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1 | 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. | |
2 | ||
3 | $HistoryLength = 0; | |
4 | ||
5 | Definition of Save and read functions | |
6 | ||
7 | SaveIt[filename_, expr_] := | |
8 | Module[{output}, | |
9 | output = Export[filename <> ".dat", ToString[expr // InputForm], | |
10 | "String"]; | |
11 | ClearMemory; | |
12 | output]; | |
13 | SaveIt[varnamestring_] := | |
14 | Module[{output}, | |
15 | output = Export[varnamestring <> ".dat", | |
16 | ToString[ToExpression[varnamestring] // InputForm], "String"]; | |
17 | ClearMemory; | |
18 | output]; | |
19 | ReadIt[filename_] := | |
20 | Module[{output}, | |
21 | output = ToExpression[ | |
22 | Import[StringReplace[filename, ".dat" -> ""] <> ".dat", "String"]]; | |
23 | ClearMemory; | |
24 | output]; | |
25 | ClearMemory := Module[{}, Unprotect[In, Out]; | |
26 | Clear[In, Out]; | |
27 | Protect[In, Out]; | |
28 | ClearSystemCache[];]; | |
29 | ||
30 | Definition of Bosonic oscillators | |
31 | ||
32 | Clear[Boson, BosonC, BosonA] | |
33 | Boson /: MakeBoxes[Boson[cr : (True | False), sym_], fmt_] := | |
34 | With[{sbox = If[StringQ[sym], sym, ToBoxes[sym]]}, | |
35 | With[{abox = | |
36 | If[cr, SuperscriptBox[#, FormBox["\[Dagger]", Bold]], #] &@sbox}, | |
37 | InterpretationBox[abox, Boson[cr, sym]]]] | |
38 | BosonA[sym_: String "a"] := Boson[False, sym] | |
39 | BosonC[sym_: String "a"] := Boson[True, sym] | |
40 | ||
41 | Aliasing NonCommutativeMultiply with CenterDot | |
42 | ||
43 | 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. | |
44 | ||
45 | Unprotect[NonCommutativeMultiply]; | |
46 | Clear[NonCommutativeMultiply, CenterDot]; | |
47 | CenterDot[a__] := NonCommutativeMultiply[a]; | |
48 | NonCommutativeMultiply /: MakeBoxes[NonCommutativeMultiply[a__], fmt_] := | |
49 | With[{cbox = ToBoxes[HoldForm[CenterDot[a]]]}, | |
50 | InterpretationBox[cbox, NonCommutativeMultiply[a]]] | |
51 | Protect[NonCommutativeMultiply]; | |
52 | Clear[CRule] | |
53 | CRule = {NonCommutativeMultiply[a_] :> a}; | |
54 | ||
55 | Definition of clean - the function which orders and simplifies expressions | |
56 | ||
57 | ClearAll@expand | |
58 | - | (*the attribute is new*) |
58 | + | |
59 | SetAttributes[expand, HoldAll] | |
60 | ||
61 | - | (*equivalent to your code but in my opinion/expectation clearer \ |
61 | + | |
62 | - | cleaner and slightly faster*) |
62 | + | |
63 | Block[{NonCommutativeMultiply (*or times*)}, | |
64 | expr //. {times[left___, cnum_ /; FreeQ[cnum, (_Boson)], | |
65 | right___] :> cnum*times[left, right], | |
66 | times[left___, | |
67 | - | expr /. {times[left___, cnum_ /; FreeQ[cnum, (_Boson)], |
67 | + | |
68 | right___] :> | |
69 | Times @@ Apply[Power, Drop[FactorList[cnum], -1], 2]* | |
70 | times[left, First[Last[FactorList[cnum]]], right], | |
71 | times[left___, Boson[False, s_], Boson[True, s_], right___] :> | |
72 | times[left, right] + | |
73 | times[left, Boson[True, s], Boson[False, s], right], | |
74 | times[left___, fst : Boson[_, s_], sec : Boson[_, t_], | |
75 | right___] :> | |
76 | times[left, sec, fst, right] /; | |
77 | FreeQ[Ordering[{s, t}], {1, 2}], times[b_Boson] :> b | |
78 | (*moved and rewrote this rule*), | |
79 | times[] -> 1}]] /. {HoldPattern[times] -> NonCommutativeMultiply} | |
80 | expand[Alternatives[NonCommutativeMultiply, CenterDot][ | |
81 | a1_, (a2_ + a3_)]] := expand[a1 ** a2] + expand[a1 ** a3] | |
82 | expand[Alternatives[NonCommutativeMultiply, CenterDot][(a1_ + a2_), | |
83 | a3_]] := expand[a1 ** a3] + expand[a2 ** a3] | |
84 | ||
85 | Clear[clean] | |
86 | ||
87 | clean = Simplify[FixedPoint[expand, Distribute //@ #] //. CRule] &; | |
88 | ||
89 | Commutator function | |
90 | ||
91 | Clear[CommOp]; | |
92 | CommOp[x_, y_] := clean[(x\[CenterDot]y - y\[CenterDot]x)]; | |
93 | Clear[ACommOp]; | |
94 | ACommOp[x_, y_] := clean[(x\[CenterDot]y + y\[CenterDot]x)]; | |
95 | ||
96 | Defining the basic variables. | |
97 | ||
98 | Clear[Ad, A, Bd, B, Xd, X, Yd, Y] | |
99 | Clear[\[Psi], \[Chi], \[CapitalGamma], \[CapitalSigma], i, j, k, l, m, n, p, \ | |
100 | q, g, T, G, P, Tr1, Tr2, Tr3, Cartan, Comm, Killing, Joseph, Commutator, \ | |
101 | Zero, zero, KillComm, KillMet, DoubleTr, GComm, Dim] | |
102 | Dim = 6; | |
103 | Ad = BosonC[a]; A = BosonA[a]; Bd = BosonC[b]; B = BosonA[b]; Xd = | |
104 | BosonC[x]; X = BosonA[x]; Yd = BosonC[y]; Y = BosonA[y]; | |
105 | ||
106 | Defining the algebra. | |
107 | ||
108 | Dirac \[CapitalGamma] matrices: | |
109 | ||
110 | \[CapitalGamma][ | |
111 | 1] = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}}; | |
112 | \[CapitalGamma][ | |
113 | 2] = {{0, 0, 0, -1}, {0, 0, -1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}; | |
114 | \[CapitalGamma][ | |
115 | 3] = {{0, 0, 0, I}, {0, 0, -I, 0}, {0, -I, 0, 0}, {I, 0, 0, 0}}; | |
116 | \[CapitalGamma][ | |
117 | 4] = {{0, 0, -1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, -1, 0, 0}}; | |
118 | \[CapitalGamma][ | |
119 | 5] = {{0, 0, I, 0}, {0, 0, 0, I}, {I, 0, 0, 0}, {0, I, 0, 0}}; | |
120 | Zero = {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}; | |
121 | ||
122 | Spinors and metric: | |
123 | ||
124 | g = DiagonalMatrix[{-1, 1, 1, 1, 1, -1}]; | |
125 | \[Psi] = {{A}, {B}, {-Xd}, {-Yd}}; | |
126 | \[Chi] = {{Ad, Bd, X, Y}}; (*This is the Dirac conjugate of \[Psi]*) | |
127 | ||
128 | Subscript[\[CapitalGamma], ij] matrices: | |
129 | ||
130 | Table[\[CapitalSigma][i, | |
131 | j] = -I ( \[CapitalGamma][i].\[CapitalGamma][j] - \[CapitalGamma][ | |
132 | j].\[CapitalGamma][i])/4, {i, 1, 4}, {j, 1, 4}]; | |
133 | ||
134 | \[CapitalSigma][5, 6] = -\[CapitalGamma][5]/2; \[CapitalSigma][6, | |
135 | 5] = \[CapitalGamma][5]/2; | |
136 | ||
137 | Table[\[CapitalSigma][i, | |
138 | 5] = -I \[CapitalGamma][i].\[CapitalGamma][5]/2; \[CapitalSigma][5, i] = | |
139 | I \[CapitalGamma][i].\[CapitalGamma][5]/2; \[CapitalSigma][i, | |
140 | 6] = -\[CapitalGamma][i]/2; \[CapitalSigma][6, i] = \[CapitalGamma][i]/ | |
141 | 2, {i, 1, 4}]; | |
142 | ||
143 | \[CapitalSigma][5, 5] = Zero; | |
144 | \[CapitalSigma][6, 6] = Zero; | |
145 | ||
146 | Generators of SO(4,2) denoted by G[i,j] | |
147 | ||
148 | Table[G[i, j] = | |
149 | Sum[\[CapitalSigma][i, j][[m, | |
150 | n]] \[Chi][[1, m]]\[CenterDot]\[Psi][[n, 1]], {m, 1, 4}, {n, 1, 4}], {i, | |
151 | 1, Dim}, {j, 1, Dim}]; | |
152 | ||
153 | Check the commutation relations of \[CapitalSigma] matrices | |
154 | ||
155 | (*Table[Print[i,",",j,",",k,",",l,"=",Simplify[(\[CapitalSigma][i,j].\ | |
156 | \[CapitalSigma][k,l]-\[CapitalSigma][k,l].\[CapitalSigma][i,j]+ I( g[[i,k]]\ | |
157 | \[CapitalSigma][j,l]- g[[i,l]]\[CapitalSigma][j,k] - \ | |
158 | g[[j,k]]\[CapitalSigma][i,l] + \ | |
159 | g[[j,l]]\[CapitalSigma][i,k]))]],{i,1,Dim},{j,1,Dim},{k,1,Dim},{l,1,Dim}];*) | |
160 | ||
161 | Check the commutation relations of group generators | |
162 | ||
163 | (*Table[Print[i,",",j,",",k,",",l,"=",clean[(G[i,j]\[CenterDot]G[k,l]-G[k,l]\ | |
164 | \[CenterDot]G[i,j]+ I( g[[i,k]]G[j,l]- g[[i,l]]G[j,k] - g[[j,k]]G[i,l] + \ | |
165 | g[[j,l]]G[i,k]))]],{i,1,Dim},{j,1,Dim},{k,1,Dim},{l,1,Dim}];*) | |
166 | ||
167 | Generators of SO(3,1) and rest of the conformal generators | |
168 | ||
169 | Clear[M, \[Eta], \[CapitalPi], \[CapitalKappa], \[CapitalDelta]] | |
170 | (*\[CapitalPi] are translations, \[CapitalKappa](this is greek capital K \ | |
171 | obtained by esc K esc)are special conformal and \[CapitalDelta] is \ | |
172 | dilatations*) | |
173 | ||
174 | \[Eta] = DiagonalMatrix[{-1, 1, 1, 1}]; | |
175 | Table[M[i, j] = G[i, j], {i, 1, 4}, {j, 1, 4}]; | |
176 | Table[\[CapitalKappa][i] = G[i, 6] - G[i, 5], {i, 1, 4}]; | |
177 | Table[\[CapitalPi][i] = G[i, 6] + G[i, 5], {i, 1, 4}]; | |
178 | \[CapitalDelta] = -G[5, 6]; | |
179 | ||
180 | Casimirs | |
181 | ||
182 | Quadratic Casmir | |
183 | ||
184 | Clear[ind1, ind2, ind3, ind4, Cas2, Z] | |
185 | ||
186 | Z = Ad\[CenterDot]A + Bd\[CenterDot]B - Xd\[CenterDot]X - Yd\[CenterDot]Y; | |
187 | ||
188 | Cas2 = clean[ | |
189 | Sum[g[[ind1, ind3]]\[CenterDot]G[ind3, ind2]\[CenterDot]g[[ind2, | |
190 | ind4]]\[CenterDot]G[ind4, ind1], {ind1, 1, 6}, {ind2, 1, 6}, {ind3, 1, | |
191 | 6}, {ind4, 1, 6}]]; | |
192 | clean[Cas2 + (3/2) (Z\[CenterDot]Z - 4)]; | |
193 | SaveIt["Cas2"]; | |
194 | Cas2sq = clean[Cas2\[CenterDot]Cas2]; | |
195 | SaveIt["Cas2sq"]; | |
196 | ||
197 | Cubic Casmir | |
198 | ||
199 | (*Clear[ind1,ind2,ind3,ind4,ind5,ind6,cas3] | |
200 | ||
201 | Timing[cas3=Sum[g[[ind1,ind4]]g[[ind2,ind5]]g[[ind3,ind6]]G[ind4,ind2]\ | |
202 | \[CenterDot]G[ind5,ind3]\[CenterDot]G[ind6,ind1],{ind1,1,6},{ind2,1,6},{ind3,\ | |
203 | 1,6},{ind4,1,6},{ind5,1,6},{ind6,1,6} ];]; | |
204 | Cas3=clean[cas3]; | |
205 | SaveIt["Cas3.dat"]; | |
206 | clean[Cas3 -2I Cas2];*) | |
207 | ||
208 | Quartic Casimir | |
209 | ||
210 | (*Clear[ind1,ind2,ind3,ind4,ind5,ind6,ind7,ind8,cas4] | |
211 | ||
212 | cas4=Sum[g[[ind1,ind5]]g[[ind2,ind6]]g[[ind3,ind7]]g[[ind4,ind8]]G[ind5,ind2]\ | |
213 | \[CenterDot]G[ind6,ind3]\[CenterDot]G[ind7,ind4]\[CenterDot]G[ind8,ind1],{\ | |
214 | ind1,1,6},{ind2,1,6},{ind3,1,6},{ind4,1,6},{ind5,1,6},{ind6,1,6},{ind7,1,6},{\ | |
215 | ind8,1,6}]; | |
216 | Cas4=clean[cas4]; | |
217 | SaveIt["Cas4"]; | |
218 | clean[Cas4-(1/6) Cas2sq+4 Cas2]*) | |
219 | ||
220 | Cas4 = ReadIt["Cas4"]; | |
221 | ||
222 | clean[Cas4 - (1/6) Cas2sq + 4 Cas2] | |
223 | ||
224 | 0 | |
225 | ||
226 | 6th order Casimir | |
227 | ||
228 | (*Clear[ind1,ind2,ind3,ind4,ind5,ind6,ind7,ind8,ind9,ind10,ind11,ind12,cas6,\ | |
229 | Cas6,dim] | |
230 | ||
231 | dim=6; | |
232 | ||
233 | Timing[cas6=ParallelSum[g[[ind1,ind7]]g[[ind2,ind8]]g[[ind3,ind9]]g[[ind4,\ | |
234 | ind10]]g[[ind5,ind11]]g[[ind6,ind12]]G[ind7,ind2]\[CenterDot]G[ind8,ind3]\ | |
235 | \[CenterDot]G[ind9,ind4]\[CenterDot]G[ind10,ind5]\[CenterDot]G[ind11,ind6]\ | |
236 | \[CenterDot]G[ind12,ind1],{ind1,1,dim},{ind2,1,dim},{ind3,1,dim},{ind4,1,dim},\ | |
237 | {ind5,1,dim},{ind6,1,dim},{ind7,1,dim},{ind8,1,dim},{ind9,1,dim},{ind10,1,dim}\ | |
238 | ,{ind11,1,dim},{ind12,1,dim}];]*) | |
239 | ||
240 | (*Cas6=clean[cas6];*) | |
241 | ||
242 | (*Clear[ind1,ind2,ind3,ind4,ind5,ind6,ind7,ind8,ind9,ind10,ind11,ind12,cas6,\ | |
243 | Cas6,dim] | |
244 | ||
245 | dim=6; | |
246 | ||
247 | AbsoluteTiming[Table[cas6[i,j]=ParallelSum[g[[i,ind7]]g[[j,ind8]]g[[ind3,ind9]\ | |
248 | ]g[[ind4,ind10]]g[[ind5,ind11]]g[[ind6,ind12]]G[ind7,j]\[CenterDot]G[ind8,\ | |
249 | ind3]\[CenterDot]G[ind9,ind4]\[CenterDot]G[ind10,ind5]\[CenterDot]G[ind11,\ | |
250 | ind6]\[CenterDot]G[ind12,i],{ind3,1,dim},{ind4,1,dim},{ind5,1,dim},{ind6,1,\ | |
251 | dim},{ind7,1,dim},{ind8,1,dim},{ind9,1,dim},{ind10,1,dim},{ind11,1,dim},{\ | |
252 | ind12,1,dim}],{i,1,dim},{j,1,dim}];]*) | |
253 | ||
254 | (*AbsoluteTiming[Table[Cas6[i,j]=clean[cas6[i,j]],{i,1,dim},{j,1,dim}]]*) | |
255 | ||
256 | 3-3 split | |
257 | ||
258 | (*Clear[Cas6133,cas6133] | |
259 | ||
260 | Table[cas6133[ind1,ind4]=Sum[g[[ind1,ind7]]g[[ind2,ind8]]g[[ind3,ind9]]G[ind7,\ | |
261 | ind2]\[CenterDot]G[ind8,ind3]\[CenterDot]G[ind9,ind4],{ind7,1,6},{ind2,1,6},{\ | |
262 | ind8,1,6},{ind3,1,6},{ind9,1,6}],{ind1,1,6},{ind4,1,6}]; | |
263 | Table[Cas6133[i,j]=clean[cas6133[i,j]],{i,1,6},{j,1,6}]; | |
264 | SaveIt["Cas6133"];*) | |
265 | ||
266 | ClearMemory[]; | |
267 | ||
268 | (*Clear[Cas6233,cas6233] | |
269 | ||
270 | Table[cas6233[ind4,ind1]=Sum[g[[ind4,ind10]]g[[ind5,ind11]]g[[ind6,ind12]]G[\ | |
271 | ind10,ind5]\[CenterDot]G[ind11,ind6]\[CenterDot]G[ind12,ind1],{ind10,1,6},{\ | |
272 | ind5,1,6},{ind11,1,6},{ind6,1,6},{ind12,1,6}],{ind4,1,6},{ind1,1,6}]; | |
273 | Table[Cas6233[i,j]=clean[cas6233[i,j]],{i,1,6},{j,1,6}]; | |
274 | SaveIt["Cas6233"];*) | |
275 | ||
276 | ClearMemory[]; | |
277 | ||
278 | (*Clear[Cas6333,cas6333] | |
279 | ||
280 | cas6333=Sum[Cas6133[ind1,ind4]\[CenterDot]Cas6233[ind4,ind1],{ind1,1,6},{ind4,\ | |
281 | 1,6}]; | |
282 | SaveIt["cas6333"]; | |
283 | Cas6333=clean[cas6333]; | |
284 | SaveIt["Cas6333"];*) | |
285 | ||
286 | Clear[Cas2cub] | |
287 | Cas2cub = clean[Cas2\[CenterDot]Cas2sq]; | |
288 | SaveIt["Cas2cub"]; | |
289 | ||
290 | $Aborted | |
291 | ||
292 | (*clean[Cas6333-\[Alpha] Cas2quart]*) | |
293 | ||
294 | Cas63 = ReadIt["Cas6333"]; | |
295 | Cas2quart = ReadIt["Cas2quart"]; | |
296 | ||
297 | clean[Cas63 - (1/36) Cas2cub + 2 Cas2sq - 16 Cas2] | |
298 | ||
299 | 0 |