Dimensional analysis with fundeps (some physics comments)
Jerzy Karczmarczuk
karczma@info.unicaen.fr
Tue, 10 Apr 2001 12:04:57 +0100
[[MOVED FROM haskell TO haskell-cafe]]
Nothing to do with Haskell. Pure computer scientists, not
interested in physics might not find anything in this posting.
Tom Pledger wrote:
//about dimensions, units...//
> If you clearly make the type system deal with dimensions rather than
> units, there's no problem with plugging in multiple unit systems. You
> just have to pick a scale for the representation.
>
> newtype Dimensioned mass length time rep = Dimensioned rep
>
> type Mass rep = Dimensioned One Zero Zero rep
>
> kg, lb :: Num a => Mass a
> kg = dm 1
> lb = dm 0.4535924
>
> Angles are dimensionless. (Think of the Taylor series for trig
> functions.)
>
> radian, degree :: Unit
> radian = dm 1
> degree = dm (pi/180)
The distinction between dimensions and units may seem for some of
you utterly trivial. Mass is a mass, kg -> lb, etc. are just conversion
factors. Angles are dimensionless.
It was not so trivial for the authors of SI, and for some speculative
physicists, who are not all dead yet. The problem is: what is a
*fundamental*, *universal* constant?
You see, those constants establish our unit systems!
Many, really many among my friends don't hesitate to work in a frame
where the speed of light is equal to 1. Yes, ONE, dimensionless. Then,
passing from length to time is just a unit conversion, and all
velocities are dimensionless.
(Like, say, steradians in SI?...)
Who needs candelas?
Who needs Amperes
What is the dimension of a mole? (In SI it is an independent unit)
Who needs Kelvins?
Everybody (within a particular mafia I happen to know) knows very
well that the temperature is/should be measured in energy units,
i.e., the Boltzmann constant k is equal to 1, dimensionless.
Now, even if you won't (probably) find people who would claim
that the status of k or c is the same as PI/180, the question
"what is a unit" has no unambiguous and non-conventional answer.
You may put the universal gravitational constant to 1. Or simply
put to 1 the Plankeon mass, and get rid of all the mass
"dimensional units".
==
And if you leave the world of classical physics, and start to analyze
the structure of quantum theories you may get some nasty surprises.
//Now I have to trivialize a bit//
(First, most people who do some calculations in QM put the Planck
constant equal to 1, so the energy is inverse time...)
Some quantities which should - for geometric reasons - have fixed
dimensions, such as the "force" of a field respecting Laplace
equation: ~1/r^2 (Newton, Coulomb),
get what is called the "anomalous dimension", instead of 2 one has
to play with r^(2+x), with x fractional (some people, I suppose that
Andrew Kennedy is one of them, don't like such things...).
Now, if something which should be - classically - dimensionless,
say p^0 acquires this anomaloud dimension p^x, then the only
computationally sane way of dealing with it is to introduce a new
"universal" (?) dimensional constant q, such that the expression
above may be written as (p/q)^x in appropriate units of p and q.
Then, no x may do any harm. This phenomenon, the "dimensional
transmutation" suggests that it is not clear which is the set of
universal dimensional quantities in quantum world.
==
I like all papers about dimensions, and it is very refreshing to
see how people squeeze them into the type systems. (Or play with
geometrical "navigation" in a dimensional world:
http://mail.collegestudent.com/gda01/GDA01.HTM
http://physics.nist.gov/cuu/Units/SIdiagram.html
)
I believe, however, that one should not forget that all that is
*conventional*.
Jerzy Karczmarczuk
Caen, France