       RE: commuting and non-commuting symbols

• To: mathgroup at smc.vnet.net
• Subject: [mg31378] RE: [mg31361] commuting and non-commuting symbols
• From: "David Park" <djmp at earthlink.net>
• Date: Wed, 31 Oct 2001 03:31:11 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Ian,

You should be able to do it. Here is a little toy system which labels
numbers and symbols as to whether they are to be used for multiplication or
as powers. 3[t] means 3 is to be used for multiplication and 3[p] means that
3 is to be used as a power. Then we can write our products using
CircleTimes. CircleTimes can be used as an infix operator and has no
attributes. CircleTimes can be entered as esc c * esc, or by \[CircleTimes].
You can make a palette if you are going to use these symbols a lot. We can
then write definitions for CircleTimes which implement the specific
operations depending on the type of quantity, t or p. In this set of
definitions, an expression is evaluated from left to right.

Clear[CircleTimes];
CircleTimes[a_[t], b_[t], c___] := CircleTimes[(a b)[t], c];
CircleTimes[a_[t], b_[p], c___] := CircleTimes[(a^b)[t], c]
CircleTimes[a_[b : (p | t)]] := a[b]

3[t]\[CircleTimes]4[t]
12[t]

3[t]\[CircleTimes]4[p]
81[t]

If there is no definition, you just get the expression back.

4[p]\[CircleTimes]3[t]
4[p]\[CircleTimes]3[t]

3[t]\[CircleTimes]x[t]\[CircleTimes]y[p]
3^y*x^y

3[t]\[CircleTimes](x[t]\[CircleTimes]y[p])
(3*x^y)[t]

This would be one approach. All the definitions on how to handle different
combinations of element types have to be specified. You will probably get

David Park

> From: Ian Swanson [mailto:swanson at theory.caltech.edu]
To: mathgroup at smc.vnet.net
> 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         |
> |____________________________|
>
>

```

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