Identify equation in "Men at Arms" by Pratchett

Question

Does this (possibly corrupted?) equation look familiar to anyone?

$$ \mathrm{10^{-3} ( M_e / M_p ) \alpha^{6} \alpha^{G} - ^{1/2}N \approx 10N} $$

The equation is from the fiction novel Men at Arms by Terry Pratchett, first published in 1993, part of his Discworld series. In the passage:

That meant he'd become stupid again, as sure as
  *equation*.
Better make the most of it, then.

I'm pretty sure that this equation is as Pratchett originally specified it: Equation in "Men at Arms" by Pratchett. In Pratchett's Discworld, magic takes the place of our world's physics. Terms like high energy magic, splitting magical particles, and quantum are used, misused, and abused.

It is very likely that the equation is nonsense. But maybe it was yanked out of some larger valid work? What might the variables represent?

Answer 1

$\mathrm{10^{-3} ( M_e / M_p ) \alpha^{6} \alpha^{G} - ^{1/2}N \approx 10N}$

In principle, someone could have created their own notation and it would have a completely valid meaning. We don't know that is not the case, but I'll explain why this doesn't work in typical physics math.

• It is ambiguous as we do not know if $N$ is the symbol for a Newton (a unit of force) or the other common usage as the number of things. Newton would be the most likely meaning, as if it was just a number then it would cancel out being on both sides like that.
• When you see $\alpha$ and $M_e$ in physics in the same equation you would tend to assume that $M_e$ is the mass of the electron and $\alpha$ is the fine structure constant (which relates to the electromagnetic field).
• Dividing by $M_p$ suggests that it is the proton mass or the Planck mass. Again we have to guess. But it's probably a mass divided by another mass.
• Now the $\alpha$ has no dimensions (i.e. it is just a number and so would two masses divided together. You can raise $\alpha$ to any power, but $G$ is not normally used for a dimensionless number in physics when it's not conveniently set to the value one - the gravitational constant in some specialized units. However you could construct a set of units that would allow $G$ to be dimensionless and something other than one (why ?). How you would arrive at a expression raising the fine structure constant to the power of a dimensionless value of $G$ is quite beyond me and not something I would expect to arise "naturally", but I can't say it's impossible. There is also a thing called the Görtler number which is dimensionless, but the value depends on a number of variables in fluid dynamics and would not seem sensible here (although idealized fluid models do crop up sometimes in cosmology - still hard to accept as a plausible meaning, IMO). On the Scifi SE forum they suggested Gauss's constant, which I cannot recall seeing myself before, but might be more plausible as it's an abstract math constant that could (in principle) arise as a result of a mathematical process.

So far we have dimensionless numbers multiplied by $10^{-3}$ and as long as that bit is in Newtons we're fine for dimensional consistency so far.

The problem part ...

Now the bit that is problematic is that $^{-1/2}N$. As a notation it's nothing I have seen before and unless it's a typo and should be $-\frac 1 2 N$ then it makes no sense. If we could consider that some kind of typo then we could have an equation linking e.g. a bunch of fundamental constants together. What that would mean is that one of them would not be fundamental - i.e. you could deduce one of them from the others, (which we certainly can't unless I missed it in the news).

The rest is just "approximately equals $10N$, but as written the last part of the left hand side would make it something I would not interpret as valid math, let alone anything else.

Now I can't claim to be a mathematician, so I can't say for sure that that problem part is really not a valid notation, but given what I know of math it's not valid so the whole thing become nonsense. Which, as Terry Pratchett would probably have said, is a shame because, really, as it was doing quite well up to then. :-)

Answer 2

$M_e$ and $M_p$ can be the electron and proton masses, $\alpha$ can be the fine structure constant, but as a whole the equation does not make any sense.