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Why the wave starts ?

Monday, 25-Jan-1999
09:14:51 by PAL


Onihsoy Ijneg's Quaternion's computing rule is quite siple. It was named after Irish mathematician Hamilton, and he called his Quaternion as commutative Hamiltonian.


i = jk = k j

j = ki = ik

-k = ij = ji

-1 = i² = j² = -k²
if your machine does not show them correctly, then you may read as

i = j x k = j x j
j = k x i = i x k
-k = i x j = j x i
-1 = i x i = j x j = -k x k

By separating this Quaternion into 2 orthogonal groups, the electro-magnetic field and Newtonian regular force field does not effect each other at all. One may wonder why this definition is new. It seems like an old-one, but it is not.


Quaternion induces new elements μ0 μ and ν0 ν.
and , since μ and ν are orthogonal we have
μxμx*** xμxμ xνx νx****xν = 0 x 1 = 0
μ and ν can be changed with μ0 or ν0
as long as at least one of the each group element exists.

p85 : reinstating the definitions of moon brackets.

μ0 = ( 1 + k )/2 ≡ μº (identity element) μ = -( i + j )/2 μ0・μ0 = μ0 μ0・μ = μ
ν0 = ( 1 - k )/2 ≡ νº (identity element) ν = -( i - j )/2 ν0ν0 = ν0 ν0ν = ν
i = jk = k j i² =-1 -i = μ + ν ω =[a+d, b+c, a-d, b-c]=(α+γ,β+δ,β-δ,α-γ)/2
j = ki = ik j² =-1 -j = μ - ν ω = [α,β,γ,δ] = α・μ0 -β・μ +γ・ν0 -δ・ν
-k = ij = ji k² = 1 k = μ0 - ν0 ω = ( a, b, c, d ) = 1・a + i・b + j・c + k・d
-1 = i² = j² = -k² 1² = 1 1 = μ0 + ν0 ω = ( x, y, z, t ] = exp( 1・x + i・y + j・z + k・t )





p86 : reinstating the transformation rules among moon brackets.

[α, β, γ, δ] = ( a, b, c, d )
= [ a+d, b+c, a-d, b-c ]
= ( α+γ, β+δ, β-δ, α-γ )/2
= ( (α+γ)/2, (β+δ)/2, (β-δ)/2, (α-γ)/2 )
[α, β, γ, δ] = ( x, y, z, t ]

= [ exp(x+t)・cos(y+z),
exp(x+t)・sin(y+z),
exp(x-t)・cos(y-z),
exp(x-t)・sin(y-z) ]

= ( { loge(α² +β²) + loge(γ² +δ²) }/4,
{ tan¯¹(β/α) + tan¯¹(δ/γ) }/2,
{ tan¯¹(β/α) - tan¯¹(δ/γ) }/2,
{ loge(α² +β²) - loge(γ² +δ²) }/4 ]




p87: explicit formulae of 4x4 matrix μ0(≡μº),μ,ν0(≡νº),ν, each and the multiplication rule of square-moon brackets.

μ0 = | 1/2 , 0 , 0 , 1/2 |
| 0 , 1/2 , 1/2 , 0 |
| 0 , 1/2 , 1/2 , 0 |
| 1/2 , 0 , 0 , 1/2 |

=(1/2)・| 1 , 0 , 0 , 1 |
| 0 , 1 , 1 , 0 |
| 0 , 1 , 1 , 0 |
| 1 , 0 , 0 , 1 |
half of identity and its
transpose forming X :
μ = | 0 , 1/2 , 1/2 , 0 |
|-1/2 , 0 , 0 , -1/2 |
|-1/2 , 0 , 0 , -1/2 |
| 0 , 1/2 , 1/2 , 0 |

=(1/2)・| 0 , 1 , 1 , 0 |
|-1 , 0 , 0 , -1 |
|-1 , 0 , 0 , -1 |
| 0 , 1 , 1 , 0 |
half of minus-horisontal and
plus-vertical : form a diamond :
ν0 = | 1/2 , 0 , 0 , -1/2 |
| 0 , 1/2 , -1/2, 0 |
| 0 , -1/2 ,1/2 , 0 |
|-1/2 , 0 , 0 , 1/2 |

=(1/2)・| 1 , 0 , 0 , -1 |
| 0 , 1 , -1 , 0 |
| 0 , -1 , 1 , 0 |
| -1 , 0 , 0 , 1 |
half of identity and its
minus-transpose forming X :
ν = | 0 , 1/2, -1/2 , 0 |
|-1/2 , 0 , 0, 1/2 |
| 1/2 , 0 , 0, -1/2 |
| 0, -1/2 , 1/2 , 0 |

=(1/2)・| 0 , 1, -1 , 0 |
| -1, 0 , 0 , 1 |
| 1 , 0 , 0 , -1 |
| 0 , -1 , 1 , 0 |
half of plus-minus twisted diamond :
[α, β, γ, δ][α´, β´, γ´, δ´] = [α・α´ - β・β´,
α・β´ + β・α´,
γ・γ´ - δ・δ´,
γ・δ´ + δ・γ´ ]

Note that μº,νº should be written μ0,ν0 to denote non-identity element. However, μ0,ν0 act just like μº,νº in the multiplication. One may use μ0,ν0 to avoid the confusion : μ0(≡μº),μ,ν0(≡νº),ν,meant just this facts.

This multiplication rule is equivalent to say that μ²ν³ = 0 , and 2, 3 can be replaced by m, n for all m, n = 0,1,2,3,4,・ ・・・・・・
It is simply saying thatμ and ν are perpendicular to each other so that they cannot interfere each other.

Thus, the square-moon brackets form an orthogonal system in the quaternion space. We may say the square-moon brackets is the orthogonalization of the full-moon bracets which dipict the regular quaternion spaces, just like an electric field and a magnetic field. Or, one could say a mass and the force acting on it perpendiculary.

And, the most prominent property of the commutative Hamiltonian (this quaternion!) is that , if x,y,z,t are continuous differentiable functions of some parameters, for example, of time τ(tau) , then ω is always an wave equation for the pair x, t , or for the pair y, z separately such that :

ω/dx² - d²ω/dt² = ω´´x x - ω´´t t = 0
ω/dy² - d²ω/dz² = ω´´y y - ω´´z z = 0

And, this fact can be most effective for the computaion of the power functions of ω, ω², ω³ ・・.



APPENDIX

p1 :

THEORY OF COMMUTATIVE HAMILTONIAN
by Õnihsôÿ Îjñëg
29th
July (mon), 1996.

This paper introduces the extention of the complex variable with complex operator 1, i, j, k such that :

i = jk = k j
i² =-1
i = | 0, -1, 0 , 0 |
| 1 , 0 , 0 , 0 |
| 0 , 0 , 0 , 1 |
| 0 , 0 ,-1, 0 |
j = ki = ik
j² =-1
j = | 0 , 0 ,-1, 0 |
| 0 , 0 , 0 , 1 |
| 1 , 0 , 0 , 0 |
| 0, -1, 0 , 0 |
-k = ij = ji
k² = 1
k = | 0, 0, 0, 1 |
| 0, 0, 1, 0 |
| 0, 1, 0, 0 |
| 1, 0, 0, 0 |
-1 = i² = j² = -k²
1² = 1
1 = | 1, 0, 0, 0 |
| 0, 1, 0, 0 |
| 0, 0, 1, 0 |
| 0, 0, 0, 1 |

so that 1, i, j, k are 4x4 matrix each, and the commutative version of Hamiltonian of 16th Oct.,1843.


Note that 1, i, j, k are commutative. Old Hamiltonian is non-commutative so that it was not so useful at all, and till now all the scientists in physics had to deal with the nature without tools. They must have been super genius to make any theory without knowing the 4 dimentional mathematics. With the wrong mathematical structure, how they could make a rockets, I wonder.


p88 : if x, y, z are the 3 dimentions and if t is the time, then

ω = ( x, y, z, t ] = exp( 1・x + i・y + j・z + k・t )
forms a wave equation
(d²ω/dx²) - (d²ω/dt²) = 0
(d²ω/dy²) - (d²ω/dz²) = 0

This gives the reason of the existence of waves in our universe.

In another word, this proves that our universe is 4 dimentional space.

p89 : general formula for the nth power of ω : exp( n・logeω ) is expanded in general form :


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