by gill1109 » Wed Oct 23, 2019 8:49 pm
Joy Christian wrote:***
I have some sympathy for Richard Gill's struggle with the arXiv moderation, even though I do not think his current paper criticizing my work is a serious paper with sound arguments. In the past, I too had difficulties with the arXiv moderation because of their blanket policy of rejecting any paper that criticizes Bell's theorem. For example, the preprint of my RSOS paper was not accepted by the arXiv moderators. They had asked me to first publish the paper in a respectable journal before submitting to the arXiv. Fortunately, my paper was accepted by RSOS and then the moderators had no choice but to accept the preprint version of it. Their policies, however, have become more coherent lately after they conducted a major survey of their readers and systematized their moderation procedures.
***
Here is a slight revision of my earlier posting on Christian's pure maths paper, in preparation for the symposium debate. The latest version of my paper containing my items for debate is
https://www.math.leidenuniv.nl/~gill/GA.pdf. This revision is being processed at viXra. No news from arXiv on the moderation.
Christian argues that the 8-dimensional real and associative Clifford algebra Cl(0, 3), which is the even subalgebra of Cl(4, 0) (see
https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively. See Christian's RSOS paper or his "pure mathematics paper"
https://arxiv.org/abs/1908.06172One can take as vector space basis for the 8 dimensional real vector space Cl(0, 3) the
scalar 1, three
vectors, three
bivectors, and the
pseudo-scalar. According to the definition of Clifford algebras, the three vectors square to -1. Take any unit
bivector . It satisfies
hence
. If the space could be given a norm such that the norm of a product is the product of the norms, we would have
hence either
or
(or both), hence either
or
(or both), implying that v = 1 or v = -1.
Recall that a normed division algebra is an algebra that is also a normed vector space and such that the norm of a product is the product of the norms; a division algebra is an algebra such that if a product of two elements equals zero, then at least one of the two elements concerned must be zero.
Perhaps Joy can tell us exactly what he disagrees with here. For instance, let's take it step by step:
1. Over the real numbers, is the even subalgebra of Cl(4, 0) isomorphic to Cl(0, 3)?
2. In Cl(0, 3), the basis vectors e_1, e_2 and e_3 anti-commute and square to -1. The algebra is associative. I can define the pseudoscalar M = e_1 e_2 e_3. It follows that M^2 = -1. I can define three bivectors v_1, v_2 and v_3 by v_i = M e_i. They anticommute and square to +1.
3. As a real vector space one can take the basis of Cl(0, 3) to be 1 (the scalar), the three basis vectors e_i, the three basis bivectors v_i, and the pseudo-scalar M.
[quote="Joy Christian"]***
I have some sympathy for Richard Gill's struggle with the arXiv moderation, even though I do not think his current paper criticizing my work is a serious paper with sound arguments. In the past, I too had difficulties with the arXiv moderation because of their blanket policy of rejecting any paper that criticizes Bell's theorem. For example, the preprint of my RSOS paper was not accepted by the arXiv moderators. They had asked me to first publish the paper in a respectable journal before submitting to the arXiv. Fortunately, my paper was accepted by RSOS and then the moderators had no choice but to accept the preprint version of it. Their policies, however, have become more coherent lately after they conducted a major survey of their readers and systematized their moderation procedures.
***[/quote]
Here is a slight revision of my earlier posting on Christian's pure maths paper, in preparation for the symposium debate. The latest version of my paper containing my items for debate is [url]https://www.math.leidenuniv.nl/~gill/GA.pdf[/url]. This revision is being processed at viXra. No news from arXiv on the moderation.
Christian argues that the 8-dimensional real and associative Clifford algebra Cl(0, 3), which is the even subalgebra of Cl(4, 0) (see https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively. See Christian's RSOS paper or his "pure mathematics paper" [url]https://arxiv.org/abs/1908.06172[/url]
One can take as vector space basis for the 8 dimensional real vector space Cl(0, 3) the [i]scalar[/i] 1, three [i]vectors[/i], three [i]bivectors[/i], and the [i]pseudo-scalar[/i]. According to the definition of Clifford algebras, the three vectors square to -1. Take any unit [i]bivector[/i] [tex]v[/tex]. It satisfies [tex]v^2 = 1[/tex] hence [tex]v^2 - 1 = (v - 1)(v + 1) = 0[/tex]. If the space could be given a norm such that the norm of a product is the product of the norms, we would have [tex]\|v - 1\|. \| v + 1\| = 0[/tex] hence either [tex]\|v - 1\| = 0[/tex] or [tex]\|v + 1\| = 0[/tex] (or both), hence either [tex]v - 1= 0[/tex] or [tex]v + 1 = 0[/tex] (or both), implying that v = 1 or v = -1.
Recall that a normed division algebra is an algebra that is also a normed vector space and such that the norm of a product is the product of the norms; a division algebra is an algebra such that if a product of two elements equals zero, then at least one of the two elements concerned must be zero.
Perhaps Joy can tell us exactly what he disagrees with here. For instance, let's take it step by step:
1. Over the real numbers, is the even subalgebra of Cl(4, 0) isomorphic to Cl(0, 3)?
2. In Cl(0, 3), the basis vectors e_1, e_2 and e_3 anti-commute and square to -1. The algebra is associative. I can define the pseudoscalar M = e_1 e_2 e_3. It follows that M^2 = -1. I can define three bivectors v_1, v_2 and v_3 by v_i = M e_i. They anticommute and square to +1.
3. As a real vector space one can take the basis of Cl(0, 3) to be 1 (the scalar), the three basis vectors e_i, the three basis bivectors v_i, and the pseudo-scalar M.