"Robert J. Bradbury" wrote:
>
> ...is quantum mechanics the bottom or string theory
> the bottom?
I'm no physicist (I'm a paid professional actor :->),
but my understanding is that string theory **is**
quantum mechanics, or rather, an attempt to construct
a mathematically consistent blend of quantum theory
and general relativity (i.e., quantum gravity). The
"strings", AFAIK, are substitutions of entities which
are extended, geometrically, for the points used
to formulate traditional quantum theory, in order to avoid
mathematical, uh, singularities [blush] that would
otherwise result. Strings have been more recently
generalized into "branes" (M-branes, not to be confused
with M-brains [;->]).
My friend Joe Fineman, who does have graduate training
in physics, has remarked that the fatal flaw in all
physics books written for a popular audience is that,
not being able to take advantage of the math that is
the heart and soul of the subject, they all degenerate
into hand-waving. However, one book which I read fairly
recently (a couple of years ago -- I **do** wish I had permanent
recall of such things, as is dreamed about here :-< ) and
enjoyed very much indeed (and which I thought was extremely lucid
and well-written) is Brian Greene's _The Elegant
Universe_ ( http://www.amazon.com/exec/obidos/ASIN/0375708111 ).
I fished up some correspondence (with my friend Joe Fineman,
who from now on I'm simply going to refer to as "F", which is
what I call him, anyway) from two years ago about the book:
Did I tell you I'm reading a physics book I picked up a couple
of weeks ago -- _The Elegant Universe_ by one Brian Greene,
who is a professor of mathematics and physics at Columbia?
It's a layman's book (of course), but it seems quite serious
to me (less flaky than the Deutsch book I bought when I visited
you, but never finished) -- it's an up-to-the-minute account of
the state of superstring theory, which seems to be quite
respectable these days. I'll send it along to you when I'm
done with it, if you like.
----------------
> Has anybody managed to _calculate_ anything with it?
I gather it's still extremely hard to calculate physical quantities
(like particle masses, etc.) with enough precision to permit direct
experimental confirmation.
However, there are apparently some very exciting (to those who
know the math!) quantitative things going on. Let me see if I can
describe one of them (it'll be a test of whether I'm following the
book at all).
In Chapter 8, "More Dimensions Than Meets The Eye", the author
describes how impossible quantum-mechanical results (like negative
probabilities) can only be eliminated from supersymmetric string
theory if there are more degrees of freedom for strings to
vibrate in than the ordinary three spatial dimensions. These
extra dimensions (first proposed as a purely philosophical
speculation by Theodor Kaluza in 1919) only make sense if they
are curled up too tightly for ordinary experimental probes of
distance to detect them, and there needs to be at least six of
them to cancel all of the nonsensical negative probabilities.
These six extra curled-up dimensions could be "attached" to
each point of ordinary 3D space in a variety of topologies, but
it was proved in 1984 (by Candelas, Horowitz, Strominger and
Witten [heard of Edward Witten? the author says some people think
he's the modern successor of Einstein, and others think he may
be the greatest physicist ever] that the physical requirements
of string theory restrict the geometrical form of the extra
dimensions to one of a large (numbering in the tens of thousands)
but not infinite class of possibilities called Calabi-Yau spaces.
These are not described much more precisely in the main body of
the text (apart from a rather impressionistic drawing), but Note 8
for this chapter says "For the mathematically inclined reader,
we note that a Calabi-Yau manifold is a complex Kahler manifold
with vanishing first Chern class. In 1957 Calabi conjectured
that every such manifold admits a Ricci-flat metric, and in
1977 Yau proved this to be true."
In Chapter 10, "Quantum Geometry", the author states that in 1988,
Lerche, Vafa, and Warner conjectured ("based on aesthetic
arguments rooted in considerations of symmetry") that it might
be possible for two **different** Calabi-Yau shapes (chosen to
be the extra six curled-up dimensions of superstring theory) to
produce identical physics. It had previously been demonstrated
by Candelas, Horowitz, Strominger and Witten that the the number
of families of elementary particles (as in the three that are known
today, where family 1 has the electron, family 2 has the muon, and
family 3 has the tau) depends on the total number of holes (of
whatever dimension) in the underlying Calabi-Yau space.
Beginning in 1988, the author himself (together with Plesser) had begun
a project to explore a process of transforming one Calabi-Yau space into
another via a mathematical technique called orbifolding (invented in the
mid-80's by Dixon, Harvey, Vafa, and Witten), which interchanges
the number of odd- and even-dimensional holes in a Calabi-Yau
space but leaves the total number of holes (and hence the predicted
number of families of fundamental particles) unchanged. I don't
know what it means for a hole to be odd- or even- dimensional, but
once again, Chapter 10 has a relevant Note 5: "For the mathematically
inclined reader, we note that, more precisely, the number of families
of string vibrations is one-half the absolute value of the Euler
characteristic of the Calabi-Yau space, as mentioned in note 16 of
Chapter 9. This is given by the absolute value of the difference
between h-super-2,1 and h-super-1,1, where h-super-p,q denotes the
(p,q) Hodge number. Up to a numerical shift, these count the number
of nontrivial homology three-cycles ('three-dimensional holes') and
the number of homology two-cycles ('two-dimensional holes'). And
so, whereas we speak of the total number of holes in the main text,
the more precise analysis shows that the number of families depends
on the absolute value of the difference between the odd- and
even-dimensional holes. The conclusion, however, is the same.
For instance, if two Calabi-Yau spaces differ by interchange of
their respective h-super-2,1 and h-super-1,1 Hodge numbers, the
number of particle families -- and the total number of 'holes' --
will not change."
Later, the author (with Plesser) was able to demonstrate the stronger
result that these pairs of Calabi-Yau spaces with the numbers of
odd- and even-dimensional holes interchanged (which the author
terms "mirror manifolds") not only give rise to the same number
of families of elementary particles, but will give rise to
**all** the same physical laws when used for the extra dimensions
of string theory (a rather strong claim, and not justified in much
more detail in the text. The author does note that Yau described
his results as "far too outlandish to be true"). Almost at the same
time, Candelas, Lynker and Schimmrigk, examining a large sample of
computer-generated Calabi-Yau spaces, found that "almost" all of
them came in pairs that differed only by the interchange of the numbers
of even- and odd-dimensional holes.
HERE'S THE QUANTITATIVE STUFF:
You remember how with differential equations, the Laplace transform
can be used to change a relatively difficult differential equation
into an equivalent algebraic equation, which can be easily solved,
following which the inverse Laplace transform can be used to turn
this solution into the solution of the original differential equation
(if I'm remembering this correctly, and not pipe-dreaming)?
Well, it turns out that these pairs of mirror-manifolds can be
used as transforms to tame some of the difficult calculations involved
with superstring theory, because it sometimes turns out that a
calculation that is difficult or impossible to perform using one
Calabi-Yau space becomes much more tractable when the Calabi-Yau
space is transformed into its mirror-manifold partner. This was
demonstrated by the author and Plesser, and by Candelas, de la Ossa,
Parkes and Green. At a 1991 physics and mathematics conference in
Berkeley, it was revealed that Norwegian mathematicians Ellingsrud
and Stromme had produced a result (the hard way) from an elaborate
and difficult calculation to determine the number of spheres
that can be packed into a particular Calabi-Yau space. The same
computation was performed by Candelas and his group via a much
simpler calculation based on the mirror-manifold method.
At first, the results did not agree, but a month later the Norwegians
discovered and corrected a bug in their computer program, after which
the results agreed perfectly (the numerical result was
317,206,375). Similar mathematical checks have been
performed many times since then, all with perfect agreement.
Also since then, mathematicians (Yau, Lian and Liu; with contributions
from Kontsevich, Manin, Tian, Li, and Givental) have elucidated
more of the mathematical foundatios of the symmetry of Calabi-Yau
spaces, and have produced a rigorous mathematical proof of the formulas
used by Candelas and the Norwegians used to count spheres inside
Calabi-Yau spaces.
This quantitative work has more to do with pure math than physics,
although it used a new technique whose discovery was motivated by
physical considerations (the author says "For quite some time,
physicists have 'mined' mathematical archives in search of tools
for constructing and analyzing models of the physical world. Now,
through the discovery of string theory, physics is beginning to
repay the debt and to provide mathematicians with powerful new
approaches to their unsolved problems.").
Cheers.
Jim
----------------
> In Chapter 8, "More Dimensions Than Meets The Eye", the author
> describes how impossible quantum-mechanical results (like negative
> probabilities) can only be eliminated from supersymmetric string
> theory if there are more degrees of freedom for strings to vibrate
> in than the ordinary three spatial dimensions.
That part I've heard about. It will be interesting to see more
details.
> "For the mathematically inclined reader, we note that a Calabi-Yau
> manifold is a complex Kahler manifold with vanishing first Chern
> class.
This. I know. From nothing. What? I am going? To do? I think of
great Lobachevsky and...
> the number of families of string vibrations is one-half the absolute
> value of the Euler characteristic of the Calabi-Yau space,
V + F - E? Wow!
> You remember how with differential equations, the Laplace transform
> can be used to change a relatively difficult differential equation
> into an equivalent algebraic equation, which can be easily solved,
> following which the inverse Laplace transform can be used to turn
> this solution into the solution of the original differential
> equation (if I'm remembering this correctly, and not pipe-dreaming)?
Yes. There was a EE professor at Caltech (R. V. Langmuir) who would
draw enormous circuit diagrams on the blackboard and envelop them with
pipe smoke & Laplace transforms. Being one-sided in time, they were
mainly good, IIRC, for calculating responses to transient
disturbances.
----------------
There is a long article in today's New York Times (in the Science
section; the business section is dominated by the Federal judge's
findings of law in the Microsoft anti-trust case) about a recent
development in superstring theory. You can access this article
yourself on-line (it's free all day today) at www.nytimes.com. The
physicists in question are from Princeton (a woman, and pretty, too --
shades of Greg Egan) and Stanford. Let's see if I can attempt a
layman's summary (condensed from a journalist's version -- a blind man
painting a picture from a description given by another blind man)!
You remember from Brian Greene's _The Elegant Universe_ how
superstring theory entails additional dimensions beyond the standard
three, on a very small linear scale (Planck length)? Greene also
mentions that recent versions of superstring theory also contain
entities of higher dimension than strings, called "branes". Well,
apparently one story about our universe suggested by all this is that
the familiar universe is a 3-brane embedded in an extended (rather
than curled up) fourth dimension, with most of the familiar particles
of physics (including the force-carrying ones) composed of strings
confined to the surface of the brane, but with gravity (gravitons)
capable of escaping from the brane. Apparently, a drawback to this
picture (up until now) is that this would entail gravity being
stronger than it is observed to be. Well, apparently these two
physicists have come up with a geometry for the embedding space which
allows gravitons to move about in it while still constraining the
gravitational force to have its observed properties.
This has two interesting consequences:
1. The "missing mass" of the universe may be matter embedded in other
branes. This matter would be dark to us (because photons can't escape
from their native brane, only gravitons).
2. Gravitons which intersect our brane but which also extend into the
embedding space are heavier than "normal" gravitons confined to our
brane, and could be detected (indirectly) by the new generation of
accelerators (like the Large Hadron Collider in Geneva). This
possiblility gives the Times article its title -- "Physicists Finally
Discover a Way to Test the Superstring Theory".
----------------
Did you get all that? :->
Jim F.
This archive was generated by hypermail 2b30 : Mon May 28 2001 - 09:59:41 MDT