Hello and welcome to this column. I hope to entertain and inform you with a useful resource relating both the methods and philosophies of theoretical physics.
The reason for this column is that there are plenty of articles on experiments and how to make observations and I thought it would be nice if there were a similar resource for those who seek to understand and explore theory. I will be presenting methods and ways of thinking about theory. I will also be adding material on mathematics.
This column then has the role of acting as guide to the landscape of theory in general and theoretical physics in particular.
Each column after this one will begin with a statement of what was done in the previous column. Each column will then have an exposition. This may take up more than one section of the column. The exposition will be followed by a list of books that I like that are relevant to the subject of the chapter, along with comments about why I like the book(s) in question. Then you will find what is called The Theory Challenge; a question that I pose for you to attempt before reading the next column (where I will answer the Theory Challenge). Then I will include a study guide of what order to read supplemental material to get the most out of your experience and what sorts of things you need to be able to do to move on. The study guide may also suggest project ideas for expanding the material in useful ways. Finally, there will be a set of exercises that will help you to mastering the techniques in the column and to expand on it.
The goal of this column is to help you to study theoretical physics and be able to apply it as the basis of our own individual scientific investigations. So, if we are interested in cell biology, we must keep in the back of our minds that we are looking for possible ways of applying what we are studying to that science.
In this column we will begin at the beginning. We will start with the study of mechanics. We will also develop the necessary mathematics as we go. We will cover the principles of conservation laws, harmonic motion, and rigid bodies before deciding to move on to the next subject.
The conservation laws of mechanics are the conservation of energy, the conservation of momentum, and the conservation of angular momentum. To understand the conservation of energy we will need to understand the geometry of motion, a subject called kinematics. That will be our actual starting point.
Physics is a science, perhaps the most fundamental. This means that we base all of our knowledge on nature, not simply on good ideas. When we have a good idea we must always compare that with nature. If our idea is not supported by nature then our idea must be discarded or changed to conform with nature. To compare with nature we must make measurements of nature and conduct experiments.
An experiment is a means of duplicating a natural phenomena in a laboratory. This can be our beginning point, we measure something in nature. If we measure enough things we will eventually be able to make general statements about what we are measuring.
So long as such general statements are based upon what is measured, they can be considered as true. Then we can make predictions using this assumption. These predictions can then be tested experimentally. This is called a model. These experimental tests are the ultimate test of any idea, and are the basis of scientific truth.
In physics, we begin by taking the simplest possible point of view for any phenomena. Then we study this simple model until we understand it. Once we understand it we can begin to make the model more realistic by adding complications that were removed by our simplification.
To answer this, we begin with the theorist. A theorist is a scientist who specializes in finding patterns in experimental or observation data and then making predictions assuming those patterns to be reality. Every theorist, like every other scientist, has a slightly different approach. Despite this, there are some things common to all of us:
But wait a minute! Don't all scientists have these traits?
True enough; what makes a theorist different is not their attitude about science—rather it is in their approach. Instead of verifying that an idea is true the theorist often invents the idea to be proven. Of course, to an extent, all scientists do this. What the theorist brings is specialization. We are skilled at putting an idea into the right language to be tested. We also develop many of the mathematical and computational methods to make sure an idea is consistent with what we already understand to be true. Once a new idea can is shown to be both self-consistent and in keeping with what we believe to be true, it can then be tested experimentally.
It is important to realize that before an idea can be called a theory, it is subjected first to proof of internal self-consistency and then is shown to be consistent with the current body of scientific knowledge. Any inconsistencies must be identified and explained, otherwise the proposed theory must be considered to be unproven.
At the heart of science there is a fundamental connection between theory and experiment. Without an experimental verification theorizing is either pure philosophy or pure mathematics; neither of which are really science. Similarly, experimentation without a theoretical background is like a ship running full speed without a rudder.
One difficulty that people experience when learning and doing science is the need for exact terms and precise ideas. We believe that ultimately the ideas of science can be expressed simply. The combination of the need for precision of expression and the fundamental simplicity we believe in allows us to use mathematics to express scientific ideas. One of my goals, then, for this book will be to provide some of these mathematical tools for you.
One of the startling things about science is how much of it is based upon approximations. The most useful type of approximation is a model. Almost all of science is based on the use of models. Some are physical, some chemical, most are mathematical, and many are performed on a computer. The theorist invents models to ask "what-if" questions of existing ideas. These lead to predictions, which in turn lead to experiments to test the predictions. These experiments lead to new data being collected, which can then be analyzed for new patterns, new models, to emerge. And science marches on...
Now that we have explored what theory is, it is reasonable to ask the question, "How is theory done?"
To actually do theoretical work you must decide on a problem to work on. Please, don't try to tackle impossibly difficult problems. There is simply no chance in the universe that you, an amateur theorist, are going to be able to adequately contribute to the cutting edge of quantum gravity, quantum cosmology, string theory, etc., without spending literally decades of your life on mastering difficult mathematics and physical concepts on a daily basis. Let's be reasonable here. You don't see amateurs trying to build large hadron colliders, or space telescopes, or the like. Anyone who claimed they were doing so would be viewed with, to put it mildly, some skepticism. It is no different for theorists; choose a problem that you have a chance of solving.
Let's say that you have, in fact, chosen a problem to work on. One that you think you can solve. How do you know that you can solve it? The smart-aleck answer would be to solve it and find out.
For problems where you are trying to calculate something, or to derive a formula, ask yourself these questions:
If you can answer all of these questions
then you can probably solve the problem.
So go ahead and solve it. The next step after
that is to verify that you have actually
solved the problem. I can highly recommend
Georgi Polya's book on problem-solving .
There are other good books on problem-solving
too; , .
For problems where you are trying to prove something to be true you need to get a little bit deeper:
There are numerous other strategies for making logical proofs. Look in some of the references listed below , , , .
If you have an idea that you have demonstrated to be true, then you should use it to make some predictions. Change some of your variables and predict how this will change your overall results. Change your variables into other expressions and study how this changes the behavior of your whole system. These predictions form the basis for new experiments.
 M. S. Longair, 2003, Theoretical Concepts in Physics, Second Edition, Cambridge University Press. I used some of the notions in this wonderful book in this chapter.
 G. Polya, 1973, How to Solve It, Princeton University Press. This is a wonderful book and one of the best on developing problem-solving strategies.
 Wayne A. Wickelgren, 1974, How to Solve Problems: Elements of a Theory of Problems and Problem Solving, W. H. Freeman and Company (reprinted by Dover Publications, Inc. in 1995 under the title How To Solve Mathematical Problems). An interesting book covering methods of problem solving and an attempt to craft a general theory of problem solving.
 Steven G. Krantz, 1999, Techniques of Problem Solving, American Mathematical Society. This book describes problem solving techniques for a bewildering array of different classes of problems.
 Bryan Bunch, 1982, Mathematical Fallacies and Paradoxes, Van Nostrand Rheinholt Company, New York (this book has been reprinted by Dover Publications, Inc. in 1997). A delightful book that will demonstrate the most common mathematical flaws in reasoning and understanding.
 V. M. Bradis, V. L. Minkovskii, A. K. Kharcheva, 1959, Lapses in Mathematical Reasoning, Permagon Press, Oxford and London, and The Macmillan Company, New York (reprinted by Dover Publications, Inc. in 1999). This has the standard Russian mathematical take-no-prisoners attitude that I like. It deals only with high school level mathematics.
 Howard Eves, 1990, Foundations and Fundamental Concepts of Mathematics, PWS Kent Company, Boston (reprinted by Dover Publications, Inc. in 1997). An excellent survey of the mathematics important for theory, from the perspective of developing logic and reasoning ability. This book is not for the faint of heart.
 John P. D'Angelo, Douglas B. West, 2000, Mathematical Thinking, Prentice-Hall, Inc. This is a very modern book covering the ideas of problem-solving and proofs at an intermediate level. This book assumes some familiarity with calculus.
The subject is too huge at this point to do much meaningful study. If you feel you must turn to some addition material and begin outside study, I can recommend:
 R. P. Feynman, Robert B. Leighton, Matthew Sands, 1964, The Feynman Lectures on Physics, Addison-Wesley (There is a new version published in 2006 by Benjamin Cummings). This is the set of books where I first learned physics back in the early-1970s. I think the treatment is wonderful, and Feynman was a genius of the first order; so many of his lectures contain really advanced concepts explained in understandable ways (mostly). Read through these and take copious notes. Be sure to ask yourself lots of questions! Then go ahead and try to answer all of them; make sure you can verify that your answers are right! If not, why not? For the subject of this chapter, read through chapters one to three of volume one.
Do any or all of these that are of interest to you.
You will find that a scientific notebook is a great tool for not only recording your work, but for developing your ideas. It is the most important tool you will have. It is hard to keep a notebook like this, at first. Once you get into the habit, you will find it a valuable resource.
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