Re: mirrors, generalized or not

[John Denker <jsd@MONMOUTH.COM>]

At 10:15 AM 8/6/00 -0500, Glenn A. Carlson wrote:

> By definition, the reflection operator R transforms as x -> -x.

That's fine, but the original question asked whether there were any other
operators -- generalizations of the reflection operator -- with similar
properties.

> > Decker 2: Suppose we have two kinds of mirrors;  this time let's call
> > them V and C respectively.  We could have
> >        V^2 = C^2 = I
> >        VC = CV
> >        VC != I


> Since the reflection operator is its own unique inverse, there are no
> reflection operators V and C which have the property in Decker 2.  In
> particular, for any reflection operators V and C, VC = CV = I.

The argument is invalid, and the conclusion is false.

In general, the fact that C has a unique inverse, combined with the fact
that V has a unique inverse, does not prove that C is the inverse of V, or
anything like that.

I defined my hypothetical operators with the following non-hypothetical
example in mind:
        V could be the usual mirror-image operator
        C could be charge conjugation (matter <--> antimatter)

So it seems G.C. has accidentally proven that antimatter cannot exist.

> If we abandon the definition of the reflection operator above, then
> what definition are we using?

That's a rhetorical question.  But Abhishek asked the non-rhetorical
version of the same question.  He asked what are the defining properties of
reflection operators, and asked if/how we know there are no other similar
operators.

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

General philosophical remark:

--  Understanding familiar things should be a starting place,
    not an ending place.

--  Understanding the familiar reflection operator should be a
    stepping stone, not an impediment, to an investigation
    of less-familiar generalizations thereof.

Some of the greatest inventions of all time have come because someone dared
to ask whether some familiar concept could be generalized.

Think about the work of
  William Rowan Hamilton
    http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html
    http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/
  Sophus Lie
    http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
  Paul Dirac
    http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html

I wouldn't want to tell Hamilton there is no such thing as quaternions,
because "everybody knows" multiplication is commutative.  I wouldn't want
to tell Dirac there's no such thing as antimatter, because "everybody
knows" reflections and rotations are the only possible isometries.

> until we define what this alternative reflection operator is, there is no
> problem to talk about.

I 100% disagree.  Open-mindedly asking about possible alternative
definitions is a fine thing to talk about.  Such questions cut to the core
of how the universe is put together.  Such questions should be cherished.