I’ve been learning some physics from Allan Adams and some physics students here at MIT, and I’ve suddenly realized that there are a lot of contextual tricks I take for granted when I’m doing physics, which make the physicists’ lives easier and sometimes really irritates the mathematicians. The first two examples I can think of is differentiating under the integral and never checking convergence, though both of these really fall under the bigger umbrella of assuming everything is well-behaved (which probably accounts for 80% of the mathematical gripes I’ve seen against physicists). Now, I’m a happy supporter of this “wishful thinking” practice: to use a programming analogy, I think of this habit as the lazy evaluation version of having good definitions, and as a lover of Python generators I totally appreciate the idea of saying “we’ll figure out the right definitions later since they actually exist.”
One of the most common physical tricks, however, is not of this category. It is the curiously natural framework: “we have a consistent idea of units.” Here’s a perfectly sound argument to get something that is not entirely obvious:
Take the integral . There is a way to get some information about it without doing the real integral:
Do the substitution . Then the integral becomes
Using a slightly physical language: f we don’t care about the actual constant, just the “order” of (though it is a similar concept, we’re not exactly doing the order of growth of ), we can deduce that the answer is in the “units” of (the complete answer is ), by “isolating” the part of the integral with dependence on .
Even though this is already somewhat trick-sy, it is not quite as far as what a physicist would do. They would (confirmed by experience!) look at this and say something like:
“Let have the units of . The exponential must be unit-less otherwise it doesn’t have a well-defined unit, which means must have units of and the integrand itself must then be unit-less. When we integrate, we then pick up a single unit of in terms of , so it must be for some constant .”
The problem is this makes perfect “sense” to me in a completely sound way (there is no approximation or heuristic here), yet I cannot argue it to my satisfaction in any mathematical matter. All I know is that most people with even elementary physics experience have picked up a very consistent language of “units” that we can use to make definite deductions, but I’m finding it hard to axiomatize them in a clear way. After trying for about half an hour, the only thing I’ve decided is that we really want some sort of valuation on a space of functions that is multiplicative, which I believe is enough to make the differentiation and integration instincts about units work, and that we limit all addition to be done with functions of homogeneous valuation. However, is that really it (for example, I don’t feel this is all that has gone into the logic above)? If so, what is the right way to formalize it? Also, I distinctly remember having seen usage of units to argue more sophisticated chains of logic than the example I’ve given here, though the exact examples don’t come to mind. If anyone has further insight and examples it would be really helpful.
Update: after an unnecessarily long discussion w/ Qiaochu (the source of the un-necessity being my muddled thinking about something irrelevant), I now agree the formalism is “easy” and can be done in several ways (though I still find the intuition to be a clearer way to think than the formalism). The method that seems most natural to me is to just think of all functions we care about as lying in a graded algebra with grades indexed by powers of units; Qiaochu prefers to think of the “physical” attributes as living in one-dimensional representations / weight spaces. Pick whatever you like. My request for more “interesting” examples of using units still holds.
(thanks to Allan, Yoni, and Josh for teaching me physics)