Quaternion norm and the invariant interval

Subject: Re: Quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/01
Message-Id: <E9I9Ip.IFu@world.std.com>
Newsgroups: sci.physics.electromag,sci.physics.particle,sci.physics.relativity
[Mo re Headers]

Chris Hillman wrote:

>There is at least one very important DIFFERENCE, namely quaternion
>multiplication preserves a positive definite quadratic form
>
>[1] t^2 + x^2 + y^2 + z^2
>
>(where we write q = t + ix + jy + kz where
>ijk = - kji = i^2 = j^2 = k^2 = -1)
>whereas the 3+1- dimensional spacetimes of GTR possess an indefinite
>quadratic form
>[2] t^2 - x^2 - y^2 - z^2

Both of these numbers are easily generated with quaternions, but they get
put into use in a manor different from standard practice. [1] is the
Euclidean norm, and is formed by multiplying the adjoint of a quaternion
by the quaternion. The adjoint of q = t + ix + jy + kz is
q = t - ix - jy - kz. This is where I start to build a Banach space (a
complete normed vector space) to lay the foundations of quantum mechanics.
It is worth noting that Adj[q] q is still a quaternion of the form
t^2 + x^2 + y^2 + z^2 + 0i + 0j + 0k.

To create [2] is even easier (but more work to interpret):

q q = t^2 - x^2 - y^2 - z^2 + 2itx + 2jty + 2ktz

The scalar part can be exploited to solve problems in special relativity.
As far as I can tell so far, the vector part currently doesn't play a role
in how people solve problems in physics. That doesn't make the vector
part wrong! It makes it new and different and fun to play with (because
it allows mixing of scalars with vectors).


Doug Sweetser
http://world.std.com/~sweetser