Re: [math-fun] Cohl Furey & the Octonians
Actually, the question I was going to ask was: "Where does she use *division*?" At 11:23 AM 7/28/2018, Mike Stay wrote:
Sedenions aren't a division algebra.
On Sat, Jul 28, 2018 at 8:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
From the end of the second introductory short at https://www.youtube.com/watch?v=ng1bMsSokgw we are informed complex octonions |C (x) |O "lead to" (via a currently undefined process) a Clifford algebra (also currently undefined). Yet the first is non-associative, the second associative.
Additionally, why insist on stopping the Cayley construction at #4 ? The next iteration has certainly been investigated, and even named --- sedenions, I think.
My BS detector is beginning rumble ominously ...
WFL
On 7/28/18, Mike Stay <metaweta@gmail.com> wrote:
https://www.youtube.com/watch?v=3BZyds_KFWM
On Fri, Jul 27, 2018 at 7:47 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand any of the physics, but this is an interesting math idea that I'd never encountered before.
I think this refers to the tensor product of the four real division algebras:
R (x) C (x) H (x) O
(where R = reals, C = complexes, H = quaternions, O = octonions, and (x) denotes tensor product over the real numbers.
Tensor products are one of the two basic ways to combine algebras over a ring. But if I recall how I learned about this in a course called 18.26, I think everything was commutative. (Which the quaternions and octonions aren't, but I'm not sure it matters.)
I will attempt to unwind the definition of tensor product of algebras, since I'd like to know what this weird thing is.
*****-----until further notice, R means any commutative ring-----*****
Let's say A and B are algebras over a commutative ring R. Then their tensor product A (x) B will be the quotient of the free algebra over R generated by the elements
{a_j (x) b_j | a_j in A, b_j in B},
i.e.,
A*B = {Sum r_j (a_j (x) b_j), a_j in A, b_j in B, r_j in R, Sum is finite}.
To get the tensor product we need to factor out by the subring Z generated by elements of the form
((ra)(x)b) - (a(x)(rb)), r in R, a in A, b in B
and the like that give the tensor product its bilinearity. -----
*****-----From now on R is just the reals again-----*****
Dimensions of tensor products multiply, so the dimension of
R (x) C (x) H (x) O
where (x) is always understood as tensor with respect to R=reals, is
1 x 2 x 4 x 8 = 64.
Hmm, weird. Anyone care to explain (anything about) the physics?
--DDan
Henry Baker wrote: ----- ... ...
Combined as $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the four number systems form a 64-dimensional abstract space.
... ...
Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
"Where does she use *division*?"
Quite so; so why stop there? Her frequent use of the weasel word "number" alerts suspicion that some basic idea involved here has been incompletely examined. WFL On 7/28/18, Henry Baker <hbaker1@pipeline.com> wrote:
Actually, the question I was going to ask was:
"Where does she use *division*?"
At 11:23 AM 7/28/2018, Mike Stay wrote:
Sedenions aren't a division algebra.
On Sat, Jul 28, 2018 at 8:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
From the end of the second introductory short at https://www.youtube.com/watch?v=ng1bMsSokgw we are informed complex octonions |C (x) |O "lead to" (via a currently undefined process) a Clifford algebra (also currently undefined). Yet the first is non-associative, the second associative.
Additionally, why insist on stopping the Cayley construction at #4 ? The next iteration has certainly been investigated, and even named --- sedenions, I think.
My BS detector is beginning rumble ominously ...
WFL
On 7/28/18, Mike Stay <metaweta@gmail.com> wrote:
https://www.youtube.com/watch?v=3BZyds_KFWM
On Fri, Jul 27, 2018 at 7:47 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand any of the physics, but this is an interesting math idea that I'd never encountered before.
I think this refers to the tensor product of the four real division algebras:
R (x) C (x) H (x) O
(where R = reals, C = complexes, H = quaternions, O = octonions, and (x) denotes tensor product over the real numbers.
Tensor products are one of the two basic ways to combine algebras over a ring. But if I recall how I learned about this in a course called 18.26, I think everything was commutative. (Which the quaternions and octonions aren't, but I'm not sure it matters.)
I will attempt to unwind the definition of tensor product of algebras, since I'd like to know what this weird thing is.
*****-----until further notice, R means any commutative ring-----*****
Let's say A and B are algebras over a commutative ring R. Then their tensor product A (x) B will be the quotient of the free algebra over R generated by the elements
{a_j (x) b_j | a_j in A, b_j in B},
i.e.,
A*B = {Sum r_j (a_j (x) b_j), a_j in A, b_j in B, r_j in R, Sum is finite}.
To get the tensor product we need to factor out by the subring Z generated by elements of the form
((ra)(x)b) - (a(x)(rb)), r in R, a in A, b in B
and the like that give the tensor product its bilinearity. -----
*****-----From now on R is just the reals again-----*****
Dimensions of tensor products multiply, so the dimension of
R (x) C (x) H (x) O
where (x) is always understood as tensor with respect to R=reals, is
1 x 2 x 4 x 8 = 64.
Hmm, weird. Anyone care to explain (anything about) the physics?
--DDan
Henry Baker wrote: ----- ... ...
Combined as $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the four number systems form a 64-dimensional abstract space.
... ...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Sat, Jul 28, 2018 at 12:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
Furey isn't claiming that this is a Theory of Everything. She's just saying that if you take the Georgi–Glashow GUT, which uses SU(5), and then factor it as ℝ⊗ℂ⊗ℍ⊗𝕆, which are the four normed division algebras, then the transition that leads to proton decay mixes the algebras. If such mixing is disallowed, maybe we can avoid the problem of proton decay with a technique like this.
"Where does she use *division*?"
Time reversal / unitarity.
Quite so; so why stop there?
https://ncatlab.org/nlab/show/normed+division+algebra "This classification turns out to closely connect to various other systems of exceptional structures in mathematics and physics: 1. The Hopf invariant one theorem says that the only continuous functions between spheres of the form S^{2n-1} -> S^n whose Hopf invariant is equal to 1 are the Hopf constructions on the four real normed division algebras, namely * the real Hopf fibration; * the complex Hopf fibration; * the quaternionic Hopf fibration; * the octonionic Hopf fibration. 2. Patterns related to Majorana spinors in spin geometry are intimately related to the four normed division algebras, and, induced by this, so is the classification of supersymmetry in the form of super Poincaré Lie algebras and super Minkowski spacetimes (which are built from these real spin representations). For more on this see at supersymmetry and division algebras." See also https://ncatlab.org/nlab/show/division+algebra+and+supersymmetry
Her frequent use of the weasel word "number" alerts suspicion that some basic idea involved here has been incompletely examined.
Huh? https://en.wikipedia.org/wiki/Complex_number
WFL
On 7/28/18, Henry Baker <hbaker1@pipeline.com> wrote:
Actually, the question I was going to ask was:
"Where does she use *division*?"
At 11:23 AM 7/28/2018, Mike Stay wrote:
Sedenions aren't a division algebra.
On Sat, Jul 28, 2018 at 8:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
From the end of the second introductory short at https://www.youtube.com/watch?v=ng1bMsSokgw we are informed complex octonions |C (x) |O "lead to" (via a currently undefined process) a Clifford algebra (also currently undefined). Yet the first is non-associative, the second associative.
Additionally, why insist on stopping the Cayley construction at #4 ? The next iteration has certainly been investigated, and even named --- sedenions, I think.
My BS detector is beginning rumble ominously ...
WFL
On 7/28/18, Mike Stay <metaweta@gmail.com> wrote:
https://www.youtube.com/watch?v=3BZyds_KFWM
On Fri, Jul 27, 2018 at 7:47 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand any of the physics, but this is an interesting math idea that I'd never encountered before.
I think this refers to the tensor product of the four real division algebras:
R (x) C (x) H (x) O
(where R = reals, C = complexes, H = quaternions, O = octonions, and (x) denotes tensor product over the real numbers.
Tensor products are one of the two basic ways to combine algebras over a ring. But if I recall how I learned about this in a course called 18.26, I think everything was commutative. (Which the quaternions and octonions aren't, but I'm not sure it matters.)
I will attempt to unwind the definition of tensor product of algebras, since I'd like to know what this weird thing is.
*****-----until further notice, R means any commutative ring-----*****
Let's say A and B are algebras over a commutative ring R. Then their tensor product A (x) B will be the quotient of the free algebra over R generated by the elements
{a_j (x) b_j | a_j in A, b_j in B},
i.e.,
A*B = {Sum r_j (a_j (x) b_j), a_j in A, b_j in B, r_j in R, Sum is finite}.
To get the tensor product we need to factor out by the subring Z generated by elements of the form
((ra)(x)b) - (a(x)(rb)), r in R, a in A, b in B
and the like that give the tensor product its bilinearity. -----
*****-----From now on R is just the reals again-----*****
Dimensions of tensor products multiply, so the dimension of
R (x) C (x) H (x) O
where (x) is always understood as tensor with respect to R=reals, is
1 x 2 x 4 x 8 = 64.
Hmm, weird. Anyone care to explain (anything about) the physics?
--DDan
Henry Baker wrote: ----- ... ...
Combined as $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the four number systems form a 64-dimensional abstract space.
... ...
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_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
On Sat, Jul 28, 2018 at 3:31 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 12:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
Furey isn't claiming that this is a Theory of Everything. She's just saying that if you take the Georgi–Glashow GUT, which uses SU(5), and then factor it as ℝ⊗ℂ⊗ℍ⊗𝕆 , which are the four normed division algebras,
SU(5) is a group (hence associative) of dimension 24 as a real manifold ( https://en.wikipedia.org/wiki/Special_unitary_group#Properties ) ℝ⊗ℂ⊗ℍ⊗𝕆 is a non-associative algebra over the reals of dimension 1*2*4*8= 64. So how can the latter be a factorization of the former? Clearly there is something else going on.
On Sat, Jul 28, 2018 at 4:24 PM, W. Edwin Clark <wclark@mail.usf.edu> wrote:
On Sat, Jul 28, 2018 at 3:31 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 12:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
Furey isn't claiming that this is a Theory of Everything. She's just saying that if you take the Georgi–Glashow GUT, which uses SU(5), and then factor it as ℝ⊗ℂ⊗ℍ⊗𝕆 , which are the four normed division algebras,
SU(5) is a group (hence associative) of dimension 24 as a real manifold ( https://en.wikipedia.org/wiki/Special_unitary_group#Properties )
ℝ⊗ℂ⊗ℍ⊗𝕆 is a non-associative algebra over the reals of dimension 1*2*4*8= 64.
So how can the latter be a factorization of the former? Clearly there is something else going on.
Sorry, I phrased that badly. Here's the abstract from the relevant paper: We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras ℝ, ℂ, ℍ, and 𝕆. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow's SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6, together with an extra U(1)X symmetry, related to B−L. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
Here's Lubos Motl's criticism of her work. He's a troll, but he's not wrong about the problems with Lisi's E8 model and similar criticisms apply to Furey's work. https://motls.blogspot.com/2018/07/cohl-furey-understands-neither-field.html On Sat, Jul 28, 2018 at 4:40 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 4:24 PM, W. Edwin Clark <wclark@mail.usf.edu> wrote:
On Sat, Jul 28, 2018 at 3:31 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 12:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
Furey isn't claiming that this is a Theory of Everything. She's just saying that if you take the Georgi–Glashow GUT, which uses SU(5), and then factor it as ℝ⊗ℂ⊗ℍ⊗𝕆 , which are the four normed division algebras,
SU(5) is a group (hence associative) of dimension 24 as a real manifold ( https://en.wikipedia.org/wiki/Special_unitary_group#Properties )
ℝ⊗ℂ⊗ℍ⊗𝕆 is a non-associative algebra over the reals of dimension 1*2*4*8= 64.
So how can the latter be a factorization of the former? Clearly there is something else going on.
Sorry, I phrased that badly. Here's the abstract from the relevant paper:
We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras ℝ, ℂ, ℍ, and 𝕆. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow's SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6, together with an extra U(1)X symmetry, related to B−L. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
Oof! Well, Motl pulls no punches; but to this physical ignoramus he appears both instructive and entertaining, and well worth visiting. And I can gracefully withdraw from this discussion, in which of my legs remains seriously out of its depth. Incidentally Adam's contribution is unlinked from this thread: his interpretation of tensor products is new to me. WFL On 7/29/18, Mike Stay <metaweta@gmail.com> wrote:
Here's Lubos Motl's criticism of her work. He's a troll, but he's not wrong about the problems with Lisi's E8 model and similar criticisms apply to Furey's work.
https://motls.blogspot.com/2018/07/cohl-furey-understands-neither-field.html
On Sat, Jul 28, 2018 at 4:40 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 4:24 PM, W. Edwin Clark <wclark@mail.usf.edu> wrote:
On Sat, Jul 28, 2018 at 3:31 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 12:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
>Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
Furey isn't claiming that this is a Theory of Everything. She's just saying that if you take the Georgi–Glashow GUT, which uses SU(5), and then factor it as ℝ⊗ℂ⊗ℍ⊗𝕆 , which are the four normed division algebras,
SU(5) is a group (hence associative) of dimension 24 as a real manifold ( https://en.wikipedia.org/wiki/Special_unitary_group#Properties )
ℝ⊗ℂ⊗ℍ⊗𝕆 is a non-associative algebra over the reals of dimension 1*2*4*8= 64.
So how can the latter be a factorization of the former? Clearly there is something else going on.
Sorry, I phrased that badly. Here's the abstract from the relevant paper:
We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras ℝ, ℂ, ℍ, and 𝕆. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow's SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6, together with an extra U(1)X symmetry, related to B−L. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
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On Sat, Jul 28, 2018 at 6:27 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Oof! Well, Motl pulls no punches; but to this physical ignoramus he appears both instructive and entertaining, and well worth visiting.
There's a reason he's no longer employed as a physicist. See for instance http://backreaction.blogspot.com/2007/08/lubo-motl.html The doc she refers to at the end can be found here: http://web.archive.org/web/20110825025033/http://prime-spot.de/Bored/bolubos... I think this is getting too far off topic, so let's end this here. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
Agreed. Although I have always regretted my ignorance of physics, I am beginning to wonder whether it might confer certain benefits ... WFL On 7/29/18, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 6:27 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Oof! Well, Motl pulls no punches; but to this physical ignoramus he appears both instructive and entertaining, and well worth visiting.
There's a reason he's no longer employed as a physicist. See for instance http://backreaction.blogspot.com/2007/08/lubo-motl.html The doc she refers to at the end can be found here:
http://web.archive.org/web/20110825025033/http://prime-spot.de/Bored/bolubos...
I think this is getting too far off topic, so let's end this here. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
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participants (4)
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Henry Baker -
Mike Stay -
W. Edwin Clark