Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Thursday, March 12, 2009

Mersenne MUBs

Consider the MUB matrices in prime dimensions $p$. If we demand that the entries form a finite field of $p + 1$ elements (that is, including $0$) it follows that $p$ must be a Mersenne prime.

It is not known if there are an infinite number of such primes or not, but Euler showed that any perfect number $n$ may be written in the form

$n = \frac{1}{2} M_i (M_i + 1)$

for some Mersenne prime $M_i$. In particular, the perfect number $6$ comes from the Mersenne prime $3$.


Anonymous PhilG said...

Interesting, and of course you mean that any even perfect number takes that form. The case for odd perfect numbers is still unsettled.

So do the MUBs form a finite field for Mersenne primes?

March 12, 2009 8:26 AM  
Blogger Matti Pitkanen said...

Really interesting. Mersenne primes define in p-adic mass calculations physically preferred p-adic mass scales L_p, p =about 2^k. M_k, k=89, 107, 127 correspond to intermediate gauge bosons, hadronic space-time sheet, and electron. k =127 is the largest p-adic Mersenne scale which is not completely super-astrophysical.

I would be happy to find mathematical and physical justifications for why the primes very near to powers of 2 are winners in fight for survival at elementary particle level.

Also Gaussian Mersennes are important. The one near 2^k, k=113 corresponds to muon mass scale. There is entire bundle of them- those near to 2^k, k=151,157,163,167 - in the biologically interesting range from 10 nm to 2.5 micrometers.

March 12, 2009 6:07 PM  
Anonymous PhilG said...

I was just thinking aboiut this a tiny bit more. If you look at the more general case where the dimension of the MUB is a prime power p^n, then you might also have examples where p is two and 2^n+1 is a Fermat prime. I think there is only one other case where prime powers differ by one which is 3^2 = 2^3+1.

But can any of these sizes of MUB give cases where the entries actaully form a finite field? There might be some interesting exceptional structures if they do.

March 18, 2009 9:18 PM  

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