Post by richard miller
Sounds a nice question, we (one) have ascribed continuous laws over a length
of the Planck scale. do we have the justifiication for this continuity or
are all the action integrals etc. not validate nowadays (things have moved
on since the 80s?), at least as continuous functions? I don't know either.
Look forward to the answer.
This is actually a deep question, I believe. It has been asked a couple of
times before on this group, if I recall correctly, in one way or another.
I am nor sure if Eric Zaslow is still collecting FAQs, maybe this should
be included in our list (Is anyone compiling attempts at giving answers
to the FAQs?):
FAQ: "How can it be that the string is a mathematical line?"
I believe one should say at least three things as comments on that
1) Elementary particles are mathemtaical points. Is that less mysterious
than being a mathematical line?
Of course one may suspect that elementary particles are not fundamentally elementary
precisley because they are mathematical points. This leads me to point 2)
1) Spacetime is emergent. What we really have in perturbative string
theory is just any superconformal field theory of central charge c=15 on abstract
2-dimensional Riemannian surfaces.
In _some_ cases this superconformal field theory can be interpreted as
describing the dynamics of "embedding fields" which describe how this
Riemannian surface sits inside a manifold which we interpret as spacetime
(the "background spacetime").
(And, BTW, it need not be an embeeding at all, there are in general lots
In other cases it may not be possible to have such a geometric
interpretation of your CFT. CFTs without such a geometric interpretation
describe "spacetimes" which are not manifolds in the classical sense.
Sometimes these are referred to as being a "non-geometric phase" of
spacetime, or something like that.
So in general it is not even true that a string is a line and that it
sweeps ot a worldsheet in spacetime!
In "most" cases however, it is.
(Hm, do we know how much is "most"?)
3) Perturbative string theory is not the last word, so much is for sure.
M-theory is the last word, by definition. (Imagine an appropriate simley
here...) Do strings still look like mathematical lines in M-theory?
Of course they become membranes, but that doesn't help us with our
question. There is the Matrix Theory description of everything, where all
things become kind of fuzzy.
A good discussion of this point requires more time than I can currently
afford, I am afraid. Lubos has to say much more about this. Maybe if you
kindly ask him he'll write a more complete FAQ entry to the above FAQ. :-)