Maths - Quaternion Notations - Product of two complex planes

There is an alternative way to think of quaternions, imagine a complex number:

n1 + i n2

but this time make n1 and n2 (the real and imaginary parts) to be themselves complex numbers (but with a different imaginary part at right angles to the first), so,

If we substitute them into the first complex number this gives,

(a + jc) + i (b + jd)

since i*j = k (see under multiplication) this can be rearranged to give the same form as above.

a + i b + jc + kd

 

 

There are a number of notations and ways to think about quaternions:

When we are using quaternions to represent rotations in 3 dimensions, then we restrict the quaternions to unit length and only use the multiplication operator, in this case there are other notations and ways to think about quaternions:

These are all equivalent and in group theory are represented by the group SU(2).

Even when normalised, there is still some redundancy when used for 3D rotations, in that the quaternions a + i b + j c + k d represents the same rotation as -a - i b - j c - k d. At least it does in classical mechanics. However in quantum mechanics a + i b + j c + k d and -a - i b - j c - k d represent different spins for particles, so a particle has to rotate through 720 degrees instead of 360 degrees to get back where it started.


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