Quaternion and Pauli matrix

  • Thread starter QMrocks
  • Start date
  • #1
85
0

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?
 

Answers and Replies

  • #2
member 11137
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
 
  • #3
selfAdjoint
Staff Emeritus
Gold Member
Dearly Missed
6,786
7
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
 
Last edited by a moderator:
  • #4
85
0
still yet to figure out.. but the web link looks pretty informative. Thanks. Will see if i can make some sense out of it.
 

Related Threads on Quaternion and Pauli matrix

  • Last Post
Replies
16
Views
8K
  • Last Post
Replies
2
Views
2K
Replies
7
Views
5K
Replies
3
Views
3K
  • Last Post
Replies
15
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
7
Views
2K
Replies
2
Views
1K
  • Last Post
Replies
3
Views
889
Top