From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Length scales in physics 4 - the Planck length
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Fri, 21 Aug 92 17:41:16 GMT

In article <1992Aug19.024824.26789@nuscc.nus.sg> matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:
>I honestly don't see the point of this kind of thing. All it does is to
>make people think that they understand things, when in fact they don't.
>I have nothing against sloppiness per se, but to discuss black holes in
>terms of escape velocities is just completely misleading and serves no
>purpose that I can see. I would really like to understand why physicists
>enjoy this kind of thing.

Hi - I just sent you email saying this post had been zonked by our
newsreader, but now I found it.  Quite briefly, physicists are often
interested in knowing roughly what order of magnitude a quantity will
be.  For example, one might ask: how big are the parity-violating
effects of the weak interaction on properties of left-
and right-handed isomers?  How small a ball would one have to squish the
earth down to in order to make it into a black hole?  How tall could
mountains on the Earth be made before they collapsed?  Are we likely to
be able to detect effects from quantum gravity using the SSC?  Etc. 
To work these things out precisely typically takes a lot of time and
effort, but a good practical physicist can often get an answer that's
okay within a factor of 10 just by scribbling on an envelope for a
couple of minutes - and often this is good enough.  Even when it's not
good enough it's often a useful preliminary step before doing a more
serious calculation.  A knowledge of the basic length, time, and energy
scales of physics is very handy for doing this sort of thing.  Moreover,
these calculations are often (though not always) remarkably robust under
variations of the detailed physical theory one uses to do them.  That's
an empirical fact that is largely explained by the principals of
dimensional analysis together with the fact that the undetermined
constants often arise from geometrical considerations (the volume or
area of the sphere, etc. etc.), hence tend not to be huge.  The fact
that one can get a decent estimate of the Schwarzschild radius of a
black hole without using GR is a good example.  

I am a mathematical physicist so I am not a real expert at this
"rough-and-ready" sort of thing; that's why I decided to rederive the
Bohr radius, the Compton wavelength of the electron, the classical
electron radius, and the Planck length for my own entertainment.
Perhaps some more practical physicists would like to comment on the uses
of this kind of thinking.

--
As to your other question - in what space do the knots of Rovelli and
Smolin live - they live in "space", that is, if one were trying to
quantize GR on R^4, they would live in R^3.  I do not think the
mathematical development of the loop variables approach to quantum
gravity is anywhere near over yet, though.  My current line of thinking
is to work with knots in a purely algebraic manner; the resulting
algebraic setup would be a kind of replacement for ordinary "space."  I
have been led to this view by certain difficulties in making the work of
Rovelli and Smolin rigorous.  

Sorry that this is a bit vague, but it's very much in a state of
ferment.  I recommend that everyone read Smolin's review paper, Recent
Developments in Non-Perturbative Quantum Gravity (approximate title),
which can be gotten off of hep-th@xxx.lanl.gov.  (Everyone make sure
they KNOW WHAT THEY'RE DOING before they get stuff off of hept-th.  Send
them a message with subject header HELP and no message body.)  

For an understanding of "working with knots in a purely algebraic
manner", I recommend that everyone read Louis Kaufmann's "Knots and
Physics".  Yes, I know this is the millionth time I've said this.  :-)