0

I'am looking for a precise name for the mathematical structure that we use to manipulate physical quantities that have the same physical dimension (mass, length, etc...).

I know it is a one dimensional vectorial space on the reals. Let's call $Q$ the ensemble of the physical quantities of the same dimension (for example the mass):

  • there are two internal composition law: the addition and the substraction $$ (\forall a\in Q) (\forall b \in Q) (a+b\in Q)$$ $$ (\forall a\in Q) (\forall b \in Q) (a-b\in Q)$$
  • multiplication by a real gives a physical quantity of the same dimension $$ (\forall a\in Q) (\forall b \in \mathbb{R}) (a\times b\in Q)$$
  • and something like the division could be defined on Q as an external compisition law: $$ (\forall a,b\in Q) (b\ne0)(\exists s \in \mathbb{R}) (a=s\times b)$$

I have found it is also an homogeneous space on wikipedia and phicists say that when two physical quantity have the same physical dimension they are homogeneous. Is there a precise name for this structure?

Is there a name for a one-dimensional vectorial space over a ring?

Oliv
  • 103

2 Answers2

2

What you're describing is just a 1-dimensional vector space $V$ over $\mathbb{R}$. Since $V$ has dimension 1, any nonzero vector in $V$ is a basis, and so any other vector can be written as a scalar multiple of this vector (in light of the last condition you mentioned). The first two conditions are simply the requirements $V$ is closed under addition and scalar multiplication.

In fact, in physics, we often want to let the underlying field be complex; e.g. impedance in an RC circuit. So really the "precise name" you're looking for is a one dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$.

ccroth
  • 916
  • I am looking for a precise name, not a rephrasing... – Oliv Jun 16 '20 at 19:25
  • Actualy I said real as an example but I suppose that for a mathematician it is just a ring. I make an edit to avoid confusion about the subject tx. – Oliv Jun 16 '20 at 19:28
  • And I want to use it where the scalar is not realy a ring, like in C programming language int or float.... I am just looking for a short name. – Oliv Jun 16 '20 at 19:43
  • @Oliv A "one dimensional vectorial space over a ring" would just be a (one dimensional) module. So if $R$ is a ring with a left action on a group $M$, then we say $M$ is a left $R$-module. – ccroth Jun 16 '20 at 19:47
  • And the special case where the ring $R$ is also a field is when you obtain a vector space over $R$. – ccroth Jun 16 '20 at 19:48
1

I think it would probably be all right to use the term biray, if you give the following reference:

H. Whitney, The Mathematics of Physical Quantities: Part I, Amer. Math. Monthly, 75, 1968, pp. 115-138.

Extract (obviously omitting much, including a construction of the real number system $R,$ and the definition of the operation of $R$ upon a biray):

${\rm D{\small EFINITION}}$ 14A. A semi-ray $L$ is a commutative semi-group such that:

($\rm{R}_1$) ${}$ For all $x$ and $y$ in $L,$ $x + y \ne x.$

($\rm{R}_2$) ${}$ For all $x$ and $y$ in $L$ with $x \ne y,$ we can find $u$ and $v$ in $L$ such that $x + u + v = y$ or $y + u + v = x.$

[$\ldots$]

${\rm D{\small EFINITION}}$ 15F. A ray is a complete semi-ray.

[$\ldots$]

${\rm D{\small EFINITION}}$ 20A. A biray $(B, B^+, +)$ is a set $B,$ a subset $B^+,$ and an operation of addition in $B,$ such that:

($\rm{B}_1$) $\ (B, +)$ is a commutative semi-group.

($\rm{B}_2$) $\ (B^+, +)$ is a ray.

($\rm{B}_3$) ${}$ For each $x, y \in B$ there is a $z \in B$ such that $x + z = y.$

($\rm{B}_4$) ${}$ If $x \ne y,$ $x + z = y,$ and $y + z' = x,$ then $z \in B^+$ or $z' \in B^+.$

[$\ldots$]

${\rm T{\small HEOREM}}$ 24D. Any biray $(B, B^+, +),$ with the operation of $R,$ is an oriented one-dimensional vector space over $R,$ and conversely.

Calum Gilhooley
  • 12,447
  • 1
    F. A. Behrend, A Contribution to the Theory of Magnitudes and the Foundations of Analysis (1955/56) defines a continuum (a non-standard use of a standard term, like his use of the adjective "complete"), and given an arbitrary "unit" element $1,$ defines the operation of multiplication on any continuum. He rather unclearly (to me right now, at least) defines a bicontinuum as "any set isomorphic to $\Gamma(<,.)$", where $\Gamma$ is a continuum. Perhaps he means a complete dense totally ordered group? He doesn't explicitly consider the vector space structure. – Calum Gilhooley Jun 16 '20 at 22:41
  • Thank you, you gave me interresting readings! It is such a pleasure to read old papers! – Oliv Jun 17 '20 at 09:36
  • 1
    I am chocked not to have read the H. Whitney paper and most importantly the second part. I did my PhD in the area of fluid mechanic and dimensional analysis was a key stone. I read books about self-similarities, dimensional analysis, but what a lost of time, compared to these two dense articles... My supervisor had never talked about those articles and nobody among the 100 researchers in fluid mechanics that were in the lab I used to belong talked to me about that! Are you more a physicist than a mathematician? How have you found those articles? – Oliv Jun 17 '20 at 15:42
  • Although it was a long time ago, I'm fairly sure I learned of the existence of those two papers from the references in Krantz, Luce, Suppes & Tversky, Foundations of Measurement, vol. I: Additive and Polynomial Representations (1971, repr. Dover 2007). Sadly, I've never been either a physicist or a mathematician, but I once had a job in which I was paid to put numbers to use, and ever since then I've shared the feeling of the author of this question: "this tears me apart for some unknown reason". – Calum Gilhooley Jun 17 '20 at 16:55
  • You never had the title or the job, but mathematician, physicist or both is certainly what you are. Thank you for the reference! – Oliv Jun 17 '20 at 19:26