Skip to main content

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.



Answer



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
*)

Comments

Post a Comment

Popular posts from this blog

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
Post a Comment
Read more

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]
Post a Comment
Read more

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1....
Post a Comment
Read more