(The nonsense of) Joy Christian Reloaded

I was preparing the first physics posts of the year when I got some comments and a question on Joy Christian on an old blog post. In the words of late Yogi Berra, this is "deja vu all over again". Probably the best description of Joy Christian is given by the Monty Python: The Dead Parrot sketch:

The question I got is the following:

"I would like to understand whether the equations (67) - (75) in Joy Christian’s paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” make any sense at all. I don't understand how the mathematical limes operation are carried out."

The paper which got past the referees by trickery is on the archive: https://arxiv.org/pdf/1405.2355v7.pdf and there you see the full derivation of the main faulty claim. Minus some obfuscation techniques, Eqs. 67-75 are nothing but the one-pager Joy preprint: https://arxiv.org/pdf/1103.1879v1.pdf

The main hand-waving trick in the "derivation" is a conversion inside of a sum of \(\lambda^k\) from a variable into an index which amounts to adding apples to oranges and obtaining the incorrect result (see the bottom of page 8 on my preprint: https://arxiv.org/pdf/1109.0535v3.pdf). 

The mistake happens on the transition from Eq 73 to Eq 74 because the L's belong to two distinct kinds of algebras: let's call them apples and oranges. Ignoring the axb, the troubled sum term is something like this:

\(L(\lambda^1) + L(\lambda^2)+L(\lambda^3)+L(\lambda^4)+L(\lambda^5)+...=\)

apple_1 + apple_2 + orange_3 +apple_4 + orange_5+...

with \(\lambda^1 = +1, \lambda^2\ = +1, \lambda^3 = -1, \lambda^4 = +1, \lambda^5 = -1...\)

and with the transformation rule: "apple = - oranges" when we convert to objects of the same kind (let's pick apples) we get:

\(apple(\lambda^1) + apple(\lambda^2)-apple(\lambda^3)+apple(\lambda^4)-apple(\lambda^5)+...=\)

\(apple(+1) + apple(+1)-apple(-1)+apple(+1)-apple(-1)+...=\)

\(apple+ apple+apple+apple+apple+...=\)

which no longer vanishes.

The preparation for this trick is on Eq. 49 which encodes the two distinct algebras (of apples and oranges) into a common formula.  In my preprint you can double check this by trying out the matrix representations of the two algebras (eqs 53-56).

Hopefully my explanation is clear enough. I know all Joy's mathematical tricks in all of his papers or in his blog debates, but I ran out of energy debunking his nonsense. Kudos to Richard Gill for pursuing this further. I was aware of the "Causality in a Friedmann-Robertson-Walker Spacetime" paper and it was on my to do list to write a rebuttal to it, but the journal withdrew it before I could get to it.