Re: Imaginary reality

["John S. Denker" <jsd@MONMOUTH.COM>]

At 02:20 PM 4/21/00 -0700, Daniel Schroeder wrote:
> But with complex numbers, it seems a little too lucky that the rule for
> multiplication, "derived" using the crazy formula i^2 = -1, would turn
> out to be just what we need.  This issue, of course, is not unique
> to quantum mechanics; I find it a bit spooky wherever it comes up in
> physics.  Probably I just don't understand the math deeply enough.

That's a very reasonable question.

See if this helps:

Forget you ever heard of complex numbers.  Rewrite your favorite wave
equation, and its solution, as vector equations using 2-dimensional vectors.

Propagation in space and time will turn out to be rotations in this vector
space.

Obviously the concept of "perpendicular" vectors will be important, so the
operator that rotates by 90 degrees will show up in interesting places.
Call it L.  In component form, it is

            0  1
      L =
           -1  0

Note that two rotations by 90 degrees will flip a vector end-for-end, which
is just like multiplying by the scalar (-1).  You might even say that L is
a fourth root of unity.

What's more, you might discover that L shows up in other interesting
places.  It is the "generator of rotations" in a Lie-derivative sense.
That is, consider
        R(theta) = exp(L theta)

(emphasize that there are NO complex numbers anywhere near here)

write out the exponential in a power series if there is any doubt about how
to exponentiate a matrix.  Show that the R constructed this way is in fact
the ordinary rotation matrix.  Show in particular that a rotation through
an infinitesimal angle (epsilon) is
        R(epsilon) = 1 + epsilon L

Imaginary numbers?  Who needs imaginary numbers?  They're strictly optional
chez moi.