Tuesday, 10 July 2012

Approaching the Truth, Term by Term

When I was younger, I had an odd misconception about science: I believed it was all about exact expressions.

Part of the reason I though this was that textbooks would have these nice expressions, and it seemed obvious to me that they could be solved exactly.  Part of it is for the same reason I sometimes call myself a failed mathematician.  Part of it is undoubtedly the same reason so many of my students would quote ten significant figures in lab classes, when their measurements were only accurate to two.

The truth is, there are very few problems in nature that can be solved, even after making reasonable simplifications.  One important tool to get around this problem is the Perturbation Series, where instead of a simple exact result we have an infinite sum that, for practical purposes, is good enough.  In my field this is ubiquitous.  For example, the recent Higgs discovery is based on perturbation theory, in that the theoretical predictions for the rates of Higgs production where calculated that way1.

Before talking about perturbation series, let me give some examples about the limitations of exact results.  The simplest is certainly the three body problem.  Consider the motion of the Earth around the Sun.  Remember that this is caused by the gravitational force between the two objects, which Newton told us is proportional to their masses m and the square of the inverse of the distance r between them:
$F_G = \frac{G_N m_S m_E}{r^2} .$
G is the constant of proportionality, called Newton's constant.

This is a nice, simple expression.  With fairly elementary calculus, it can be solved exactly.  The solutions are the set of curves known as the conic sections: circles, ellipses, parabolas and hyperbolas. This must have been very satisfying to Newton, as it explained Kepler's First Law of Planetary Motion (that the planet's orbit the sun on ellipses, not circles).  With a little more work, all three of Kepler's laws can be explained by these solutions.

Except it kind of doesn't.  Consider instead of the just the Sun and Earth, trying to include the Moon as well.  Now we have a slightly more complex situation.  Newton's theory of gravity was constructed in the late seventeenth century, so when was this problem solved?

Never.

The second-simplest problem in one of the oldest theories in physics remains unsolved.  Quite likely no exact solution exists, except in the trivial sense (using the problem to define the answer).  And the Solar System itself contains many more than three objects; the Sun, eight planets, four dwarf planets, dozens of moons, thousands of asteroids and comets...

Now, going back to the idea of adding the Moon to the Sun-Earth situation, you might have the intuition that this is a small change.  The Moon is much smaller than the Sun and Earth, surely it doesn't change things that much?  And on short time-scales, you'd be right.2  This type of thinking is the essence of perturbation theory (and how we make predictions about planetary orbits in practice).

The essence of perturbation theory is as follows.  Start with a problem that you cannot solve (in our case, the Sun-Earth-Moon system).  Identify it as being close to a problem that you can solve (the Sun-Earth system).  Specifically, the difference between the two should depend on a small parameter (the ratio of the Moon-Earth force to the Sun-Earth force).  It should then be possible to right the full answer as the approximate answer, plus a series of terms proportional to powers of the small parameter:
Full Solution = Approx. Solution + g x First Correction + g2 x Second Correction + ...
Here g is the small number.

A big reason why perturbation theory is so important in particle physics is because there are almost no theories that can be solved.  Indeed, the only ones that can be are trivial ones where nothing interacts.  There are some extra complications with perturbation series in this domain, but I'll defer a discussion of those for a later post.

1. This is actually a slight lie, in that the computations were not done solely by perturbation theory.  A full explanation of what is going on there is to long for this footnote, however.
2. Short in this context is millions of years.  Eventually, however, the system becomes chaotic and fundamentally unpredictable. The planets don't look chaotic because everything happens slowly.  By analogy, consider that weather forecasts can be reasonable for a few days, but useless for more than a week.  Predictions of the solar system are reasonable for a few billion years, which is enough for most purposes.