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Physics, Astronomy & Cosmology / Re: The Hopf gauge
« on: 21/01/2024 06:30:56 »
Hi.
It's not entirely clear if you ( @varsigma ) want any replies or discussion. Maybe you do. It seems to be an interesting collection of ideas that you are considering.
Some way back you said this:
Frobenius' Theorem classifies the finite dimensional associative division algebras over the Reals. There are only 3:
(i) The Reals.
(ii) The Complex Numbers.
(iii) The Quaternions.
The Quaternions are not commutative, while R and C are. So if you wish to have commutativity then there are only 2 possible finite dimensional division algebras over the Reals.
The Octonions would not be included until you relax the associativity demand. The Octonions satisfy only a weaker form of associtivity usually called alternativity.
Mathematicians Kervaire and Milnor are credited with demonstrating that, even if you drop the demand for asscociativity and commutativity completely and just demand finite dimensionality then that dimension must be 1,2,4 or 8 only. Further work demonstrated that all finite dimensional division algebras over R would have to be isomorphic to one of these:
R , C , H (the quaternions), or O (the octonions) and there are no others.
There are still some properties that you may choose to drop or alter. If you drop the demand for finite dimensionality then ..... I know of few theorems that place a lot restrictions on what you can get.
You may also decide that there was nothing very special about R and you could have taken some other field as the scalar field for the algebra. It is known that if you choose a field K that is neither algebraically closed or Real closed the dimensionality of a finite dimensional division algebra over K is not limited to just 1,2,4 or 8. Specifically, there is an infinite set of different (non isomorphic) finite dimensional division algebras over K.
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As it happens there are some applications for octonions in physics: Wikipedia lists some..... https://en.wikipedia.org/wiki/Octonion#Applications - not many of these are demonstrably practical or have yielded anything that could be used, tested and verified except some applications in robotics and machine learning.
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The various theorems about division algebras you've referred to are not marking "division" as being special. They are just limiting the behaviour of the structure under "the second binary operation" (whatever that may be).
Best Wishes.
It's not entirely clear if you ( @varsigma ) want any replies or discussion. Maybe you do. It seems to be an interesting collection of ideas that you are considering.
Some way back you said this:
they (the octonions) are a division ring with four dimensions.....One of a handful of such division rings. I don't think you get any further than the octonions (I could be wrong about that).You seem to be taking division algebras over the Reals. It's not clear which additional properties (such as associativity, commutativity and dimensionality over R) you wish to have.
Frobenius' Theorem classifies the finite dimensional associative division algebras over the Reals. There are only 3:
(i) The Reals.
(ii) The Complex Numbers.
(iii) The Quaternions.
The Quaternions are not commutative, while R and C are. So if you wish to have commutativity then there are only 2 possible finite dimensional division algebras over the Reals.
The Octonions would not be included until you relax the associativity demand. The Octonions satisfy only a weaker form of associtivity usually called alternativity.
Mathematicians Kervaire and Milnor are credited with demonstrating that, even if you drop the demand for asscociativity and commutativity completely and just demand finite dimensionality then that dimension must be 1,2,4 or 8 only. Further work demonstrated that all finite dimensional division algebras over R would have to be isomorphic to one of these:
R , C , H (the quaternions), or O (the octonions) and there are no others.
There are still some properties that you may choose to drop or alter. If you drop the demand for finite dimensionality then ..... I know of few theorems that place a lot restrictions on what you can get.
You may also decide that there was nothing very special about R and you could have taken some other field as the scalar field for the algebra. It is known that if you choose a field K that is neither algebraically closed or Real closed the dimensionality of a finite dimensional division algebra over K is not limited to just 1,2,4 or 8. Specifically, there is an infinite set of different (non isomorphic) finite dimensional division algebras over K.
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what the hell are (the octonions)? Why are they there? Are they any use for anything or just mathematical?All mathematics is valuable and worth having. I may be a little biased in making that statement.
As it happens there are some applications for octonions in physics: Wikipedia lists some..... https://en.wikipedia.org/wiki/Octonion#Applications - not many of these are demonstrably practical or have yielded anything that could be used, tested and verified except some applications in robotics and machine learning.
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This must mean that the Rubik's cube and indeed any 3-dimensional puzzle that has rotating parts is "quaternionic".I'm sure most mathematicians will be pleased that you use Mathematics to describe real world objects and phenomena and not the other way around. For example, you don't say "the quaternions have properties that mimic rotations of a 3-D puzzle". However, the quaternions (and any mathematical object) may be abstract and it's a bit of a leap to assume the world (or universe) could only follow some pattern or behaviour that exists in some algebraic structure - but maybe that is how it is, I don't know. It's more common to imagine that the world does what it does, there is then some mathematical structure that can be constructed which will represent that behaviour.
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I guess we can consider these guys (sedenions) as the doorkeepers of division. Then why is division such a special operation?In many ways it isn't division that is special. The simplest algebraic structure is a Group, it has only one binary operation and you can use a + symbol and addition notation to represent it. However, you can just as easily use a x sign, or a dot . and multiplicative notation to represent it. Rings are a more complicated algebraic structure where you have two binary operations and we naturally choose to use addition notation for one of these and multiplication notation for the other. We put a greater restriction on the properties and behaviour under one binary operation and the Ring is actually an abelian Group just under that operation. By convention, we mark this first binary operation with + and talk about it as if it was addition. There are less restrictions placed on the other binary operation and the Ring is not a group under the other binary operation because various things were not demanded. The use of additive and multiplicative notation is just convenient. It is arbitrary which of these operations you might consider to be analagous to addition in the Real numbers and which one is analagous to multiplication. Indeeed it's un-necessary to make any analogy to operations on ordinary numbers, they are just "the first" and "the second" binary operation acting on some set of elements.
The various theorems about division algebras you've referred to are not marking "division" as being special. They are just limiting the behaviour of the structure under "the second binary operation" (whatever that may be).
Best Wishes.
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