What's the term for a "physical vector space"?

Question

In physics, we often use the term "vector space" (or just "space," or other similar terms) to refer to a vector space in which the different dimensions are "compatible," i.e. that it "makes sense" to rotate dimensions into one another. Let me label this sort of space a "physical vector space." Unfortunately nobody seems to give a proper definition of what a "physical vector space" is, or what it means for a rotation to "make sense" in this context. Typically the idea is only touched on by way of a couple examples:

• the coordinates (x, y, z) of points in real space (neglecting the curvature described by general relativity) do form a "physical vector space"
• the possible triples of numbers giving the counts of different kinds of fruit in a fruit basket (# of apples, # of oranges, # of bananas) is often given as the canonical example of something that is not a "physical vector space" (although it's been pointed out to me that this is not really a vector space either)
• the possible values of an arbitrary tuple of physical quantities, like potential energy and inverse temperature $(U, 1/T)$, constitute a vector space and even an inner product space, but not a "physical vector space"

Maybe I can try to come up with a couple more, if it would help.

My question: is there a mathematical object that corresponds to what I'm describing as a "physical vector space" here? If so, what is it called?

I think, but I'm not sure, that the thing I'm looking for may be a standard vector space $V$ equipped with an inner product $(\cdot,\cdot)$ and a continuous group $G$ such that $V$ is closed under $G$ and $(v_1,v_2) = (g v_1, g v_2)$ for any $v_1,v_2\in V$ and any $g\in G$. The purpose of the rotation group $G$ would be to disqualify things like the "fruit vector space" I described above. But I'm not entirely convinced that it does that. (And even if it does, I don't know of a name for this kind of space.)

Answer 1

There probably isn't quite a term for this, but I agree that probably what you want to talk about is not naked vector spaces but vector spaces $V$ equipped with an action of a suitable group $G$, namely the group of "physically meaningful" symmetries. "Physical" means something like $V$ being an irreducible representation of $G$. In the case of physical space we might have $V = \mathbb{R}^3$ and $G = \text{SO}(3)$ (this representation is irreducible), for example, while in the case of something like fruits in a fruit basket we might have $V = \mathbb{R}^3$ and $G$ at best is some abelian group acting by scaling on each coordinate individually (this representation is reducible).

Inner products don't seem to play any role in this discussion though.

Answer 2

The variables (scalar or vectorial) occuring in physics have "types" or "dimensions" (in the physical sense), like "length", "time", "velocity", "pressure", etc. A data vector may very well contain variables of different types, like $(p,V, T)$ in thermodynamics, or $({\bf x}, \dot{\bf x})$ in dynamical systems. But note that a "data vector" is not an element of an overriding vector space $X$, but just a list of numbers (or of vectors living in some constituent vector spaces). Nobody would think of a rotation in $(p,V,T)$-space, or of an orthogonal rotation in $4$-dimensional Minkowski space. Usually each entry in such a data vector lives in some vector space (say ${\mathbb R}$, or ${\mathbb R}^3$), and for each entry the group operations available in its space can be applied separately. A particular instance of this principle are the transformations induced by "changing the physical units".

Answer 3

I think you're looking at the problem backwards. It's not that your "physical vector space" requires extra structure to distinguish it from the other types: it's that the extra distinguishing information belongs to the other types of vector spaces.

For example, your vector space of fruit (once you fix up the technicalities) has extra information: we attach significance to a particular basis of the vector space, so that the coordinates with respect to this basis pick out how many of each particular kind of fruit we have.

Aside: note that "coordinate with respect to this particular basis we have chosen" is a basis-independent notion. We can "mix" the coordinates of this vector space all we like, and it doesn't change the result when we do the calculation to extract the coordinates with respect to our chosen basis.

The same deal with your energy-temperature example; you have a vector space along with a significant basis for the vector.

Your "physical vector space", incidentally, doesn't allow freely mixing the coordinates either, since you attach significance to the values of an inner product.

Your intuition to look at a symmetry group is a reasonable one; one of the standard methods to look at structure is to think instead in terms of operations that preserve the structure. In each of the cases above, you can look at the group that preserves the form of the relevant structure. e.g.

• The symmetry group of your fruit vector space together with its canonical basis is the trivial group.
• The symmetry group of your energy-temperature space is the group $\mathbf{R}^* \times \mathbf{R}^*$, since you put significance on keeping the two coordinates separate, but you're probably not attaching significance to scale, so the symmetry group allows you to rescale the two coordinates independently
• The symmetry group of your "physical" vector space is $SO(3)$, since that preserves the inner product as well as orientation (which you probably find significant)

And the symmetry group of just a 3-dimensional vector space, of course, is $GL(3)$: the group of invertible $3 \times 3$ matrices.