General Relativity's Frame Dragging Confirmed

Earth's spin warps space around the planet, according to a new study that confirms a key prediction of Einstein's general theory of relativity. After 11 years of watching the movements of two Earth-orbiting satellites, researchers found each is dragged by about 6 feet (2 meters) every year because the very fabric of space is twisted by our whirling world. The results, announced today, are much more precise than preliminary findings published by the same group in the late 1990s.

[Thanks to DVD for bringing this link to my attention.]

The Ph.D. gap

http://discusseducation.blogspot.com/2004/10/more-on-phd-gap.html

On the black hole information paradox

After nearly 30 years of arguing that a black hole destroys everything that falls into it, Stephen Hawking is saying he was wrong. It seems that black holes may after all allow information within them to escape. Hawking will present his latest finding at a conference in Ireland next week.

Other articles:
http://www.newscientist.com/news/news.jsp?id=ns99996193
http://www.spacedaily.com/news/blackhole-04c.html

I didn't see this story back when it was published. The conference was on July 21st, and I'm just now learning about it through the American Physical Society's coverage. It's not yet clear that this new explanation is correct. John Preskill has not yet accepted the encyclopedia in part because he says "I didn't understand the talk." Kip Thorne (Hawking's betting partner) is apparently also unconvinced.

A very accessible (but still technical) discussion of the information paradox can be found on Baez's website.

Field Quantization

I'm currently refreshing my understanding of quantum field theory by reading Field Quantization by Greiner & Reinhardt. I hope to follow this up by tackling those parts of Relativistic Quantum Fields (Bjorken & Drell) and Quantum Field Theory (Itzykson & Zuber) that I did not adequately master during my graduate training. Field Quantization covers things in much the same level of detail as Greiner's Relativistic Quantum Mechanics and Quantum Electrodynamics, although some steps are skipped in derivations, as befits a book meant for advanced graduate students. Still, the book covers the mathematical details of relativistic quantum field theory in a simple way, contains more details than most other standard treatments and includes many instructive examples that most other authors just gloss over or exclude altogether.

One thing worth noting about this book is that although it has far too many typos, it has nowhere near the ridiculous number contained in Greiner's RQM or QED books. The publisher should be completely ashamed of himself for marring such an excellent pedagogical exposition in those other books with typographic blunders so numerous that the inexpert reader is thoroughly distracted from the clarity of the exposition in those two books. Field Quantization at least keeps the number of errors down low enough where the text is still readable and maybe even usable in a classroom environment.

So what is the Poincaré Conjecture?

Wolfram's site explains it quite well.

In its original form, the Poincaré conjecture stated that every simply connected closed three-manifold is homeomorphic to the three-sphere. More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by Poincaré.

Soon thereafter this specialized conjecture for the case n=3 was generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere, which is logically equivalent to the earlier weaker conjecture in the case n=3. The n=1 case is trivial, the n=2 case was known to 19th century mathematicians, n=4 was proved by Freedman in 1982 for which he was awarded the 1986 Fields medal, n=5 was proved by Zeeman in 1961, n=6 was proved by Stallings in 1962, and n>6 was proved by Smale in 1961 (although Smale subsequently extended his proof to include all n>4).

The n=3 case has baffled mathematicians since it was first proposed, spawning a multitude of incorrect proofs by many mathematicians, including Poincare and Whitehead. However, it now appears that this remaining case has finally been proved by Perelman although the proof has not yet been fully verified. In fact, Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow.

Perelman's two papers are The entropy formula for the Ricci flow and its geometric applications and Ricci flow with surgery on three-manifolds. For the non Fields Medal candidates among us, a more accessible (but still highly technical) discussion can be found on the AMS site.

2004 Nobel Prize in Physics

The winners of the 2004 Nobel Prize in Phyiscs have just been announced.

"for the discovery of asymptotic freedom in the theory of the strong interaction"

David J. Gross - Kavli Institute for Theoretical Physics, University of California at Santa Barbara
H. David Politzer - California Institute of Technology
Frank Wilczek - Massachusetts Institute of Technology

Most Interesting Open Math Questions

1. Twin Prime Conjecture: There are an infinite number of twin primes. (This problem was known to the ancient Greeks; it’s related to Hilbert’s Eighth Problem.)

2. Goldbach Conjecture: Every even integer larger than 2 is the sum of two primes. (This problem was known to the ancient Greeks; it’s related to Hilbert’s Eighth Problem.)

3. Riemann Hypothesis/Conjecture: All of the non-trivial zeros of the Zeta Function are located on the line Re(s) = ½. (This is the central point of Hilbert’s Eighth Problem; also one of the Millennium Problems.)

4. Birch & Swinnerton-Dyer Conjecture: If a given elliptic curve has an infinite number of solutions, then the associated L-series has value 0 at a certain fixed point. (One of the Millennium Problems)

5. Hodge Conjecture: For projective algebraic varieties, Hodge cycles are actually rational linear combinations of algebraic cycles. (One of the Millennium Problems)

6. Jacobian Conjecture: If det[F’(x)]=1 for a polynomial map F, then F is bijective with polynomial inverse. (One of Steve Smale’s 1998 problems)

7. Under what conditions does a solution exist to the Navier-Stokes non-linear partial differential equation? Do these equations have solutions that last for all time, given arbitrary sufficiently nice initial data, or do singularities develop in the fluid flow that prevent the solution from continuing? (One of the Millennium Problems)

8. How many limit cycles are possible for a given ODE? (Half of Hilbert’s Sixteenth Problem)

9. Describe the shapes possible for the graphs of algebraic functions with only real numbers allowed as solutions. (Half of Hilbert’s Sixteenth Problem)

10. Does P = NP? (One of the Millennium Problems)

The Poincare Conjecture is not included in this list because based on what I've read (not that I'm qualified to evaluate a lot of what's being written about it) it seems that the solution by Perelman is correct.

A plethora of additional unsolved mathematics problems can be found at mathworld.wolfram.com.

Most Interesting Open Physics Questions

I make no pretense of originality in the following. This list of what I consider the most interesting and important unanswered physics questions borrows heavily from Warren Siegel, John Baez and Lee Smolin. However, this compilation does introduce one or two new ideas not contained in any of their writings. It is worth noting that nine questions (with multiple subparts) is almost certainly excessive. A greater mind than mine could probably collapse these into fewer questions, and it is likely that identifying the right relations between these questions would in and of itself lead to some sort of breakthrough.

Question 1 – The Standard Model
(a) Are all 61 fundamental particles in the Standard Model truly elementary, or do they have some substructure?  Can the various particles be explained as manifestations of a more fundamental entity? Do quarks or leptons have any substructure, or are they truly elementary?
(b) What is the mechanism of CP violation? Is it explicable entirely within the Standard Model, or is some new force or mechanism required? Related to this, is there more matter than antimatter? If so, why?
(c) Are the measurable dimensionless parameters that characterize Nature calculable in principle or are they merely determined by historical or quantum mechanical accident and incalculable?
(d) Are there exactly three generations of leptons and quarks? Why do the generations have the structure they do?
(e) Why do these particles have the precise masses they do? Or is this an unanswerable question? How do we understand neutrino mass?
(f) Is it possible to calculate masses of hadrons from QCD?

Question 2 – String Theory
(a) Does string theory actually work? Do problems fixed at the perturbative level return with the non-perturbative 11th dimension described by non-renormalizable membrane theory? Is a 10-dimensional perturbation expansion reasonable for an 11-dimensional theory?
(b) Are there any other strings than the D=10(11) and 26 ones, not counting dual theories?
(c) Do the four forces really become unified at sufficiently high energy?
(d) What are the fundamental degrees of freedom of M-theory? Does M-theory describe Nature? Does M-theory give specific predictions about elementary particles? If so, are they correct?

Question 3 - Gravity
(a) What is gravity? Can we merge quantum theory and general relativity to create a quantum gravity?
(b) What, if anything, do gravity waves (now that they have been discovered) teach us about Nature?
(c) Does the graviton exist? If so, is it fundamental?
(d) Can quantum gravity help explain the origin of the Universe?

Question 4 – Cosmology
(a) What happened at or before the Big Bang? Was there really an initial singularity? Does the history of the Universe go back in time forever or only a finite amount? Will the future of the Universe go on forever or not?
(b) Is the Universe infinite in spatial extent? More generally, what is the topology of space?
(c) Why is there an arrow of time? Why is the future so much different from the past?
(d) How can we understand the cosmological horizon problem? Why is the Universe almost, but not quite, homogeneous, on the very largest distance scales? Is this the result of an inflationary epoch? If so, what caused this inflation?
(e) Why does the cosmological constant have the value that it has? Is it zero? How do we reconcile the requirements of cosmology with the predictions of quantum field theory or string theory? Is the cosmological constant actually constant?
(f) What is the real solution to the "dark matter" and "dark energy" problems? Do they actually exist? If so, what are they? If not, how and why is gravity modified on large scales?
(g) Do we live in a false vacuum (i.e., not the lowest possible energy state)?

Question 5 – Black Holes
(a) Do black holes exist? What happens inside a black hole? What do you do with the singularities? Doesn't a singularity signify a breakdown of the theory? Do naked singularities exist, or is the Cosmic Censorship Hypothesis true?
(b) Do black holes evaporate through Hawking radiation? If so, what happens when they radiate completely away?
(c) Has the information paradox really been resolved? Was Hawking right in 1975, or was Hawking right in 2005?

Question 6 - Confinement & the Mass Gap
(a) Does confinement work? Can we calculate the observed linear Regge trajectories and see what happens to the bound states as their excitation energy increases?
(b) Is there a mathematically rigorous formulation of a relativistic quantum field theory describing interacting fields in four dimensions?
(c) Can we rigorously solve the SU(2) Yang-Mills theory in four dimensions so that we can quantitatively predict quark and gluon confinement as well as the existence of a mass gap between the Planck scale and the electroweak unification scale? (This is actually one of the seven Millenium mathematics problems.)
(d) Does QCD predict that quarks and gluons become deconfined and form plasma at high temperature? If so, what are the characteristics of the deconfinement phase transition? Does this really happen in Nature?

Question 7 – Nature’s Dimensionality
(a) Is Nature four-dimensional? If not, then why does Nature appear to have one time and three space dimensions?
(b) Does compactification work? What forces the extra dimensions to hide and prevents them from reappearing? Does compactification destroy predictability? Do the extra dimensions really do anything we couldn't reproduce without them?

Question 8 - Supersymmetry
(a) Is Nature supersymmetric? If so, how is supersymmetry broken? If not, is supersymmetry still useful?
(b) Is fine tuning really that much worse than any other kind of tuning? Are superpartners any worse than non-minimal Higgs?

Question 9 – Quantum Mechanics
How should we think about the foundations of quantum mechanics? What is meant by a "measurement" in quantum mechanics? Does "wavefunction collapse" actually happen as a physical process? If so, how and under what conditions? If not, what happens instead? Can we make sense of the theory as it stands today? If not, can we invent a new theory that does make sense?

Master's Level Mathematics Curriculum

Algebra:
This recommendation assumes that a year-long undergraduate course at the level of A First Course in Abstract Algebra by Fraleigh has been completed.

• Algebra by Lang
• Algebra by Hungerford

Analysis:
This recommendation assumes that a year-long undergraduate course at the level of "Baby Rudin" has been completed.

• Real and Functional Analysis and Complex Analysis by Lang
• Real and Complex Analysis and Functional Analysis by Rudin

Topology:
This recommendation assumes that a year-long undergraduate course in point-set topology at the level of Topology by Munkres has been completed.

• A Basic Course in Algebraic Topology by Massey, Algebraic Topology by Fulton, and Elements of Algebraic Topology by Munkres
• Differential Topology by Guillemin & Pollack

Master's Level Physics Curriculum

Mathematical Physics:
This recommendation assumes that a text at the level of Arfken or Butkov has been completed as an undergraduate.

• Principles of Advanced Mathematical Physics Volumes I & II by Robert Richtmyer

Mechanics:

• Mechanics by Landau & Lifshitz is a good place to start. Brilliant exposition, but too short; this can serve as a supplement to the standard Goldstein text.
• Classical Mechanics (3rd Edition) by Herbert Goldstein et al
• Theoretical Mechanics of Particles and Continua by Fetter & Walecka. The first half (on particles) is standard, the material is the same as (but the exposition inferior to) Goldstein or L&L. The second part (on continua) provides excellent coverage of a great deal of material that is not covered in ANY of the other standard graduate mechanics texts.
• Problems and Solutions on Mechanics: Major American Universities Ph.D. Qualifying Questions and Solutions by Yung-Kuo Lim

Electrodynamics:

• Classical Electrodynamics by John David Jackson
• Electrodynamics of Continuous Media by Landau & Lifshitz
• Problems and Solutions on Electromagnetism (Major American Universities Ph.D. Qualifying Questions and Solutions) and Problems and Solutions on Optics (Major American Universities Ph.D. Qualifying Questions and Solutions) by Yung-Kuo Lim

Quantum Mechanics:

• Unfortunately, undergraduate quantum mechanics does not adequately prepare you for the material you'll have to tackle in graduate quantum mechanics. Invariably there are (sometimes amazingly large) gaps which must be filled in before moving on. Two outstanding texts for doing this are Quantum Mechanics (3rd Edition) by Leonard Schiff and Quantum Mechanics: Non-Relativistic Theory, Third Edition by Landau & Lifshitz
• Once that's accomplished, you can move on to "relativistic quantum mechanics" and quantum field theory. The standard treatment is in Relativistic Quantum Mechanics and Relativistic Quantum Fields by Bjorken & Drell, which is a bit dated but still quite valuable. A much more accessible (and consequently less thorough) treatment can be found in Relativistic Quantum Mechanics: Wave Equations, Quantum Electrodynamics and Field Quantization by Walter Greiner.
• Quantum Electrodynamics by Landau & Lifshitz contains material not covered in either of the treatments mentioned in the previous bullet point, in particular with respect to phenomenology and calculational techniques.
• Quantum Field Theory by Itzykson & Zuber rounds out and brings up to the date the material in the previously mentioned books.
• Problems and Solutions on Quantum Mechanics: Major American Universities Ph. D. Qualifying Questions and Solutions (Major American Universities Ph. D. Qualifying Questions and Solutions) and Problems and Solutions on Atomic, Nuclear and Particle Physics: Major American Universities Ph.D. Qualifying Questions and Solutions by Yung-Kuo Lim, just to prove you know what you're talking about.

Statistical Mechanics:

• Statistical Mechanics by Donald McQuarrie
• Statistical Mechanics: A Set of Lectures by Richard Feynman
• Statistical Physics and Physical Kinetics by Landau & Lifshitz
• Problems and Solutions on Thermodynamics and Statistical Mechanics (Major American Universities Ph.D. Qualifying Questions and Solutions) by Yung-Kuo Lim

General Relativity:
I'm not sure why so few graduate programs include general relativity. It's really hard to consider yourself a fully educated physicist without having a grounding in this subject.

• Gravitation by Misner, Thorne & Wheeler for unparalleled breadth
• General Relativity by Robert Wald for added depth in the more important parts and a more modern treatment than MTW
• The Classical Theory of Fields by Landau & Lifshitz. This book has several very good chapters on electrodynamics, but it is for its discussion of relativity that you should devour this book.
• Problems and Solutions on Relativity and Miscellaneous Topics (Major American Universities Ph. D. Qualifying Questions and Solutions) by Yung-Kuo Lim

An Undergraduate Mathematics Curriculum

The indispensable lower division core consists of the following three courses.

• Calculus by Howard Anton
• Linear Algebra and Its Applications by David Lay
• Elementary Differential Equations by Derrick & Grossman

A standard problem in all mathematics curricula is how to transition from lower-division problem-based courses such as the three above to the upper-division proof-based courses. The following trio of books accomplishes this quite well.

• How to Prove It: A Structured Approach by Daniel Velleman
• Introduction to Analysis by Edward Gaughan
• Set Theory and Logic by Robert Stoll

The following constitute the standard trio of upper-division courses that all mathematics majors should cover to be considered mathematically mature (as well as mathematically literate).

• A First Course in Abstract Algebra by John Fraleigh
• Principles of Mathematical Analysis by Walter Rudin
• Topology by James Munkres

Some standard elective courses in the area of applied mathematics are the following.

• Freund's Mathematical Statistics by Miller & Miller
• Numerical Analysis by Burden & Faires
• Partial Differential Equations : An Introduction by Walter Strauss

Some standard elective courses in the area of pure mathematics are the following.

• Introduction to Modern Set Theory by Judith Roitman
• Introduction to Mathematical Logic by Mendelson & Mendelson
• Introductory complex analysis and applications by William Derrick

One topic that is seldom covered in the undergraduate curriculum is geometry. This course is required for most mathematics education majors, but not mathematics majors. This is a shame since this course, even though it doesn't lead into any of the major fields of mathematics research, provides a most insightful foray into how an axiomatic mathematical system should function. I found it utterly fascinating and am very glad I took the opportunity to include it in my coursework.

• Euclidean and non-Euclidean geometries: Development and history by Marvin Greenberg

An Undergraduate Physics Curriculum

Introductory Physics:

• Sears and Zemansky's University Physics by Hugh Young. This book is much better than the standard text by Halliday & Resnick. You’ll probably want to pick up the study guide and solution manual to help you navigate through the text.
• Modern Physics by Llewellyn & Tipler
• Feynman Lectures On Physics by Feynman. You will never learn the material just from this book, but in conjunction with a more accessible book such as Young, Feynman’s Lectures will yield an insight on every page.

Mathematical and Computational Physics:

• Mathematical Physics by Eugene Butkov. This book is better than the standard text by Arfken.
• A First Course in Computational Physics by David Yevick. In this day and age, computational physics is no longer a luxury.
  

Classical Mechanics:

• Classical Dynamics of Particles and Systems by Marion & Thornton
• Classical Mechanics and Classical Mechanics: Systems of Particles by Walter Greiner. Not good enough to stand alone but a great supplement to Marion & Thornton.

Classical Electromagnetism:

• Electromagnetic Fields and Waves by Lorrain, Corson & Lorrain. This book contains two fascinating chapters covering the relativistic description of electricity and magnetism as a unified phenomenon that I have not seen in any other undergraduate text.
• Classical Electrodynamics by Walter Greiner. Not good enough to stand alone but a great supplement to Lorrain, Corson & Lorrain.

Quantum Mechanics:

• Quantum Mechanics (2nd Edition) by Bransden & Joachain. Good quantum mechanics textbooks are hard to find; this is one of the better ones out there. (Griffiths is a good alternative.)
• Quantum Mechanics: An Introduction, Quantum Mechanics: Symmetries, and Quantum Mechanics: Special Chapters by Walter Greiner is an excellent series.
  

Thermodynamics & Statistical Mechanics:

• Fundamentals of Statistical and Thermal Physics by Frederick Reif. Good thermodynamics textbooks are hard to find; this is one of the better ones out there.
• Thermodynamics and Statistical Mechanics by Walter Greiner. Not good enough to stand alone but a great supplement to Reif.

Relativity:

• A First Course in General Relativity by Bernard Schutz. Special relativity is usually sufficiently covered in Modern Physics classes (and the suggested book by Llewellyn & Tipler does it well), but general relativity seems to be never covered in the undergraduate curriculum. This book is a good introduction for the undergraduate. (Hartle is a good alternative.)

Started another blog

Decided to create a second blog for my mathematics and physics observations. I have much more of a vision for this one. I'll probably post something new in it later today or tomorrow at the latest.