06 September 2002

A Word About Amateur Theorists

by George E. Hrabovsky, President, MAST

News from MAST

Hello, this is the big week! The announcement of the winner of the Theory Challenge!

The winner is: Harry Keller. Congrats Harry, thank you for responding!

As I have been saying the current prize is your choice of either Riley, Hobson, Bence, Mathematical Methods for Physics and Engineering or your choice of any three Schaum's Outline Books.

Doing Amateur Theory

Let me start by saying that I enjoy thinking about problems from a theoretical perspective. I have been doing this for so long that it seems like second nature at times. As I have said before, all scientists need to do at least a little bit of theory. To that end I would like to make a few comments that can be of assistance.

Let me begin by saying that theory is probably the hardest aspect of science to do well. Why? It is simple, to do theory you have to know a little bit about every aspect of your chosen subject. I do not mean that you have heard a few words written or spoken about each area; I mean that you have worked at least one problem through to completion in each area of your chosen field.

If you do not intend to specialize in theory, what do you need to know to be able to do the portions of experimental/observational science that you will need? There are two main things: Statistical data analysis and calculus. If you know how to do these things, then you will be in good shape.

If you intend to do theory, then things get a bit more complicated. In a previous column I discussed what I considered to be a good self-study course in the mathematics of theory (Coursing Through Mathematics.) Here are the questions to ask in learning theoretical science:

1. What are the primary objects considered by the science?

2. What are the properties of these objects?

3. What are the interactions between these objects?

4. What quantities among these objects are conserved?

5. What is the central goal of the science you are interested in, with regard to the objects and interactions in question?

6. What are the principle methods of achieving this goal?

Since so many people seem to be interested in solving problems of quantum gravity I will examine three sciences: Molecular physical chemistry, quantum mechanics, and classical gravity theory.

For molecular physical chemistry the primary objects are molecules and atoms. The relevant properties of these objects are molecular weight, atomic weight, atomic number, and the various so-called quantum numbers. Molecules and atoms interact through electromagnetic fields and collisions. All of the classical and quantum conservation laws are in force: Energy, momentum, angular momentum, and charge. The central goal of molecular physical chemistry is to predict chemical structure and reaction results on the molecular level. The principle methods for achieving this goal is to solve the Schrödinger equation, though for most chemical systems this is too hard to do, so approximations are made.

For quantum mechanics the primary objects are particles. For quantum mechanics the properties of particles are mass, position, momentum, angular momentum, spin, polarization, kinetic and potential energies, charge, magnetic moment, and interaction cross section. Particles interact by collisions and electromagnetic interactions. All of the conservation laws are in force. The central goal of quantum mechanics is to predict the behavior of quantum particles. There are many ways of reaching this goal: solving the Schrödinger equation, so-called matrix mechanics, the so-called path integral approach, and numerous approximation methods.

For classical gravity theory (often called general relativity) the primary objects are hypothetical globs of matter called test particles and spacetime. The properties of test particles are their mass, position, velocity, and charge; the property of spacetime is its curvature. Particles interaction with one another by collisions or electromagnetic interactions, and they interact with spacetime by generating curvature; spacetime interacts with the test particles by defining the paths that they move along (called worldlines). All of the conservation laws are in force. The central goal of general relativity is to predict the effect of matter on spacetime and vice-versa. The method for determining these effects are equating the curvature tensor (arrived at through differential geometry) with the stress-energy tensor (arrived at through Newtonian, Lagrangian, or Hamiltonian field theory).

Here is what I would consider to be a good course of reading for molecular physical chemistry:

1. Work through Linus Pauling's book General Chemistry while studying Thompson and Gardner's calculus book.

2. Work through Riley, Hobson, and Bence's book after you have done with Thompson-Gardner (you will likely still be working through Pauling).

3. Have a general physics text on hand for reference as you plunge into the Schaum's Outline of Physical Chemistry.

4. Work through Garrity's All the Mathematics You Missed.

5. Pilar's book on Quantum Chemistry.

6. Cantrell's book on Mathematical Methods.

Once you have worked through these books you will be ready to do some research and make some real contributions. I think you could get through to step 6 in two years if you worked very hard.

Here is what I would consider to be a good course of reading in quantum mechanics:

1. Work through Thompson and Gardner's calculus book.

2. Work through Riley, Hobson, and Bence's book while working through a physics book (for budget-minded people I recommend the Schaum's Outline of Physics for Scientists and Engineers, if you don't mind spending a lot of money I recommend Sears and Zemansky).

3. Work through a book on classical mechanics (I like Symon's book and that of Kibble and Berkshire).

4. Work through a book on electrodynamics (I don't know of any really good inexpensive books on the subject, I like Griffiths).

5. Work through Garrity's All the Mathematics You Missed.

6. Work through a book on quantum mechanics (for the budget-minded I highly recommend the Schaum's outline of Quantum Mechanics, otherwise I really like Griffiths or Shankar-though Shankar might be best as a second text).

7. Cantrell's book on Mathematical Methods.

Once you have worked through these books you will be ready to do some light research and make some real contributions; though you should work through another set of graduate-level mechanics, electrodynamics, and quantum mechanics books to make this realistic. I think you could get through to step 7 in three years if you worked very hard.

Here is what I would consider to be a good course of reading in general relativity:

1. Work through Thompson and Gardner's calculus book.

2. Work through Riley, Hobson, and Bence's book while working through a physics book (for budget-minded people I recommend the Schaum's Outline of Physics for Scientists and Engineers, if you don't mind spending a lot of money I recommend Sears and Zemansky).

3. Work through a book on classical mechanics (for the budget-minded I recommend the Schaum's outline, otherwise I like Symon's book and that of Kibble and Berkshire).

4. Work through a book on electrodynamics (I don't know of any really good inexpensive books on the subject, I like Griffiths).

5. Work through Garrity's All the Mathematics You Missed.

6. Cantrell's book on Mathematical Methods.

7. Work through a book on fluid mechanics (I recommend Aris' book).

8. Work through an elementary relativity book (no good inexpensive books I am afraid, I recommend d'Inverno's book).

9. Work through a graduate mechanics text (no good modern treatments that are cheap, I recommend Landau and Lifshitz, José and Saletan, or McCauley).

10. Work through a graduate electrodynamics text (same problem as step 9, I recommend Landau and Lifshitz, or Schwinger, et al for traditional texts, or Baylis since it uses the same math as relativity).

11. Work through a graduate relativity text (same problem as step 9, I recommend Misner-Thorne-Wheeler).

12. Work through an advanced relativity text (I recommend Wald).

Once you have worked through these books you will be ready to do basic research in gravity, or you can study additional books on black holes, cosmology, or gravitational radiation to specialize. To work on quantum gravity you will need to combine this list with that of quantum mechanics, and add books on quantum field theory and perhaps string theory. You should be able to start doing research in relativity in four years of hard work. Quantum gravity would require another two or three years of intense study.

You might be tempted to say, "You're just towing the standard line, what if we came up with a truly original approach?" My response is that until you understand why we believe what we believe, you are in no position to criticize it. It is tempting to look at all of this stuff and its seemingly endless complexity and say, "It has got to be simpler than that!" It might be, but it is unlikely that it will be obviously simpler that we think. You are not going to solve the problem of quantum gravity by applying simple algebra or calculus to Newtonian physics.

Think of it another way; if you want to play football you have to follow the rules. You have to be able to run, to catch the ball, to tackle, to pass the ball, etc. If you don't have skill in these areas you must acquire it. It is no different in science, and theory is the most demanding aspect of science, since it involves knowing what is consistent with what is known.

Enough preaching! Have a great week!

Theory Challenge

The theory challenge will return next time.

Books That I Like

All the books mentioned in this column are available through the Internet.

Linus Pauling (1970), General Chemistry, W. H. Freeman and Co (Reprinted by Dover Publications Inc. in 1988). This is a really good and in-depth text on chemistry, despite its age it is one of the best.

Silvanus P. Thompson, Martin Gardner (1998), Calculus Made Easy, St. Martin's Press. This is the classic text by Silvanus with updated notation, notes to the text added by Gardner, and three initial chapters on functions, limits, and derivatives also by Gardner. This is the best text on elementary techniques in calculus that I have EVER SEEN!!!! It is a little short on practice problems (but you can get a Schaum's outline book for that). This book explains calculus.

K. F. Riley, M. P. Hobson, S. J. Bence, (1997), Mathematical Methods for Physics and Engineering, Cambridge University Press. After you have mastered elementary calculus from Silvanus and Gardner this will teach you the techniques of advanced mathematics (including complex analysis, partial differential equations, group theory, and tensors). A new edition is coming out that will add some new things that will be quite interesting.

Clyde R. Metz (1989), Physical Chemistry, McGraw-Hill Inc. (Schaum's Outline Series). This is a problem-oriented text that covers all aspects of physical chemistry.

Thomas A. Garrity (2002), All the Mathematics You Missed [But Need to Know for Graduate School], Cambridge University Press. A fantastic introduction to higher mathematics.

Frank L. Pilar (1990), Elementary Quantum Chemistry, McGraw-Hill Publishing Company, New York (republished in 2001 by Dover Publications Inc.) A nice introduction to the physics and chemistry of atoms and molecules.

C. D. Cantrell, (2000), Modern Mathematical Methods for Physicists and Engineers, Cambridge University Press. This book emphasizes ideas of abstract algebra and functional analysis that are absent from the other books. Thus it rounds out the mathematics you will need for doing just about any theory.

Dare A. Wells, Harold S. Slusher (1983), Physics for Engineering and Science, McGraw-Hill Inc. (Schaum's Outline Series). This is a problem-oriented text that covers all aspects of general physics.

Hugh Young, Roger A. Freedman, (2000), Sears and Zemansky's University Physics with Modern Physics, 10th Ed., Addison-Wesley. This is an immensely good book that is also immensely expensive (I think I paid more than $100 for the damn thing)! It is a very good book, including covering common conceptual problems that students have.

Keith R. Symon (1971), Mechanics, Addison-Wesley Publishing Company. This is a very conventional book, but it also covers just about everything you will need at this level.

T. W. B. Kibble, F. H. Berkshire (1996), Classical Mechanics, Addison-Wesley Longman, Limited. This is a more modern treatment than Symon with a good introduction to chaos theory.

David J. Griffiths (1999), Introduction to Electrodynamics, Prentice-Hall, Inc. This is an excellent treatment of electrodynamics and has a good introduction to special relativity.

Yoav Peleg, Reuven Pnini, Elyahu Zaarur (1998), Quantum Mechanics, McGraw-Hill Inc. (Schaum's Outline Series). This is a problem-oriented text that covers the fundamental aspects of quantum mechanics quite well.

David J. Griffiths (1995), Introduction to Quantum Mechanics, Prentice-Hall, Inc. This is an excellent treatment of quantum mechanics and some of its applications.

R. Shankar (1994), Principles of Quantum Mechanics, Plenum Press. This text is very good for self-study, it goes through a lot of the details of derivations and explains a lot of the math. A very good second text on quantum mechanics.

Rutherford Aris (1962), Vectors, Tensors, and the Equations of Fluid Mechanics, Prentice-Hall, Inc. (Reprinted by Dover Publications in 1989). This text is a very good introduction to the equations of fluid dynamics presented in a way that corresponds with much of relativity.

Ray D'Inverno (1992), Introducing Einstein's Relativity, Oxford University Press. This text is a very good introduction to general relativity and the mathematics it uses.

L. D. Landau, E. M. Lifshitz (1976), Mechanics, Permagon Press (reprinted in 1989 with corrections, this is the first volume in the famous Course of Theoretical Physics from the famed Landau Institute). An elegant is sometimes vague treatment of classical mechanics.

Jorge V. José, Eugene J. Saletan (1998), Classical Dynamics, Cambridge University Press. My favorite text on mechanics, it treats the subject from a differential geometry perspective. This is very good preparation for relativity.

Joseph L. McCauley (1997), Classical Mechanics, Cambridge University Press. A very good textbook from a more tradition point of view. This is very good preparation for quantum mechanics. This also has extensive sections on chaos theory.

L. D. Landau, E. M. Lifshitz (1975), The Classical Theory of Fields, Permagon Press (reprinted in 1994 with corrections, this is the second volume in the famous Course of Theoretical Physics from the famed Landau Institute). An elegant is sometimes vague treatment of classical electrodynamics with an introduction to general relativity including some cosmology.

Julian Schwinger, Lester L. DeRaad, Jr., Kimball A. Milton, Wu-yang Tsai (1998), The Classical Theory of Fields, Perseus Books. This is what is commonly called a posthumous collaboration. The book consists of 52 short chapters, so progress can seem very quick. It is also well-written and engaging in its style.

William E. Baylis (1999), Electrodynamics A Modern Geometric Approach, Birkhäuser Boston. This book focuses on the application of Clifford Algebras to formulate electrodynamics; this is an area of current research in both classical field theory and quantum mechanics.

Julian Schwinger (2001), Quantum Mechanics, Springer-Verlag. An interesting approach that makes a good second book in Quantum Mechanics. The book begins by focusing on measurement problems and ends the first part with a discussion of symmetries of motion. The second part begins with a discussion of the quantum equations of motion and ends with discussions of the harmonic oscillator and Hydrogen atom. The final part treats systems of particles, many-electron atoms, and radiation.

Leslie E. Ballantine (1998), Quantum Mechanics, World Scientific Publishing Company. A solid treatment of graduate-level quantum mechanics (not quantum field theory). I like this book, I find it easy to read and engaging.

L. D. Landau, E. M. Lifshitz (1977), Quantum Mechanics, Permagon Press (reprinted in 2000, this is the third volume in the famous Course of Theoretical Physics from the famed Landau Institute). An elegant is sometimes vague treatment of non-relativistic quantum mechanics. Another good second quantum mechanics book.

Charles W. Misner, Kip S. Thorne, John A. Wheeler (1973), Gravitation, W. H. Freeman and Sons. Even after this many years this is still (in my opinion) the best intermediate textbook on general relativity (there are better beginning texts and better advanced texts). I still find myself coming to this book after almost twenty years. I would seek elsewhere for an introduction to tensors, once you are used to them this will seem an elegant depiction.

Robert M. Wald (1984), General Relativity, The University of Chicago Press. After a brief review section (about a quarter of the book that covers the first two-thrds of Gravitation) the books tackles some very heavy stuff; causal structures, singularities, and even an introduction to quantum effects.

Michael E. Peskin, Daniel V. Schroeder (1995), An Introduction to Quantum Field Theory, Perseus Books. Beginning with a review of creation and annihilation operators this books delves into several of the "classical" quantum fields and the study of Feynman diagrams, then it covers renormalization, non-Abelian gauge theories, QCD, anomalies, and symmetry breaking. The book has a nice feel to it and is written in an engaging way.

Nirmala Prakash (2000), Mathematical Perspectives on Theoretical Physics, Tata McGraw-Hill. An excellent blending of mathematics and physics, culminating in a good introduction to string theory.

Joseph Polchinski (1998), String Theory, vol. 1 and vol. 2, Cambridge University Press. In my opinion this is the starting place for studying the physics of strings. Very heavy and scary stuff here!