Why Octonions 6: Octonion Work-Force and Conservation Equations
Rick
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