Re: String Theory, Gravitons, Spacetime (especially for Lubos Motl)

Lubos (if I may)

First of all, thank you very much for taking the time to write your
long responses, they are very much appreciated! I have some basic
knowledge of QFT but I still find it difficult to grasp some of the
concepts of string theory. Posts like yours are printed and read
carefully!

I'd like to ask a few follow-up questions (I'm sure they will reveal
how dumb I am but I am past the stage where I was avoiding asking
questions in fear of revealing my ignorance :-) First, let me quote a
few of your remarks and then I'll ask my questions.

You wrote 

> The exchange of
> virtual gravitons is a specific way how the processes are calculated
> perturbatively; it does not change the classical field limit of the theory
> which is nothing else than general relativity.
> 


and then 


> If the curvature is
> weak, its particle-like nature must be taken into account and the process
> really looks like the exchange of gravitons. Quantum mechanics usually
> affects physics in similar ways. For example, a weak enough
> electromagnetic wave looks like a stream of discrete photons 

and

> In  quantum theory, spacetime curvature can be reinterpreted as a condensate
> of gravitons of some sort, and the gravitational force resulting from this
> curvature can be calculated via the exchange of gravitons, but it is still
> the same thing.


First, let me make sure I understand correctly.

There are two limits involved here,  the "weak limit" in which case
the graviton picture is required, and the "classical limit". The
classical picture corresponds to a "condensate" of gravitons (right?).

Now my questions:

Question #1: what does one mean exactly by "condensate of gravitons".
At first, it sounds like simply meaning "a bunch of gravitons" but
there must be more to it (see another question below).  And what is
the physical meaning of the "classical limit"? By that I mean what is
the dynamical process that would explain why two macroscopic object
(let's say a comet and the Sun, to use your example) would necessarily
be plunged into such a condensate of gravitons? Presumably one could
start perturbatively by considering the exchange of one, two, three,
etc gravitons and, lo and behold, we *must* end up with the comet and
the Sun moving through a "condensate", because we must recover the
spacetime curvature picture. What forces this to happen??

   

( I have often read about this business of "condensate". I know that
when one writes the path integral as the exponential of the actions
times an infinite sum of terms containing products of graviton vertex
factors, one can exponentiate the sum and see that the infinite
expansion in the graviton vertex operator amounts to a small
perturbation of the flat spacetime metric (if that's what's used as
the background))

Question #2:  Going back to the exponentiation trick, one really needs
some kind of "conspiracy" in the coefficients of the higher order
terms with more and more graviton vertex operators to really get an
exponential. So do people really mean that the coefficients must be
chosen to exactly get the expansion of an exponential? (which is maybe
what defines a "condensate" the PI approach?!?)


Or do people really mean that 

exp ( - Action in flat spacetime ) ( 1 + one graviton vertex operator
+ ...)

is  *approximately* equal to 


exp (- Action in spacetime with small curvature),

the equality being true modulo terms with two or more graviton vertex
operators? (AT first, it might seem like it's obvious that people mean
this (that the equality holds only to first order) but then there
would not be any need to talk about "condensates" and one could even
say that the exchange of a single graviton can be interpreted as
slightly changing the metric! I'll come back to this in my follow up
question.)


  I have a follow up question, depending on the answer to the previous
question.


Question #3: If the answer is that the coefficients must indeed be the
exact coefficients needed to produce exactly and exponential to all
orders, the obvious question is why? What is the dynamical process
that ensures this?


(An aside: If I recall correctly (you are more than welcome to correct
me!), insuring the vanishing of the Weyl anomaly forces the background
metric to obey Einstein's equation, which is something I find really
neat. But here we the graviton exchanges are creating a correction to
the background metric so I am not sure if this is relevant in my
discussion. Maybe imposing the vanishing of the Weyl anomaly
constrains the interactions with the gravitons and that answer of my
question??)
   
Question #4: Well, I've already asked it above but let me repeat it
here. If one only means that adding the exchange of one graviton is
*approximately* equal to modifying the metric through the exponential
trick (modulo higher order terms in number of gravitons exchanged),
then another obvious question arises. If that's the case, even the
exchange of a single graviton could be seen as equivalent to changing
a bit the metric (mdulo higher order terms which are presumably
neglected in this approach). But then why the need to talk about a
condensate?  That would not be required at all and this is why I am
guessing this is not the correct way to understand the exponentiation
trick.





That's it for now. I hope you (or someone else) will find the time to
help me understand.

  

     Best regards,
                     Patrick