Why the wave starts ?
Monday, 25Jan1999
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.
By separating this Quaternion into 2 orthogonal groups, the electromagnetic 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 oldone, but it is not.
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  ν0・ν0 = ν0  ν0・ν = ν 
i = j ・ k = k ・ j  i² =1  i = μ + ν  ω =[a+d, b+c, ad, bc]=(α+γ,β+δ,βδ,αγ)/2 
j = k ・ i = i ・ k  j² =1  j = μ  ν  ω = [α,β,γ,δ] = α・μ0 β・μ +γ・ν0 δ・ν 
k = i ・ j = j ・ i  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, ad, bc ] = ( α+γ, β+δ, βδ, αγ )/2 = ( (α+γ)/2, (β+δ)/2, (βδ)/2, (αγ)/2 ) 
[α, β, γ, δ] = ( x, y, z, t ] = [ exp(x+t)・cos(y+z), exp(x+t)・sin(y+z), exp(xt)・cos(yz), exp(xt)・sin(yz) ] = ( { 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 squaremoon 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 minushorisontal and plusvertical : 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 minustranspose 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 plusminus 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 nonidentity 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 μ²・ν³
= 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 squaremoon brackets form an orthogonal system in the quaternion
space. We may say the squaremoon brackets is the orthogonalization of
the fullmoon 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 :
d²ω/dx²  d²ω/dt² = ω´´x
x  ω´´t t
= 0
d²ω/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 ・・.
p1 :
This paper introduces the extention of the complex variable with complex
operator 1, i, j, k such that :


i =  0, 1, 0 , 0   1 , 0 , 0 , 0   0 , 0 , 0 , 1   0 , 0 ,1, 0  


j =  0 , 0 ,1, 0   0 , 0 , 0 , 1   1 , 0 , 0 , 0   0, 1, 0 , 0  


k =  0, 0, 0, 1   0, 0, 1, 0   0, 1, 0, 0   1, 0, 0, 0  


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 noncommutative 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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