Posted by: yanzhang | August 3, 2011

On Physical Units

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 \int_{-\infty}^{\infty} e^{-ax^2} dx. There is a way to get some information about it without doing the real integral:

Do the substitution y = \sqrt{a} x. Then the integral becomes \int e^{-y^2} (1/\sqrt{a}) dy = (1/\sqrt{a}) \int e^{-y^2} dy = C a^{-1/2}.

Using a slightly physical language: f we don’t care about the actual constant, just the “order” of a (though it is a similar concept, we’re not exactly doing the order of growth of a), we can deduce that the answer is in the “units” of a^{-1/2} (the complete answer is \sqrt{\pi / a}), by “isolating” the part of the integral with dependence on a.

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 x have the units of [\text{length}]. The exponential must be unit-less otherwise it doesn’t have a well-defined unit, which means a must have units of [\text{length}^{-2}] and the integrand itself must then be unit-less. When we integrate, we then pick up a single unit of [\text{length}] in terms of a, so it must be C a^{-1/2} for some constant C.”

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)

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  1. I first learned about how to reason with units in high school chemistry and learned more in my university physics courses. Over time, I have learned how to incorporate the same logic in my math. I find it extremely useful when doing calculations in differential geometry and PDE’s. You learn that certain functions or variables have “units” associated with them, and the basic idea is that the units transfer to formulas that are linear or homogeneous in these quantities but if you want to stick any of these quantities with units into a nonlinear nonhomogeneous function, you have to first divide by the appropriate constant or function to make the quantity unitless. Try it out. Also, whenever you try to prove an inequality or formula in differential geometry or PDE’s,, you always do a sanity check by asking how the two sides of the equation or inequality scale if you change either the function or the co-ordinates by a constant scalar factor. If the two sides scale differently, you automatically know that it is wrong.

  2. What’s non-rigorous here? All we’re talking about is various actions of the multiplicative group on spaces of functions. I am really annoyed when people treat this like it’s some magical trick that only physical intuition can pull off.

  3. @Deane: yeah thanks for the comments. I really like the scaling sanity check – it had helped me a lot in many things.

    @Qiaochu: as stated, I think I’ve seen more interesting uses of units, though my memory fails me to come up with an example (which is why I wrote this post). I don’t think anyone thinks this stuff is magical; though I think it may have the property that axiomatizing it obfuscates intuition, so it may be better off treated as “slightly magical.” =)

  4. To be precise, let G be the group of positive reals under multiplication. This story is about actions of G. Let V_{\lambda} be the one-dimensional representation where a \in G acts by multiplication by a^{\lambda}.

    To say that x has units of length is to assert that it lies in V_1 (and the action of G then corresponds to changing the unit). To say that v has units of inverse length squared is to assert that it lies in V_{-2}. Now x^2 naturally lies in S^2(V_1) \cong V_2, and the unitless quantity ax^2 naturally results from the natural G-equivariant isomorphism V_{-2} \otimes V_2 \cong \mathbb{R}.

    The expression e^{-ax^2} \, dx is an element of V_1, since G acts trivially on e^{-ax^2} by design. Its integral depends on a \in V_{-2} and is well-defined whenever a \neq 0, so we have a G-equivariant map from V_{-2} - \{ 0 \} to V_1 - \{ 0 \}, and taking the inverse square root is (up to a multiplicative constant) the unique such G-equivariant map.

  5. @qiaochu: so why “should” ax^2 be unitless? Or was that arbitrary? I like everything else, though I’d have probably rephrased all our functions as lying in a graded algebra graded over $\RR$ instead of setting up the group action your way, which is totally fine.

  6. Because the exponential of a unital quantity isn’t homogeneous, so it doesn’t itself have a well-defined unit.

  7. @qiaochu: wait, is your argument: “it can’t be well-defined unless the exponent is unitless, thus since we assume it can be well-defined, the exponent must be unitless?” This is really the single statement I had problem convincing myself (mathematically, and not physically) rigorous… am I really missing something trivial?

  8. @qiaochu: well, I guess one way I’d be satisfied is to say: “if we force the exponent to be unitless, then this is well-defined as a unitless quantity…”

  9. Well, it depends on what you think the exponential is. If you think it’s a thing that takes in real numbers and spits out real numbers, then these things have to be unitless by definition. (Note that the argument works with any sufficiently rapidly-decaying function replacing the exponential.) If you specifically think it’s \sum \frac{x^n}{n!}, then this thing can’t be homogeneous unless x is unitless.

  10. @qiaochu: I agree with everything you said, though I think you were missing my point, which is that this may have been more than a dichotomy; there could have been a third option “we don’t have ‘units’ defined for this thing,” which is more of a “we can define it but we won’t.” I thought this was the problem until I realized you can just say “we force a proper definition of unit onto this quantity, in which case it must be unitless.”

  11. I mean, yes, that’s an option, but then the result isn’t a number, it’s a non-homogeneous element of a graded ring or whatever.

  12. Hey guys, this is really interesting and something I’ve thought about before. I still don’t think I fully understand what’s going on. My confusion seems similar to what Yan’s was, and every time I think I’ve made complete rigorous sense of what Qiaochu is saying, it falls apart. I don’t have time right now to think about it carefully enough, since I’m actually working on another post, but if one of you wants to spell things out really slowly, once and for all, in a mathematical way, I would love it. If not, I will do it sometime soon.

  13. [...] Yan Zhang: On Physical Units [...]

  14. @lewallen: can you clarify what part of my explanation you found confusing?

  15. You can find an interesting example of the use of units in this paper by Duane from 1923

    “The transfer in quanta of radiation momentum to matter.”

  16. This is many months after the original post, but I came across this post via Google. Coming from a physics perspective, I’d say that the quantity x in exp(x) has to be unitless because the exponential function is not scale invariant. I think this is what Qiaochu Yuan means when he says “the exponential of a unital quantity isn’t homogeneous, so it doesn’t itself have a well-defined unit,” but I’m not sure.

    The most important point of units is that they are arbitrary and we assign a real number to the parameter by choosing an arbitrary basis unit. So, physical equations should be scale invariant. Take the function f(x) = x^2, and let’s say x has units [length]. Tentatively call the units of f(x) [length]^2. Now what happens when we change scale – say the basis [length] to [length'] = 2[length]. Then the real number assigned to x is halved so f(x) goes down by a factor of 4. So scale invariance is preserved by letting the basis [length']^2 = 4[length]^2 = (2 [length])^2.

    You can’t do this with functions like exp(x). Say x has units [length] and say that exp(x) has hypothetical units exp[x]. Then what happens when we rescale x to have units [length'] = 2[length]? This halves the numerical part of x. Then what happens to exp(x)? It’s numerical part does not go down in a way that can be accounted for by rescaling the unit exp[length]. So, there can be no unit exp[x] – the exponential function exp(x) is not scale invariant since it behaves fundamentally differently when x is given different (arbitrary and equally valid) units.

    Most physicists don’t see the need to “axiomatize” units, but I’m somewhat interested in the problem. A professor of mine (if I understood him correctly) said that he thought of units of being the generators of a representation in real numbers or complex numbers of the actual physical quantities, and that all physical laws are really defined on on the physical quantities, and must be invariant with respect to the particular representation (units) used. I was thinking that a better template for axiomatizing units would be category theory, since I know that has something to do with talking about allowed transformations that preserve some structure. I don’t know how you’d formulate such a category though. I tried but wasn’t satisfied with what I cam up with.

  17. Coming into the discussion even later…

    In case anyone is trying to figure out what phrases to Google for more information, the engineers call this kind of reasoning “Dimensional Analysis”. Historically it’s been useful in applications like fluid mechanics where the underlying PDEs can be intractable and nondimensionalization makes it easier to quantify observed patterns. Practically it’s been useful as a way for people doing complicated algebra to get a little bit of error-detection on their results.

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