pseudo-Riemannian
manifold with one timelike direction", no matter which coordinates I put on
it and hence no matter if the metric has the diag(-1,1,...,1) form or that
of polar coordinates or whatever. Apparently this is also what Robert has in
mind.
I believe this allows us to understand each other's use of terminology.
With that out of the way I would still like to understand what Robert had in
mind in his original comment on non-perturbative quantization of gravity. I
still don't quite see what he is getting at.
Lubos Motl
> I think you clarify this further below. When saying "Minkowski space" you
> are thinking of a manifold with coordinates such that the metric is in the
> form diag(-1,1,...,1) and this is indeed not preserved by all
> diffeomorphisms, but only by the isometries (which form the Poincare group
> in this case).
Right.
> When I say "Minkowski space" I usually mean "a flat pseudo-Riemannian
> manifold with one timelike direction", no matter which coordinates I put on
> it and hence no matter if the metric has the diag(-1,1,...,1) form or that
> of polar coordinates or whatever. Apparently this is also what Robert has in
> mind.
I mean the same thing.
> I believe this allows us to understand each other's use of terminology.
Not really. Regardless of your choice of the coordinates for the Minkowski
space - or any other manifold, for that matter - its metric is only
invariant under a group that is isomorphic to the isometry group. What
Robert seems to mean is that the metric is invariant under *any*
transformation (of coordinates), up to a transformation (of coordinates).
which is a vacuous tautology. Everything would be symmetric under
everything if this notion of "invariance" were adopted, and I think that
my definition is the only definition of "being invariant" that makes any
sense. So I don't really have understanding for any other terminology. Am
I missing anything?
Urs Schreiber
news:Pine.LNX.4.31.04091...@feynman.harvard.edu...
> Not really. Regardless of your choice of the coordinates for the Minkowski
> space - or any other manifold, for that matter - its metric is only
> invariant under a group that is isomorphic to the isometry group. What
> Robert seems to mean is that the metric is invariant under *any*
> transformation (of coordinates), up to a transformation (of coordinates).
> which is a vacuous tautology. Everything would be symmetric under
> everything if this notion of "invariance" were adopted, and I think that
> my definition is the only definition of "being invariant" that makes any
> sense. So I don't really have understanding for any other terminology. Am
> I missing anything?
Agreed. For this reason I said before that I don't understand what Robert
means when saying "a manifold is diff invariant". Either it is meant in "my"
sense, that the abstract geometry does not care about the coordinates (in
which case it is pretty tautologous) or it is meant in "your" sense saying
that the metric tensor is invariant under all diffeos, in which case it is
not correct.
But for something different: You mentioned recently the failure of SFT to
capture certain non-perturbative degrees of freedom. Is it conceivable that
one can somehow "augment" SFT in a nice way to include these?
Lubos Motl
independence" thread):
> But for something different: You mentioned recently the failure of SFT to
> capture certain non-perturbative degrees of freedom. Is it conceivable that
> one can somehow "augment" SFT in a nice way to include these?
This is a very interesting question, I think. Let me say a couple of
related opinions of mine plus facts.
SFT used to be a very natural candidate for a full formulation of string
theory. It is the closest thing to "field theory" that one can have - a
field theory with infinitely many fields, if you decompose the string
field into component fields. From this point of view, it looks "background
independent" to many critics - and it is "off-shell", which means that it
sort of *has* local Green's functions, not just the S-matrix, which is
something that has extensively been used in the study of tachyon
condensation.
This ambitious program has only been partially successful. First of all, a
non-perturbatively complete theory should be defined with an exact form of
the action etc., not just a perturbative approximation of it. However,
closed string field theory requires to add correcting terms to the action
at each order (as well as the BV machinery). The action is therefore an
expansion itself, and it can have various non-perturbative completions.
Not a good starting point for a non-perturbatively complete theory.
All these reasons led the people to focus on the open string field theory
whose action can be well-defined - e.g. the cubic (polynomial) Witten's
action; it is enough to get the full amplitudes and cover the full Riemann
surface moduli spaces. Can one see all physics of string theory in it?
Well, the first problem are the closed string states. They can be seen as
poles in open string scattering, but as far as I know, no one has made a
convincing construction of the closed strings as composites of the open
string fields so far. The understanding of the closed strings would have
to improve a lot so that one could also construct non-trivial geometric
configurations including NS21-branes (or NS5-branes) etc. in open SFT.
Another question are the D-branes. Using the modern perspective, the open
strings themselves describe dynamics of a spacetime filling D-brane. Sen's
insights made it expected that one can construct the lower dimensional
branes as classical solutions of open string field theory.
String field theory has nevertheless been made less natural by the results
of the Duality revolution - its degrees of freedom are made of strings,
but at a generic point in the moduli space, there should be brane
democracy and strings are equally (non-)fundamental as other objects. If
string field theory becomes a good starting point for a full formulation,
one must ask several obvious questions.
Are the S-dualities and the strong coupling limit derivable from this
generalized SFT? For example, can one derive that the strong coupling
limit of a type IIA string theory is eleven-dimensional, and type IIB is
S-self-dual?
The answers must be yes if the generalized SFT is gonna become
non-perturbatively successful. Well, there are still two basic pictures
how this could happen:
1. One would still be using the same string fields, even at strong
coupling, and there are non-trivial functions or transformations
of these string fields that can be used to define the S-dual strings,
or the 11-dimensional physics, and so on.
I think that this viewpoint has become a bit obsolete after the
strong coupling revolutions of the 1990s. At strong coupling, the
original degrees of freedom are strongly coupled, physics becomes
obscure if we use them. They are not too useful, and moreover we
have learned that there can be better degrees of freedom that
become weakly coupled. They are typically infinitely heavy in the
weakly coupled limit, and therefore they are absent.
In field theory it is legitimate to imagine - for example in QCD
- that the fundamental UV fields are the gluons and quarks, and
the IR physics is whatever is implied. The gluons are superuseful
in the UV - because of the asymptotic freedom - but their physics
can be extrapolated to low energies. But we know that there is an
asymmetry - the IR can be derived from the UV, but not quite the
other way around. Therefore the analogy with strings, that are good
variables at the weak coupling, is not quite perfect - because
the strong and weak coupling may be totally equivalent.
The lessons of the 1990s seem to indicate that we should not try to
push the validity of some degrees of freedom to too strong couplings.
2. Of course, the second choice is that at generic coupling, there could
be new generalized degrees of freedom, whose structure itself is
determined, together with the action or whatever replaces it, by some
self-consistent rules. These degrees of freedom, determined by the
deeper rules, would have to reduce to the usual perturbative strings
in each weakly-coupled stringy limit.
While this second option is highly unusual, I believe that it is
plausible and attractive. It is unusual because we have not constructed
a single theory whose degrees of freedom are themselves determined by
deeper rules, dynamically. We always start with some well-defined
degrees of freedom, with a well-defined action or something equivalent.
Such theories can have many interesting regimes and behaviors, but
they cannot be quite universal.
In the perturbative limit we kind of know what are the rules that
determine the allowed degrees of freedom and the action: the rules are
the usual axioms of conformal field theory. The conformal symmetry
constrains both the worldsheet field content as well as the action.
But is there a non-perturbative generalization of this nice structure?
What happens with the worldsheet as you increase the coupling? Well,
it transmutes into a M-volume, which is the worldvolume of M, which
is the non-perturbative generalization of a string. ;-) The worldsheet
becomes a bit fuzzy, non-local, its dimension may effectively grow
(strings become membranes, but don't imagine quite local membranes). I
think that its internal dynamics is itself target space dynamics of
some other string theory; I have the N=2 and N=(2,1) string in mind.
We know that this complicated structure of the worldsheet theory *does*
occur in some context: the worldsheet of a D-string at weak coupling,
in which the D-string is superheavy, is described by open string theory
- all open strings attached to the D-string with the whole Hagedorn
tower of excitations are relevant. Nevertheless this D-string can be
continued to something we call the fundamental string.
There should be some more general
description of the allowed worldvolume theories of objects, including
non-geometrical ones - and the rules would non-perturbatively
generalize conformal field theory.
I've spent some time with thinking about the form of such a possible
generalization. Try to think about a more general theory that has a
BRST operator and the state-operator correspondence, but you relax
the assumption that it is a local two-dimensional theory. It can be
a theory of any dimension, with fuzzy dynamics, matrices, whatever
you want. Just try to require that something as strong as the
requirement of conformal symmetry applies, and the conformal symmetry
itself appears as a limit of this requirement for the special case
of weakly coupled backgrounds...
... One more comment. There have been some Japanese papers that studied
the behavior of the boundary states under the closed-string
Kyoto-group-like SFT star product; the boundary states act as projectors,
roughly speaking. This sort of thinking, even though it is formal, looks
like an important step towards obtaining the non-perturbative
generalization of CFT mentioned above. Today, our consistency requirements
for closed strings and open strings follow similar logic, but technically
they are different.
The allowed spectrum of D-branes must follow Cardy's rules, and so forth.
What I would like to see is to derive Cardy's rules as something like the
(generalized) closed string (M) equations of motion applied on the closed
string field whose vev happens to be the (total) boundary state. Adding a
D-brane is a deformation of the background, and it does correspond to a
change of the two-dimensional CFT. Well, the change is that we allow some
new boundaries. Formally, it is analogous to adding the vertex operator
for the boundary state into the 2D action although I realize that there
are technical difficulties in making this procedure well-defined (but
definitely, this is how the D-brane is seen from far away, as a
deformation of the closed string background; in this case, we can
restrict the boundary state to its lowest components).
Now imagine that the coupling becomes larger. Adding a D1-brane in type
IIB becomes equivalent to adding a light string if the coupling is really
large - by S-duality. But adding a fundamental string is a local change of
the 2D action. Recall that adding the D-brane was a non-local change: we
allowed the worldsheets to have boundaries. The goal is to describe a
structure that interpolates between this local modification of the 2D
action (adding a fundamental string) and a non-local modification
(allowing D-branes). The worldsheet itself should become fuzzy; the
distinction between local and non-local must go away at the generic
coupling. But what is exactly the theory at the generic point, and how do
you constrain it?
This is a sort of bootstrap thinking, but maybe not so impossible - it may
be just a generalization of CFTs.
All the best
Ilja Schmelzer
> If you add new fields, you can make virtually any theory be invariant
> under virtually any group. The fields that you must add to Newton's theory
> to make it generally covariant are non-dynamical and they can easily be
> erased, together with these symmetries, and therefore Newton's theory did
> not morally satisfy Einstein's requirement of a diff. invariant theory.
I would be interested to understand how distinguishing "non-dynamical"
from "dynamical" fields is supposed to work.
To describe a flat metric eta_mn, I can write down equations like R_ijkl=0
which look quite dynamical. I don't see how identify R_ijkl=0 as
non-dynamical but R_ij=0 for GR in the vacuum as dynamical.
[Moderator's note: \eta_{mn} constrained to satisfy R_{ijkl}=0 is a
textbook example of non-dynamical degrees of freedom because once you
fix the general covariance in the well-known way, the constraint forces
you to pick \eta_{mn}=diag(-1,1,1,1), for example. Henceforth, there
are no wave solutions of these constraints. Once again, the only allowed
solution is - up to symmetry transformations - static, and "static" means
"completely non-dynamical". In fact, it is completely constant here,
not just static.
In the quantized version
of the theory, there are no physical particle-like excitations coming
from your constrained version of \eta_{mn}, which is why it is called
an auxilliary, non-dynamical degree of freedom. The situation is
completely analogous to a gauge field A_\mu whose fields strength is
strongly required to vanish - F_{\mu\nu}=0. In this case, one can also
fix the gauge so that A_\mu=0 to see that this gauge field has no
physical polarizations - it would become non-dynamical. (If you set
the coupling constant in usual gauge theories strictly to zero, you get
a decoupled gauge field - although the equations are not quite F=0 -
which is still equivalent to its becoming unphysical.)
On the other hand, the Yang-Mills equations or the Einstein equations
R_{ij}=0 (or G_{ij}=T_{ij}) are dynamical because they lead to
propagating wave-like solutions for the gauge bosons and/or gravitons.
Although it may be unclear to you a priori, a simple procedure of solving
the constraints shows that R_{ijkl}=0 is completely non-dynamical, while
R_{ij}=0 is about dynamical metric if the spacetime dimension is greater
than two. Is it clear now? Adding a metric tensor and requiring
the complete flatness R_{ijkl}=0 is adding *nothing* to the Euclidean
geometry - it is about allowing different coordinates which people
had done for centuries - while adding the metric tensor with
Einstein's equations is equivalent to discovery of GR, with all the
new dynamics and curvature of space(time) that guarantees the
gravitational force. If you think that these are equally dynamical
and physical situations, you must be doing something wrong. ;-) LM]
I also don't see what means "easily erased". Of course we can simplify the
theory using a non-covariant formulation. But this is true in GR too,
the Einstein equations are much simpler in harmonic coordinates.
Ilja
[Moderator's note: The procedure above showed that in the theory
R_{ijkl}=0 you can always set the metric to the usual flat metric - all
other solutions are gauge transformations of this one. Once you know
that the constraint is physically (up to symmetries) equivalent to requiring
\eta_{mn}=diag(-1,1,1,1), there is nothing that can be varied in
in \eta_{mn} anymore, and therefore its degrees of freedom have been
erased. Of course, the whole point of GR is that such an elimination
is impossible in GR - there are solutions of R_{ij}=0, for example
the gravitational waves, that are *not* equivalent to the flat space.
The number of degrees of freedom in the metric is larger than the amount
of symmetry, and therefore some propagating degrees of freedom survive.
The same holds for gauge theories in d>2 dimensions.
Incidentally, your flawed thinking, combined with Mach's principle,
was one of the reason why the relativists could not understand that
there exist gravitational waves for such a long time - they spent
literally years trying to prove that these solutions are pure gauge.
Well, they are not! Every excitation of the R_{ijkl}=0 constraint -
which by the way does not follow from a nice action, except for
an action with a Lagrange multiplier - is easily seen to be pure gauge. LM]
Robert C. Helling
> Agreed. For this reason I said before that I don't understand what Robert
> means when saying "a manifold is diff invariant". Either it is meant in "my"
> sense, that the abstract geometry does not care about the coordinates (in
> which case it is pretty tautologous) or it is meant in "your" sense saying
> that the metric tensor is invariant under all diffeos, in which case it is
> not correct.
I never claimed to say anything deep. Of course I ment the first
version of the statement. You change coordinates and then have to
dress tensors with the appropriate powers of the Jacobian. Lubos calls
this undoing the diffeomorphism. The difference between GR and say
Newtonian physics is that in GR only dynamical fields get these powers
of Jacobians whereas in theories with background, there are tensors
that get these Jacobians that are not dynamical (eta or dt in the
example).
d70yxj
> ...we can study the quantum regime by studying the CFT on the world
> sheet. The spectrum of
> this CFT includes both perturbative and non-perturbative states and
> encodes the (quantum) geometry of target space. Surely if you
> formulate the CFT on the world-sheet general enough it will have
> excitations in its spectrum that correspond to target spaces of
> different topolgy in the low-energy effective description.
Hi Rufus
So I think this is what I had thought. I presume you mean above that if
the CFT on the worldsheet is defined appropriately, then in addition to
graviton excitations corresponding to perturbative changes in the
geometry of the target space, there will also be some CFT states
corresponding to changes in the homology groups of the target space.
(For example, the massless D3 branes that Lubos discussed above, I
guess).
[Moderator's note: Well, if there are massless D3-branes, then the
CFT breaks down. The CFT can perturbatively treat the perturbative
string states only, and they are too "light" and unable to change
the topology too much. Perturbative string theory is only OK if the
states that it neglects - such as D-branes - are heavy or otherwise
decoupled. It's not just a matter of calculational
complexity: it is also difficult "physically" to change the topology
of space. LM]
Am still not sure what this has to do with equivalence classes of the
worldsheet metric, though, as you implied much earlier on in this
thread. Possibly I am missing something obvious. Do you mean that
choosing a different equivalence class of worldsheet metric is
equivalent (apologies for mixed usage) to choosing a target space with
different homology groups?
Robert, do you really not expect it to be possible for a classical
manifold to be a ground state of (quantum) gravity? Or were you just
saying that such a conclusion reflects sloppy definitions of
diffeomorphism invariance and gauge invariance?
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Urs Schreiber
news:Pine.LNX.4.31.04091...@feynman.harvard.edu...
> All these reasons led the people to focus on the open string field theory
> whose action can be well-defined - e.g. the cubic (polynomial) Witten's
> action; it is enough to get the full amplitudes and cover the full Riemann
> surface moduli spaces. Can one see all physics of string theory in it?
> Well, the first problem are the closed string states. They can be seen as
> poles in open string scattering, but as far as I know, no one has made a
> convincing construction of the closed strings as composites of the open
> string fields so far. The understanding of the closed strings would have
> to improve a lot so that one could also construct non-trivial geometric
> configurations including NS21-branes (or NS5-branes) etc. in open SFT.
What looks plausible and weird at the same time is that in the
"nonperturbative vacuum" of OSFT (as it is sometimes called), meaning the
point where the tachyon is sitting in its minimum and the D25 brane has
completely decayed, the BRST operator of the OSFT with that background is
pure ghost and has trivial cohomology. On the one hand side this is
plausible, because at this point the open strings must have disappeared, so
that it makes sense that no non-trivial physical states are left. On the
other hand *something* should be left, namely physical states of closed
strings. Where are they in this picture? I have once seen a paper arguing
how these might arise from that pure-ghost BRST operator, but I didn't
understand the construction and forget which paper that was, unfortunately.
> Another question are the D-branes. Using the modern perspective, the open
> strings themselves describe dynamics of a spacetime filling D-brane. Sen's
> insights made it expected that one can construct the lower dimensional
> branes as classical solutions of open string field theory.
And this has been checked in many examples, hasn't it?
> I think that its internal dynamics is itself target space dynamics of
> some other string theory; I have the N=2 and N=(2,1) string in mind.
That's one of the most intriguing things that I have ever heard of, which
was when you first told me about it at the SCT
http://golem.ph.utexas.edu/string/archives/000265.html#c000386.
It seems to imply that what one needs is a "field theory of target space
theories" (as opposed to an ordinary target space field theory ,-) of the
N=(2,1) string. It would automatically contain the ordinary string as well
as membrane degrees of freedom and the like.
Hm, so maybe the ordinary cubic vertex of OSFT must be replaced by some sort
of vertex of objects living in a delPezzo or something?
> ... One more comment. There have been some Japanese papers that studied
> the behavior of the boundary states under the closed-string
> Kyoto-group-like SFT star product; the boundary states act as projectors,
> roughly speaking. This sort of thinking, even though it is formal, looks
> like an important step towards obtaining the non-perturbative
> generalization of CFT mentioned above. Today, our consistency requirements
> for closed strings and open strings follow similar logic, but technically
> they are different.
So maybe one can make a change of variables from string degrees of freedom
to brane-degrees of freedom in the SFT formulaiton: Usually the string field
Phi is expanded in terms of worldhseet oscilllation with respect to the
ordinary worldsheet vacuum. On the other hand, as you point out, there are
string fields Phi, the boundary states, which are far from that worldsheet
vacuum and describe offshell states that encode various D-branes. Maybe
there are enough of these boundary states that one can expand any other
string field in terms of them? hat would replace the ordinary expansion in
terms of string oscialltions by something like an expansion in terms of
D-brane states. The question would be: Do the boundary states of SFT form a
"complete" set in an appropriate sense?
I think this might even be true. To me the fact that boundary states which
encode a space-filling brane with arbitrary gauge field excitations are just
a recombination of DDF states hints at precisely such a reformulation.
> This is a sort of bootstrap thinking, but maybe not so impossible - it may
> be just a generalization of CFTs.
BTW, your exposition has generated some reactions over here:
http://www.math.columbia.edu/~woit/blog/archives/000080.html .
d70yxj
> [Moderator's note: Well, if there are massless D3-branes, then the
> CFT breaks down. The CFT can perturbatively treat the perturbative
> string states only, and they are too "light" and unable to change
> the topology too much. Perturbative string theory is only OK if the
> states that it neglects - such as D-branes - are heavy or otherwise
> decoupled. It's not just a matter of calculational
> complexity: it is also difficult "physically" to change the topology
> of space. LM]
I'm not sure if you're disagreeing with Rufus as I've quoted him in the
post above, or just with my interpretation of what he was saying....
Also, I thought you said the D3-branes in the target space can be
described by adding a boundary to the world sheet CFT. Is this what you
mean by break down?
[Moderator's note: No. Infinitely extended D3-branes or D3-branes wrapped on
finite volumes are, indeed, represented by adding all possible
worldsheets that can also have the boundaries with the boundary
conditions associated with these D3-branes. The conformal field theory
is "generalized", according to Polchinski's recipe, and it accounts
for the physics of the original background plus the D3-branes.
However, I was talking about *massless* D3-branes. If you consider
a topology change similar to the conifold transition, you will find out
that there are 3-dimensional cycles at the conifold point - the singular
point in the moduli space where the manifold pinches off. D3-branes can
be wrapped on these "vanishing cycles" (3-dimensional submanifolds of
vanishing volume), and because their total mass is proportional to the
volume and the volume goes to zero, these D3-branes are massless.
That's a disaster for the conformal field theory. The "well-behaved",
finite-mass D3-branes can be represented by boundaries of the
worldsheets, and the worldsheets with too many boundaries become
increasingly irrelevant. However, if the D3-branes are massless,
the worldsheets with very many boundaries cannot be neglected - in
fact, they are at least as important as the worldsheet with a few
boundaries. Consequently, the sum over Riemann surfaces with boundaries
does not converge at all, and you can't get any finite results out of
it.
By "breaking down" in physics, we always mean that the theory
does not work at all anymore, and your calculations lead to nonsensical
(divergent) results. This is what happens with a CFT if some D-branes
become massless. More generally, this breakdown occurs for any
type of calculation that only considers a subset of particles and states,
as soon as you try to describe a situation in which some other
(neglected) particles become important enough (light and sufficiently
strongly coupled). New massless particles that were not accounted for
always mean a disaster for your original description.
You may object by saying that the D3-branes *were* accounted for because
we *wanted* to add the boundaries. That might have been true, but they
were not treated as *perturbative* objects but rather as "solitons"
(a generalization of the magnetic monopole; a classical solution
interpreted as a very heavy object). This solitonic description is not
good enough if they are light; if the D3-branes become light, i.e. if
the volume of the 3-cycle shrinks to zero, the D3-branes should be
considered on equal footing with the fundamental strings, which is
certainly not what the conformal field theory is doing. However, this
democratic treatment *can* be achieved if we use a spacetime effective
field theory - with light perturbative string states *and* the new fields
arising from the light/massless D3-branes. This strategy was chosen
by Andy Strominger, and he was the first one who understood that the
existence of D3-branes exactly accounts for all singularities seen in
the field theoretical description. He found out that the full string
theory, including all the predicted D3-branes etc., is smooth and
non-singular. A week later, he and Brian Greene and David Morrison
extended this fact and showed that topology can be changed at this point.
I wrote what I wrote simply because I had a feeling that neither you nor
Rufus understand that the pure conformal field theory is meaningless
at the point where the D3-branes become massless, i.e. at the very
moment when the topology change occurs. LM]
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