Thursday, January 19, 2012

Guide to math needed to study physics

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.

In defense of Shakespeare—a conversation with veteran Australian actor and director John Bell

By David Walsh
13 December 2011
William Shakespeare


The recent film Anonymous, directed by Roland Emmerich and written by John Orloff, which argues that the Earl of Oxford, Edward de Vere, was the actual author of the three dozen plays attributed to William Shakespeare (1564-1616), raises a host of questions. (See “Anonymous: An ignorant assault on Shakespeare” and “An exchange: More on the contemporary assault on Shakespeare”) At the center of the debate is the figure of Shakespeare himself and the enduring character of his work.

The arguments in favor of the Earl of Oxford are not serious ones and his champions largely attempt to take advantage of the generally low level of historical knowledge at present to gain a hearing. More serious, however, is the thrust of the effort, aimed, in our view, against the plays and Shakespeare’s extraordinary contribution as a whole.

In the assault on Shakespeare, incomprehension in the face of genuine artistic genius combines with hostility toward the universality of the plays and the playwright, including the confidence of the Elizabethan playwright that he could cognize every corner of reality and bring it to life in a poetic manner. There is something threatening and disturbing to a certain contemporary social type, self-involved and self-centered, often obsessed with gender or ethnic identity, about an artistic figure of Shakespeare’s depth and breadth.

Out of a concern with some of these general questions, I recently spoke to John Bell, the distinguished Australian actor and director who founded the Bell Shakespeare theatre company in 1990.

In his lively memoir, The Time of My Time, Bell (born 1940 in Maitland, New South Wales) describes his first eye-opening encounter with Shakespeare at a Catholic high school, when one of his teachers introduced an English class to A Midsummer Night’s Dream. On top of that, a viewing of Laurence Olivier’s Henry V at a local cinema left Bell “stunned and blinking.” He goes on, “I couldn’t believe what I’d just experienced so I went back in and watched it all over again. … In the years following, Olivier’s Hamlet and Richard III appeared and my fate was sealed.”

At the time Bell went off to university there was no full-time professional theatre company in Sydney. He told an interviewer, “If you wanted a career in the theatre, there simply wasn't one [in Australia].” Bell traveled to England in 1964, and after six months, was invited to join the prestigious Royal Shakespeare Company.

He returned to Australia in 1970, taught at the National Institute of Dramatic Art and co-founded the Nimrod Theatre Company in Sydney.

For the Shakespeare company he established 21 years ago, Bell has played Shylock, Richard III, Macbeth, Malvolio, Coriolanus, Leontes, Prospero, King Lear and Ulysses, among other roles. In the last five years alone, he has directed productions of Shakespeare’s Romeo and Juliet, The Comedy of Errors, Macbeth, As You Like It and Pericles, along with an adaptation of Melville’s Moby Dick, Heiner Müller’s reworking of Titus Andronicus, Gogol’s The Government Inspector, Ben Jonson’s The Alchemist and John Webster’s The Duchess of Malfi.
John Bell

Bell Shakespeare is Australia's only national touring theatre company. It currently tours three mainstage productions to each Australian state every year, in addition to a variety of educational programs.

In 2011, Bell published On Shakespeare, his thoughts and reminiscences of playing Shakespeare over the course of half a century.

We spoke by telephone.

David Walsh: Anonymous is the immediate occasion for this conversation, but our more general concern is the appeal of Shakespeare, his universality: what it is that still draws audiences, actors, and directors to the plays.

You write in your memoir about your first encounters with Shakespeare. Can you recall the experience and some of the elements that were attractive or forceful to you at the time?

John Bell: Yes, I can. I think my very first epiphany, if you like, was hearing Julius Caesar on the radio when I was 12 or 13 years old. I was struck by the language, the poetry was what moved me most. I’d never heard language like that.

I was very fortunate in having two very good English teachers in high school, when I was about 14 or 15. They didn’t simply pass the book around the classroom and say, ‘Please, paraphrase this.’ The first one I had actually acted out the play for us, in the room, and took on all the parts, described the sets, the costumes, and lighting, the whole lot.

I guess the next thing, around the same time, when I was 14 or 15, was seeing Laurence Olivier’s movie of Henry V, which had all those elements together: the language was so thrilling, the spectacle, the sheer excitement, and generally the rough nature of it. It opens in a reconstruction of the Globe Theatre, which is full of horseplay and the actors and audience adlibbing with each other, with a great sense of improvisation about it. This reminded me of my very first encounters with theatre, which were pantomime and circus.

At the age of 15 or so, I was an absolute convert. All I wanted to do was be an actor and perform Shakespeare.

DW: I remember your comment about the circus. This is something that still strikes people: the remarkable combination in Shakespeare of vulgarity and poetry, of high-flown ideas and low-flown ideas, the mixture of genres and conceptions, personalities and social types, the rich and varied presentation of life.

JB: That’s absolutely right. It’s total theatre. You look at almost every other playwright, they’re working within relatively narrow boundaries, whether it’s Tennessee Williams or Noel Coward, Harold Pinter or Samuel Beckett. Shakespeare, as you say, crosses all genres, can go from the most vulgar to the most sublime within a single scene, in The Winter’s Tale, for instance.

One is at the same time always aware that one is in the theatre, enhanced by the character of the Globe Theatre itself. The new one in London gives you some sense of what is was like: the audience surrounding the actor, this direct contact in broad daylight, no tricks, no scenery, no fancy lighting, no illusion …

DW: Shakespeare was apparently obliged to draw 1,500 to 2,000 people a day to the Globe.

JB: We tend perhaps to underestimate Shakespeare’s audience. An audience that simply wanted to throw vegetables at the stage would have gone to the bear-baiting next door.

Also, Shakespeare is continually raising the bar, making it more difficult for his audience. If you go from the rough and readiness of The Comedy of Errors and Henry VI through to Troilus and Cressida, it’s an extraordinary escalation in demands made on the audience’s sophistication, listening power, patience and intelligence.

DW: In your memoir you suggest that the language transcended the purely rational and touched all the senses, I wonder if you have any thoughts about that.

JB: Language that captures the whole body and soul. The rhythms, the cadences, the after-effect of poetry is not just literal. The juxtaposition of words that you would not normally juxtapose, the rhythm that stays with you after you’ve heard the line, which all genuine poetry does, I think. A good many of the other prose plays of the period are just that, they are literal, they are prosaic, they even concoct a story, but it’s the cadences in Shakespeare, the qualities of sound that stay with you in a good line of poetry.

DW: Can you speak about your experience with the Royal Shakespeare Company in Britain in the 1960s?

JB: I was there nearly five years and I’d had very little acting training up to that point. I had about six months at the Bristol Old Vic school, and then I was invited to join the Stratford company, and it was during the five years I was there that I really learned what the craft of acting was about, working with very good directors of course, people like Peter Hall, John Barton, Peter Brook, at their peak. And working with some very fine actors, like Paul Scofield (my very favorite), Ian Richardson, Ian Holm, Glenda Jackson …

This was a troupe of very fine actors, and it was through being in the rehearsal room, watching them at work, then being on stage with them, that I really learned what the craft of acting was. I learned a great deal about directing as well. So I found that experience absolutely invaluable. And the fact that one was at Stratford for most of the year, so that there were relatively few distractions and you gave yourself up entirely to the work … you had the leisure, the time to simply watch rehearsals and ruminate on the work and the things that were happening.

DW: Did you have the opportunity to be in any of Peter Brook’s productions?

JB: I was only in one, unfortunately: The Investigation [by Peter Weiss], a dramatizing of the transcript of the Auschwitz trials. It was only a public reading [in 1965]. That was the only time I encountered him as a director, but he was around the company a lot, and, of course, I saw all his productions.

DW: What did you think of Paul Scofield’s King Lear [directed on film by Peter Brook in 1971]?

JB: I never got to see his Lear on stage, I only got to see the movie, which I don’t think does him justice. I don’t think it’s a particularly good movie. It’s too tricksy and too affected, I think. From what I heard about the stage production, the director wasn’t getting in the way, he simply had Scofield in that space. I was so conscious of the camera work in the movie, that it undermined the power of Scofield.

DW: Were you there at the time of [Peter Weiss’s] Marat/Sade?

JB: Yes, I saw Marat/Sade several times. Again, much, much better than the film. I think the film lost a lot of the sheer, raw presence. It was at the Aldwych Theatre in London and I saw it a number of times, I was in the company by then, and the visceral impact of that piece on stage was astonishing. That was watered down considerably in the film.

DW: You mention in your book at one point that there was no permanent professional theatre in Sydney when you began to be interested in being an actor.

JB: That was when I first joined the profession, that was in 1960. Actually, I was fortunate, because a company was established then and I worked with it for two years. There had been other companies previously that had been set up and failed.

When I came back from England, in 1970, there was a theatre industry, but it was nowhere near what it is today. I think that Australia has a pretty healthy theatre culture, and that has been a product of the last 40 years, quite extraordinarily.

DW: Obviously, the city and the country have undergone extraordinary changes and you belong to a generation that was responsible in many ways for those changes. How do you see your generation, what its challenges were and what it accomplished?

JB: I think the main thing I felt when I came back from England was that Australia was too hidebound, it wasn’t creating its own theatre, we were still imitating the English system. So that each of the major companies would have one Shakespeare, one Bernard Shaw, one Neil Simon, one Feydeau farce, or whatever, and very little in terms of Australian content. So my generation was determined to turn that around and create an Australian theatre.

There was a group in Melbourne, the Australian Performing Group, starting that work, and then in 1970, myself and a partner, set up a little theatre called the Nimrod Theatre in King’s Cross and we started producing Australian work, which I then carried on for the next 14 years. And that kind of took off. There was also the beginning of a fledgling film industry here, and I think the theatre fed the film industry with new scripts, new talent, and also an awareness of making our own voice heard. That was a significant development, it hadn’t happened much before. We also produced international talent, like Cate Blanchett, who runs a theatre company here, and many other actors who got exported to Hollywood, unfortunately, but it was the beginning of the recognition of Australian talent that we hadn’t had before.

DW: The presence of Australian actors is quite remarkable in the global film industry.

JB: It is quite striking. My only regret is that they’re all putting on American accents and pretending to be Americans. Most of the world wouldn’t even know they were Australian performers. I long for the moment when we can start using our own voice more and stop being phony Americans.

DW: That’s a problem, and not just in Australia. Back to Shakespeare. You indicate the influence his work had on you, and you’ve established a company devoted to his work, so you obviously believe in the power of these plays to have that same sort of impact on other people, and not simply those from more privileged backgrounds. I’m curious whether you find it more or less difficult for contemporary audiences to respond to Shakespeare than it was, say, several decades ago, or what sort of changes, if any, there have been in audiences.

JB: I think 40 years ago we were very hooked on a traditional way, so-called, of performing Shakespeare, which meant imitating what people thought was the Old Vic or the Royal Shakespeare Company. This meant people putting on period costumes and a very English-sounding sort of accent. And so productions were very conservative, and people who went to the opera and ballet liked those sorts of productions, they were in line with their cultural expectations. Rather lavish, decorative, rather escapist, and obviously ‘high art,’ Culture with a capital C.

And what we’ve been trying to do is break all that down over the last 40 years. So that my company, which is now 21 years old, has always performed Shakespeare in modern dress, using Australian accents throughout, or whatever accent you happen to have, whether it was Polish, Russian, Chinese, whatever. You don’t hide your voice or where you come from. And we focused very much on contemporary issues in the plays, racism, anti-Semitism, gender conflict, anti-war sentiment, whatever is in the plays that can be brought out to make the plays resonate with an audience now. We’ve concentrated on that. We’re hardly alone, that’s been a global concern. Shakespeare Our Contemporary, as [Polish critic] Jan Kott called him.

I think Shakespeare is generally taught in school as badly as he always has been. We try to counteract that by having an education wing in our company, eight young actors who spend all year performing in schools all over Australia to some 60,000 students, bringing Shakespeare into the classroom as performance. We also run workshops and seminars so we can help teachers communicate Shakespeare, especially for classrooms full of children where English is the second language, a lot of Middle Eastern students, for instance.

We find when we go into Aboriginal communities, remote communities, and play Shakespeare for them, they take to it very readily, they have no problem. They have three or four languages of their own, this is just one more language. They pick it up quite easily and respond to the big issues, the symbolism, the mythology. When we work in those communities, we translate the plays into the various Aboriginal languages and they teach us their languages in return. So there’s a lot of activity overall in education, in theatre practice, rescuing Shakespeare from the traditionalists and the conservatives, who want to keep the plays in a museum context.

DW: This is a sweeping question. At its best, in your opinion, what sort of impact does a Shakespeare or any major artistic figure have on an audience member?

JB: I don’t want to sound complacent or self-congratulatory, but I think we have made quite an impact on people. We’ve played before more than two million people, in this company, and the kind of feedback we get from younger and older people alike is gratifying. They didn’t know Shakespeare could be so entertaining, so much fun, we didn’t think we could ever understand it. People ask, who did the translation? Nobody, that was Shakespeare. So I think we’ve achieved something in performing it in a way that makes it very accessible and clear to audiences. That was my aim, that’s what I set out to do. So I think we’ve had a record of success with that.

DW: Are there plays or parts that are favorites of yours, as an actor, director or spectator?

JB: I’m often asked that, and it’s the old question: which is your favorite child? One has to love the play that one is directing or acting in, because you put about twelve months into every one, in terms of thinking about it, casting, rehearsing, getting it up. One devotes a year of one’s life to each play, and one falls in love with each one.

I think the play I most admire and that I’m most in awe of is King Lear. I’ve done that several times, and never really gotten very far up the mountain with that one. I think of all the plays, my very favorites would be Henry IV, Parts 1 and 2, in terms of the sheer scope and range of those plays, and the appeal they have.

DW: You mention Henry IV, and I was thinking about that the other day in relation to Anonymous and the Earl of Oxford. The scene of the carriers preparing to load their horses at the inn [Part I, Act II, Scene I], talking about the fleas and the price of oats… The notion that the Earl of Oxford could have written some of those scenes is so preposterous.

JB: Absolutely ludicrous. That scene you mention, I think, is a piece of verbatim theatre. I think Shakespeare was lying in bed and heard those two guys outside the window talking about the price of oats and the horses with the shakes, or whatever. It’s so authentic, and, again, I don’t see the Earl of Oxford knowing people like Bottom and Quince and Bardolph and Pistol and Mistress Quickly. This was totally out of his realm, it is so ludicrous, I agree.

DW: In our view, this is not just an attack on who wrote the plays, it’s an attack on the plays. I think there’s something offensive to certain people about the grandness and universality of the plays, they are so titanic, and certainly we reject the notion that women should only write about women, and Jews about Jews and Australians about Australians … There’s something about the universal figure of this artist that is very powerful, I think.

JB: I totally agree.

DW: Do you have any thoughts about the ‘authorship controversy,’ or is it something you simply ignore?

JB: I tend to ignore it. It’s been around for so long now, I heard about it at university, whether it was Christopher Marlowe, or Oxford, or Pembroke, or even Queen Elizabeth. Crazy, crazy notions. Look, people like parlor games, they like conspiracy theories, but I think there’s such a body of good writing now from good scholars, like Jonathan Bate, James Shapiro, Stephen Greenblatt, even Bill Bryson, and they all have good answers to the conspiracy theories, and it’s such nonsense. I guess there’s always a certain intrigue for people, and there’s a frustration because Shakespeare is so enigmatic.

Actually, we know a good deal more about his life than most of the playwrights of the period. We know nothing about John Webster, for instance, at all. He was up there with Shakespeare in terms of popularity. It’s remarkable we know as much about Shakespeare as we do. But because he remains enigmatic as a personality, and his character is so hard to pin down, people want to create their own Shakespeare, someone they would like to see as the author of those plays. Gay people will say he must have been gay, Catholics say he must have been a Catholic, atheists insist he must obviously have been an atheist, we all want to create a Shakespeare who appeals to us. That’s why people have this romantic notion of some English nobleman, rather than someone they find too shadowy to connect with.

DW: Any final words on why you continue to direct and act in Shakespeare, what the continuing appeal is?

JB: Sometimes, oddly enough, I feel it keeps me young, because it’s a continual exploration, with the acting, directing and researching. I guess like some sort of crazy scientist, someone who’s fascinated by one aspect of life, you go on exploring, researching, getting excited by it. I feel sorry for actors and directors who are jaded by their careers and say, ‘I’m doing this crap just for the money,’ ‘I’m in this TV soap or this B-grade movie because I’ve got to earn a living.’ I’ve never felt that. If you’re working on Shakespeare, I think it’s a privilege to devote your life to working alongside such a great mind. It can never be boring or exhausting, it’s always revitalizing. I still feel like the 15-year-old I was when I first discovered Shakespeare.

Tuesday, January 3, 2012

Did you know?

The German-born physicist Albert Einstein developed the first of his groundbreaking theories while working as a clerk in the Swiss patent office in Bern. After making his name with four scientific articles published in 1905, he went on to win worldwide fame for his general theory of relativity and a Nobel Prize in 1921 for his explanation of the phenomenon known as the photoelectric effect. An outspoken pacifist who was publicly identified with the Zionist movement, Einstein emigrated from Germany to the United States when the Nazis took power before World War II. He lived and worked in Princeton, New Jersey, for the remainder of his life.

Sunday, November 28, 2010

Pair of Aluminum Atomic Clocks Reveal Einstein's Relativity at a Personal Scale

ScienceDaily (Sep. 24, 2010) — Scientists have known for decades that time passes faster at higher elevations -- a curious aspect of Einstein's theories of relativity that previously has been measured by comparing clocks on the earth's surface and a high-flying rocket.

Now, physicists at the National Institute of Standards and Technology (NIST) have measured this effect at a more down-to-earth scale of 33 centimeters, or about 1 foot, demonstrating, for instance, that you age faster when you stand a couple of steps higher on a staircase.Described in the Sept. 24 issue of Science, the difference is much too small for humans to perceive directly -- adding up to approximately 90 billionths of a second over a 79-year lifetime -- but may provide practical applications in geophysics and other fields.

Similarly, the NIST researchers observed another aspect of relativity -- that time passes more slowly when you move faster -- at speeds comparable to a car travelling about 20 miles per hour, a more comprehensible scale than previous measurements made using jet aircraft.

NIST scientists performed the new "time dilation" experiments by comparing operations of a pair of the world's best experimental atomic clocks. The nearly identical clocks are each based on the "ticking" of a single aluminum ion (electrically charged atom) as it vibrates between two energy levels over a million billion times per second. One clock keeps time to within 1 second in about 3.7 billion years and the other is close behind in performance. The two clocks are located in different laboratories at NIST and connected by a 75-meter-long optical fiber.

NIST's aluminum clocks -- also called "quantum logic clocks" because they borrow logical decision-making techniques from experimental quantum computing -- are precise and stable enough to reveal slight differences that could not be seen until now. The clocks operate by shining laser light on the ions at optical frequencies, which are higher than the microwave frequencies used in today's standard atomic clocks based on the cesium atom.

Optical clocks could someday lead to time standards 100 times more accurate than today's standard clocks.

The aluminum clocks can detect small relativity-based effects because of their extreme precision and high "Q factor" -- a quantity that reflects how reliably the ion absorbs and retains optical energy in changing from one energy level to another -- says NIST postdoctoral researcher James Chin-Wen Chou, first author of the paper.

"We have observed the highest Q factor in atomic physics," Chou says. "You can think about it as how long a tuning fork would vibrate before it loses the energy stored in the resonating structure. We have the ion oscillating in sync with the laser frequency for about 400 thousand billion cycles."

The NIST experiments focused on two scenarios predicted by Einstein's theories of relativity. First, when two clocks are subjected to unequal gravitational forces due to their different elevations above the surface of the Earth, the higher clock -- experiencing a smaller gravitational force -- runs faster.
Second, when an observer is moving, a stationary clock's tick appears to last longer, so the clock appears to run slow. Scientists refer to this as the "twin paradox," in which a twin sibling who travels on a fast-moving rocket ship would return home younger than the other twin. The crucial factor is the acceleration (speeding up and slowing down) of the travelling twin in making the round-trip journey.

NIST scientists observed these effects by making specific changes in one of the two aluminum clocks and measuring the resulting differences in the two ions' relative ticking rates, or frequencies.

In one set of experiments, scientists raised one of the clocks by jacking up the laser table to a height one-third of a meter (about a foot) above the second clock. Sure enough, the higher clock ran at a slightly faster rate than the lower clock, exactly as predicted.

The second set of experiments examined the effects of altering the physical motion of the ion in one clock. (The ions are almost completely motionless during normal clock operations.) NIST scientists tweaked the one ion so that it gyrated back and forth at speeds equivalent to several meters per second.
That clock ticked at a slightly slower rate than the second clock, as predicted by relativity. The moving ion acts like the traveling twin in the twin paradox.

Such comparisons of super-precise clocks eventually may be useful in geodesy, the science of measuring the Earth and its gravitational field, with applications in geophysics and hydrology, and possibly in space-based tests of fundamental physics theories, suggests physicist Till Rosenband, leader of NIST's aluminum ion clock team.

NIST scientists hope to improve the precision of the aluminum clocks even further, as much as 10-fold, through changes in ion trap geometry and better control of ion motion and environmental interference. The aim is to measure differences in timekeeping well enough to measure heights to an accuracy of 1 centimeter, a performance level suitable for making geodetic measurements. The paper suggests that optical clocks could be linked to form a network of "inland tidal gauges" to measure the distance from the earth's surface to the geoid (the surface of the earth's gravity field that matches the global mean sea level). Such a network could be updated far more frequently than current techniques.

(For the rest of this article, please click on the link in this blog title's post above it.)

Friday, September 17, 2010

Physicists Are Close to Uncovering the Fundamental Rules of Reality

String theorist Brian Greene on his hopes for science over the next 30 years

By Andrea Ventura
Discover October 2010 issue
Published online September 14, 2010


This article is part of DISCOVER's 30th anniversary special section, including 11 eminent scientists' predictions about the next 30 years. Share your thoughts on the future of science at the Science Not Fiction blog.

"My great hope is that we will figure out how to meld gravity and quantum mechanics, realizing a dream that can be traced back to Einstein and that, in its more modern form, has captivated two generations of theoretical physicists. Such a theory might provide us the tools for gaining a clear understanding of the origin of the universe.

A vital part of such progress requires making contact between experiment and our theoretical attempts to quantize gravity and unify nature’s forces. For a long time we have been pursuing theoretical ideas like string theory without input from experiment or observation, and that is an unusual way for a science to evolve. In three decades—perhaps sooner with the help of the Large Hadron Collider and satellite-based astronomical observations—I would hope this changes. Should the observations support the theory, great; should they rule it out, that’s great too, because we’d be able to move on, full throttle, to other ideas. A big puzzle now facing string theory is that there are many possible forms for the extra dimensions that the mathematics requires. In the mid-1980s there were dozens. Today that number has soared, by some estimates to 10500 if not more.

There’s no way theorists can possibly examine all of them; 10500 dwarfs the number of particles in the observable universe! So we will continue to search for some mathematical equation that pinpoints a handful or even one specific form for the extra dimensions, allowing us to determine a single universe that string theory predicts. Alternatively, we may establish that there is not a unique universe but many. Each universe would make use of a different form for the extra dimensions, with our universe being just one of many in a grand multiverse. That would be one of the most profound revolutions in thinking we have ever sustained.

I am confident that well before 2040 we will nail down what dark matter is. Identifying dark energy will be harder, but we might nail that, too. And if I allow my imagination to run wild, I would love it if we had some deep insight that let us understand what space and time actually are. We know a lot about the features of space and time, what they can do—but many of us believe these are not fundamental. Identifying the constituents of space and time would be a grand insight."

Brian Greene is a string theorist at Columbia University, author of the best seller The Elegant Universe, and cofounder of the World Science Festival in New York.

Space, Time and String Theory

The Official String Theory Website


If string theory is a theory of gravity, then how does it compare with Einstein's theory of gravity? What is the relationship between strings and spacetime geometry?

Strings and gravitons

The simplest case to imagine is a single string traveling in a flat spacetime in d dimensions, meaning that it is traveling across space while time is ticking, so to speak. A string is a one-dimensional object, meaning that if you want to travel along a string, you can only go forwards or backwards in the direction of the string, there is no sideways or up and down on a string. The string can move sideways or up and down in spacetime, though, and as the string moves around in spacetime, it sweeps out a surface in spacetime called the string worldsheet, a two-dimensional surface with one dimension of space and one dimension of time.

The string worldsheet is the key to all the physics of the string. A string oscillates as it travels through the d-dimensional spacetime. Those oscillations can be viewed from the two-dimensional string worldsheet point of view as oscillations in a two-dimensional quantum gravity theory. In order to make those quantized oscillations consistent with quantum mechanics and special relativity, the number of spacetime dimensions has to be restricted to 26 in the case of a theory with only forces (bosons), and 10 dimensions if there are both forces and matter (bosons and fermions) in the particle spectrum of the theory.

So where does gravity come in?

If the string traveling through spacetime is a closed string, then the spectrum of oscillations includes a particle with 2 units of spin and zero mass, with the right type of interactions to be the graviton, the particle that is the carrier of the gravitational force.

Where there are gravitons, then there must be gravity. Where is the gravity in string theory?

Strings and spacetime geometry

The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime. But how do the Einstein equations come out of string theory?

If a closed string is traveling in a curved spacetime, then the coordinates of the string in spacetime feel this curvature as the string propagates. Once again, the answer lies on the string worldsheet. In order for their to be a consistent quantum theory in this case, the curved space in which the string travels must be a solution to the Einstein equations.
Now this is really something! This was a very convincing result for string theorists. Not only does string theory predict the graviton from flat spacetime physics alone, but string theory also predicts the Einstein equation will be obeyed by a curved spacetime in which strings propagate.

What about strings and black holes?

Black holes are solutions to the Einstein equation, therefore string theories that contain gravity also predict the existence of black holes. But string theories give rise to more interesting symmetries and types of matter than are commonly assumed in ordinary Einstein relativity. So black holes are more interesting to study in the context of string theory, because there are more kinds to study.

Is spacetime fundamental?

Note that there is a complication in the relationship between strings and spacetime. String theory does not predict that the Einstein equations are obeyed exactly. String theory adds an infinite series of corrections to the theory of gravity. Under normal circumstances, if we only look at distance scales much larger than a string, then these corrections are not measurable. But as the distance scale gets smaller, these corrections become larger until the Einstein equation no longer adequately describes the result.

In fact, when these correction terms become large, there is no spacetime geometry that is guaranteed to describe the result. The equations for determining the spacetime geometry become impossible to solve except under very strict symmetry conditions, such as unbroken supersymmetry, where the large correction terms can be made to vanish or cancel each other out.
This is a hint that perhaps spacetime geometry is not something fundamental in string theory, but something that emerges in the theory at large distance scales or weak coupling. This is an idea with enormous philosophical implications.

Random Acts of Science

By GRAHAM FARMELO
Published: June 11, 2010
The New York Times


Quantum mechanics is the most revolutionary scientific theory to appear in the past 150 years. In the atomic domain, it superseded laws first set out by Isaac Newton a quarter of a millennium earlier and has since had an unbroken string of successes. Today, it continues to give an utterly reliable account of the behavior of the subatomic world, yet there are nagging doubts that there is something rotten at its core.

In his lively new book, “Quantum,” the science writer Manjit Kumar cites a poll about the interpretation of quantum mechanics, taken among physicists at a conference in 1999. Of the 90 respondents, only four said they accepted the standard interpretation taught in every undergraduate physics course in the world. Thirty favored a modern interpretation, laid out in 1957 by the Princeton theoretician Hugh Everett III, while 50 ticked the box labeled “none of the above or undecided.” Almost a century after a few physicists first set out the basic theory, quantum mechanics is still a work in progress.

It was the German theoretician Max Planck who first presented the idea that energy is fundamentally granular. In a lecture given in the closing weeks of 1900, he described his bizarre proto-theory that when light and matter interact, energy cannot be transferred in arbitrary amounts, as would be expected on the basis of Newton’s account. Rather, Planck suggested, energy transfers take place only in discrete chunks, which he called “quanta.” A deeply conservative thinker, he was never comfortable with this notion, which he saw as a “purely formal assumption,” and was unconvinced when the young Albert Einstein suggested — in what he considered to be his only revolutionary contribution to science — that it was possible to think of light in terms of particles, later called photons. Planck died, almost 50 years later, unwilling to believe the picture of light that he himself had introduced. This is a classic example of the adage that physics progresses through a succession of funerals — of the pioneers who could not live with the consequences of their most radical work.

In resisting the photon concept, Planck was in good company. Another influential skeptic was the Danish physicist Niels Bohr, a remarkably profound thinker and inveterate mumbler who continually struggled to find coherent expressions of his ideas. (“You should never express more clearly than you can think,” he would whisper to often-baffled colleagues.) Bohr at first refused to believe in the reality of photons, even after the American experimenter Arthur Compton first found compelling evidence for them in 1922. For a short time, Einstein was in the vanguard of quantum theory, while Bohr lagged ­behind.

For many of the physicists who forged the first comprehensive quantum theory in the second half of the 1920s, Bohr was a kind of intellectual godfather. Through cajoling and persistent tactful criticism, he helped them to do their best work and produce the components of the theory, whose coherence and unity emerged only gradually. One of its creators, the taciturn English physicist Paul Dirac, liked to point out that quantum mechanics was the first mathematical theory in science in which the discoverers did not fully understand the meaning of the terms in their own equations.

“Quantum” is a wide-ranging account, written for readers who are curious about the theory but want to sidestep its mathematical complexities. It’s full of a surprising amount of detail, perhaps rather more than most readers will want. The story is chock-full of colorful characters, including the two physicists who independently set out the first two versions of the theory, which initially appeared to be quite different. The first was the young Werner Heisenberg, not two years past his doctorate, fun-seeking and intensely competitive, not least at the Ping-Pong table. The other was the older Erwin Schrödinger, an Austrian polymath who scandalized his conservative colleagues by showing up at conferences in his climbing gear, sometimes accompanied by an adolescent lover.

Kumar will not win prizes for historical originality. This is an unapologetically orthodox account, largely derived from the standard sources and without the benefit of some of the latest scholarship. Occasionally, the narrative appears to be driven by a wish to thread together every amusing story, anecdote and famous quotation. There is, however, no doubt about the author’s skill in making accessible the philosophical controversies in his story, especially the debates between Bohr and Einstein. For Bohr, physics was not about finding out what nature is, but about what can be said about it. Quantum mechanics was a complete theory of the behavior of matter and light, and we just have to come to terms with the limitations it places on what can be known, for example as illustrated by the Heisenberg uncertainty principle. Einstein was having none of it. He believed that there is an objective world out there and that it is the job of scientists to describe it. The appearance of probabilities in the theory was, for him, evidence of its incompleteness.

In 1964, after both Einstein and Bohr had died, the Irish physicist John Bell did something they had failed to do; he found a way of testing experimentally which of their opposing viewpoints most accurately described nature, by laying out a mathematical theorem. In the denouement of “Quantum,” Kumar describes the result of the experiment, which I shall not reveal, though I think it fair to say it leaves us feeling that the story of quantum mechanics is not yet over.

In the late 1970s, I had the pleasure of talking with John Bell about the Bohr-­Einstein debates during a train journey from Oxford to London. Every seat was taken, so we had to stand. Pressed against me by sullen commuters, Bell summarized his apparently reluctant conclusion as we pulled into Paddington station: “Bohr was inconsistent, unclear, willfully obscure and right. Einstein was consistent, clear, down-to-earth and wrong.”

Einstein, always his own man, never cared whether his colleagues regarded him as wrongheaded. While the public all over the world regarded him as a kind of sage, he knew that his fellow physicists — especially younger ones — saw him as eccentric or even senile. In the early 1940s, the theorist John Wheeler visited him at his home in Princeton to brief him on a new development in quantum theory and to ask if he would now accept it. “I still can’t believe that the good Lord plays dice,” Einstein replied. After a pause, he added, “Maybe I have earned the right to make my mistakes.” Yet Einstein never publicly accepted that he was mistaken; nothing was going to persuade him to change his way of looking at the world. A few years later, he told a friend that he believed “in a world that objectively exists, and which I, in a wildly speculative way, am trying to capture.” It seemed he could not live with the consequences of his most revolutionary idea.

Kumar ends his fascinating book with the verdicts of some of today’s leading physicists on Bohr’s and Einstein’s contrasting views on quantum mechanics. It is clear from this that quite a few of Einstein’s most distinguished successors believe he was right to say that the theory is fundamentally unsatisfactory and that we need a deeper account of reality. The sage of Princeton may yet have the last chuckle.

Graham Farmelo is the author of “The Strangest Man,” a biography of Paul Dirac.

A version of this review appeared in print on June 13, 2010, on page BR25 of the Sunday Book Review.