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Commutative version of Hamiltonian
ハミルトニアンの可換版
by Õnihsôÿ Îjñëg
（ オニソイ・イーネグ ） in 1996
i=j×k=k×j
j=k×i=i×k
-k=i×j=j×i
-1=i2=j2=-k2
| i= |
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| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
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| j= |
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| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
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| k= |
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| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
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| 1= |
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| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
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| w+xi+yj+zk= |
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| w | -x | -y | z |
| x | w | z | y |
| y | z | w | x |
| z | -y | -x | w |
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Q=A×B=B×A
=(Aw+Axi+Ayj+Azk)・(Bw+Bxi+Byj+Bzk)
=+AwBw +AwBxi +AwByj +AwBzk
+AxiBw+AxiBxi+AxiByj+AxiBzk
+AyjBw+AyjBxi+AyjByj+AyjBzk
+AzkBw+AzkBxi+AzkByj+AzkBzk
=+AwBw +AwBxi+AwByj+AwBzk
+AxBwi-AxBx -AxByk+AxBzj
+AyBwj-AyBxk-AyBy +AyBzi
+AzBwk+AzBxj+AzByi+AzBz
Qw = +AwBw -AxBx -AyBy +AzBz
Qx = +AwBx +AxBw +AyBz +AzBy
Qy = +AwBy +AxBz +AyBw +AzBx
Qz = +AwBz -AxBy -AyBx +AzBw
| w+xi+yj+zk= |
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| w | -x | -y | z |
| x | w | z | y |
| y | z | w | x |
| z | -y | -x | w |
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Hamiltonian
i=j×k=-k×j
j=k×i=-i×k
k=i×j=-j×i
-1=i2=j2=k2=i×j×k
| i= |
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| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
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| j= |
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| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
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| k= |
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| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
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| 1= |
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| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
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| w+xi+yj+zk= |
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| w | -x | -y | -z |
| x | w | z | -y |
| y | -z | w | x |
| z | y | -x | w |
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Q=A×B ( ≠ B×A)
...
Qw = +AwBw -AxBx -AyBy -AzBz
Qx = +AwBx +AxBw +AyBz -AzBy
Qy = +AwBy +AyBw -AxBz +AzBx
Qz = +AwBz +AzBw +AxBy -AyBx
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