# Quaternion and Pauli matrix

## Main Question or Discussion Point

i am learning Quaternion now for my EM course. Can someone enlighten me on the correspondence between Quaternion and Pauli Matrix algebra?

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Not so easy to explain;
metric tensor of the Minkowski's space <=> introduction of the quaternions;
a proposition from Dirac to discuss the Schrödinger equation => introduction of (4-4) matrices built in fine with the (2-2) Pauli's matrices;
Let us call m(a) for a = 0, 1, 2, 3 the different (4-4) matrices; the discussion shows that following relation must hold: m(a). m(b) + m(b). m(a) = 2. g(ab)
where g(ab) is the metric tensor for a Minkowski’s space.

So: not a real good explanation (sorry) but a short exposé of the connections between the actors

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From this site: http://home.pcisys.net/~bestwork.1/HamiltonQ/hamilton.htm [Broken]

This quote:
The Hamilton multiplication rules differ from the Pauli matrix rules only by a factor of i. It is possible to formulate special relativity with Hamilton quaternions having complex coefficients(called biquaternions) and indeed it was first done that way(Silberstein). It turns out that the formulae of general relativity are simpler with the Pauli quaternions. There is also a very interesting (and possibly significant) relation between the Pauli quaternions and three dimensional Clifford Algebra

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still yet to figure out.. but the web link looks pretty informative. Thanks. Will see if i can make some sense out of it.