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Aug 9, 2008, 12:05:02 AM8/9/08

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Now that we know there are two separate non-isomorphic

Octonion Algebras; Left and Right types, and that each type

may be expressed in eight isomorphic ways, it may seem we

are closer to showing Octonion Algebras are too problematic

for physics than we are to showing their virtues.

Octonion Algebras; Left and Right types, and that each type

may be expressed in eight isomorphic ways, it may seem we

are closer to showing Octonion Algebras are too problematic

for physics than we are to showing their virtues.

After all, any physical system will be described by

functional expressions that are formed by one or more

multiplication operations. We just saw that the definitions

for Octonion multiplication vary between choice of

representation, but we must insist that the mathematical

results representing physically measurable entities are

completely consistent.

The previous discussion on the variations allowed for the

multiplication rules for Octonions did not indicate that

any of the sixteen possible choices are special enough to

forget about the others. In fact, they are all quite

similar to each other. But all is not lost, since as we

will soon see, it will be possible to form product terms

that actually are consistent across all possible algebras.

It would appear that all expected representations of

observable phenomena will need to be consistent across all

possible Octonion Algebra representations. I will indeed

demand this and see where it leads. This is what I call

"The Law of Octonion Algebraic Invariance".

Any time we form one or more product of Octonion

quantities, we form a number of resultant product terms

using the rules of our singular choice of Octonion Algebra.

It must be a singular choice because you can not change the

rules in the middle of the game. Since I have elected to

keep consistent unit triplets within each permutation,

replaying the entire product generation using a different

form of Octonion Algebra will retain the same components in

every product term. The only thing that might change is

their sign.

A product term is an algebraic invariant if its sign does

not change when the multiplication rules are changed to

those of any other Octonion Algebra. A product term is an

algebraic variant if its sign does change when the

multiplication rules are changed to those of any other

Octonion Algebra. Octonion product expressions may be full

algebraic invariant, mixed variant and invariant, or fully

variant.

When the variant product terms between two algebra choices

are compared to those between two other choices, the

intersection of the two sets may not be empty. Considering

all possible changes, there is a way to separate the

variant product terms into sets without intersection. I

call these irreducible sets minimum distances. I have

devised an algorithm to sift out invariant product

terms and irreducible variant product terms into separate

sets. I will not repeat its description here, it is

available inside a PDF downloadable from my website. It is

called "The Octonion Variance Sieve Process".

If an Octonion product expression is expected to be an

algebraic invariant yet it is not explicitly invariant, it

can be forced invariant by insisting each irreducible

variant product set sums to zero. This is an assignment of

algebraic equations of constraint, and may be very

important to the application of Octonions to physics.

The question of algebraic variance and invariance is fully

determined by the product history of a product term. Since

we have lost associativity for multiplication, this history

is registered in the order performed to reach the final

product term configuration. For every fundamental

multiplication operation, there is an effective unit number

on both sides of the multiplication symbol. They may be

straight up single units or a composite of earlier

multiplications. Either way, it is the effective unit

numbers that will determine the multiplication rule for

that particular multiplication operation, and it will be

the set of all applied multiplication rules that will

define the algebraic variance properties of the product

term.

It is convenient to have an invariant starting point for

the eventual product history. If F = Fn un for n = 0 to 7,

and Fn is a real number, a real valued function, or a

differential operator then F is an algebraic invariant

since there are no products of Octonion basis units in

its definition.

The question of algebraic variance only manifests itself

through the product of unlike vector basis units. The

scalar u0 unit multiplies all other terms exactly the same

for all Octonion Algebras. The same goes for like basis

vector products. Neither of these product rules participate

in the algebraic variance character of an Octonion product

term.

The sign of the product of two unlike vector basis units is

determined by the multiplication rules set within a single

permutation. This single product is an algebraic variant,

since there are a number of algebra changes that will

negate its rule. If a second multiplication is performed

that uses the same permutation rule, any negation from a

change in algebra will be performed twice, leaving the

final sign unchanged. This type of double product is an

algebraic invariant. In fact, it is the only way to

generate an algebraic invariant from a double product

including unlike vector basis units. The even negation

count is the mechanism by which algebraic invariants are

created, although not necessarily requiring the same

permutation rule. Invariant quad product terms can be

formed using the rules of four separate permutations.

Any product term with an odd number of unlike vector basis

unit products is an algebraic variant, since there will be

a choice of algebra change that will produce an odd number

of negations. This is not to say any Octonion triple

product is an algebraic variant, for there are scalar and

like basis unit products in the mix that will allow

formation of invariant product terms.

Lets apply some of this to Octonion extensions to things we

know from Electrodynamics. The 8-potential "A" and

8-differential operator "D" are assumed to be fundamental

algebraic invariants, not formed by any Octonion product

operations.

The decoupled 8-current density for Octonion unit [i] is

- uj uj DjDj(Ai) summed j = 0 to 7

This is an algebraic invariant, since the basis unit

products are all like unit number. We would have expected

this. Scalar charge density and vector current density must

be algebraic invariants.

The irrotational field form for vector unit [i] is

ui u0 Di(A0) + u0 ui D0(Ai)

This field is an algebraic invariant, since it is a single

product with one unit being scalar. This is a good thing,

since we would demand the product of this field and the

invariant scalar component of the 8-current yielding the

central force form to also be invariant.

The rotational field form for vector unit [i] is

uj uk Dj(Ak) summation jk over (ijk)

This is an algebraic variant, since it is a single product

defined by the multiplication rules of the single

permutation (ijk). No big deal, since detectable force

forms involve another vector unit multiplication. Lets look

at this for the extended cross product of current density

and rotational field. Since the 8-current is an algebraic

invariant, we can simply call its kth component Jk. This

force may then be written

Fj = Jk {Dj(Ak)-Dk(Aj)} uk(uj uk) ijk: (ijk)

This force is indeed an algebraic invariant, being the

double application of the multiplication rules of (ijk).

Field contributions to energy density are straight forward

algebraic invariants, like unit product only for

irrotational fields, and like unit product with a double

application of a single permutation rule for the rotational

fields.

How about scalar work? It is the like unit product of two

algebraic invariants, the irrotational field and the

8-current. Again, an algebraic invariant as expected.

Finally, look at the Octonion extension of the Poynting

vector. It is the extended cross product of irrotational

field and rotational field. Since the irrotational field

is an algebraic invariant, the argument is identical to

the cross product force above, just substitute the

irrotational field for the 8-current. Again, Octonion

Algebraic Invariance is there every time it needs to be.

I have fit Electrodynamics within Octonion Algebra from

fields, to current density, to forces, to work-force

action, to the analogy of the stress-energy-momentum

tensor. Every Electrodynamic form needing to be an

algebraic invariant actually is such.

So far, I have only shown that the multiplicity of

definitions for the algebra of Octonions are not much of a

problem for the description of physical phenomenon. All

places where the variation would matter are shown to be

algebraic invariants, consistent across all valid forms of

the algebra. While this is certainly important for

justifying the connection between Octonion Algebra and

physics, it is only part of the story on Octonion Algebraic

Invariance.

One of the best things we might hope for would be for our

mathematical framework to provide a guiding light to

greater understanding of the physical world. The brightest

light would be mandated structure. We do not guess then

verify, we are told how it must be up front. We have such

with the concept of Octonion Algebraic Invariance. If we

look inside Octonion Algebra for things we understand well

like Electrodynamics, we find them as well as other

invariant forms we may never know how to piece together

without the insight provided by the full fundamental

structure of Octonion Algebra and what it mandates

algebraic invariant forms look like.

Algebraic Invariance principles can also be used to modify

Octonion differential equations to other equivalent forms.

The analogous Octonion stress-energy-momentum "tensor"

formation requires changing the work-force action form,

which is the algebraic invariant portion of the Octonion

product of 8-current and field, to an integrable form which

requires an outside differentiation on every product term.

I tried the guess and try method on this for longer than I

would like to admit. Then I got the bright idea to put my

trust in the fundamental truth of Algebraic Invariance.

Since the action expression was the full complement of

invariance of an Octonion product, I looked for the full

complement of algebraic invariant forms for the product

string

ui [ (uj uk) (ul um) ]

Within part time days, I was able to find the identity. It

is published in a PDF on my website and is laid out in a

discussion thread to follow. The form has a factor of >16

more product terms than the work-force product of field and

the sum of 8-current and 8-gradient of the Octonion

equivalent of the Lorentz Condition, which as with standard

classical Electrodynamics is present explicitly in the

identity. This is what made it so difficult to do without

the guiding light of Algebraic Invariance.

If one would run the associator: 1/2( a*(b*c)-(a*b)*c )

through the Octonion Variance Sieve Process, they would

find the portion sieved out as invariant is identically

zero. This is so because as I mentioned, the only way to

get an invariant two-product is a double application of

the rules from a single permutation. This will require two

of the three basis units involved to be identical. It is a

fact that any Octonion subalgebra with only two basis units

is associative. This does not imply all Octonion

algebraically invariant forms are associative, they are

not. However, it would appear the "concept" of Algebraic

Invariance is associative. By this, I mean if one finds a

string of unit products that is an algebraic invariant, any

order changes created by insertion of parenthesis will also

be an algebraic invariant. The permutation rule set may

change, as might the sign of the product term, but it will

be an algebraic invariant. In the interest of full

disclosure, I have not proved this is always the case, I

have only inspected a good number of cases. Some may ask,

so what? I am just planting seeds, that is all.

In conclusion, the variability within the full definition

of Octonion Algebra is one of its most important

attributes. For when we try to reconcile it with the easily

acceptable Law of Octonion Algebraic Invariance, we are

given structure to exploit. Recalculation using a

different selection for the applied algebra results in

product terms that either change sign or do not. When all

sixteen algebras are considered, algebraic invariants take

on mandated form, which matches expectations when we have

them, and shows the way when we may not have enough

information. The algebraic variants can be sieved into

irreducible sets that can provide overall Algebraic

Invariance when their sums are individually assigned a

value of zero. When we are dealing with differential forms,

the irreducible sets provide homogeneous equations of

algebraic constraint.

Rick Lockyer

For more information see

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

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