Pauli matrices versus quaternions

Subject: Re: pi_k and group representations
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/04/27
Message-Id: <5k012g$f7f$1@agate.berkeley.edu>
Newsgroups: sci.physics.research,sci.physics,sci.math
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John Baez wrote:

>(Of course, if you were raised on quaternions, you would say, with
>more historical precedent, that the Pauli matrices are really just
>another way of thinking about quaternions!)

I want to make sure I have the relationship between these close relatives
clear :-) Quaternions and Pauli matrices are isomorphic but they are not
identical (they are identical if the standard Pauli matrices are
multiplied by i). This difference does have a consequence. Work by
Birkhoff and von Neumann on the axioms underlying quantum mechanics states
(according to Stephen Adler in "Quantum mechanics of fundamental systems
1")

"...one really wants to have a division algebra. A simple heuristic
way of seeing this is, if one wants to get propagators, everything
in one's algebra of scalars must have an inverse, so it is not
possible to form propagators without a division algebra."

Pauli matrices do not form a division algebra. Perhaps when used in
quantum mechanics, there is often an extra factor of i around that
multiplies the Pauli matrix and makes it identical to a quaternion. It is
helpful then to understand what the role of that i is.


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

Call a duck a duck, and an i Pauli matrix a quaternion.