Dimension in mathematics and physics

Question

I have studied linear algebra and commutative algebra, there are two kinds of dimension there : the vector space's dimension and the Krull dimension.

Also, in physics, dimension is also a very intuitive concepts.

My question is : What is the nearest mathematical definition of dimension to the physical one ?

In the wikipedia page, they also list some mathematical types of dimension : Dimension

Answer 1

The relevant notion of dimension to physics as I understand it is the dimension of a manifold (e.g. a (3+1)-dimensional Lorentzian manifold). If the manifold is smooth (which is, as I understand it, the case in physics) then the dimension of a manifold agrees with the dimension of its tangent spaces, so morally it is the linear-algebraic notion that is relevant here.

(Of course, in nice cases the Krull dimension ought to also equal the dimension of Zariski tangent spaces. But as far as I know, the universe is not profitably modeled as an algebraic variety.)

Answer 2

I think the appropriate way to feel about dimension is the analogous of degree of freedom. A physical system having $N$ degree of freedom usually can be described in phase space by a $N$ dimensional trajectory. Differential Physical systems can be described by different systems of equations, but we can always 'count' the degree of freedom in some sense, even if some occasions it is infinite.

Answer 3

The answers and comments so far indicate that we are talking about two completely different kinds of "dimension" here:

1. There is the notion of dimension of a real vector space $V$ or manifold $M$. This is an integer $d\geq0$ and has the same meaning in physics as in mathematics. The intuitive physical interpretation of $d$ is the "number of degrees of freedom" in the physical system under study. – In a space of dimension $d$ (infinitesimal) volumes scale like $\lambda^d$ under a linear scaling by a factor $\lambda>0$. This property can be used to envisage sets $S\subset{\mathbb R}^d$ whose "volume" scales like $\lambda^\alpha$ with a noninteger $\alpha\leq d$. This value $\alpha$ is called the Hausdorff dimension of $S$; but this is a dimension in a measure theoretical, not in a topological sense.

2. Physical quantities have a "dimension" of length, time, degree Kelvin, etc. This dimension is not a number, but a quality. It's up to a physics member of the community to give an exact definition. Tentatively I would say that (at least in the realm of mechanics) the set of physical dimensions is the multiplicative abelian group generated by the three elements $L$ (for length), $T$ (for time) and $M$ (for mass). To any physical quantity an element of this group is associated. Two physical quantities can be sensefully compared or added only if the associated dimensions are the same. Furthermore, the dimension of a quantity determines how the numerical value of this quantity changes when the base units for $L$, $T$ and $M$ are changed.