# Physical dimensions in math

I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need:

• Scalar multiplication over the reals (so you can get "2 kg" from "2 * kg")
• Addition within the same dimension (so you can have "kg + kg = 2 kg")
• Multiplication of any two elements (so you can have "J = N m = N * m")
• Inverses (so you can have "m/s = m * s^(-1)")

A tensor algebra could formalise this system -- but then you'd get all sorts of objects like "1 kg + 1 m", which make no sense.

A group would make sense -- with sub-groups like "mass measurements", "time measurements", "real numbers", "units" -- but then you can't have zero. Plus, I'd like to have some notion of units or "unit vectors"/"unit tensors".

What's a good way to formalise this?

• Also consider that some physical measurements are unitless. – David K Nov 29 '18 at 18:20
• You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/… – awkward Nov 29 '18 at 19:55
• I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension. – Berci Nov 29 '18 at 23:25

## 1 Answer

You could just use a multivariate polynomial ring $$\mathbb R[x_1, x_2,x_3,x_4,\ldots]$$ and map units of measure to a specific $$x_i$$. To obtain inverses you can go one step further to the field of fractions of this ring $$\mathbb R(x_1,x_2,x_3,x_4\ldots)$$.

That allows you to do addition and $$\mathbb R$$ scaling inside the subspace $$\langle x_i\rangle$$. Inversions are obviously possible in the field of fractions.

This allows you to freely construct different products of different units, and you can tell what combination is going on by the makeup of its monomial coefficient. If $$x_1$$ is kg and $$x_2$$ is s, then $$\frac{x_1}{x_2^2}$$ is kg/s^2.

Until you're a little more specific about what you hope to achieve, and why just using $$\mathbb R$$ with conversions between units doesn't satisfy your problem, I'm not sure what else can be said.

I'm also don't know why you are worried about addition between objects where addition isn't appropriate. Why do you want to eliminate it? It happens all the time in physical interpretations of Clifford algebra, and Clifford algebra is still useful. Extra nonsense that we don't use isn't a justification to throw out the stuff that works.