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Re: commuting and non-commuting symbols

• To: mathgroup at smc.vnet.net
• Subject: [mg31377] Re: [mg31361] commuting and non-commuting symbols
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 31 Oct 2001 03:31:10 -0500 (EST)
• References: <200110300935.EAA23905@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Ian Swanson wrote:
> 
> Hi
> 
> My research group is trying to use Mathematica to simplify and verify some
> complicated expressions that combine various parameters living in
> different spaces.  Basically, we need to be able to label certain symbols
> as commuting and certain others as non-commuting (ie. Grassman numbers).
> We then need to perform standard matrx multiplication with this mixture of
> commuting and non-commuting variables -- and have Mathematica simplify the
> resulting expressions as much as possible.  Can anyone help??
> 
> Thank you,
> Ian
> 
> ______________________________
> | Ian J Swanson              |
> | Theoretical Physics        |
> | 253 Lauritsen              |
> | swanson at theory.caltech.edu |
> | office: 626.395.2615       |
> | cell: 626.230.1882         |
> |____________________________|

You will probably want to use Inner, which generalizes Dot, to handle
the matrix products. You will need to write your own myTimes or some
such to handle a mix of scalars and variables. If they satisfy something
along the lines of anticommuting relations, all the better in that one
can easily canonicalize. Some related ideas, with code, have appeared on
MathGroup and are at the below location in the archives.

http://library.wolfram.com/mathgroup/archive/1999/Dec/msg00105.html
with some corrections in:
http://library.wolfram.com/mathgroup/archive/1999/Dec/msg00167.html
(I've been told even the corrected version had a flaw or two).

Here are two other URLs that may be relevant.

http://library.wolfram.com/mathgroup/archive/2000/Nov/msg00299.html
http://library.wolfram.com/mathgroup/archive/1999/Mar/msg00510.html

Several others have also contributed posts regarding various aspects of
implementation of noncommutative algebra, so you may want to search
MathGroup archives. Moreover there are packages on MathSource that may
have functionality of use here, for example NCAlgebra.


Daniel Lichtblau
Wolfram Research

• References:
  • commuting and non-commuting symbols
    • From: Ian Swanson <swanson@theory.caltech.edu>