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Reply to Pessoa, P.; Arderucio Costa, B. Comment on “Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17”
Correction to Entropy 2020, 22(1), 61.
Correction

Correction: Gill, R.D. Does Geometric Algebra Provide a Loophole to Bell’s Theorem? Entropy 2020, 22, 61

Mathematical Institute, Leiden University, Rapenburg 70, 2311 EZ Leiden, The Netherlands
Entropy 2021, 23(5), 631; https://doi.org/10.3390/e23050631
Received: 21 April 2021 / Accepted: 8 May 2021 / Published: 19 May 2021
Corrections are made to my paper “Gill, R.D. Does Geometric Algebra Provide a Loophole to Bell’s Theorem? Entropy 2020, 22, 61”. Firstly, there was an obvious and easily corrected mathematical error at the end of Section 6 of the paper. In the Clifford algebra under consideration, the basis bivectors M e i do not square to the identity, but to minus the identity. However, the trivector M does square to the identity and hence non-zero divisors of zero, M 1 and M + 1 , can be found by the same argument as was given in the paper.
Secondly, in response to a complaint about ad hominem and ad verecundam arguments, a number of scientifically superfluous but insulting sentences have been deleted, and other disrespectful remarks have been rendered neutral by omission of derogatory adjectives. I would like to apologize to Dr. Joy Christian for unwarranted offence.
The end of Section 6 of Gill (2020) [1] discussed the even sub-algebra of C 4 , 0 , isomorphic to C 0 , 3 :
One can take as basis for the eight-dimensional real vector space C 0 , 3 the scalar 1, three anti-commuting vectors e i , three bivectors v i , and the pseudo-scalar M = e 1 e 2 e 3 . The algebra multiplication is associative and unitary (there exists a multiplicative unit, 1). The pseudo-scalar M squares to 1 . Scalar and pseudo-scalar commute with everything. The three basis vectors e i , by definition, square to 1 . The three basis bivectors v i = M e i square to + 1 . Take any unit bivector v. It satisfies v 2 = 1 hence v 2 1 = ( v 1 ) ( v + 1 ) = 0 . If the space could be given a norm such that the norm of a product is the product of the norms, we would have v 1 . v + 1 = 0 hence either v 1 = 0 or v + 1 = 0 (or both), implying that either v 1 = 0 or v + 1 = 0 (or both), implying that v = 1 or v = 1 , neither of which are true.
But the bivectors v i square to 1 and the trivector M squares to + 1 . Still, it then follows that ( M + 1 ) ( M 1 ) = 0 , and by the argument originally given, it follows that M = 1 or M = 1 , a contradiction.

Reference

  1. Gill, R.D. Does Geometric Algebra Provide a Loophole to Bell’s Theorem? Entropy 2020, 22, 61. [Google Scholar] [CrossRef] [PubMed]
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