**Next message:**Eliezer S. Yudkowsky: "Re: Tragedy and Boredom was Re: Why just simulation? (fwd)"**Previous message:**Robert J. Bradbury: "Re: How To Live In A Simulation"**In reply to:**Robert J. Bradbury: "Re: How To Live In A Simulation"**Next in thread:**CurtAdams@aol.com: "Re: How To Live In A Simulation"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

"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.

**Next message:**Eliezer S. Yudkowsky: "Re: Tragedy and Boredom was Re: Why just simulation? (fwd)"**Previous message:**Robert J. Bradbury: "Re: How To Live In A Simulation"**In reply to:**Robert J. Bradbury: "Re: How To Live In A Simulation"**Next in thread:**CurtAdams@aol.com: "Re: How To Live In A Simulation"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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