complex numbers

Polar Representations of Quaternions

Posted by doug

QMN1009.2352

Doug Sweetser

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latex pdf:

polar_representation_tex.pdf

nb.pdf:

polar_representation_master.pdf

Mathematica Notebook:

polar_representation_master.nb

youtube link:

Polar Rep

abstract:

Complex numbers have a polar representation. Three complex numbers that share the same real number are subgroups of a quaternion. Therefore a polar representation of a quaternion must exist. The amplitude is the absolute value of the whole quaternion. The imaginary number i expands to i, j, k for quaternions. The remaining question is how to handle the angle. Two ways work. The first is to take the inverse cosine of the scalar over the absolute value of the quaternion. The second method takes the inverse tangent of the absolute values of the 3-vector over the scalar. A right triangle is animated so the connection between velocity and the polar representation is more apparent.

Calculate the various parts that go in to the polar representation:

$A = (1, 2, 3, 4) \quad eq. 1$

$|A|=\sqrt{30} \quad eq. 2$

$\theta =\arccos \left(\frac{A+A^*}{2|A|}\right)=\arccos \left(\frac{scalar(A)}{|A|}\right)=\arccos \left(\frac{1}{\sqrt{30}}\right) \quad eq. 3$

$\vec{I}=\frac{vector(A)}{|vector(A)|}=\left(0,\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}}\right) \quad eq. 4$

The polar representation of a quaternion is the magnitude of the quaternion times the exponential of angle theta in the I direction:

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Change Log:

2010 Nov. 3: Added animated right triangle.

Complex Numbers Move Straight

Posted by doug

summary:

Complex numbers are constrained to move with their basis vectors, unable to explore all of spacetime.

description:

Complex numbers have some freedom to move in space. The 3 straight lines indicate a choice of Cartesian basis vectors, but other basis vectors could have been chosen. Complex numbers are used extensively in quantum mechanics. Yet the obvious limitations in the animations suggest we should rebuild the foundations of complex-valued quantum mechanics. Sounds like a lot of work!

command:

q_graph -out complex -dir vp -loop 0 -box 5 -command 'q_add_n -3 4 0 0 .002 -0.008 0 0 1000' -color yellow -command 'q_add_n 4 0 4 0 -.006 0 -0.008 0 1000' -color blue -command 'q_add_n -6 0 0 -6 .012 0 0 .012 1000' -color green

youtube link:

Complex Numbers

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