From: Vaughan Pratt <pratt@cs.Stanford.EDU>
To: categories@mta.ca
Subject: Re: Modeling infinitesimals with 2x2 matrices
Date: Sat, 24 Apr 2004 15:46:40 -0700 [thread overview]
Message-ID: <200404242246.i3OMkeOq031856@coraki.Stanford.EDU> (raw)
Correction to my suggestion id+di = 0. Don't do it. id = di is fine
as it stands for refined complex numbers, which should be represented in
C(2) = 2x2 complex matrices (embeddable in R(4) - 4x4 real matrices)
as the obvious extension of the refined reals x+yd.
I shouldn't have been so smug about 4D Clifford algebras, this algebra of
refined complex numbers doesn't satisfy d^4 = 1, needed if d is to be a
Clifford generator.
And in fact although di =
1 0
0 0
we have id =
0 0
0 1
(I should have checked that more carefully.)
I thought about trying to make the infinitesimals points on the "light cone"
of R(2) (the singular matrices) but couldn't get that to work. So 2x2
complex matrices with id = di is the best I could think of. This works
for modeling the refined complex numbers (barring any other errors), but
with nothing left to motivate id+di = 0.
The representation x+iy+dv+idw is fine, with idw = diw = wid etc., all is
commutative. (I was hoping too hard for the excitement of noncommutativity,
this is boringly noninteractive as it stands.)
Vaughan
next reply other threads:[~2004-04-24 22:46 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2004-04-24 22:46 Vaughan Pratt [this message]
-- strict thread matches above, loose matches on Subject: below --
2004-04-29 0:54 John Baez
2004-04-28 5:13 Vaughan Pratt
[not found] <s.j.vickers@cs.bham.ac.uk>
[not found] ` <408CCCAA.9090404@cs.bham.ac.uk>
2004-04-26 16:54 ` Vaughan Pratt
2004-04-25 6:58 Vaughan Pratt
2004-04-24 6:45 Vaughan Pratt
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