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Re: Commutators with boson operators

• To: mathgroup at smc.vnet.net
• Subject: [mg97875] Re: Commutators with boson operators
• From: Simon <simonjtyler at gmail.com>
• Date: Tue, 24 Mar 2009 05:32:55 -0500 (EST)
• References: <200903211016.FAA14634@smc.vnet.net> <gq557g$87v$1@smc.vnet.net>

On Mar 22, 7:51 pm, Filippo Miatto <mia... at gmail.com> wrote:
> For the creation and annihilation operators you could use their matrix =
 
> form, you can find it in The Quantum Theory of Fields Vol. 1 by Weinberg
>
> On Mar 21, 2009, at 11:16 AM, Volodymyr wrote:
>
> > Hello,
> > I need some help.
>
> > I want to write code that Mathematica would calculate commutators  
> > [A,B], [A,[B,C]] and so on...,
>
> > where A,B,C,D,... are functions like
>
> > A = a+ b*creat + c*annih + d*creat**annih + e*annih**creat +  
> > f*creat^2+g*annih^2+...
>
> > where a,b,c,d,... are usual complex numbers;
>
> > and annih & creat are boson noncommutative operators
> > that satisfy commutation relation:
> > [annih, creat] = 1.
>
> > Thank you in advance!

You could define a function com[a_,b_] and make it bilinear and
satisfy the Jacobi identity and make it a derivation, [a,bc]=[a,b]c+b[a,c],
so that it recursively reduces any expression down to the basic
commutators
[a,a]=0=[a^\dag,a^\dag] , [a^\dag,a]=1.

Simon

• References:
  • Commutators with boson operators
    • From: Volodymyr <ktpist@ukr.net>