What mathematical structure models arithmetic with physical units?

Question

In physics we deal with quantities which have a magnitude and a unit type, such as 4m, 9.8 m/s², and so forth. We might represent these as elements of $\Bbb R\times \Bbb Q^n$ (where there are $n$ different fundamental units), with $(4, \langle1,0,0\rangle)$ representing 4m, $(9.8, \langle1,0,-2\rangle)$ representing 9.8 m s⁻², and so forth, with the following rules for arithmetic:

1. $(m_1, u_1) + (m_2, u_2)$ is defined if and only if $u_1 = u_2$, in which case the sum is $(m_1 + m_2, u_1)$.
2. $(m_1, u_1) \cdot (m_2, u_2)$ is always defined, and is equal to $(m_1m_2, u_1+u_2)$, where $u_1+u_2$ is componentwise addition of rationals.

This structure has a multiplicative identity, namely $(1, 0)$, and a family of additive quasi-identities, namely $(0, u)$ for each $u$. We have $$(1, 0)\cdot (m, u) = (m,u)\cdot(1,0) = (m, u)$$ and $$(0, u) + (m, u) = (m,u)+(0,u) = (m, u)$$ for every $(m, u)$, but in the latter case the quasi-identity $(0,u)$ isn't a constant; it depends on the $u$ part of $(m,u)$.

Every element with $m\ne 0$ has a multiplicative inverse, and every element $(m, u)$ has an additive quasi-inverse $(-m, u)$ with $(m,u) + (-m, u) = (0, u)$, where $(0,u)$ is an additive quasi-identity. Multiplication distributes over addition. If either side of $$p\cdot(q+r) = (p\cdot q)+ (p\cdot r)$$ is defined, then so is the other, and they are equal.

All taken together this is very much like a field, except that the additive identity is peculiar. There is a $0_\text{m}$, a $0_\text{s}$ and a $0_\text{kg}$, represented as $(0, \langle1,0,0\rangle), (0, \langle0,1,0\rangle), $and $ (0, \langle0,0,1\rangle)$, and they can be multiplied but not added.

We might generalize this slightly, and define the same sort of structure over a ring $\langle G,+, \cdot\rangle$ and a group $\langle H,\star\rangle$: $G❄H$ is an algebraic structure whose elements are elements of $G\times H$, where $(g_1, h_1) + (g_2, h_2)$ is defined to be $(g_1+g_2, h_1)$ if and only if $h_1 = h_2$, and $(g_1, h_1) \times (g_2, h_2)$ is defined to be $(g_1\cdot g_2, h_1\star h_2)$ always. (Or we might relax the condition on $H$ and make it a monoid, or whatever.)

Does this thing have a name? Is it of any interest? Are there any interesting examples other than the one I started with?

I did observe that this structure is also a bit like floating-point numbers, where the left component is the mantissa and the right component the exponent, except that floating-point numbers also have a normalization homomorphism that allows one to add $(m_1, e_1)$ and $(m_2, e_2)$ even when $e_1\ne e_2$, and to understand $(m_1, e_1)$ and $(b\cdot m_1, e_1 - 1)$ as different representations of the same thing.

Answer 1

Tensor product of 1-dimensional graded vector spaces.

A number with units (as defined in the question) is an element of such a product and the grading carries the type of unit and its dimension. The product of spaces with different gradings gets a multiple grading, or you can see all the gradings as living in one large commutative group of all possible dimensions. The group does not have to be free, it can have relations between different dimensions, as are discovered from time to time in science.

There is also Arnol'd remark that dimensional analysis has been re-packaged as Toric Varieties, and internet postings by Terence Tao on units which he takes to be elements of a dual space.

• Terry Tao: A mathematical formalisation of dimensional analysis

Answer 2

Most of these approaches don't allow you to express, for example, $\ln(T/T_0) = \ln T - \ln T_0$.

Here's a more general approach: a unit system assigns positive real numbers to kilogramme, metre, second etc. An expression of a physical quantity is dimensionless if it is the same for all unit systems. An equation or inequality is true if it is true for all unit systems. And so forth.

Answer 3

Physical quantities are categories from the mathematical field category theory. The elements of same categories are physical values.

A physical unit is a functor

$$\mathbb{R} \to\mathrm{Length}$$

that allows to write

$$1\ \mathrm{m} + 2\ \mathrm{m} = (1 + 2)\ \mathrm{m}$$

by basic functor law and $1\ \mathrm{m}$ is just a notation for $m(1)$, with $m$ going from $\mathbb{R}$ to $\mathrm{Length}$. $\mathrm{Length}$ is a category on its own and there exists a function $$f: \mathrm{Length}\times\mathrm{Time}\to\mathrm{Velocity} \\f(l, t) = \frac{l}{t}.$$

$\mathrm{Velocity}$ is also a category on its own. I would argue, the fact that

$$\frac{1\ \mathrm{m}}{2\ \mathrm{s}} = \frac{1}{2}\ \frac{\mathrm{m}}{\mathrm{s}}$$

can not be expressed by some general notion of multiplication of units is because physical quantities cannot be combined by multiplication or division in general. That is contrary to the suggestion in the question, which says physical units are $\mathbb{R}\times\mathbb{Q}^n$.

E.g. $1 \frac{s}{m}$ is not a physical quantity even though it has been constructed according to the implied rules, making the model inadequate.

Rather $\frac{m}{s}$ is merely a convenient choice for the unit symbol for velocity because of the relation of the three involved physical quantities in reality.

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We can furthermore state that two physical quantities $C$ and $D$ have the same dimension, iff there exists a function $s: C\to D$ and some constant $s_f \in \mathbb{R}$ (the scale factor) such that $s(x) = s_f \cdot x$.

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And there is no problem with $t\in \mathrm{Time}, \log{t}$, either. If the physics add up, we can always implicitly rely on the functor law.