Re: [Phys-l] math notation question

[John Denker <jsd@av8n.com>]

On 12/09/2006 06:17 PM, Larry Smith wrote:

> I agree with your general principle.  But there is a difference between
> Euclidean 4-space and hyperbolic Minkowski 4-space where the sign is not
> the same on all terms

That's true ... and things are even worse than that.  I once
saw a paper that defined \delta to be the negative of \box,
i.e. they had both a timelike and a spacelike d'Alembertian.

I understand the temptation for doing things like that ... but
I think they should have resisted the temptation.

> --at least that seems to be the justification at
> http://en.wikipedia.org/wiki/Laplacian and
> http://en.wikipedia.org/wiki/D%27Alembert_operator.

Yeah, but notice that they used \Delta to symbolize both the
Laplacian and the d'Alembertian.

========================

We have some _bona fide_ dilemmas here.

One whole line of argument revolves around the fact that we don't
have (and will never have) enough symbols to permit separate
notation for all the relevant operators in each dimensionality of
interest.  At some point practicality requires us to say that "·"
denotes whatever version of dot product is appropriate to the
space we are working in.

Other considerations include:
 -- Sometimes we do physics in 2D, 3D, 4D, and occasionally
  other dimensionalities.
 -- Many of the laws of the physics are exactly the same
  regardless of dimensionality.
 -- It is tempting to say that some laws change "only slightly"
  as we change dimensionality ... but that's tricky.  Some
  would argue that there is no such thing as a "slight" change
  in a law.  It's either a law or it isn't.  If all you've
  studied is rotations in two dimensions, then rotations in
  three dimensions are going to bring a rude awakening.
 -- Others take a more lenient view:  If you properly understand
  rotations in three dimensions, it is easy to see rotations
  in two dimensions as a natural special case.  But even that
  is slightly tricky, because if you are unfortunate enough to
  formalize 3D rotations in terms of cross products, the
  formalism fails miserably in 2D (and 4D).  Hint:  don't use
  cross products.  Ever.  Use wedge products instead.
 -- The same can be said for Minkowski space (4D or otherwise).
  The geometry and trigonometry of Minkowski space is similar
  to ordinary Euclidean space -- far more similar than most
  people realize -- but it is obviously not exactly the same.
  If you start with a good understanding of Minkowski space,
  you can see the Euclidean subspace as a natural corollary
  ... but the converse is not true;  if you start with Euclidean
  ideas and try to re-invent Minkowski space, you will have a
  few rude awakenings.


I say again, there are genuine dilemmas here, and some genuine
pedagogical nightmares.  One problem is that we, as professional
physicists, often see these issues one way (3D as a natural subspace
of 4D) while students start out seeing them the other way (4D as
a totally nontrivial generalization of 3D).  As the saying goes,
learning proceeds from the known to the unknown.  The students
heartily wish for a distinction between the "3D dot" and the
"4D dot", at least temporarily while they are learning what
generalizes and what doesn't.  I sympathize with their wish ...
but it is not in my power to grant this wish!  There are several
different definitions of dot product, all of which are equivalent
in Euclidean space ... some of which generalize to Minkowski space
and some of which don't.  I can't change that.  About all I know
how to do is to tell students it is OK to keep using dot products
/provided/ they use this definition and not that definition.

In general I hate putting students in situations where they
have to unlearn something.  But sometimes that seems to be the
cost of doing business.  Constructive suggestions are always
welcome.