Re: A question about mirrors

[John Denker <jsd@MONMOUTH.COM>]

At 02:07 PM 8/2/00 +0530, Abhishek Roy wrote:
> Along the same line, how do you show that apart from L and R forms there is
> not a third - X form of the hand which is 'opposite' to both the others.

Wow, that's a deep question.

The answer has to do with the relationship between the reflection operator
and the rotation operator.  Actually there are several possible answers:

1)  In a world where rotation is a symmetry, you can pick a single
"standard" reflection (such as the one that takes X to -X) and then show that
 -- any odd number of reflections is equivalent to one standard reflection
plus a rotation
 -- any even number of reflections is equivalent to a rotation.

See outline of proof below.

====

2) In a world where rotation is _not_ a symmetry, there are infinitely many
different reflection operators.  You can hold the mirror at this angle or
that angle, and they're all different.

In this case you cannot prove that L and R are the only outcomes, because
it's not true!

====

3) Item 2 is not a nit-pick.  It is not to be taken lightly.  In physics we
_do_ have other operations that are reflection-like, in the sense that
applying the operator twice gets you back to where you started.  These include
  -- charge conjugation (positive <--> negative. matter <--> antimatter)
  -- time reversal
  -- and numerous more obscure ones

Again in this case you cannot prove that L and R are the only outcomes,
because it's not true!

Time reversal (the "reflection" that takes t <--> -t) is special because
rotations in the (t, x) plane (i.e. boosts, i.e. changes in velocity) have
a different structure than rotations in the (x, y) plane.

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

It is not hard to prove item 1:
  a) Imagine you have an operator V(0) which represents reflection in a
mirror held in some "standard" orientation.  For instance, I choose V(0) to
take X to -X.
  b) Define V(A, theta) to be the operator which represents reflection in
a mirror rotated from the standard orientation by an angle theta around the
axis A.
  c) Temporarily restrict things so we consider only infinitesimal angles
theta, and consider only the principal axes (A = X, Y, or Z).  Convince
yourself that for axes lying in the plane of the mirror, the double
reflection V(0) V(A, theta) corresponds to R(A, 2*theta) -- rotation by
twice the angle around the same axis.  For axes perpendicular to the
mirror, rotating the mirror does nothing and the double-reflection is a no-op.
  d) Show that reflections with unrestricted A and theta can be built by
compounding the infinitesimal version.  Mutter something about Lie algebra
if you want.
  e) The rest follows immediately.