       Re: Commutators with boson operators

• To: mathgroup at smc.vnet.net
• Subject: [mg97878] Re: Commutators with boson operators
• From: Sotirios Bonanos <sbonano at inp.demokritos.gr>
• Date: Tue, 24 Mar 2009 05:33:29 -0500 (EST)

```Hi there,

This is precisely the type of problems that the symbolic matrix capabilities of my EDC package (http://www.inp.demokritos.gr/~sbonano/EDC/) can handle. It is rather slow but does what you want. Download the code (matrixEDC.m), put it in an appropriate directory, and evaluate the attached notebook.

Sotirios Bonanos

>
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.
>

CreatAnnih.nb

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Cell[BoxData[
\(<< matrixEDC.m\)], "Input"],

Cell[BoxData[
\(DeclareMatrixForms[{0, annih[_], annih[_]}, {0, creat[_],
creat[_]}]\)], "Input"],

Cell[BoxData[
\(annih[n]\ and\ creat[
n]\ will\ denote\ the\ nth\ powers\ of\ your\ operators . \ \ \
\[IndentingNewLine]Wedge \((\[Wedge])\)\ is\ used\ to\ denote\ non -
commutative\ \ multiplication\)], "Text",
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Cell[BoxData[
\(Wedge[creat, annih]\)], "Input"],

Cell["\<\
Give rules for simplifying Wedge products of powers of creat, annih\
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\>", "Text"],

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\(creat = 1; annih = 1;\)], "Input"],

Cell[BoxData[
\(Wedge[creat[m_], creat[n_]] := creat[m + n]\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
The last rule implements the commutator \
creat\[Wedge]annih-annih\[Wedge]creat = -h0; (h0 = 1 in your \
case) to put annih[] operators before creat[] ones. If you prefer to have \
creat[] before annih[] use:
Wedge[annih[m_],creat[n_]]:=Wedge[annih[m-1], \
creat\[Wedge]annih+h0, creat[n-1]]/;m>0&&n>0\
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Cell[BoxData[
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Sum[aa[i, j] Wedge[creat[i], annih[j]], {i, 0, 2}, {j, 0,
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Cell["\<\
The function reWrite collects identical annih, creat powers / \
products\
\>", "Text"],

Cell[BoxData[
\(A = reWrite[A]\)], "Input"],

Cell[BoxData[
\(B =
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bb[i, j] Wedge[creat[i], annih[j]], {i, 0, 2}, {j, 0,
3}]]\)], "Input"],

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```

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