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From: Deane Yang (deane_yang_at_[hidden])
Date: 2004-01-04 14:39:03

Andy Little wrote:

>>In any case, physical dimensions are completely consistent
>>with the purely mathematical description referred to above.
> I think you will get a linker error here... judging from your statements
> two/three paragraphs up :-)
>>BUT the purely mathematical description is universally useful
>>anytime one has a linear measure of quantity of any kinds
>>(oranges, coupon periods, bytes, etc.). Physical dimensions
>>provide just one particular set of examples.
> Again ... might I suggest spending some time finding out about the subject
> :-)

Let me try to be more precise. I have a precise understanding
of the mathematical properties of what a "physical dimension"
is (1-dimensional vector spaces and all that). What I do not understand
is how physicists decide which quantities qualify as a new dimension
and which do not. I understand that time, distance, and electrical
charge are physical dimensions. But whenever you encounter a new
physical concept or quantity, how do you know whether it qualifies
as a dimension or not? (Mathematically, any physical quantity qualifies
as a dimension. It seems like physicists have some other set of rules.)

>>I am surprised. Fractional powers are useful even for physicists.
> I keep asking for real world examples... what are these secret formulas?
> Usually they are 'rules of thumb' (ie baloney) not proofs.
> (OK physicists can see an open goal... I am talking general purpose
> engineering style physics)
> To some extent this is trivial... its not a great problem to add IF there
> is a need.
>>In fact, it arises in finance because finance uses Brownian motion,
>>which of course is originally a physical concept.
> Would need to see the proofs... but The WARNING BELLs are deafening :-)

Physical application:
Any time you have a diffusion process (heat, for example), there is a
diffusion coefficient that measures how quickly the thing being diffused
spreads. The natural units for this are (units of diffused
quantity)/(square root of time unit).

Financial application:
In the Black-Scholes formula, an asset price is assumed to follow
geometric Brownian motion. The (say, daily) variability of the price
is measured by a quantity called volatility (equivalent to the diffusion
coefficient above), whose natural units are per-square-root-of-time.

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