Comments on: A mathematical formalisation of dimensional analysis

By: Mauricio Calvao

Mauricio Calvao — Tue, 21 May 2013 12:32:45 +0000

Dear Terry

Excellent; thank you very much. I guess now I have a lot of food for thought and if anything worthwhile strikes my mind I will post back.

By: Terence Tao

Terence Tao — Tue, 21 May 2013 01:55:52 +0000

(1) Given a real vector space V, one can define the projective space to be the equivalence classes of under the equivalence relation if for some non-zero t. Thus each non-zero vector v in V gets assigned a class [v] in the associated projective space which can be viewed as capturing the “unsigned” direction of v. (If one wants the “signed” direction of v, one should restrict t to be positive rather than non-zero real, but this is a minor detail which I will ignore here.)

(2) The identification between projective spaces in this context comes from the following observation: if V is a real vector space and R is a one-dimensional vector space, and one forms the tensor product (which is a vector space of the same dimension as V) then the projective spaces and are canonically isomorphic, by identifying with for any non-zero (one can check that the class of does not depend on r).

In the case of F=ma, acceleration takes values in the three-dimensional vector space , where is the three-dimensional space of displacement vectors, and is the one-dimensional vector space of dimensionality equal to time raised to the negative two power. Similarly force takes values in where is the one-dimensional vector space of dimensionality equal to that of mass. By the previous discussion, the projective space and are canonically isomorphic, thus allowing one to assign meaning to the statement that force and acceleration have the same direction.

(3) one can indeed work in coordinates if desired, though of course the co-ordinate independence of one’s constructions becomes less clear when doing so.

By: Mauricio Calvao

Mauricio Calvao — Tue, 21 May 2013 00:32:53 +0000

Sorry Terry, but I did not quite get it:
1) how do I build or go from vector spaces to projective spaces?
2) what does it mean for projective spaces to be canonically (naturally?) identifiable (isomorphic?)??
3) is this related to the expression of any (dimensionful) vector as a sort of linear combination of dimensionless unit vectors () and then being able to naturally identify the corresponding unit vectors somehow? If that is the case, I do not here intuitively get what it means to say they are naturally isomorphic, as opposed, for instance, to the identification we make between a given vector space and the dual of its dual! Here it is easy to see the independence of the starting basis!

By: Terence Tao

Terence Tao — Mon, 20 May 2013 01:25:24 +0000

The vector space of forces and the vector space of accelerations are indeed distinct (and not identifiable in a dimensionally consistent fashion), but the projective spaces and are canonically identifiable, and this is where the direction of both force and acceleration live. (To put it another way, while and do not have the same dimensional units, and does have the same units.)

By: Mauricio Calvao

Mauricio Calvao — Sun, 19 May 2013 23:55:47 +0000

I would like to call attention to a particular consequence of this topic of dimensional analysis which has particularly bothered me for some time. I will just give an example of it. We use to say that, in the context of classical single particle point mechanics, “force has the same direction as acceleration” (. Well, both and are vectors from necessarily distinct three-dimensional vector spaces, since we are not allowed to add and , if only because of the difference of their dimensions (“units”). Thus I really do not understand what it means to assert that they have the same direction. For if they belong to distinct vector spaces, how can I unambiguously compare their directions? If there were some sort of natural isomorphism between the corresponding spaces, it might help, but I do not see where/how it comes from/about and this is obviously related to the presence of “units”. In usual, abstract, formal linear algebra the scalar field and the vector spaces are constituted by purely dimensionless objects. Somehow related to that: the quantity mass in the above second law of mechanics is not a pure number scalar… These things really intrigue me and, naively, seem to be completely basic and so I must be missing something very simple…

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By: Count Iblis

Count Iblis — Wed, 27 Mar 2013 15:47:42 +0000

I think this statement is a bit misleading: “Note that once we sacrifice dimensional consistency, though, we cannot then transfer back to the dimensionful setting; the identity c = 1 does not hold for all choices of units, only the special choice of units for which c = 1.”

It hides the fact that the units were arbitry to begin with, and the physics doesn’t depend on that choice. This means that the results of dimensional analysis must be reproducable while working in natural units. Dimensional anysis can be interpreted as considering some scaling limit of a theory.

E.g., if we start with special relativity formulated in c = 1 units, one can ask how to obtain the classical limit. I give a rough outline of how to do that here:

http://en.wikipedia.org/wiki/User:Count_Iblis/Speed_of_light

By: timur

timur — Wed, 06 Mar 2013 03:42:48 +0000

I think in the 6th paragraph of part 2, and should be and , respectively.

[Corrected, thanks - T.]

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