# Sorting function for non commuting bosons

I am trying to write a sorting function which will sort expressions involving products of bosonic objects which do not commute. For example, I can have objects like $a,\ a^\dagger,\ b,\ b^\dagger$ where $[a,a^\dagger] = 1$, $[b,b^\dagger] = 1$ and all other commutators vanish. So the sorting function should for example sort a term like $a a^\dagger b a b^\dagger$ to $a + a^\dagger a^2 + ab^\dagger b+a^\dagger a^2 b^\dagger b$. So I want to sort them first lexicographically and then such that dagger objects will be to the left.

I have been using a function from the notebook provided by Simon Tyler at arXiv. So I have created a bosonic object using

Clear[Boson, BosonC, BosonA]
Boson /: MakeBoxes[
Boson[cr : (True | False), sym_], fmt_] :=
With[{sbox = If[StringQ[sym], sym, ToBoxes[sym]]},
With[{abox = If[cr, SuperscriptBox[#, FormBox["\[Dagger]", Bold]], #] &@sbox},
InterpretationBox[abox, Boson[cr, sym]]
]
]
BosonA[sym_:"a"] := Boson[False, sym]
BosonC[sym_:"a"] := Boson[True, sym]


Next following the code, I want to turn the atomic expressions into lists and use Mathematica's ordering after that. So I have functions like

NCOrder[Boson[cr : True | False, sym_]] :=
NCOrder[Boson[cr, sym]] = {{sym, Boole[cr]}}

NCSort[expr_] := Module[{temp, NCM},
temp = expr /. NonCommutativeMultiply -> NCM;
temp = temp /. (NCM[a : _Boson ..] :> (NCM[a][[Ordering[#]]] &[NCOrder /@ {a}]));
temp = temp /. NCM[singleArg_] :> singleArg;
temp = temp /. NCM -> NonCommutativeMultiply
]


This NCSort function right now is just ordering it lexicographically. I want to modify it so that it would sort and order the expressions as I explained above.

-
Ordering of creation/annihilation operators isn't a sort in a traditional sense as you are not just rearranging terms, but modifying them in the process. Most sorting assumes a stable length, and this process does not have that. – rcollyer Dec 13 '12 at 19:41
See section "Some noncommutative algebraic manipulation" in this nb. It has a couple of examples that use commutators and do canonicalization. – Daniel Lichtblau Dec 13 '12 at 20:51

I suggest the following function, which looks rather ugly and is not really efficient, but seems to do the job:

expand[expr_] :=
Module[{times},
expr /.
NonCommutativeMultiply[b___] :> times[b] //.
{
times[left___, cnum_ /; FreeQ[cnum, _Boson], right___] :>
cnum*times[left, right],
times[left___, fst : Boson[_, s_], middle__, sec : Boson[_, s_],right___] /;
FreeQ[{middle}, Boson[_, s]] &&
OrderedQ[{Last@Union[Cases[{middle}, Boson[_, sym_] :> sym]], s}] :>
times[left, middle, fst, sec, right],
times[left___, Boson[False, s_], Boson[True, s_], right___] :>
times[left, right] + times[left, Boson[True, s], Boson[False, s], right],
times[b_Boson] :> b
} //.
times[left___, p : Longest[PatternSequence[(b : Boson[type_, s_]) ..]],right___] :>
times[left, b^Length[{p}], right] /.
times[arg__] :> NonCommutativeMultiply[arg] /. times[] -> OverHat[1]
];


You can use it as

expand[NonCommutativeMultiply[BosonA["a"], BosonC["a"], BosonA["b"], BosonA["a"], BosonC["b"]]]

(*
a+a^\[Dagger]**a^2+a**b^\[Dagger]**b+a^\[Dagger]**a^2**b^\[Dagger]**b
*)

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@Jens Works for me expand[NonCommutativeMultiply[BosonA["a"],BosonC["a"]]] gives NonCommutativeMultiply[]+a^\[Dagger]**a – Leonid Shifrin Dec 13 '12 at 21:52
@Jens Right, I forgot to mention - should be If instead of f. I corrected it in the post. – Leonid Shifrin Dec 13 '12 at 22:05
Thanks - I just saw that it had already been edited but I must have had the old definition. Miscommunication. – Jens Dec 13 '12 at 22:11
@Jens No, I edited already after your comment on this, so you could not have used the edited version when you tried. – Leonid Shifrin Dec 13 '12 at 22:14
Leonid, here's a suggestion for improvement: how about replacing the last rule in your expand module by times[arg__] :> NonCommutativeMultiply[arg] /. times[] -> OverHat[1] so that the identity operator is displayed in a more readable form? – Jens Dec 13 '12 at 22:50