Why Octonions 3: Octonion Algebraic Variance and Invariance
Rick
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