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Having the table of ordered permutation triplet basis product rules for Right Octonion Algebra in the body of my essay, it can be used to visualize my algebraic variance/invariance sieve, and demonstrate that all product terms for any number of Octonion algebraic element products will fall into an algebraically invariant set, or one of 14 algebraically variant sets.

To make sure everyone...

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Having the table of ordered permutation triplet basis product rules for Right Octonion Algebra in the body of my essay, it can be used to visualize my algebraic variance/invariance sieve, and demonstrate that all product terms for any number of Octonion algebraic element products will fall into an algebraically invariant set, or one of 14 algebraically variant sets.

To make sure everyone gets the drift on the ordered permutation triplet basis product rule, going cyclically left to right, the consecutive product of two basis elements is + the third element, and commuting the product order going cyclically right to left, the consecutive product of two basis elements is - the third element. For any three unlike non-scalar basis elements there are two possible definitions, starting with one order then forming another rule exchanging any two elements which is fully equivalent to ordering them in the opposite direction. This singular rule change has the effect of changing the sign on all 6 basis product pairs. Very important that it is just a sign change.

It will not be of any importance that I use a table for Right Octonions and not Left Octonions, nor that the table is relative to algebra R0. Any other choice will sieve the same product terms into the same sets, the only difference will be the relative signs within the variant sets.

I will do a proof by example using a sequence of basis element products describing the product history of the final resultant basis element. The product e1 * e2 is ruled by the ordered permutation triplet including the set {e1 e2 e3}, using curly braces to not imply a sign rule quite yet. For R0, R1, R2 and R3 the sign rule is (e1 e2 e3) and for R4, R5, R6 and R7 the rule is the opposite: (e3 e2 e1). The correspondence between algebra enumerations and triplet enumerations is no accident, they were purposefully enumerated to make it such. Anyway, for the former 4 the result is +e3, and for the latter 4 it is -e3. This would be a simple example of an algebraically variant product term; it can change signs when a change of algebra definition is made.

Now take the e3 result and multiply on the left by e2, effectively doing e2 * (e1 * e2). This is based on the same triplet rule {e1 e2 e3}, so any algebra change induced negation will be done twice, meaning for every possible Octonion Algebra choice, the result will always be +e1. This is an example of a non-trivial (not singularly defined e0 * en, ej * ej etc.) algebraically invariant product term.

If instead we did e5 * (e1 * e2), the second product would be using the rule for {e6 e5 e3} so the sign on the final result basis element e6 would be dependent on both {e1 e2 e3} and {e6 e5 e3} rules, and they can change in different ways for specific changes in algebra. Relative to R0, which has all +1 values in its column and results in +e6, the e6 sign for some other Right algebra will be determined by the product of the row {e1 e2 e3} and {e6 e5 e3} values for that algebra's column. If the row product is +1, the result in that algebra will be the same as in R0: +e6, and if the row product is -1, the result for that algebra will have opposite sign as R0 indicates, or -e6. These column products are precisely the compositions mentioned in my essay. {e1 e2 e3} and {e6 e5 e3} have e3 in common, and the only other triplet including e3 is {e7 e4 e3} which indeed is the resultant row from the composition operation. So we can look at this row and see for R0, R3, R4 and R7 the final result for e5 * (e1 * e2) will be +e6, and for R1, R2, R5 and R6 the result will be -e6. Doing the same row composition on our first example above, the composition ends up on the all +1 row, which is where every algebraic invariant product term will end up, independent of the number of products. If along the way we needed to multiply by the scalar e0 or the very same current result basis, these rules are singularly defined, so one must stay on the current row.

You can see now 7 algebraically variant rows to land on, and might be questioning how I get to my claim of twice this number. We must bring in the Left Octonion Algebras into the discussion, or more precisely the anti-automorphism Right to Left morph which negates all seven triplet rules. If you ended up on a particular variant row through the application of an even number of variant products, changing every one would not change the final result. However, if you did an odd number of variant products, the anti-automorphism would change the result sign. Thus the variant count parity doubles the 7 to 14. My V+{abc} is even count, and V-{abc} is odd count. Since the composition rule is closed, any number of basis products may be done with comparable results.

Now, what happens in each of these variant sets is that for any legitimate Octonion Algebra definition change, every product term in a set will either change sign, or not change sign. My "Law of Octonion Algebraic Invariance" states any observable described by Octonion Algebra must be an algebraically invariant form. The corollary to this is any algebraically variant form is not observable. If we individually add/subtract per sign all product terms in a variant set and force a zero result for each set, a mixed bag of variant and invariant product terms now becomes fully invariant since +0 = -0. These are my "Homogeneous Equations of Algebraic Constraint". It is easy to believe experimentation will not show everything that needs to be seen. I am thinking these equations of constraint are extremely important.

Octonion Algebra is not talking softly here, it is shouting!

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My earlier post on my Algebraic Invariance and Variance Sieve seems to have missed the mark of being presented in an understandable way since Geoffrey Dixon did not follow it, telling me I did a poor job. Let me try to improve things. First, what the heck is this all about?

In my essay I state the different ways Octonion Algebra can be defined is fully covered by select orientation changes...

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My earlier post on my Algebraic Invariance and Variance Sieve seems to have missed the mark of being presented in an understandable way since Geoffrey Dixon did not follow it, telling me I did a poor job. Let me try to improve things. First, what the heck is this all about?

In my essay I state the different ways Octonion Algebra can be defined is fully covered by select orientation changes within the seven Quaternion subalgebra triplets, a significant simplification over changes to 64 element multiplication tables. There is no reason to believe one of the 16 is preferred over the others, but the outcome of changes between definitions is the possibility of a result sign change. So if the result is our attempt to describe something we can observe to be one sign and never the other sign, the algebra change would be problematic if the result might change sign.

Fortunately there are Octonion product terms that maintain consistent signage for all algebra definitions. These should be considered invariant symmetries of Octonion Algebra. If we can’t assign a preference to one algebra definition over another we must insure the results of any theoretical application of Octonion Algebra to physical reality must fully reside inside the set of invariant product terms. But how can we simply determine whether or not a product term is an Algebraic Invariant?

Product terms that are not invariant to Octonion Algebra definition changes will have one sign for eight of the definitions and the opposite sign for the other 8. These should be considered anti-symmetric variants of Octonion Algebra. All Algebraically Variant product terms can be sorted into one of 14 sets, where set members all change signs in exactly the same way for all Octonion Algebra definition changes.

Every product term for any possible series of Octonion algebraic element products can be classified as a member of the invariant set or one of the variant sets. Which set is fully determined by the product history from initial native algebraic element definitions through the final results after some number of multiplications have been performed. Product order matters, and at each step of the product history there is a multiplication between two basis elements with an algebra specific rule determining the result. After the first multiplication in a sequence, one of these elements is a composite result of all earlier products and the other is the next up in the history. The algebraic rule is going to be determined by one variable definition Quaternion triplet rule, or the consistently defined products including the scalar basis or between like basis elements.

For the sieve algorithm applied to a product history, at each basis element pair product, a move is prescribed from the current row in the Hadamard Matrix table in my essay to either itself or a new row. Which move is entirely determined by the next basis pair product. If the rule this product is governed by is not a variant triplet rule, remain at the current row. If not, the new row is found by doing the row composition described in my essay between the Hadamard elements of the current row and the Hadamard elements for the row labeled by the triplet called out by the basis element product. One could consider the stay on the same row as a composition move specified between the current row and the “none” row which is the composition identity element.

Procedurally you start a product history on the identity “none” row. The first basis element product rule composition is applied, and you change rows or don’t as described above. Next determine the rule between the result basis element from the last product with the next basis element up in the product history sequence, use it to make the next row change or stay. Repeat until the product history sequence completes. If you end up on the “none” row, the product sequence is an Algebraic Invariant. Otherwise you will be on a row with a triplet label and the product sequence is an Algebraic Variant partially described by the label, let’s call it {abc}. If the sequence involved an even number of triplet compositions, the variant set will be what I call V+{abc}, if odd count, it will be in the set V-{abc}.

Hopefully this clears things up.

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Dear Rick Lockyer

Wonderful words about truth in science..."The most fundamental concept in physical reality we generally refer to as Nature is truth. Nature’s truths are independent of our existence. Some of Nature’s truths are readily apparent, others are hidden by obscurity. Some truths are before us yet await our emergence from ignorance or bias. Pure science is the pursuit of...

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Dear Rick Lockyer

Wonderful words about truth in science..."The most fundamental concept in physical reality we generally refer to as Nature is truth. Nature’s truths are independent of our existence. Some of Nature’s truths are readily apparent, others are hidden by obscurity. Some truths are before us yet await our emergence from ignorance or bias. Pure science is the pursuit of understanding Natute’s truths"......

And you are exactly correct in saying that' True science has no political agenda, is not about corroborating preconceived notions, and does not necessarily require conformance with group think or so called “settled science”.

I hope you will not mind that I am not following main stream physics...

By the way…Here in my essay energy to mass conversion is proposed...……..….. yours is very nice essay best wishes …. I highly appreciate hope your essay ….You may please spend some of the valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance

Some of the Main foundational points of Dynamic Universe Model :

-No Isotropy

-No Homogeneity

-No Space-time continuum

-Non-uniform density of matter, universe is lumpy

-No singularities

-No collisions between bodies

-No blackholes

-No warm holes

-No Bigbang

-No repulsion between distant Galaxies

-Non-empty Universe

-No imaginary or negative time axis

-No imaginary X, Y, Z axes

-No differential and Integral Equations mathematically

-No General Relativity and Model does not reduce to GR on any condition

-No Creation of matter like Bigbang or steady-state models

-No many mini Bigbangs

-No Missing Mass / Dark matter

-No Dark energy

-No Bigbang generated CMB detected

-No Multi-verses

Here:

-Accelerating Expanding universe with 33% Blue shifted Galaxies

-Newton’s Gravitation law works everywhere in the same way

-All bodies dynamically moving

-All bodies move in dynamic Equilibrium

-Closed universe model no light or bodies will go away from universe

-Single Universe no baby universes

-Time is linear as observed on earth, moving forward only

-Independent x,y,z coordinate axes and Time axis no interdependencies between axes..

-UGF (Universal Gravitational Force) calculated on every point-mass

-Tensors (Linear) used for giving UNIQUE solutions for each time step

-Uses everyday physics as achievable by engineering

-21000 linear equations are used in an Excel sheet

-Computerized calculations uses 16 decimal digit accuracy

-Data mining and data warehousing techniques are used for data extraction from large amounts of data.

- Many predictions of Dynamic Universe Model came true….Have a look at

http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.h
tml

I request you to please have a look at my essay also, and give some of your esteemed criticism for your information……..

Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.

In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from “http://vaksdynamicuniversemodel.blogspot.in/ ”

I request you to please post your reply in my essay also, so that I can get an intimation that you replied

Best

=snp

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http://www.scirp.org/journal/PaperInformation.aspx?PaperID=7
9959

The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors

Full-Text HTML XML Download Download as PDF (Size:572KB) PP. 727-753

DOI: 10.4236/jhepgc.2017.34055 93 Downloads 140 Views

Author(s) Leave a comment Andrew Walcott...

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http://www.scirp.org/journal/PaperInformation.aspx?PaperID=7
9959

The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors

Full-Text HTML XML Download Download as PDF (Size:572KB) PP. 727-753

DOI: 10.4236/jhepgc.2017.34055 93 Downloads 140 Views

Author(s) Leave a comment Andrew Walcott Beckwith Affiliation(s)

Physics Department, College of Physics, Chongqing University Huxi Campus, Chongqing, China.

ABSTRACT

The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial Octonionic geometry embedded in a larger, nonlinear “pilot model” (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a “big crunch” is picked as the embedding structure, so as to avoid the “instants” of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards “Schrodinger equation” states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to; making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry.

KEYWORDS

Octonionic Geometry, Cyclic Conformal Cosmology (Penrose), Modified HUP

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Hi Rick , thank you for sharing your passion for algebra and in particular octonians.

At the beginning you state "mathematics is a language of unambiguous truth", I used to think that the certainty of mathematical solutions do make it more reliable than other kinds of communication. Though statistics is notorious for misleading. Having read "Creating perspective on set theory" by...

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Hi Rick , thank you for sharing your passion for algebra and in particular octonians.

At the beginning you state "mathematics is a language of unambiguous truth", I used to think that the certainty of mathematical solutions do make it more reliable than other kinds of communication. Though statistics is notorious for misleading. Having read "Creating perspective on set theory" by Christophe Tournayre, in this competition, I have been re-considering my opinion. I found this paper "When is one thing equal to some other thing?" Barry Mazur (2007) www.math.harvard.edu/~mazur/preprints/when_is_one.pdf Which showed me that even something as common place as use of the equality symbol is not simple and not immune from ambiguity. Also a statement, mathematical or otherwise, can be true, as in not false; and yet not be the whole truth. A more complete truthfulness painting an entirely different picture. An algebraic description can give a singular viewpoint of a phenomenon but does not give all viewpoints, which would be the whole truth. Something Peter is approaching with different observations of the same rotating phenomenon. None of the singular values being the whole truth.

In a universe where there is sequential passage of time, as needed by QM and fitting the arrows of time, there is non-commutivity. As new Now follows previous Now definitely and so when things happen in the representation is important. Which is something in favour of quaternions ( and maybe their big bother). Though as I see it the real number line is not an actual dimension of the universe where physics is happening, but a history time line without physical counterpart. You mention 4 dimensional space from primary sense perceptions. That has to be a separate 'map' space from the former representing where physics is happening. The sense space has a time dimension, and non simultaneity of events can happen as the time is related to signal transmissions and not the foundational sequential passage of time. The two different forms of universe can not be described by just the one form of the algebra because of the fundamental difference in what time is in each. If quaternions were used for both in one version there is the non physical history time line and in the other the geometric time dimension.

Thank you for food for thought. I am going to think on about non associativity in nature and whether it is needed in a representation, and about representation of the different kinds of field, that you discussed. Kind regards Georgina

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I posted this on Geoffrey Dixon’s essay thread, may be interesting to readers coming here:

Geoffrey,

Knowing you are not big on the Cayley-Dickson doubling algorithm, I did neglect to mention another reason my Quaternion triplet enumeration algorithm is cooler than yours (no offense intended). That would be for enumerating the triplets for the sedenions. For Octonion Algebra I used...

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I posted this on Geoffrey Dixon’s essay thread, may be interesting to readers coming here:

Geoffrey,

Knowing you are not big on the Cayley-Dickson doubling algorithm, I did neglect to mention another reason my Quaternion triplet enumeration algorithm is cooler than yours (no offense intended). That would be for enumerating the triplets for the sedenions. For Octonion Algebra I used the binary numbers 1 through 7 and partitioned the triplets with the bit wise exclusive or logic function. This can be extended to the sedenions by going to binary 1 through 15, of course representing directly the 15 non-scalar basis elements enumerations. Doing the exclusive or logic operation on binary 1 through 15 partitions them into 35 unique closed triplet sets, and since there are (n-1)*(n-2) = 15*14 = 210 basis products not e0 * en, en * e0 or en * en fixed product rules, and each triplet does 6 rules, 35*6 = 210, covering all 210 basis products with 35 Quaternion triplets.

For each Octonion subalgebra using this enumeration technique the seven non-scalar basis element indexes will exclusive or to a zero result, as well each Quaternion index triplet. If you take away from the seven Octonion non-scalar basis indexes any set of three which are Quaternion triplet indexes, the remaining four basis elements are called “basic quads” and their indexes will also exclusive or to zero. They are basic quads because given just them, the exclusive or of any two will be the index of one of the companion Quaternion basis triplets and all are provided 2 up with this.

This gives a method to determine how many Octonion subalgebra candidates there would be, just go through all quads of distinct indexes 1 through 15 for combinations that exclusive or to zero. A simple computer program will show there are 105 such quads. Any Octonion definition has seven Quaternion triplets each with a unique basic quad, so we would expect to see 105/7 = 15 candidate Octonion subalgebras. Each of the 35 Quaternion triplets occurs in 3 Octonion subalgebras 3*35 = 105.

The rub is you can’t make all 15 Octonion subalgebra candidates represent valid normed composition algebras, e.g. division algebras. An algebraic proof the sedenions are not a division algebra? Maybe so.

Thought you might find this interesting.

Rick

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Dear Rick

If you are looking for another essay to read and rate in the final days of the contest, will you consider mine please? I read all essays from those who comment on my page, and if I cant rate an essay highly, then I don’t rate them at all. Infact I haven’t issued a rating lower that ten. So you have nothing to lose by having me read your essay, and everything to...

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Dear Rick

If you are looking for another essay to read and rate in the final days of the contest, will you consider mine please? I read all essays from those who comment on my page, and if I cant rate an essay highly, then I don’t rate them at all. Infact I haven’t issued a rating lower that ten. So you have nothing to lose by having me read your essay, and everything to gain.

Beyond my essay’s introduction, I place a microscope on the subjects of universal complexity and natural forces. I do so within context that clock operation is driven by Quantum Mechanical forces (atomic and photonic), while clocks also serve measure of General Relativity’s effects (spacetime, time dilation). In this respect clocks can be said to possess a split personality, giving them the distinction that they are simultaneously a study in QM, while GR is a study of clocks. The situation stands whereby we have two fundamental theories of the world, but just one world. And we have a singular device which serves study of both those fundamental theories. Two fundamental theories, but one device? Please join me and my essay in questioning this circumstance?

My essay goes on to identify natural forces in their universal roles, how they motivate the building of and maintaining complex universal structures and processes. When we look at how star fusion processes sit within a “narrow range of sensitivity” that stars are neither led to explode nor collapse under gravity. We think how lucky we are that the universe is just so. We can also count our lucky stars that the fusion process that marks the birth of a star, also leads to an eruption of photons from its surface. And again, how lucky we are! for if they didn’t then gas accumulation wouldn’t be halted and the star would again be led to collapse.

Could a natural organisation principle have been responsible for fine tuning universal systems? Faced with how lucky we appear to have been, shouldn’t we consider this possibility?

For our luck surely didnt run out there, for these photons stream down on earth, liquifying oceans which drive geochemical processes that we “life” are reliant upon. The Earth is made up of elements that possess the chemical potentials that life is entirely dependent upon. Those chemical potentials are not expressed in the absence of water solvency. So again, how amazingly fortunate we are that these chemical potentials exist in the first instance, and additionally within an environment of abundant water solvency such as Earth, able to express these potentials.

My essay is attempt of something audacious. It questions the fundamental nature of the interaction between space and matter Guv = Tuv, and hypothesizes the equality between space curvature and atomic forces is due to common process. Space gives up a potential in exchange for atomic forces in a conversion process, which drives atomic activity. And furthermore, that Baryons only exist because this energy potential of space exists and is available for exploitation. Baryon characteristics and behaviours, complexity of structure and process might then be explained in terms of being evolved and optimised for this purpose and existence. Removing need for so many layers of extraordinary luck to eventuate our own existence. It attempts an interpretation of the above mentioned stellar processes within these terms, but also extends much further. It shines a light on molecular structure that binds matter together, as potentially being an evolved agency that enhances rigidity and therefor persistence of universal system. We then turn a questioning mind towards Earths unlikely geochemical processes, (for which we living things owe so much) and look at its central theme and propensity for molecular rock forming processes. The existence of chemical potentials and their diverse range of molecular bond formation activities? The abundance of water solvent on Earth, for which many geochemical rock forming processes could not be expressed without? The question of a watery Earth? is then implicated as being part of an evolved system that arose for purpose and reason, alongside the same reason and purpose that molecular bonds and chemistry processes arose.

By identifying atomic forces as having their origin in space, we have identified how they perpetually act, and deliver work products. Forces drive clocks and clock activity is shown by GR to dilate. My essay details the principle of force dilation and applies it to a universal mystery. My essay raises the possibility, that nature in possession of a natural energy potential, will spontaneously generate a circumstance of Darwinian emergence. It did so on Earth, and perhaps it did so within a wider scope. We learnt how biology generates intricate structure and complexity, and now we learn how it might explain for intricate structure and complexity within universal physical systems.

To steal a phrase from my essay “A world product of evolved optimization”.

Best of luck for the conclusion of the contest

Kind regards

Steven Andresen

Darwinian Universal Fundamental Origin

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