[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: Explanation of M-theory
In article <3A9F950C@MailAndNews.com>, Squark <squark@MailAndNews.com>
wrote:
>Can anyone explain me what's the story of M-theory? I heard various things
>about it, but I haven't managed to understand what stands behind it
>formally
>(at least semi-formally). Thx in advance!
Unfortunately, the term "M-theory" is commonly used in two related, but
different senses. In both cases, phrase is used as a placeholder for
something that has not really been formulated yet.
1) M-theory is the 11-dimensional theory whose low-energy limit is
11-dimensional supergravity. In the same way, (for instance), Type IIB
string theory is the 10-dimensional theory whose low-energy limit is
10-dimensional type IIB supergravity.
And just as string theory contains various extended objects (fundamental
strings, D-branes, . . .) M-theory has a 2-brane (world-volume is 2+1
dimensional) and a 5-brane (world-volume is 5+1 dimensional).
Now, (the various) string theories have the advantage that we understand
not just the low-energy limit, but actually have a well-defined
perturbation expansion, which makes sense at arbitrary energies. As has
been discussed in another thread here, perturbation theory is not the
whole story [In string theory, the g'th order term grow like (2g)!,
rather than the g! we are familiar with in field theory. This means that
nonperturbative effects in string theory are *stronger* than in field
theory.] but, whatever its limitations, the terms in the perturbation
series themselves are under good control.
M-theory, by contrast, is under much poorer control; we have very little
idea of how to compute the quantum corrections. The best candidate for a
formulation of M-theory is called "Matrix Theory". For various reasons,
it is not a very satisfactory formulation. But it is, heretofore, the
best we've got. Foremost among its drawbacks is that it is far less
*computable* than string perturbation theory and so we have learned far
less about quantum effects in M-Theory than we have in string theory.
2) M-theory is the mysterious quantum theory that subsumes all of the
known string theories as well as the aforementioned 11-dimensional
theory. [Personally, I prefer the name "nonperturbative string theory"
for this mysterious Ur-theory.] This mysterious quantum theory has
various *different* semiclassical limits, in which it has the
interpretation as one of the various string theories or as the
11-dimensional theory. Away from these semiclassical limits, we don't
really understand it at all. But it is clearly the "holy grail" of this
field.
Now, some people (and this is where the confusion starts) conflate these
two definitions. They believe that the 11-dimensional semiclassical
limit is somehow "more fundamental" than the various 10-dimensional
semiclassical limits. That goes against the grain of everything we've
learned about string dualities in the past 6 years. All of these
limiting theories (the various string theories and 11-dimensional
M-theory) are on an equal footing. None is more fundamental than the
others.
In some limit, one may be a better description than the others. In the
most interesting cases, two (or more) descriptions are valid and they
tell you complementary things about the physics.
<Pine.LNX.4.10.10102201214180.9342-100000@offshell.phys.ndsu.nodak.edu>,
Terry Pilling <terry@offshell.phys.ndsu.nodak.edu> wrote:
>On Mon, 19 Feb 2001, Squark wrote:
>I know that it is supposed to be the `meta' theory in 11 dimensions
>which reduces to (type IIA ?) string theory (in 10 dimensions!) at strong
>coupling and to 11-dimensional supergravity at weak coupling.
>So they are saying that at strong coupling, M-theory `loses a dimension'!
The other way around. At strong coupling, Type IIA string theory "gains"
a dimension.
M-theory does not have a dimensionless coupling constant. Compactified
on a circle, there is a dimensionless parameter (the radius of the
circle in 11-dimensional Planck units). When that parameter is small,
the theory is supposed to be identical to (perturbative) type IIA string
theory.
The fundamental type IIA string is the M2-brane wrapped on the circle.
When the radius of the circle is small, the tension of that string is
much less than the (square of) the 10-dimensional Planck mass, which is
the situation of weakly-coupled string theory. "Miraculously", a nice
perturbative description is supposed to arise, given by quantizing the
fluctuations of this string.
>It also seems that it is a theory of `branes' since whenever I see people
>talking about M-theory they are usually talking about strings on branes
>and then they seem to just slap a little circular dimension on everything
>and low and behold, M-theory!
Again, since quantum M-theory is not under good control, that
description is useful only when the resulting M-theory description is
semiclassical -- ie, the volume is large and the Riemann curvature is
small.
>One thing that confuses me is that `way back when' everybody was saying
>"Oh - the universe has to be 10 dimensional since that is the only way we
>can get anomaly cancellation in superstring theory" and it was a really
>big deal. Now they say that it is indeed possible to slap on another
>dimension? What is going on? How does this work? Can we trust the other
>claims made in the 80's about string theory?
10 dimensions is, indeed, the critical dimension for string theory. The
11-dimensional theory is not a string theory.
And you don't "slap on" an extra dimension, you manufacture it
quantum-mechanically.
To show you how this goes, let's manufacture M-theory a different way.
In type IIB string theory, there is a fundamental string with tension
T_F and a D1-brane (D-string) with tension T_D. Let us work in a limit
where we hold T_F and T_D fixed and finite. This is not quite
perturbative (in the perturbative limit, we hold T_F fixed and take T_D
to be much, much larger), but let's press on.
Now let us compactify on a circle of radius R. A fundamental string,
wrapped on the circle is a particle in the 9-dimensional effective
theory, of mass
m = 2 pi R T_F
and the D-string wrapped on the circle is a particle of mass
m = 2 pi R T_D
Moreover, there are particles corresponding to bound states of n_1
fundamental and n_2 D-strings and these have a mass
m^2 = (2 pi R T_F n_1)^2 + (2 pi R T_D n_2)^2
[I should say at this point, that even though the string theory is not
quite perturbative, all of these masses are protected by a BPS mass
formula and don't receive any corrections.]
Recognize this spectrum? It's the spectrum of a particle in a
rectangular box (with periodic boundary conditions) of sides
L_1= 1/(T_F R)
L_2= 1/(T_D R)
That is to say, we have the Kaluza-Klein spectrum of an 11-dimensional
theory compactified on a 2-torus.
In the limit R->0, the area of that 2-torus is large and we recover an
approximate 11-dimensional Lorentz-invariance. [I haven't really been to
careful here; these particles come in supermultiplets, and what we have
actually constructed is Kaluza-Klein spectrum of the full 11-dimensional
supergravity multiplet.]
Note that we "constructed" spacetime dimensions (two of them) out of the
bound state spectrum of strings and D-strings.
That's another theme: spacetime (even its dimension) is an emergent
property of this mysterious theory (Definition 2 of "M-theory"). Away
from any semiclassical limit, even the notion of spacetime (let alone
its dimension) is ambiguous.
In article <97c1du$lb9$1@glue.ucr.edu>, baez@galaxy.ucr.edu (John Baez)
wrote:
>I think it's important for the nonexperts to be informed that M-theory is
>not a "theory" in the traditional sense. . . . But M-theory is
>not like this: there is no "fundamental equation of M-theory" or "Lagrangian
>for M-theory".
In that sense, "String Theory" isn't a "theory" either. However, it is
"closer" to being a theory than M-Theory is at present.
It's mildly annoying that string theorists have taken to calling
structures that have not really been formulated yet (but which must
exist if string theory as a whole makes sense) "theories".
Perhaps we need a new name for such things. But, aside from being less
of a trap for the unwary, would a new name really make a difference?
--
PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc