From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2665 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Modeling infinitesimals with 2x2 matrices Date: Sat, 24 Apr 2004 15:46:40 -0700 Message-ID: <200404242246.i3OMkeOq031856@coraki.Stanford.EDU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018816 5221 80.91.229.2 (29 Apr 2009 15:26:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:56 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Apr 25 18:07:55 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 25 Apr 2004 18:07:55 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BHqk7-0005dZ-00 for categories-list@mta.ca; Sun, 25 Apr 2004 18:00:51 -0300 X-Mailer: exmh version 2.6.3 04/04/2003 with nmh-1.0.4 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 44 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:2665 Archived-At: 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