What is Fundamental? by Geoffrey Dixon

peter cameron

both fine now, dunno what that was about. Glad to see what appear to me to be fairly strong connections between your work and what Michaele and I are doing. Sure wish we had your group theory expertise, tho.

Jonathan Dickau

I had the same issue...

Underline characters in the PDF links were turned into spaces, by acrobat or my browser, in the first example. And some part of the https:// got lost too. But I later got the links to work as well.

All the Best,

Jonathan

peter cameron

Geoffrey/Cristi/Jonathan/...

Looking at Geoffrey's comment

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Spinor wavefunction is scalar plus bivector if i understand correctly (please explain if wrong).

Handedness comes from the bivector, of which there are three in the Pauli algebra of 3D space. However it is the two component spinor that comprises wavefunction, not just the bivector (Bohr magneton).

Seems like both the non-commutative and the non-associative properties would come from the bivector, and don't quite understand how the scalar enters into it from consideration of those two broken properties (symmetries?). Is it nothing more than just the 'gauge', not playing any role as an additional topological object (the singularity) with the bivector in the wavefunction that is somehow involved in understanding what's going on?

coming back to my opening question

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Does this mean that the eight component Pauli wavefunction Michaele and I are working with has chiral symmetry breaking built in?

and ditto the 16 component Dirac algebra of the eight by eight geometric representation of the S-matrix generated by geometric products of Pauli wavefunctions?

peter cameron

darn. another issue with formatting getting scrambled when working thru this interface. Trying again:

Geoffrey/Cristi/Jonathan/...

Looking at Geoffrey's comment nO = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Spinor wavefunction is scalar plus bivector if i understand correctly (please explain if wrong). Handedness comes from the bivector, of which there are three in the Pauli algebra of 3D space. However it is the two component spinor that comprises wavefunction, not just the bivector (Bohr magneton).

coming back to my opening question

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Does this mean that the eight component Pauli wavefunction Michaele and I are working with has chiral symmetry breaking built in?

and ditto the 16 component Dirac algebra of the eight by eight geometric representation of the S-matrix generated by geometric products of Pauli wavefunctions?

peter cameron

ok giving up on this for the moment,

gonna restart my computer,

see if that helps.

Jonathan Dickau

Seeing that the character n now replaces all carriage returns, I think it is a system wide problem with the FQXi forum platform software. I think they may be trying to make returns appear as an en-dash for compactness but this is ridiculous.

All the Best, JJD

Geoffrey Dixon

First a test.

If this looks good, I'll respond.

Geoffrey Dixon

This is another test.

The yellow banana grows on the sun. OL

[deleted]

Ok, so no carriage returns. My starting position is T = RâŠ--CâŠ--HâŠ--O. It's just a mathematical object. If it is an essential part of any viable mathematical model of reality, then I suggest it is required because fermion fields require parallelizable spheres in some manner as yet to be determined. Lacking a clear understanding of that I start with T. Like C, H and O, individually, and P =RâŠ--CâŠ--H, T is a spinor space. P is the spinor space of PL, which is essentially the Pauli algebra, so the associated geometry is 3-space. (Still just math.) It a very similar fashion T is the spinor space of the Clifford algebra TL, and its associated geometry is 9-space. In my first book I used the fact that a Dirac spinor is a pair of Pauli spinors (so P2, although this is actually an SU(2) doublet of Dirac spinors) to motivate basing my theory building on T2, which is an SU(2) doublet of Dirac spinors for a 1,9-spacetime. T itself is not only a spinor, it is also an algebra, and its identity can be decomposed into a set of orthogonal projection operators (a la GÃ¼rsey and company at Yale in the 1970s) with respect to which the bivectors of the 1,9-Clifford algebra, which is also the Lie algebra so(1,9), gets decomposed to so(1,3) x u(1) x su(3), and the spinor T2 is decomposable into a collection of su(2) doublet ordinary 1,3-Dirac spinors. Again, this is just pure mathematics, with some physics words thrown in because the maths has a kind of obvious interpretation in the physics context. Anyway, I think I won't try to duplicate all the details of two books and a number of papers in this comment, because I need breakfast. Since the neutrino was mentioned in the initial comment, I'll just add that interpreted as a basis for physics modelling, the neutrino that pops out of the maths is a Dirac neutrino, and it has the potential to have a Dirac mass. So, in conclusion: there is maths; there is physics (standard model); everything I've done is simply to show that one can use the maths as a skeleton onto which all of your preferred QFT-flesh can be attached. If the maths has something to say about the deeper quantum side of things, I am not competent to say what it is. If some other idea proves eventually to be the correct mathematical model of reality, T-maths will still have all these properties that look a lot like they ought to have something to do with our presently accepted theories of physics.

Geoffrey Dixon

The above post is me. Thought I was logged in.

peter cameron

restart was no help, and i see Jonathan's post has the same issue

Rick Lockyer

As I said over on my essay thread, liked your essay very much. We are in agreement in many ways, but not all. For me, the progression of fundamentalness is physical reality -> a subset of mathematics-> algebraic structure. I would put the algebra that allows the paralizeable spheres as more fundamental rather than the other way around as you seem to imply. Chicken or the egg though to some degree. Then there is that pesky issue of whether or not two algebras with demonstrably different structure can be truly considered isomorphic. It is not "Greek" to me.

I have learned a lot from you over the years, along with from John Baez. Can't thank you enough for championing division algebras. You are a legend in my mind.

I just put up a discussion on Octonion Algebraic Invariance/Variance referencing the table in my essay. Might be worth a look for you.

Rick

Geoffrey Dixon

I think I'm logged in, but if not, this is GD again.

As to spheres vs algebra, years ago I had a confab with Baez in which I posited the idea that octonion maths is in a sense holographic, as most of the bits and pieces we like can be used to derive the remaining bits and pieces. That is, no given bit is any more fundamental than any other. Parallelizable spheres yield division algebras; and visa versa. As you do, I start with algebra, but in my heart I view algebra as a kind of intellectual microscope, devised by humans to better see structures that exist without us. I mean, in the total absence of any intellect in the universe, planets and stars still take roughly spherical form. However, without intellect algebra does not exist. But again, I start with algebra, because it allows me to play with things more fundamental.

You say: "Then there is that pesky issue of whether or not two algebras with demonstrably different structure can be truly considered isomorphic." I assume you are referring to the left/right octonion thing. If so, isomorphism is demonstrable. I think I even did it in my windmill book. There is only one octonion algebra, but many ways to organize it. Really, one needn't use integers to to label the units. One could use fruit, or puppies, or anime characters. Integers just allows us to see some structures that are otherwise hidden. Like my fav multiplication table with e1 e2 = e4, imposing invariance with respect to cycling and doubling of integer subscripts (yielding a finite invariance group of order 21).

Anyhum, give me any two multiplication tables for O, and I'll build an isomorphism. None of the best minds in the fields (like Conway and Sloane and others) ever discuss two versions of O that are not isomorphic. There is one O. Until proven otherwise, of course.

Rick Lockyer

I don't really want to belabor the point, but in my way of thinking, sure you can call the basis elements any names you want and index them anyway. I draw my line at calling any element -name from +name, and thinking this negation does not modify fundamental Algebraic structure. You can get away with it for Quaternions since a minus sign in front of 1 or 3 non-scalar bases is the same as a simple transposition, that implying both forms are ordered triplet multiplication rules so same basic structure.

Not so much for Octonion Algebra. You can stick a minus sign in front of an even number of non-scalar bases and demonstrate this new rule set is equivalent to a number of transpositions. This would map Right to Right and Left to Left. But if you do it on an odd number of negations, this would be the map between Right and Left, and no possible set of transpositions without sticking in a minus sign will do the deed. It is not that you CAN do the negations, you MUST if you go between Right and Left.

As far as others not believing this, I have yet to see anyone else deal with Octonion definition variation the way I do. They bought the story all Octonion Algebras are the same so haven't bothered to look at the differences, which are extremely important. Not surprised by the bias and lack of effort checking closely. Their loss, for Algebraic Invariance and variance dealing explicitly with this variation is quite important.

Rick

Geoffrey Dixon

I'm trying to interpret words in terms of old equations in my head. Inevitably it's coming out looking like something I'm familiar with. If it isn't, then I'd need to see equations and multiplication tables. Without that I can't judge. Not that it's important that I do. What's important is that you carry on. Ignore my quibbles. I don't really know if they are valid.

Geoffrey Dixon

Generic octonion algebra. Begin with 3 imaginary units, I, J, L, that antiassociate (so not a quaternion triple). Then a full basis for this copy of O is:

I, J, L, IJ, IL, JL, (IJ)L = -I(JL).

Now start with ANY copy of O with using imaginary basis units e1, ..., e7. Choose 3 that do not associate:

±ea, ±eb, ±ec.

The sign is irrelevant. Map them to I, J, K above. This uniquely determines the remaining 4 unit assignments, and the full multiplication table. Any such multiplication table can be mapped to the I, J, L table, and therefore visa versa. This provides an isomorphism between any two tables you wish to start with. All copies of O are isomorphic.

I suspect I'm missing something. But it is unhealthy for me to carry on with this, so I'm dropping out now.

Rick Lockyer

I know, your out.

"The sign is irrelevant" is an incorrect assumption.

Rick

Geoffrey Dixon

Prediction for those serious about applying division algebras:

So, I encountered the following online ...

https://hackaday.com/2018/02/10/all-the-stuff-you-wished-you-knew-about-fourier-transforms-but-were-afraid-to-ask/

This is an excellent description on the Fourier transform, described as taking a signal and wrapping it around a circle, the circle (S1) being represented as the set of unit elements in C. I then did a little googling and discovered a number of papers on Fourier transforms over H and O, which means over the parallelizable spheres, S3 and S7, including discussions of corresponding uncertainty relations. This kind of thing is the analytical future of this field.

Rick Lockyer

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

Geoffrey Dixon

"Exclusive or" is this, right?

0+0 = 1+1 = 0;

1+0 = 0+1 = 1.

The same thing that I used in my Hadamard matrix extension of the octonions. Anyway, it sounds interesting, but I'm still light years away from thinking Cayley-Dickson and sedenions are interesting ... to me. As to "Quaternion triplet enumeration algorithm": I have no problem with this being cooler than mine (although I've no idea how the word "algorithm" applies to anything I do with the octonions).

Meanwhile I have a new bee in my bonnet: Fourier transforms over H and O. So little time.

[deleted]

Yes, at the bit level, the exclusive or operation is equivalent to modulo 2 addition. You can consider modulo 2 polynomials and addition instead of binary numbers and the exclusive or logic function. I have used both in the past.

Rick

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