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?
Best Answer
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 $\mathbb{C}$.