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Aug 9, 2008, 12:09:01 AM8/9/08

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The next turn given by the Electrodynamics roadmap for

Octonion application to physics requires us to form all

expected forces in the vector Action, and the work done in

the scalar Action. We must demand this Action Function be

an algebraic invariant, since it is a physical observable

that must be consistently represented.

Octonion application to physics requires us to form all

expected forces in the vector Action, and the work done in

the scalar Action. We must demand this Action Function be

an algebraic invariant, since it is a physical observable

that must be consistently represented.

Following the classical path, we might expect this form to

be the Octonion product of our newly found 8-current and

the Octonion field. The 8-current is an algebraic

invariant, so it takes on a singular form. The field form

however involves a combination of both algebraic variant

and invariant. We would be correct to think the product of

8-current and field might also contain both algebraic

variant and invariant.

Rather than search out an algebraic invariant form, we

could plunge right in with the anticipated form and run it

through the Octonion Variance Sieve Process available on my

website.

If we run F*j through the sieve process, we get the

following

Invariant (F*j)

[0]

-j1 [D0(A1)+D1(A0)]

-j2 [D0(A2)+D2(A0)]

-j3 [D0(A3)+D3(A0)]

-j4 [D0(A4)+D4(A0)]

-j5 [D0(A5)+D5(A0)]

-j6 [D0(A6)+D6(A0)]

-j7 [D0(A7)+D7(A0)]

[1]

+j0 [D0(A1)+D1(A0)]

-j2 [D1(A2)-D2(A1)] +j3 [D3(A1)-D1(A3)]

-j4 [D1(A4)-D4(A1)] +j5 [D5(A1)-D1(A5)]

-j7 [D1(A7)-D7(A1)] +j6 [D6(A1)-D1(A6)]

[2]

+j0 [D0(A2)+D2(A0)]

-j3 [D2(A3)-D3(A2)] +j1 [D1(A2)-D2(A1)]

-j4 [D2(A4)-D4(A2)] +j6 [D6(A2)-D2(A6)]

-j5 [D2(A5)-D5(A2)] +j7 [D7(A2)-D2(A7)]

[3]

+j0 [D0(A3)+D3(A0)]

-j1 [D3(A1)-D1(A3)] +j2 [D2(A3)-D3(A2)]

-j4 [D3(A4)-D4(A3)] +j7 [D7(A3)-D3(A7)]

-j6 [D3(A6)-D6(A3)] +j5 [D5(A3)-D3(A5)]

[4]

+j0 [D0(A4)+D4(A0)]

-j5 [D4(A5)-D5(A4)] +j1 [D1(A4)-D4(A1)]

-j6 [D4(A6)-D6(A4)] +j2 [D2(A4)-D4(A2)]

-j7 [D4(A7)-D7(A4)] +j3 [D3(A4)-D4(A3)]

[5]

+j0 [D0(A5)+D5(A0)]

-j1 [D5(A1)-D1(A5)] +j4 [D4(A5)-D5(A4)]

-j3 [D5(A3)-D3(A5)] +j6 [D6(A5)-D5(A6)]

-j7 [D5(A7)-D7(A5)] +j2 [D2(A5)-D5(A2)]

[6]

+j0 [D0(A6)+D6(A0)]

-j1 [D6(A1)-D1(A6)] +j7 [D7(A6)-D6(A7)]

-j2 [D6(A2)-D2(A6)] +j4 [D4(A6)-D6(A4)]

-j5 [D6(A5)-D5(A6)] +j3 [D3(A6)-D6(A3)]

[7]

+j0 [D0(A7)+D7(A0)]

-j2 [D7(A2)-D2(A7)] +j5 [D5(A7)-D7(A5)]

-j3 [D7(A3)-D3(A7)] +j4 [D4(A7)-D7(A4)]

-j6 [D7(A6)-D6(A7)] +j1 [D1(A7)-D7(A1)]

These algebraic invariant forms appear to be just what we

need. Extracting just the Electrodynamic terms we have

[0]

-j5 [D0(A5)+D5(A0)]

-j6 [D0(A6)+D6(A0)]

-j7 [D0(A7)+D7(A0)]

[5]

+j0 [D0(A5)+D5(A0)]

+j6 [D6(A5)-D5(A6)] - j7 [D5(A7)-D7(A5)]

[6]

+j0 [D0(A6)+D6(A0)]

+j7 [D7(A6)-D6(A7)] - j5 [D6(A5)-D5(A6)]

[7]

+j0 [D0(A7)+D7(A0)]

+j5 [D5(A7)-D7(A5)] - j6 [D7(A6)-D6(A7)]

We can recognize the work -( j dot E) scalar in [0]. The

two familiar forces, j0 E and jXB are the vector forms.

The irreducible algebraic variants are shown below. They

would need to be assigned values of zero to make the full

expression F*j an algebraic invariant.

For minimum distance 1: 1/2[SL(123)+SR(321)]

[4]

-j5 [D7(A6)-D6(A7)] -j6 [D5(A7)-D7(A5)] -j7 [D6(A5)-D5(A6)]

[5]

+j4 [D7(A6)-D6(A7)] +j6 [D4(A7)-D7(A4)] -j7 [D4(A6)-D6(A4)]

[6]

+j4 [D5(A7)-D7(A5)] -j5 [D4(A7)-D7(A4)] +j7 [D4(A5)-D5(A4)]

[7]

+j4 [D6(A5)-D5(A6)] +j5 [D4(A6)-D6(A4)] -j6 [D4(A5)-D5(A4)]

For minimum distance 1: 1/2[SL(123)-SR(321)]

[0]

-j1 [D2(A3)-D3(A2)] -j2 [D3(A1)-D1(A3)] -j3 [D1(A2)-D2(A1)]

[1]

+j0 [D2(A3)-D3(A2)] -j2 [D0(A3)+D3(A0)] +j3 [D0(A2)+D2(A0)]

[2]

+j0 [D3(A1)-D1(A3)] +j1 [D0(A3)+D3(A0)] -j3 [D0(A1)+D1(A0)]

[3]

+j0 [D1(A2)-D2(A1)] -j1 [D0(A2)+D2(A0)] +j2 [D0(A1)+D1(A0)]

For minimum distance 2: 1/2[SL(761)+SR(167)]

[2]

-j3 [D4(A5)-D5(A4)] -j4 [D5(A3)-D3(A5)] -j5 [D3(A4)-D4(A3)]

[3]

+j2 [D4(A5)-D5(A4)] -j4 [D2(A5)-D5(A2)] +j5 [D2(A4)-D4(A2)]

[4]

+j2 [D5(A3)-D3(A5)] +j3 [D2(A5)-D5(A2)] -j5 [D2(A3)-D3(A2)]

[5]

+j2 [D3(A4)-D4(A3)] -j3 [D2(A4)-D4(A2)] +j4 [D2(A3)-D3(A2)]

For minimum distance 2: 1/2[SL(761)-SR(167)]

[0]

-j1 [D7(A6)-D6(A7)] -j6 [D1(A7)-D7(A1)] -j7 [D6(A1)-D1(A6)]

[1]

+j0 [D7(A6)-D6(A7)] +j6 [D0(A7)+D7(A0)] -j7 [D0(A6)+D6(A0)]

[6]

+j0 [D1(A7)-D7(A1)] -j1 [D0(A7)-D7(A0)] +j7 [D0(A1)+D1(A0)]

[7]

+j0 [D6(A1)-D1(A6)] +j1 [D0(A6)+D6(A0)] -j6 [D0(A1)+D1(A0)]

For minimum distance 3: 1/2[SL(572)+SR(275)]

[1]

+j3 [D4(A6)-D6(A4)] -j4 [D3(A6)-D6(A3)] +j6 [D3(A4)-D4(A3)]

[3]

-j1 [D4(A6)-D6(A4)] -j4 [D6(A1)-D1(A6)] -j6 [D1(A4)-D4(A1)]

[4]

+j1 [D3(A6)-D6(A3)] +j3 [D6(A1)-D1(A6)] -j6 [D3(A1)-D1(A3)]

[6]

-j1 [D3(A4)-D4(A3)] +j3 [D1(A4)-D4(A1)] +j4 [D3(A1)-D1(A3)]

For minimum distance 3: 1/2[SL(572)-SR(275)]

[0]

-j2 [D5(A7)-D7(A5)] -j5 [D7(A2)-D2(A7)] -j7 [D2(A5)-D5(A2)]

[2]

+j0 [D5(A7)-D7(A5)] -j5 [D0(A7)+D7(A0)] +j7 [D0(A5)+D5(A0)]

[5]

+j0 [D7(A2)-D2(A7)] +j2 [D0(A7)+D7(A0)] -j7 [D0(A2)+D2(A0)]

[7]

+j0 [D2(A5)-D5(A2)] -j2 [D0(A5)+D5(A0)] +j5 [D0(A2)+D2(A0)]

For minimum distance 4: 1/2[SL(653)+SR(356)]

[1]

-j2 [D4(A7)-D7(A4)] -j4 [D7(A2)-D2(A7)] -j7 [D2(A4)-D4(A2)]

[2]

+j1 [D4(A7)-D7(A4)] -j4 [D1(A7)-D7(A1)] +j7 [D1(A4)-D4(A1)]

[4]

+j1 [D7(A2)-D2(A7)] +j2 [D1(A7)-D7(A1)] -j7 [D1(A2)-D2(A1)]

[7]

+j1 [D2(A4)-D4(A2)] -j2 [D1(A4)-D4(A1)] +j4 [D1(A2)-D2(A1)]

For minimum distance 4: 1/2[SL(653)-SR(356)]

[0]

-j3 [D6(A5)-D5(A6)] -j5 [D3(A6)-D6(A3)] -j6 [D5(A3)-D3(A5)]

[3]

+j0 [D6(A5)-D5(A6)] +j5 [D0(A6)+D6(A0)] -j6 [D0(A5)+D5(A0)]

[5]

+j0 [D3(A6)-D6(A3)] -j3 [D0(A6)-D6(A0)] +j6 [D0(A3)+D3(A0)]

[6]

+j0 [D5(A3)-D3(A5)] +j3 [D0(A5)+D5(A0)] -j5 [D0(A3)+D3(A0)]

For minimum distance 5: 1/2[SL(145)+SR(541)]

[2]

-j3 [D7(A6)-D6(A7)] +j6 [D7(A3)-D3(A7)] +j7 [D3(A6)-D6(A3)]

[3]

+j2 [D7(A6)-D6(A7)] -j6 [D7(A2)-D2(A7)] +j7 [D6(A2)-D2(A6)]

[6]

-j2 [D7(A3)-D3(A7)] +j3 [D7(A2)-D2(A7)] +j7 [D2(A3)-D3(A2)]

[7]

-j2 [D3(A6)-D6(A3)] -j3 [D6(A2)-D2(A6)] -j6 [D2(A3)-D3(A2)]

For minimum distance 5: 1/2[SL(145)-SR(541)]

[0]

-j1 [D4(A5)-D5(A4)] -j4 [D5(A1)-D1(A5)] -j5 [D1(A4)-D4(A1)]

[1]

+j0 [D4(A5)-D5(A4)] -j4 [D0(A5)+D5(A0)] +j5 [D0(A4)+D4(A0)]

[4]

+j0 [D5(A1)-D1(A5)] +j1 [D0(A5)+D5(A0)] -j5 [D0(A1)+D1(A0)]

[5]

+j0 [D1(A4)-D4(A1)] -j1 [D0(A4)+D4(A0)] +j4 [D0(A1)+D1(A0)]

For minimum distance 6: 1/2[SL(246)+SR(642)]

[1]

+j3 [D5(A7)-D7(A5)] +j5 [D7(A3)-D3(A7)] -j7 [D5(A3)-D3(A5)]

[3]

-j1 [D5(A7)-D7(A5)] +j5 [D1(A7)-D7(A1)] +j7 [D5(A1)-D1(A5)]

[5]

-j1 [D7(A3)-D3(A7)] -j3 [D1(A7)-D7(A1)] -j7 [D3(A1)-D1(A3)]

[7]

+j1 [D5(A3)-D3(A5)] -j3 [D5(A1)-D1(A5)] +j5 [D3(A1)-D1(A3)]

For minimum distance 6: 1/2[SL(246)-SR(642)]

[0]

-j2 [D4(A6)-D6(A4)] -j4 [D6(A2)-D2(A6)] -j6 [D2(A4)-D4(A2)]

[2]

+j0 [D4(A6)-D6(A4)] -j4 [D0(A6)+D6(A0)] +j6 [D0(A4)+D4(A0)]

[4]

+j0 [D6(A2)-D2(A6)] +j2 [D0(A6)+D6(A0)] -j6 [D0(A2)+D2(A0)]

[6]

+j0 [D2(A4)-D4(A2)] -j2 [D0(A4)+D4(A0)] +j4 [D0(A2)+D2(A0)]

For minimum distance 7: 1/2[SL(347)+SR(743)]

[1]

-j2 [D6(A5)-D5(A6)] +j5 [D6(A2)-D2(A6)] +j6 [D2(A5)-D5(A2)]

[2]

+j1 [D6(A5)-D5(A6)] -j5 [D6(A1)-D1(A6)] +j6 [D5(A1)-D1(A5)]

[5]

-j1 [D6(A2)-D2(A6)] +j2 [D6(A1)-D1(A6)] +j6 [D1(A2)-D2(A1)]

[6]

-j1 [D2(A5)-D5(A2)] -j2 [D5(A1)-D1(A5)] -j5 [D1(A2)-D2(A1)]

For minimum distance 7: 1/2[SL(347)-SR(743)]

[0]

-j3 [D4(A7)-D7(A4)] -j4 [D7(A3)-D3(A7)] -j7 [D3(A4)-D4(A3)]

[3]

+j0 [D4(A7)-D7(A4)] -j4 [D0(A7)+D7(A0)] +j7 [D0(A4)+D4(A0)]

[4]

+j0 [D7(A3)-D3(A7)] +j3 [D0(A7)+D7(A0)] -j7 [D0(A3)+D3(A0)]

[7]

+j0 [D3(A4)-D4(A3)] -j3 [D0(A4)+D4(A0)] +j4 [D0(A3)+D3(A0)]

The next turn on the Electrodynamics roadmap is to recast

the work-force equation in an integrable form such that

every product term contains an outside differentiation.

Then we integrate this over the spatial 7-volume. Terms

with outside scalar differentiation become time derivatives

of volume integrals. Terms with outside spatial

differentiations are converted to surface integrals for the

surfaces enclosing the 7-volumes. We can equate the also

7-volume integrated original work-force action with this

equivalent but different integral form. The result is an

expression for the conservation of energy in the scalar

portion, and conservation of momentum in the vector

portion.

In classical Electrodynamics, this process is expressing

in part the work-force as the differential contraction of

the stress-energy-momentum tensor. We will require the

resultant differentiations to be identical in the Octonion

representation to claim victory expressing Electrodynamics

within the algebra of Octonions.

Examining terms in the classical Electrodynamics

stress-energy-momentum tensor, we see that every term is

the product of two field components. This suggests we look

to change

Di(Aj)DkDl(Am) (ui uj)(uk (ul um))

to a form

Di [Dj(Ak)Dl(Am)] ui [(uj uk)(ul um)]

where ijklm in the first has no connection to ijklm of the

second, they are just indices. Does it look easy? Well it

is not when you change the basis unit product history.

Without major clues, this would be a daunting task even

with a computer running a symbolic algebra program highly

tuned to do the heavy lifting for the Octonion products.

Been there, done that. The job becomes much easier when we

put our faith in the Law of Octonion Algebraic Invariance.

The full measure of algebraic invariant content in F*j was

just what we were looking for to fulfill our expectations

for work-force based on classical Electrodynamics. We will

be well served by expecting a match from the full

compliment of algebraic invariant basis unit products for

the product history

ui [(uj uk)(ul um)]

There are five ways to roll algebraic invariants with this

basis unit product progression. I will not repeat them

here. They are available in a PDF on my website. Instead,

I will present something not covered in the PDF, a "C"

like pseudo-code procedural outline of how to crank them

all out with proper signs to produce an identity with

the work-force form already presented.

The pseudo-code has two loops. One cranks out

differentiations of forms found off-diagonal in the classic

stress-energy-momentum tensor. The second covers

differentiations of forms found on-diagonal in the tensor.

Of course there is more than this involved in the Octonion

formation.

Take Ti as the "i"th basis unit component for the modified

Action Function. For the off-diagonal do this

For all unequal i,j,k

{

if j = 0 or (k != 0 and (uj uk) = +/- ui)

s = -1

else

s = +1

Ti = Ti + s*Dj[ {Dj(Ak)uj uk + Dk(Aj)uk uj}

* {Di(Ak)ui uk + Dk(Ai)uk ui} ]

}

The on-diagonal forms are expressed by

For i=0 to 7

{

For j=0 to 6

{

For k=j+1 to 7

{

if i=0

s = +1

else

{

if j=0

{

if k=i

s = +1

else

s = -1

}

else

{

if j=i or k=i

s = -1

else

s = +1

}

}

Ti = Ti + s*Di[ {Dj(Ak)uj uk + Dk(Aj)uk uj}

*{Dj(Ak)uj uk + Dk(Aj)uk uj} ]

}

}

}

Since we have algebraic invariants in every product term,

it does not matter which algebra is used, all give the same

result.

Lets identify some of the terms in the off-diagonal

process.

If j=0 we have the time rate of change in the extended

Poynting Vector for component ui. If i=0 we have the seven

terms in the divergence of the Poynting Vector for unit u0.

If k=0, we have the product of two irrotational field

components, like the familiar ExEy. If none of ijk are 0,

we have the product of two rotational field components,

like BxBy.

The on-diagonal terms are all differentiations of products

of like field components representing partial energy

densities. For i=0 this is simply the negative of the time

rate of change in total energy density. For i not zero, it

is a bit more than the gradient of energy density as it is

for classical Electrodynamics.

One only gets a taste of the eloquence of the integrable

form for the work-force equations. I will present them in

their full glory in the next installment.

Rick Lockyer

For more information see

http://www.octospace.com/files/Octonion_Algebra_and_its_Connection_to_Physics.pdf

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