Physicists are abnormal. We read and write in an esoteric language that is readable by other abnormal people who speak different natural languages. That is an odd trait. But after years of uncomfortable studenthood, we start to feel comfortable to ‘speak’ in equations. I think many, but not all, physicists who write books for lay audiences forget how difficult it is to read equations if you are not abnormal. There are about six physics pop-sci books that make me think this.
I believe that if we break with some conventions, we can make equations more welcoming and more understandable. In this blog post, without mentioning much physics, I detail some of the thought processes I went through when I broke with some mathematical and typographical conventions when writing equations in my book Physicists at Fault.
Have a look at the following equation. An equation with the same basic information appears in a chapter near the end of my book: $$ |\Psi_{\text{AB}}\, (x_1, x_2)\rangle=\frac{1}{\sqrt{2}} |\text{A}(x_1),\text{B}(x_2)\rangle-\frac{1}{\sqrt{2}} |\text{B}(x_1),\text{A}(x_2)\rangle$$ To a physicist, this is a very simple equation. To a novice, there is a lot going on here.
Rather than have an equation like this with a few brief remarks following it, I think most normal people would rather have a simple equation as a visual reference with a whole paragraph telling them what they should read into it. So, let’s make it simpler.
In my book, I use one-dimensional examples with $x$ as the variable repetitively, so by this point in the book, the readers know that the $x$ variable is there whether I write it down or not. Additionally, because I use images and explain the existence of \(x_1\) and \(x_2\) in the text, we can drop these two variables. We then have a less crowded equation:
$$ |\Psi_{\text{AB}\,}\rangle=\frac{1}{\sqrt{2}} |\text{AB}\rangle-\frac{1}{\sqrt{2}} |\text{BA}\rangle$$
What can we do next? The fractions with the \(\sqrt{2}\) are only useful for those who will do actual calculations, so they can get dropped. I do this by writing the equation like this:
$$|\Psi_{\text{AB}\,}\rangle\text{“=”} |\text{AB}\rangle- |\text{BA}\rangle$$ This equation is more aesthetically pleasing as no fractions are jutting out from the main reading line. The “air quotes equal sign” tells the reader that the equation is not really true, and it also signals to them that they can relax a bit because I am not taking the equation too seriously.
At a few points in the book, I tell the readers that mathematics can be read out loud as English sentences (with varying degrees of difficulty). This latest version can be read as “the ket labeled with Psi-AB ‘is equal to’ the ket labeled AB minus the ket labeled BA.” It’s not poetry, but no one expects it to be.
At this point, I could be finished with the tweeking of the equation. It’s in a pretty simple form. However, I don’t feel like it’s as inviting as it could be. I feel like the typographic convention of packing the symbols together with little whitespace between them makesitreadlikethis. More than that, the novice reader can’t consistently tell where one grouping of symbols ends and the next begins in more complex equations. Adding just a bit more whitespace will show the groupings and make the equation slightly more inviting to the eyes:
$$|\Psi_{\text{AB}\,}\rangle\;\,\text{“=”}\;\, |\text{AB}\rangle\,-\, |\text{BA}\rangle$$ This equation appears in my book. Compare it to the first equation of this blog post. The two equations have the same essential information. But with supporting images and explanations in the text, in my opinion, this latest equation is more understandable to a normal reader. There are a few more tricks I use to make the math more inviting when writing other, more complicated equations, but I’ll end the post here.