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

On Sat, 29 Nov 2003, David Hillman wrote:

> My confusion is this: there seem to be two ways in which strings can
> collide. One is that they collide in T in one of those smooth pants-type
> collisions that is one of the famous hallmarks of string theory. ...

Dear David, there is one type of interaction of the closed strings only.
There are many technicalities that make the explanation more complicated,
but let us try to resolve these questions in the path integral formalism
involving Euclidean worldsheets embedded in a Euclidean spacetime.

The path integral formulation of any quantum theory (due to Richard
Feynman) forces you to sum over all possible histories. All possible
histories happen, and all of them contribute to the probability amplitudes
of individual outcomes. In the case of (perturbative) closed string
theory, these histories involve all the closed two-dimensional worldsheets
embedded into spacetime.

In the regions of the spacetime where the closed strings *don't* interact,
the worldsheet looks like pieces of an infinite cylinder. Once you allow
the strings to interact, there will be also histories where the strings
join or split: two strings can become one (or the other way around) if
they collide at a single point.

Note that so far you could not have determined whether I described the
"random collision" or the "smooth pants". The topology of both these cases
is identical. OK, let us now ask the question whether the vicinity of the
collision - the appropriate region of the pants - is smooth or not. This
question is the only way to distinguish what you call "two" different
interactions.

The answer is, of course, that the behavior of X(sigma,tau) is very
unsmooth. The fields X(sigma,tau) are quantum objects living on the
worldsheet, and therefore they are subject to the quantum jittery
behavior. The better resolution (in sigma, tau) you use to determine the
values of X(sigma,tau), the more oscillating the fields X(sigma,tau) will
be. Imagine that it is a consequence of the uncertainty principle: if
X(sigma,tau) is almost exactly determined (and it is, if you imagine a
particular history), its "velocity" or sigma/tau-derivatives must be very
uncertain i.e. they must be oscillating over a large interval.

In fact, if you make the resolution perfect, the typical configuration -
vector-valued function X(sigma,tau) - will be so heavily fluctuating that
it won't have a derivative (almost) anywhere and it will resemble the
Brownian motion.

This behavior is not a characteristic property of string theory; it is
rather a general feature of quantum mechanics. The same behavior is found
for quantum mechanics of ordinary point-like particles as well as for
quantum field theories of any kind.

Nevertheless it is possible to sum up the contributions of these chaotic
quantum histories (or trajectories), and we will find out that the path
integral is "localized" near smooth curves. You might ask: how do the
two strings know that they will join into one string, and therefore they
should prepare to draw a nice and smooth pants diagram in spacetime?

The answer is that they don't need to get prepared, but it just happens -
because of mathematics of the problem - that the histories in which the
strings behave smoothly will dominate the path integral. More precisely,
quantum fluctuations of these smooth histories provide us with the most
important contribution to the path integral and to all physical
observables.

The explanation in the Minkowski spacetime would be different because the
pants embedded into the Minkowski spacetime can *never* be smooth. If you
go along the pants, up and down, you must change your arrow of time, and
it can never happen smoothly in the Minkowski spacetime.

> Yes, I do not know QFT so I am probably looking at it all wrong. (I'm
> imagining a quantum state in which each worldsheet is assigned an
> amplitude. Is that correct?)

Yes, it is essentially correct. But in all quantum theories, the amplitude
of a history is very important even if this history is wildly oscillating
and un-smooth. It is only a result of a mathematical trick that we can
calculate the contribution of such a class of frenetic histories as a
number associated with a smooth history somewhere in the middle.
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