SO(4,2) Casimirs in doubleton representation. In this notebook I am constructing a representation of SO(4,2) or SU(2,2) group using 2 pairs of annihilation/creation operators and then I am computing the Casimir invariants for this group in that representation. If you encounter any Fermion/InveX objects, please ignore them as they are not relevant for this particular representation.

$HistoryLength = 0;

Definition of Save and read functions

SaveIt[filename_, expr_] := 
  Module[{output}, 
   output = Export[filename <> ".dat", ToString[expr // InputForm], 
     "String"];
   ClearMemory;
   output];
SaveIt[varnamestring_] := 
  Module[{output}, 
   output = Export[varnamestring <> ".dat", 
     ToString[ToExpression[varnamestring] // InputForm], "String"];
   ClearMemory;
   output];
ReadIt[filename_] := 
  Module[{output}, 
   output = ToExpression[
     Import[StringReplace[filename, ".dat" -> ""] <> ".dat", "String"]];
   ClearMemory;
   output];
ClearMemory := Module[{}, Unprotect[In, Out];
   Clear[In, Out];
   Protect[In, Out];
   ClearSystemCache[];];

Definition of Bosonic oscillators

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_: String "a"] := Boson[False, sym]
BosonC[sym_: String "a"] := Boson[True, sym]

Aliasing NonCommutativeMultiply with CenterDot

CenterDot can be entered by pressing Esc.Esc and this is how the non commuting objects should be multiplied and will be displayed in output.

Unprotect[NonCommutativeMultiply];
Clear[NonCommutativeMultiply, CenterDot];
CenterDot[a__] := NonCommutativeMultiply[a];
NonCommutativeMultiply /: MakeBoxes[NonCommutativeMultiply[a__], fmt_] := 
 With[{cbox = ToBoxes[HoldForm[CenterDot[a]]]}, 
  InterpretationBox[cbox, NonCommutativeMultiply[a]]]
Protect[NonCommutativeMultiply];
Clear[CRule]
CRule = {NonCommutativeMultiply[a_] :> a};

Definition of clean - the function which orders and simplifies expressions

ClearAll@expand

SetAttributes[expand, HoldAll]

Unevaluated[
  expand[expr_] := 
   Block[{NonCommutativeMultiply (*or times*)}, 
    expr //. {times[left___, cnum_ /; FreeQ[cnum, (_Boson)], 
        right___] :> cnum*times[left, right], 
      times[left___, 
        cnum_ /; (! FreeQ[cnum, Times[n___?NumericQ, ___Boson]]), 
        right___] :> 
       Times @@ Apply[Power, Drop[FactorList[cnum], -1], 2]*
        times[left, First[Last[FactorList[cnum]]], right], 
      times[left___, Boson[False, s_], Boson[True, s_], right___] :> 
       times[left, right] + 
        times[left, Boson[True, s], Boson[False, s], right], 
      times[left___, fst : Boson[_, s_], sec : Boson[_, t_], 
        right___] :> 
       times[left, sec, fst, right] /; 
        FreeQ[Ordering[{s, t}], {1, 2}], times[b_Boson] :> b
      (*moved and rewrote this rule*), 
      times[] -> 1}]] /. {HoldPattern[times] -> NonCommutativeMultiply}
expand[Alternatives[NonCommutativeMultiply, CenterDot][
   a1_, (a2_ + a3_)]] := expand[a1 ** a2] + expand[a1 ** a3]
expand[Alternatives[NonCommutativeMultiply, CenterDot][(a1_ + a2_), 
   a3_]] := expand[a1 ** a3] + expand[a2 ** a3]

Clear[clean]

clean = Simplify[FixedPoint[expand, Distribute //@ #] //. CRule] &;

Commutator function

Clear[CommOp];
CommOp[x_, y_] := clean[(x\[CenterDot]y - y\[CenterDot]x)];
Clear[ACommOp];
ACommOp[x_, y_] := clean[(x\[CenterDot]y + y\[CenterDot]x)];

Defining the basic variables. 

Clear[Ad, A, Bd, B, Xd, X, Yd, Y]
Clear[\[Psi], \[Chi], \[CapitalGamma], \[CapitalSigma], i, j, k, l, m, n, p, \
q, g, T, G, P, Tr1, Tr2, Tr3, Cartan, Comm, Killing, Joseph, Commutator, \
Zero, zero, KillComm, KillMet, DoubleTr, GComm, Dim]
Dim = 6;
Ad = BosonC[a]; A = BosonA[a]; Bd = BosonC[b]; B = BosonA[b]; Xd = 
 BosonC[x]; X = BosonA[x]; Yd = BosonC[y]; Y = BosonA[y];

Defining the algebra.

Dirac \[CapitalGamma] matrices: 

\[CapitalGamma][
   1] = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}};
\[CapitalGamma][
   2] = {{0, 0, 0, -1}, {0, 0, -1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}};
\[CapitalGamma][
   3] = {{0, 0, 0, I}, {0, 0, -I, 0}, {0, -I, 0, 0}, {I, 0, 0, 0}};
\[CapitalGamma][
   4] = {{0, 0, -1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, -1, 0, 0}};
\[CapitalGamma][
   5] = {{0, 0, I, 0}, {0, 0, 0, I}, {I, 0, 0, 0}, {0, I, 0, 0}};
Zero = {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}};

Spinors and metric:

g = DiagonalMatrix[{-1, 1, 1, 1, 1, -1}];
\[Psi] = {{A}, {B}, {-Xd}, {-Yd}};
\[Chi] = {{Ad, Bd, X, Y}};   (*This is the Dirac conjugate of \[Psi]*)

Subscript[\[CapitalGamma], ij] matrices:

Table[\[CapitalSigma][i, 
    j] = -I ( \[CapitalGamma][i].\[CapitalGamma][j] - \[CapitalGamma][
         j].\[CapitalGamma][i])/4, {i, 1, 4}, {j, 1, 4}];

\[CapitalSigma][5, 6] = -\[CapitalGamma][5]/2; \[CapitalSigma][6, 
  5] = \[CapitalGamma][5]/2;

Table[\[CapitalSigma][i, 
    5] = -I \[CapitalGamma][i].\[CapitalGamma][5]/2; \[CapitalSigma][5, i] = 
   I \[CapitalGamma][i].\[CapitalGamma][5]/2; \[CapitalSigma][i, 
    6] = -\[CapitalGamma][i]/2; \[CapitalSigma][6, i] =  \[CapitalGamma][i]/
    2, {i, 1, 4}];

\[CapitalSigma][5, 5] = Zero;
\[CapitalSigma][6, 6] = Zero;

Generators of SO(4,2) denoted by G[i,j]

Table[G[i, j] = 
   Sum[\[CapitalSigma][i, j][[m, 
      n]] \[Chi][[1, m]]\[CenterDot]\[Psi][[n, 1]], {m, 1, 4}, {n, 1, 4}], {i,
    1, Dim}, {j, 1, Dim}];

Check the commutation relations of \[CapitalSigma] matrices

(*Table[Print[i,",",j,",",k,",",l,"=",Simplify[(\[CapitalSigma][i,j].\
\[CapitalSigma][k,l]-\[CapitalSigma][k,l].\[CapitalSigma][i,j]+ I( g[[i,k]]\
\[CapitalSigma][j,l]-  g[[i,l]]\[CapitalSigma][j,k] -  \
g[[j,k]]\[CapitalSigma][i,l] +  \
g[[j,l]]\[CapitalSigma][i,k]))]],{i,1,Dim},{j,1,Dim},{k,1,Dim},{l,1,Dim}];*)

Check the commutation relations of group generators

(*Table[Print[i,",",j,",",k,",",l,"=",clean[(G[i,j]\[CenterDot]G[k,l]-G[k,l]\
\[CenterDot]G[i,j]+ I( g[[i,k]]G[j,l]-  g[[i,l]]G[j,k] -  g[[j,k]]G[i,l] +  \
g[[j,l]]G[i,k]))]],{i,1,Dim},{j,1,Dim},{k,1,Dim},{l,1,Dim}];*)

Generators of SO(3,1) and rest of the conformal generators

Clear[M, \[Eta], \[CapitalPi], \[CapitalKappa], \[CapitalDelta]] 
(*\[CapitalPi] are translations, \[CapitalKappa](this is greek capital K \
obtained by esc K esc)are special conformal and \[CapitalDelta] is \
dilatations*)

\[Eta] = DiagonalMatrix[{-1, 1, 1, 1}];
Table[M[i, j] = G[i, j], {i, 1, 4}, {j, 1, 4}];
Table[\[CapitalKappa][i] = G[i, 6] - G[i, 5], {i, 1, 4}];
Table[\[CapitalPi][i] = G[i, 6] + G[i, 5], {i, 1, 4}];
\[CapitalDelta] = -G[5, 6];

Casimirs

Quadratic Casmir

Clear[ind1, ind2, ind3, ind4, Cas2, Z]

Z = Ad\[CenterDot]A + Bd\[CenterDot]B - Xd\[CenterDot]X - Yd\[CenterDot]Y;

Cas2 = clean[
   Sum[g[[ind1, ind3]]\[CenterDot]G[ind3, ind2]\[CenterDot]g[[ind2, 
      ind4]]\[CenterDot]G[ind4, ind1], {ind1, 1, 6}, {ind2, 1, 6}, {ind3, 1, 
     6}, {ind4, 1, 6}]];
clean[Cas2 + (3/2) (Z\[CenterDot]Z - 4)];
SaveIt["Cas2"];
Cas2sq = clean[Cas2\[CenterDot]Cas2];
SaveIt["Cas2sq"];

Cubic Casmir

(*Clear[ind1,ind2,ind3,ind4,ind5,ind6,cas3]

Timing[cas3=Sum[g[[ind1,ind4]]g[[ind2,ind5]]g[[ind3,ind6]]G[ind4,ind2]\
\[CenterDot]G[ind5,ind3]\[CenterDot]G[ind6,ind1],{ind1,1,6},{ind2,1,6},{ind3,\
1,6},{ind4,1,6},{ind5,1,6},{ind6,1,6} ];];
Cas3=clean[cas3];
SaveIt["Cas3.dat"];
clean[Cas3 -2I Cas2];*)

Quartic Casimir

(*Clear[ind1,ind2,ind3,ind4,ind5,ind6,ind7,ind8,cas4]

cas4=Sum[g[[ind1,ind5]]g[[ind2,ind6]]g[[ind3,ind7]]g[[ind4,ind8]]G[ind5,ind2]\
\[CenterDot]G[ind6,ind3]\[CenterDot]G[ind7,ind4]\[CenterDot]G[ind8,ind1],{\
ind1,1,6},{ind2,1,6},{ind3,1,6},{ind4,1,6},{ind5,1,6},{ind6,1,6},{ind7,1,6},{\
ind8,1,6}];
Cas4=clean[cas4];
SaveIt["Cas4"];
clean[Cas4-(1/6) Cas2sq+4 Cas2]*)

Cas4 = ReadIt["Cas4"];

clean[Cas4 - (1/6) Cas2sq + 4 Cas2]

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6th order Casimir

(*Clear[ind1,ind2,ind3,ind4,ind5,ind6,ind7,ind8,ind9,ind10,ind11,ind12,cas6,\
Cas6,dim]

dim=6;

Timing[cas6=ParallelSum[g[[ind1,ind7]]g[[ind2,ind8]]g[[ind3,ind9]]g[[ind4,\
ind10]]g[[ind5,ind11]]g[[ind6,ind12]]G[ind7,ind2]\[CenterDot]G[ind8,ind3]\
\[CenterDot]G[ind9,ind4]\[CenterDot]G[ind10,ind5]\[CenterDot]G[ind11,ind6]\
\[CenterDot]G[ind12,ind1],{ind1,1,dim},{ind2,1,dim},{ind3,1,dim},{ind4,1,dim},\
{ind5,1,dim},{ind6,1,dim},{ind7,1,dim},{ind8,1,dim},{ind9,1,dim},{ind10,1,dim}\
,{ind11,1,dim},{ind12,1,dim}];]*)

(*Cas6=clean[cas6];*)

(*Clear[ind1,ind2,ind3,ind4,ind5,ind6,ind7,ind8,ind9,ind10,ind11,ind12,cas6,\
Cas6,dim]

dim=6;

AbsoluteTiming[Table[cas6[i,j]=ParallelSum[g[[i,ind7]]g[[j,ind8]]g[[ind3,ind9]\
]g[[ind4,ind10]]g[[ind5,ind11]]g[[ind6,ind12]]G[ind7,j]\[CenterDot]G[ind8,\
ind3]\[CenterDot]G[ind9,ind4]\[CenterDot]G[ind10,ind5]\[CenterDot]G[ind11,\
ind6]\[CenterDot]G[ind12,i],{ind3,1,dim},{ind4,1,dim},{ind5,1,dim},{ind6,1,\
dim},{ind7,1,dim},{ind8,1,dim},{ind9,1,dim},{ind10,1,dim},{ind11,1,dim},{\
ind12,1,dim}],{i,1,dim},{j,1,dim}];]*)

(*AbsoluteTiming[Table[Cas6[i,j]=clean[cas6[i,j]],{i,1,dim},{j,1,dim}]]*)

3-3 split

(*Clear[Cas6133,cas6133]

Table[cas6133[ind1,ind4]=Sum[g[[ind1,ind7]]g[[ind2,ind8]]g[[ind3,ind9]]G[ind7,\
ind2]\[CenterDot]G[ind8,ind3]\[CenterDot]G[ind9,ind4],{ind7,1,6},{ind2,1,6},{\
ind8,1,6},{ind3,1,6},{ind9,1,6}],{ind1,1,6},{ind4,1,6}];
Table[Cas6133[i,j]=clean[cas6133[i,j]],{i,1,6},{j,1,6}];
SaveIt["Cas6133"];*)

ClearMemory[];

(*Clear[Cas6233,cas6233]

Table[cas6233[ind4,ind1]=Sum[g[[ind4,ind10]]g[[ind5,ind11]]g[[ind6,ind12]]G[\
ind10,ind5]\[CenterDot]G[ind11,ind6]\[CenterDot]G[ind12,ind1],{ind10,1,6},{\
ind5,1,6},{ind11,1,6},{ind6,1,6},{ind12,1,6}],{ind4,1,6},{ind1,1,6}];
Table[Cas6233[i,j]=clean[cas6233[i,j]],{i,1,6},{j,1,6}];
SaveIt["Cas6233"];*)

ClearMemory[];

(*Clear[Cas6333,cas6333]

cas6333=Sum[Cas6133[ind1,ind4]\[CenterDot]Cas6233[ind4,ind1],{ind1,1,6},{ind4,\
1,6}];
SaveIt["cas6333"];
Cas6333=clean[cas6333];
SaveIt["Cas6333"];*)

Clear[Cas2cub]
Cas2cub = clean[Cas2\[CenterDot]Cas2sq];
SaveIt["Cas2cub"];

$Aborted

 (*clean[Cas6333-\[Alpha] Cas2quart]*)

Cas63 = ReadIt["Cas6333"];
Cas2quart = ReadIt["Cas2quart"];

clean[Cas63 - (1/36) Cas2cub + 2 Cas2sq - 16 Cas2]

0