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String Theory, at 20, Explains It All (or Not)
The New York Times ^ | December 7, 2004 | Dennis Overbye

Posted on 12/07/2004 10:01:55 AM PST by snarks_when_bored

The New York Times


December 7, 2004

String Theory, at 20, Explains It All (or Not)

By DENNIS OVERBYE

ASPEN, Colo. - They all laughed 20 years ago.

It was then that a physicist named John Schwarz jumped up on the stage during a cabaret at the physics center here and began babbling about having discovered a theory that could explain everything. By prearrangement men in white suits swooped in and carried away Dr. Schwarz, then a little-known researcher at the California Institute of Technology.

Only a few of the laughing audience members knew that Dr. Schwarz was not entirely joking. He and his collaborator, Dr. Michael Green, now at Cambridge University, had just finished a calculation that would change the way physics was done. They had shown that it was possible for the first time to write down a single equation that could explain all the laws of physics, all the forces of nature - the proverbial "theory of everything" that could be written on a T-shirt.

And so emerged into the limelight a strange new concept of nature, called string theory, so named because it depicts the basic constituents of the universe as tiny wriggling strings, not point particles.

"That was our first public announcement," Dr. Schwarz said recently.

By uniting all the forces, string theory had the potential of achieving the goal that Einstein sought without success for half his life and that has embodied the dreams of every physicist since then. If true, it could be used like a searchlight to illuminate some of the deepest mysteries physicists can imagine, like the origin of space and time in the Big Bang and the putative death of space and time at the infinitely dense centers of black holes.

In the last 20 years, string theory has become a major branch of physics. Physicists and mathematicians conversant in strings are courted and recruited like star quarterbacks by universities eager to establish their research credentials. String theory has been celebrated and explained in best-selling books like "The Elegant Universe," by Dr. Brian Greene, a physicist at Columbia University, and even on popular television shows.

Last summer in Aspen, Dr. Schwarz and Dr. Green (of Cambridge) cut a cake decorated with "20th Anniversary of the First Revolution Started in Aspen," as they and other theorists celebrated the anniversary of their big breakthrough. But even as they ate cake and drank wine, the string theorists admitted that after 20 years, they still did not know how to test string theory, or even what it meant.

As a result, the goal of explaining all the features of the modern world is as far away as ever, they say. And some physicists outside the string theory camp are growing restive. At another meeting, at the Aspen Institute for Humanities, only a few days before the string commemoration, Dr. Lawrence Krauss, a cosmologist at Case Western Reserve University in Cleveland, called string theory "a colossal failure."

String theorists agree that it has been a long, strange trip, but they still have faith that they will complete the journey.

"Twenty years ago no one would have correctly predicted how string theory has since developed," said Dr. Andrew Strominger of Harvard. "There is disappointment that despite all our efforts, experimental verification or disproof still seems far away. On the other hand, the depth and beauty of the subject, and the way it has reached out, influenced and connected other areas of physics and mathematics, is beyond the wildest imaginations of 20 years ago."

In a way, the story of string theory and of the physicists who have followed its siren song for two decades is like a novel that begins with the classic "what if?"

What if the basic constituents of nature and matter were not little points, as had been presumed since the time of the Greeks? What if the seeds of reality were rather teeny tiny wiggly little bits of string? And what appear to be different particles like electrons and quarks merely correspond to different ways for the strings to vibrate, different notes on God's guitar?

It sounds simple, but that small change led physicists into a mathematical labyrinth, in which they describe themselves as wandering, "exploring almost like experimentalists," in the words of Dr. David Gross of the Kavli Institute for Theoretical Physics in Santa Barbara, Calif.

String theory, the Italian physicist Dr. Daniele Amati once said, was a piece of 21st-century physics that had fallen by accident into the 20th century.

And, so the joke went, would require 22nd-century mathematics to solve.

Dr. Edward Witten of the Institute for Advanced Study in Princeton, N.J., described it this way: "String theory is not like anything else ever discovered. It is an incredible panoply of ideas about math and physics, so vast, so rich you could say almost anything about it."

The string revolution had its roots in a quixotic effort in the 1970's to understand the so-called "strong" force that binds quarks into particles like protons and neutrons. Why were individual quarks never seen in nature? Perhaps because they were on the ends of strings, said physicists, following up on work by Dr. Gabriele Veneziano of CERN, the European research consortium.

That would explain why you cannot have a single quark - you cannot have a string with only one end. Strings seduced many physicists with their mathematical elegance, but they had some problems, like requiring 26 dimensions and a plethora of mysterious particles that did not seem to have anything to do with quarks or the strong force.

When accelerator experiments supported an alternative theory of quark behavior known as quantum chromodynamics, most physicists consigned strings to the dustbin of history.

But some theorists thought the mathematics of strings was too beautiful to die.

In 1974 Dr. Schwarz and Dr. Joel Scherk from the École Normale Supérieure in France noticed that one of the mysterious particles predicted by string theory had the properties predicted for the graviton, the particle that would be responsible for transmitting gravity in a quantum theory of gravity, if such a theory existed.

Without even trying, they realized, string theory had crossed the biggest gulf in physics. Physicists had been stuck for decades trying to reconcile the quirky rules known as quantum mechanics, which govern atomic behavior, with Einstein's general theory of relativity, which describes how gravity shapes the cosmos.

That meant that if string theory was right, it was not just a theory of the strong force; it was a theory of all forces.

"I was immediately convinced this was worth devoting my life to," Dr. Schwarz recalled "It's been my life work ever since."

It was another 10 years before Dr. Schwarz and Dr. Green (Dr. Scherk died in 1980) finally hit pay dirt. They showed that it was possible to write down a string theory of everything that was not only mathematically consistent but also free of certain absurdities, like the violation of cause and effect, that had plagued earlier quantum gravity calculations.

In the summer and fall of 1984, as word of the achievement spread, physicists around the world left what they were doing and stormed their blackboards, visions of the Einsteinian grail of a unified theory dancing in their heads.

"Although much work remains to be done there seem to be no insuperable obstacles to deriving all of known physics," one set of physicists, known as the Princeton string quartet, wrote about a particularly promising model known as heterotic strings. (The quartet consisted of Dr. Gross; Dr. Jeffrey Harvey and Dr. Emil Martinec, both at the University of Chicago; and Dr. Ryan M. Rohm, now at the University of North Carolina.)

The Music of Strings

String theory is certainly one of the most musical explanations ever offered for nature, but it is not for the untrained ear. For one thing, the modern version of the theory decreed that there are 10 dimensions of space and time.

To explain to ordinary mortals why the world appears to have only four dimensions - one of time and three of space -string theorists adopted a notion first bruited by the German mathematicians Theodor Kaluza and Oskar Klein in 1926. The extra six dimensions, they said, go around in sub-submicroscopic loops, so tiny that people cannot see them or store old National Geographics in them.

A simple example, the story goes, is a garden hose. Seen from afar, it is a simple line across the grass, but up close it has a circular cross section. An ant on the hose can go around it as well as travel along its length. To envision the world as seen by string theory, one only has to imagine a tiny, tiny six-dimensional ball at every point in space-time

But that was only the beginning. In 1995, Dr. Witten showed that what had been five different versions of string theory seemed to be related. He argued that they were all different manifestations of a shadowy, as-yet-undefined entity he called "M theory," with "M" standing for mother, matrix, magic, mystery, membrane or even murky.

In M-theory, the universe has 11 dimensions - 10 of space and one of time, and it consists not just of strings but also of more extended membranes of various dimension, known generically as "branes."

This new theory has liberated the imaginations of cosmologists. Our own universe, some theorists suggest, may be a four-dimensional brane floating in some higher-dimensional space, like a bubble in a fish tank, perhaps with other branes - parallel universes - nearby. Collisions or other interactions between the branes might have touched off the Big Bang that started our own cosmic clock ticking or could produce the dark energy that now seems to be accelerating the expansion of the universe, they say.

Toting Up the Scorecard

One of string theory's biggest triumphs has come in the study of black holes. In Einstein's general relativity, these objects are bottomless pits in space-time, voraciously swallowing everything, even light, that gets too close, but in string theory they are a dense tangle of strings and membranes.

In a prodigious calculation in 1995, Dr. Strominger and Dr. Cumrun Vafa, both of Harvard, were able to calculate the information content of a black hole, matching a famous result obtained by Dr. Stephen Hawking of Cambridge University using more indirect means in 1973. Their calculation is viewed by many people as the most important result yet in string theory, Dr. Greene said.

Another success, Dr. Greene and others said, was the discovery that the shape, or topology, of space, is not fixed but can change, according to string theory. Space can even rip and tear.

But the scorecard is mixed when it comes to other areas of physics. So far, for example, string theory has had little to say about what might have happened at the instant of the Big Bang..

Moreover, the theory seems to have too many solutions. One of the biggest dreams that physicists had for the so-called theory of everything was that it would specify a unique prescription of nature, one in which God had no choice, as Einstein once put it, about details like the number of dimensions or the relative masses of elementary particles.

But recently theorists have estimated that there could be at least 10100 different solutions to the string equations, corresponding to different ways of folding up the extra dimensions and filling them with fields - gazillions of different possible universes.

Some theorists, including Dr. Witten, hold fast to the Einsteinian dream, hoping that a unique answer to the string equations will emerge when they finally figure out what all this 21st-century physics is trying to tell them about the world.

But that day is still far away.

"We don't know what the deep principle in string theory is," Dr. Witten said.

For most of the 20th century, progress in particle physics was driven by the search for symmetries - patterns or relationships that remain the same when we swap left for right, travel across the galaxy or imagine running time in reverse.

For years physicists have looked for the origins of string theory in some sort of deep and esoteric symmetry, but string theory has turned out to be weirder than that.

Recently it has painted a picture of nature as a kind of hologram. In the holographic images often seen on bank cards, the illusion of three dimensions is created on a two-dimensional surface. Likewise string theory suggests that in nature all the information about what is happening inside some volume of space is somehow encoded on its outer boundary, according to work by several theorists, including Dr. Juan Maldacena of the Institute for Advanced Study and Dr. Raphael Bousso of the University of California, Berkeley.

Just how and why a three-dimensional reality can spring from just two dimensions, or four dimensions can unfold from three, is as baffling to people like Dr. Witten as it probably is to someone reading about it in a newspaper.

In effect, as Dr. Witten put it, an extra dimension of space can mysteriously appear out of "nothing."

The lesson, he said, may be that time and space are only illusions or approximations, emerging somehow from something more primitive and fundamental about nature, the way protons and neutrons are built of quarks.

The real secret of string theory, he said, will probably not be new symmetries, but rather a novel prescription for constructing space-time.

"It's a new aspect of the theory," Dr. Witten said. "Whether we are getting closer to the deep principle, I don't know."

As he put it in a talk in October, "It's plausible that we will someday understand string theory."

Tangled in Strings

Critics of string theory, meanwhile, have been keeping their own scorecard. The most glaring omission is the lack of any experimental evidence for strings or even a single experimental prediction that could prove string theory wrong - the acid test of the scientific process.

Strings are generally presumed to be so small that "stringy" effects should show up only when particles are smashed together at prohibitive energies, roughly 1019 billion electron volts. That is orders of magnitude beyond the capability of any particle accelerator that will ever be built on earth. Dr. Harvey of Chicago said he sometimes woke up thinking, What am I doing spending my whole career on something that can't be tested experimentally?

This disparity between theoretical speculation and testable reality has led some critics to suggest that string theory is as much philosophy as science, and that it has diverted the attention and energy of a generation of physicists from other perhaps more worthy pursuits. Others say the theory itself is still too vague and that some promising ideas have not been proved rigorously enough yet.

Dr. Krauss said, "We bemoan the fact that Einstein spent the last 30 years of his life on a fruitless quest, but we think it's fine if a thousand theorists spend 30 years of their prime on the same quest."

The Other Quantum Gravity

String theory's biggest triumph is still its first one, unifying Einstein's lordly gravity that curves the cosmos and the quantum pinball game of chance that lives inside it.

"Whatever else it is or is not," Dr. Harvey said in Aspen, "string theory is a theory of quantum gravity that gives sensible answers."

That is no small success, but it may not be unique.

String theory has a host of lesser known rivals for the mantle of quantum gravity, in particular a concept called, loop quantum gravity, which arose from work by Dr. Abhay Ashtekar of Penn State and has been carried forward by Dr. Carlo Rovelli of the University of Marseille and Dr. Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, among others.

Unlike string theory, loop gravity makes no pretensions toward being a theory of everything. It is only a theory of gravity, space and time, arising from the applications of quantum principles to the equations of Einstein's general relativity. The adherents of string theory and of loop gravity have a kind of Microsoft-Apple kind of rivalry, with the former garnering a vast majority of university jobs and publicity.

Dr. Witten said that string theory had a tendency to absorb the ideas of its critics and rivals. This could happen with loop gravity. Dr. Vafa; his Harvard colleagues, Dr. Sergei Gukov and Dr. Andrew Neitzke; and Dr. Robbert Dijkgraaf of the University of Amsterdam report in a recent paper that they have found a connection between simplified versions of string and loop gravity.

"If it exists," Dr. Vafa said of loop gravity, "it should be part of string theory."

Looking for a Cosmic Connection

Some theorists have bent their energies recently toward investigating models in which strings could make an observable mark on the sky or in experiments in particle accelerators.

"They all require us to be lucky," said Dr. Joe Polchinski of the Kavli Institute.

For example the thrashing about of strings in the early moments of time could leave fine lumps in a haze of radio waves filling the sky and thought to be the remains of the Big Bang. These might be detectable by the Planck satellite being built by the European Space Agency for a 2007 launching date, said Dr. Greene.

According to some models, Dr. Polchinski has suggested, some strings could be stretched from their normal submicroscopic lengths to become as big as galaxies or more during a brief cosmic spurt known as inflation, thought to have happened a fraction of a second after the universe was born.

If everything works out, he said, there will be loops of string in the sky as big as galaxies. Other strings could stretch all the way across the observable universe. The strings, under enormous tension and moving near the speed of light, would wiggle and snap, rippling space-time like a tablecloth with gravitational waves.

"It would be like a whip hundreds of light-years long," Dr. Polchinski said.

The signal from these snapping strings, if they exist, should be detectable by the Laser Interferometer Gravitational Wave Observatory, which began science observations two years ago, operated by a multinational collaboration led by Caltech and the Massachusetts Institute of Technology.

Another chance for a clue will come in 2007 when the Large Hadron Collider is turned on at CERN in Geneva and starts colliding protons with seven trillion volts of energy apiece. In one version of the theory - admittedly a long shot - such collisions could create black holes or particles disappearing into the hidden dimensions.

Everybody's favorite candidate for what the collider will find is a phenomenon called supersymmetry, which is crucial to string theory. It posits the existence of a whole set of ghostlike elementary particles yet to be discovered. Theorists say they have reason to believe that the lightest of these particles, which have fanciful names like photinos, squarks and selectrons, should have a mass-energy within the range of the collider.

String theory naturally incorporates supersymmetry, but so do many other theories. Its discovery would not clinch the case for strings, but even Dr. Krauss of Case Western admits that the existence of supersymmetry would be a boon for string theory.

And what if supersymmetric particles are not discovered at the new collider? Their absence would strain the faith, a bit, but few theorists say they would give up.

"It would certainly be a big blow to our chances of understanding string theory in the near future," Dr. Witten said.

Beginnings and Endings

At the end of the Aspen celebration talk turned to the prospect of verification of string theory. Summing up the long march toward acceptance of the theory, Dr. Stephen Shenker, a pioneer string theorist at Stanford, quoted Winston Churchill:

"This is not the end, not even the beginning of the end, but perhaps it is the end of the beginning."

Dr. Shenker said it would be great to find out that string theory was right.

From the audience Dr. Greene piped up, "Wouldn't it be great either way?"

"Are you kidding me, Brian?" Dr. Shenker responded. "How many years have you sweated on this?"

But if string theory is wrong, Dr. Greene argued, wouldn't it be good to know so physics could move on? "Don't you want to know?" he asked.

Dr. Shenker amended his remarks. "It would be great to have an answer," he said, adding, "It would be even better if it's the right one."


Copyright 2004 The New York Times Company | Home |

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TOPICS: Science
KEYWORDS: oopsnotyet; physics; quantumgravity; science; stringtheory; superstringtheory; theoryofeverything; workinprogress
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String Theory, at 20, Explains It All (or Not)

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1 posted on 12/07/2004 10:01:55 AM PST by snarks_when_bored
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To: snarks_when_bored

I prefer the String cheese theory.

Anyway I'll have to read this later


2 posted on 12/07/2004 10:06:12 AM PST by escapefromboston (manny ortez: mvp)
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To: snarks_when_bored
the string theorists admitted that after 20 years, they still did not know how to test string theory

I happen to think string theory shows great promise - but it is this admission which, so far, makes it tantamount to religion. Here's hoping they find a way to test it.
3 posted on 12/07/2004 10:08:33 AM PST by beezdotcom (I'm usually either right or wrong...)
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To: snarks_when_bored
the string theorists admitted that after 20 years, they still did not know how to test string theory

I happen to think string theory shows great promise - but it is this admission which, so far, makes it tantamount to religion. Here's hoping they find a way to test it.
4 posted on 12/07/2004 10:08:55 AM PST by beezdotcom (I'm usually either right or wrong...)
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To: escapefromboston

Mmmmmmm...String cheese.


5 posted on 12/07/2004 10:09:12 AM PST by snarks_when_bored
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To: beezdotcom

whoops...danged proxy made me post it twice....


6 posted on 12/07/2004 10:09:28 AM PST by beezdotcom (I'm usually either right or wrong...)
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To: beezdotcom

They're looking pretty hard for ways to test Superstring Theory. And there are many critics of the field, a loyal opposition who think that physics ought to be testable. But what's testable now and what might be testable a few years from now are very different things. We're still on the way to something, but what it is we don't quite know (kind of describes life, too, huh?).


7 posted on 12/07/2004 10:15:50 AM PST by snarks_when_bored
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To: snarks_when_bored
The famous one-sided bottle....
8 posted on 12/07/2004 10:23:23 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: snarks_when_bored

Hey...this article didn't help Muttly at all.

What does it have to do with his Yo-Yo ?!!

Well...maybe something...may have too many dimensions in there or something.

When do we eat.


9 posted on 12/07/2004 10:24:33 AM PST by PoorMuttly ("The right of the People to be Muttly shall not be infringed,")
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To: Ernest_at_the_Beach

That's vaguely disturbing.


10 posted on 12/07/2004 10:25:38 AM PST by Tijeras_Slim (I'm here because I'm not all there.)
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To: snarks_when_bored

From top, Rick Friedman; Laura Pedrick; Emilio Flores, all for The New York Times
Scientists around the country leading the study of string theory include Dr. Andrew Strominger and Dr. Cumrun Vafa, photo at top; Dr. Edward Witten, middle; and Dr. Joe Polchinski, above.
11 posted on 12/07/2004 10:33:06 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: snarks_when_bored
Einstein was tutored.........he read Bible (Jesus) truth and rejected blood redemption...

But,.......accepted the 'relativism' of light to creation concept in the bible.

.......NEVER giving God and His Bible Truth credit!

(....stringing Darwin along....)

/sarcasm?

(Romans 10:17)

......the demons believe (too) and tremble (too).....

12 posted on 12/07/2004 10:36:48 AM PST by maestro
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To: Ernest_at_the_Beach
Could'a done it, would'a done it, should'a done it—didn't think of it.
13 posted on 12/07/2004 10:36:53 AM PST by snarks_when_bored
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To: All
The Official String Theory Web Site:--> Mathematics

Basics

Guide to math needed to study physics
  What mathematics do string theorists need to learn? Not surprisingly, it's a long list! And getting longer every day, it seems.

I. Math for learning undergraduate level physics

II. Math background suggested for graduate students in theoretical physics especially those who want to work in string theory

III. Math topics that are hot topics in development right now in string theory

Home
Basics
Mathematics
Experiment
Cosmology
Black Holes
People
History
Theatre
Links
Forum

Some basic books for further reading.

Some advanced books for further reading.


<< Previous

Next >>


Guide to math I // Guide to math II // Guide to math III

And we have an official website for learning the math required to understand this theory....

Link:

http://superstringtheory.com/math/


14 posted on 12/07/2004 10:39:14 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: All
The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.
Algebra
  Algebra provides the first exposure to the use of variables and constants, and experience manipulating and solving linear equations of the form y = ax + b and quadratic equations of the form y = ax2+bx+c.
Geometry
  Geometry at this level is two-dimensional Euclidean geometry, Courses focus on learning to reason geometrically, to use concepts like symmetry, similarity and congruence, to understand the properties of geometric shapes in a flat, two-dimensional space.
Trigonometry
  Trigonometry begins with the study of right triangles and the Pythagorean theorem. The trigonometric functions sin, cos, tan and their inverses are introduced and clever identities between them are explored.
Calculus (single variable)
  Calculus begins with the definition of an abstract functions of a single variable, and introduces the ordinary derivative of that function as the tangent to that curve at a given point along the curve. Integration is derived from looking at the area under a curve,which is then shown to be the inverse of differentiation.
Calculus (multivariable)
  Multivariable calculus introduces functions of several variables f(x,y,z...), and students learn to take partial and total derivatives. The ideas of directional derivative, integration along a path and integration over a surface are developed in two and three dimensional Euclidean space.
Analytic Geometry
  Analytic geometry is the marriage of algebra with geometry. Geometric objects such as conic sections, planes and spheres are studied by the means of algebraic equations. Vectors in Cartesian, polar and spherical coordinates are introduced.
Linear Algebra
  In linear algebra, students learn to solve systems of linear equations of the form ai1 x1 + ai2 x2 + ... + ain xn = ci and express them in terms of matrices and vectors. The properties of abstract matrices, such as inverse, determinant, characteristic equation, and of certain types of matrices, such as symmetric, antisymmetric, unitary or Hermitian, are explored.
Ordinary Differential Equations
  This is where the physics begins! Much of physics is about deriving and solving differential equations. The most important differential equation to learn, and the one most studied in undergraduate physics, is the harmonic oscillator equation, ax'' + bx' + cx = f(t), where x' means the time derivative of x(t).
Partial Differential Equations
  For doing physics in more than one dimension, it becomes necessary to use partial derivatives and hence partial differential equations. The first partial differential equations students learn are the linear, separable ones that were derived and solved in the 18th and 19th centuries by people like Laplace, Green, Fourier, Legendre, and Bessel.
Methods of approximation
  Most of the problems in physics can't be solved exactly in closed form. Therefore we have to learn technology for making clever approximations, such as power series expansions, saddle point integration, and small (or large) perturbations.
Probability and statistics
  Probability became of major importance in physics when quantum mechanics entered the scene. A course on probability begins by studying coin flips, and the counting of distinguishable vs. indistinguishable objects. The concepts of mean and variance are developed and applied in the cases of Poisson and Gaussian statistics.

15 posted on 12/07/2004 10:45:13 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: snarks_when_bored
"They all laughed 20 years ago. It was then that a physicist named John Schwarz jumped up on the stage during a cabaret at the physics center here and began babbling about having discovered a theory that could explain everything."

We're still laughing; here's why:

String Theory attempts to imagine a universe in which *both* Quantum Mechanics and General Relativity are valid.

The problem with such a fantasy, however, is that QM and GR are mututally exclusive theories. They fundamentally contradict each other on rather large areas such as universal Gravity, etc.

What has happened is that we have two theories, QM and GR, that our modern, effeminate, politically correct scientists can't choose between. One theory is correct, the other is not.

But no one wants to say that one of those theories is wrong.

So rather than make the hard choice, today's limp-wristed researchers are wasting enormous amounts of time and money pursuing a String Theory that somehow makes both QM and GR valid.

...And that's why we're laughing. Unifying GR with QM is not feasible. The two theories contradict each other. Yet on go the Strong Theory adherents, unwilling to say that one King (either GR or QM) has no clothes...even if it means publishing ridiculous nonsense about String Theories.

16 posted on 12/07/2004 10:46:08 AM PST by Southack (Media Bias means that Castro won't be punished for Cuban war crimes against Black Angolans in Africa)
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To: All
Here are some of the topics in mathematics that a person who wants to learn advanced topics in theoretical physics, especially string theory, should become familiar with.
Real analysis
  In real analysis, students learn abstract properties of real functions as mappings, isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants and homeomorphisms.
Complex analysis
  Complex analysis is an important foundation for learning string theory. Functions of a complex variable, complex manifolds, holomorphic functions, harmonic forms, Kähler manifolds, Riemann surfaces and Teichmuller spaces are topics one needs to become familiar with in order to study string theory.
Group theory
  Modern particle physics could not have progressed without an understanding of symmetries and group transformations. Group theory usually begins with the group of permutations on N objects, and other finite groups. Concepts such as representations, irreducibility, classes and characters.
Differential geometry
  Einstein's General Theory of Relativity turned non-Euclidean geometry from a controversial advance in mathematics into a component of graduate physics education. Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors. Students should learn about metrics and covariant derivatives, and how to calculate curvature in coordinate and non-coordinate bases.
Lie groups
  A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groups have been especially important in modern physics. The study of Lie groups combines techniques from group theory and basic differential geometry to develop the concepts of Lie derivatives, Killing vectors, Lie algebras and matrix representations.
Differential forms
  The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory. Students begin with antisymmetric tensors, then develop the concepts of exterior product, exterior derivative, orientability, volume elements, and integrability conditions.
Homology
  Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has no edges, and hence no boundary. In homology this case is generalized to "The boundary of a boundary is zero." Students learn about simplexes, complexes, chains, and homology groups.
Cohomology
  Cohomology and homology are related, as one might suspect from the names. Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups.
Homotopy
  Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important in string theory because closed strings can wind around donut holes and get stuck, with physical consequences. Students learn about paths and loops, homotopic maps of loops, contractibility, the fundamental group, higher homotopy groups, and the Bott periodicity theorem.
Fiber bundles
  Fiber bundles comprise an area of mathematics that studies spaces defined on other spaces through the use of a projection map of some kind. For example, in electromagnetism there is a U(1) vector potential associated with every point of the spacetime manifold. Therefore one could study electromagnetism abstractly as a U(1) fiber bundle over some spacetime manifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps, covariant derivatives, curvature, and the connection to gauge field theories in physics.
Characteristic classes
  The subject of characteristic classes applies cohomology to fiber bundles to understand the barriers to untwisting a fiber bundle into what is known as a trivial bundle. This is useful because it can reduce complex physical problems to math problems that are already solved. The Chern class is particularly relevant to string theory.
Index theorems
  In physics we are often interested in knowing about the space of zero eigenvalues of a differential operator. The index of such an operator is related to the dimension of that space of zero eigenvalues. The subject of index theorems and characteristic classes is concerned with
Supersymmetry and supergravity
  The mathematics behind supersymmetry starts with two concepts: graded Lie algebras, and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x times y = –y times x. The mathematical technology needed to work in supersymmetry includes an understanding of graded Lie algebras, spinors in arbitrary spacetime dimensions, covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication, derivation and integration, and Kähler potentials.

17 posted on 12/07/2004 10:48:04 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: PatrickHenry; Physicist; RadioAstronomer; Right Wing Professor

Physics ping.


18 posted on 12/07/2004 10:49:34 AM PST by Lurking Libertarian (Non sub homine, sed sub Deo et lege)
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To: snarks_when_bored

NYT still has the disinterested ability to get a story more or less right.


19 posted on 12/07/2004 10:51:50 AM PST by RightWhale (Destroy the dark; restore the light)
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To: Ernest_at_the_Beach
Under the "Characteristic Classes" heading you will find mentioned "Chern classes". I've just posted Shiing-Shen Chern's obituary from today's New York Times:

Shiing-Shen Chern

20 posted on 12/07/2004 10:52:13 AM PST by snarks_when_bored
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To: snarks_when_bored

The "Chern class", I should have said.


21 posted on 12/07/2004 10:54:56 AM PST by snarks_when_bored
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To: Lurking Libertarian; VadeRetro; jennyp; Junior; longshadow; RadioAstronomer; Physicist; ...
Science list Ping! This is an elite subset of the Evolution list.
See the list's description in my freeper homepage. Then FReepmail me to be added or dropped.
22 posted on 12/07/2004 10:56:38 AM PST by PatrickHenry (The List-O-Links for evolution threads is at my freeper homepage.)
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To: All
Some very serious mathematics research going on :........

_________________________________________________________________

These are topics in mathematics at the current cutting edge of superstring research.

K-theory
 

    Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.
    Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory.

Overview of K-theory Applied to Stringssby Edward Witten

D-branes and K-theory by Edward Witten

Noncommutative geometry (NCG for short)
 

    Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.
    Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.
    In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.
    The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes.

Two Lectures on D-Geometry and Noncommutative Geometry by Michael R. Douglas
Noncommutative Geometry and Matrix Theory: Compactification on Tori by Alain Connes, Michael R. Douglas, Albert Schwarz

String Theory and Noncommutative Geometry by Edward Witten and Nathan Seiberg.

Non-commutative spaces in physics and mathematics by Daniela Bigatti

Noncommutative Geometry for Pedestrians by J.Madore


23 posted on 12/07/2004 10:58:10 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: Southack

Do you know of any web sites out there that presents string theroy in laymans' terms?


24 posted on 12/07/2004 11:05:54 AM PST by jmc813 (J-E-T-S JETS JETS JETS)
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To: maestro

May I ask what the &*)! you are talking about?


25 posted on 12/07/2004 11:06:18 AM PST by Shryke
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To: Southack

Physicists who work at the fundamental level fully understand that General Relativity breaks down at the Planck scale and also that Quantum Field Theory isn't feeling so good at that scale, either. That's why they're searching for an over-arching theory that transcends (and subsumes) both GR and QFT. Superstring Theory, it was hoped, was such a theory. That hasn't proved to be the case...yet.


26 posted on 12/07/2004 11:07:38 AM PST by snarks_when_bored
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To: snarks_when_bored
"Physicists who work at the fundamental level fully understand that General Relativity breaks down at the Planck scale..."

It's more than that. GR contradicts QM in regards to universal Gravity and energy.

...And even Einstein himself had to delete his Universal Constant from GR after 4 decades of scientific denial...

27 posted on 12/07/2004 11:11:36 AM PST by Southack (Media Bias means that Castro won't be punished for Cuban war crimes against Black Angolans in Africa)
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To: snarks_when_bored
If true, it could be used like a searchlight to illuminate some of the deepest mysteries physicists can imagine, like the origin of space and time in the Big Bang and the putative death of space and time at the infinitely dense centers of black holes.

Majikthise: I mean, what's the use of our sitting around half the night arguing about whether there may...

Vroomfondel: Or may not.

Majikthise: - be a God, if this machine only goes and gives you his phone number in the morning?

28 posted on 12/07/2004 11:12:01 AM PST by Junior (FABRICATI DIEM, PVNC)
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To: snarks_when_bored
An interesting reference here:

Noncommutative Geometry for Pedestrians

29 posted on 12/07/2004 11:12:35 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: PatrickHenry

Thanks for the ping!


30 posted on 12/07/2004 11:13:58 AM PST by Alamo-Girl
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To: All
Some links:

Leonard Susskind (one of the discoverers of String Theory) talks about the String Theory Landscape and the Anthropic Principle

Responses to Susskind's talk on the Landscape and the Anthropic Principle (scroll down)

Susskind stomps on poor Lee Smolin, the Loop Quantum Gravity guy

The entire Edge site is embarrassingly fun to read:

Edge


31 posted on 12/07/2004 11:21:40 AM PST by snarks_when_bored
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To: Alamo-Girl

Alexander's Horned Sphere

The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere. It is homeomorphic with the ball , and its boundary is therefore a sphere. It is therefore an example of a wild embedding in . The outer complement of the solid is not simply connected, and its fundamental group is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander's horned sphere is a Cantor set.

The horned sphere as originally drawn by Alexander (1924) is illustrated above.

The complement in of the bad points for Alexander's horned sphere is simply connected, making it inequivalent to Antoine's horned sphere. Alexander's horned sphere has an uncountable infinity of wild points, which are the limits of the sequences of the horned sphere's branch points (roughly, the "ends" of the horns), since any neighborhood of a limit contains a horned complex.

A humorous drawing by Simon Frazer (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander's horned sphere growing from his head.

Antoine's Horned Sphere

Links




References

Albers, D. J. Illustration accompanying "The Game of 'Life."' Math Horizons, p. 9, Spring 1994.

Alexander, J. W. "An Example of a Simply Connected Surface Bounding a Region Which Is Not Simply Connected." Proc. N. A. S. 10, 8-10, 1924.

Guy, R. "Conway's Prime Producing Machine." Math. Mag. 56, 26-33, 1983.

Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 80-81, 1976.

Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 58, 1991.


32 posted on 12/07/2004 11:24:48 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: All
And here's a link to a widely read technical article by Leonard Susskind:

The Anthropic Landscape of String Theory (PDF format)

33 posted on 12/07/2004 11:25:56 AM PST by snarks_when_bored
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To: snarks_when_bored
Yeah, but I want to differentiate between things that we know HOW to test, but CAN'T until we achieve some new technological milestone (like a more powerful supercollider, or probes that are light-years away), vs. those things for which we can't even devise a good gedanken experiment to validate/invalidate.
34 posted on 12/07/2004 11:27:48 AM PST by beezdotcom (I'm usually either right or wrong...)
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To: Ernest_at_the_Beach
Throwing in the kitchen sink horned sphere, eh?
35 posted on 12/07/2004 11:27:48 AM PST by snarks_when_bored
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To: snarks_when_bored

That was the universal counter example to a lot of stuff in topology of simple surfaces ( proves 3 dimensions is very different than two dimensions..... )


36 posted on 12/07/2004 11:31:45 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: Southack
...And even Einstein himself had to delete his Universal Constant from GR after 4 decades of scientific denial...

The cosmological constant is back...it seems to account for the accelerating cosmic expansion being attributed to 'dark energy'. Check out the Sepetember, 2004, issue of Scientific American.

It's more than that. GR contradicts QM in regards to universal Gravity and energy.

I'm not sure what you mean by this.

37 posted on 12/07/2004 11:35:12 AM PST by snarks_when_bored
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To: snarks_when_bored

'September' (of course)


38 posted on 12/07/2004 11:37:22 AM PST by snarks_when_bored
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To: snarks_when_bored

Thanks.


39 posted on 12/07/2004 11:37:52 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: snarks_when_bored

"A day without string is chaos."

Rudolph Smuntz


40 posted on 12/07/2004 11:39:06 AM PST by Francis McClobber
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To: snarks_when_bored

May the Schwarz be with you!


41 posted on 12/07/2004 11:40:34 AM PST by mikrofon (Space Balls)
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To: escapefromboston
A string walks into a bar. The bartender says "We don't serves strings in here. Get out."
The string walks outside, musses himself up, makes a couple loops around himself and reenters the bar.
The bartender says "Hey, aren't you that string I just threw out?"
The string replies,"No, I'm a frayed knot."
42 posted on 12/07/2004 11:45:52 AM PST by bayourod (Bush said. "Let's see if I can say it as plainly as I can: I am for the intelligence bill.")
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To: snarks_when_bored
The lesson, he said, may be that time and space are only illusions or approximations, emerging somehow from something more primitive and fundamental about nature...

I like this. It allows for the view that the fundamental reality and nature of existence is that it is spiritual. This material, mortal, earthly perception of existence is an illusion and when we allow the influence of spiritual reality to come into our experience (the reality that Jesus Christ demonstrated - overcoming every human problem and limitation) we realize our true nature and existence as spiritual "children of God" - spiritual offspring of a spiritual Source.

43 posted on 12/07/2004 11:45:52 AM PST by Semper
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To: snarks_when_bored

Thanks for posting this. Lubos Motl at Harvard has written about this article at his blog http://motls.blogspot.com/

Lubos regularly writes comments on String theory, but politics as well, he is a conservative (Sic!)


44 posted on 12/07/2004 11:46:52 AM PST by AdmSmith
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To: beezdotcom
Yeah, but I want to differentiate between things that we know HOW to test, but CAN'T until we achieve some new technological milestone (like a more powerful supercollider, or probes that are light-years away), vs. those things for which we can't even devise a good gedanken experiment to validate/invalidate.

I sympathize with that point of view. But I've recently been reading the history of the Dirac equation. That equation, which predicts the existence of anti-electrons (positrons), was developed by Dirac from just fooling around with trying to put special relativity together with the new quantum mechanics of Heisenberg and Schrödinger. Dirac played around until he got something interesting, then was able to predict some spectral deviations that non-relativistic QM couldn't predict...and he also got the prediction of anti-matter, too, which was a neat plus (er, minus?). Anyway, the Dirac equation turns out not to be right, but it was right for its time, and now it has been re-interpreted in Relativistic Quantum Field Theory as an operator equation, and all is well (within its range of applicability).

The point? We don't always know where our mathematical imagination, guided as far as possible by physical insight, is going to lead. Sometimes the mathematics and the physics get out of phase (which seems to be happening now); but I trust that that will be taken care of in the future.

45 posted on 12/07/2004 11:48:01 AM PST by snarks_when_bored
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To: beezdotcom
...but it is this admission which, so far, makes it tantamount to religion.

Not religion, it keeps it on the level of mathematics with no evidence that the equations describe reality.
46 posted on 12/07/2004 11:50:27 AM PST by BikerNYC
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To: snarks_when_bored
...and he also got the prediction of anti-matter, too, which was a neat plus (er, minus?).

Actually, in the case of the positron, it was a plus!

47 posted on 12/07/2004 11:51:47 AM PST by snarks_when_bored
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To: snarks_when_bored
That Edge site is interesting.....taking a serious branch here:

"WHAT KIND OF A THING IS A NUMBER?"

48 posted on 12/07/2004 11:52:53 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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To: Semper
You wrote:

The lesson, he said, may be that time and space are only illusions or approximations, emerging somehow from something more primitive and fundamental about nature...

I like this. It allows for the view that the fundamental reality and nature of existence is that it is spiritual. This material, mortal, earthly perception of existence is an illusion and when we allow the influence of spiritual reality to come into our experience (the reality that Jesus Christ demonstrated - overcoming every human problem and limitation) we realize our true nature and existence as spiritual "children of God" - spiritual offspring of a spiritual Source.

Please note that the quote you cited ends with the words "something more primitive and fundamental about nature", where by 'nature' is meant 'the physical world'.

49 posted on 12/07/2004 11:56:31 AM PST by snarks_when_bored
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To: BikerNYC

You want reality......ever see a real Alexander Horned Sphere?


50 posted on 12/07/2004 11:57:28 AM PST by Ernest_at_the_Beach (A Proud member of Free Republic ~~The New Face of the Fourth Estate since 1996.)
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