4D-Dirac Killing operators in Cartesian charts with standard Dirac-matrices restart; with(linalg): n:=4;

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eta_5:=<<1|0|0|0|0>,<0|-1|0|0|0>,<0|0|-1|0|0>,<0|0|0|-1|0>,<0|0|0|0|-1>>;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNXFfZ0dWMlduVz0=

eta:=<<1|0|0|0>,<0|-1|0|0>,<0|0|-1|0>,<0|0|0|-1>>;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNSEqZmdHVjJXblc9

Id:=<<1|0|0|0>,<0|1|0|0>,<0|0|1|0>,<0|0|0|1>>: Coordinates and local frames Zu:=array(1..5,[Z0,Z1,Z2,Z3,Z5]);

PTYiNiM7IiIiIiImRVxbbCZGJkkjWjBHRiMiIiNJI1oxR0YjIiIkSSNaMkdGIyIiJUkjWjNHRiNGJ0kjWjVHRiM=

dZu:=array(1..5);

PTYiNiM7IiIiIiImRVxbbCE=

Xu:=array(1..4,[t,x1,x2,x3]);

PTYiNiM7IiIiIiIlRVxbbCVGJkkidEdGIyIiI0kjeDFHRiMiIiRJI3gyR0YjRidJI3gzR0Yj

dXu:=array(1..4,[dt,dx1,dx2,dx3]);

PTYiNiM7IiIiIiIlRVxbbCVGJkkjZHRHRiMiIiNJJGR4MUdGIyIiJEkkZHgyR0YjRidJJGR4M0dGIw==

Dd:=array(1..4,[D0,D1,D2,D3]);

PTYiNiM7IiIiIiIlRVxbbCVGJkkjRDBHRiMiIiNJI0QxR0YjIiIkSSNEMkdGI0YnSSNEM0dGIw==

xD:=x1*D1+x2*D2+x3*D3;

LCgqJkkjRDFHNiIiIiJJI3gxR0YlRiZGJiomSSNEMkdGJUYmSSN4MkdGJUYmRiYqJkkjRDNHRiVGJkkjeDNHRiVGJkYm

r:=sqrt(x1^2+x2^2+x3^2);

KiQsKCokSSN4MUc2IiIiIyIiIiokSSN4MkdGJkYnRigqJEkjeDNHRiZGJ0YoI0YoRic=

Z0:=-1/2/t/omega^2*(1-omega^2*(t^2-r^2));

LCQqKEkidEc2IiEiIkkmb21lZ2FHRiUhIiMsJiIiIkYqKiZGJyIiIywqKiRGJEYsRioqJEkjeDFHRiVGLEYmKiRJI3gyR0YlRixGJiokSSN4M0dGJUYsRiZGKkYmRiojRiZGLA==

Z5:=-1/2/t/omega^2*(1+omega^2*(t^2-r^2));

LCQqKEkidEc2IiEiIkkmb21lZ2FHRiUhIiMsJiIiIkYqKiZGJyIiIywqKiRGJEYsRioqJEkjeDFHRiVGLEYmKiRJI3gyR0YlRixGJiokSSN4M0dGJUYsRiZGKkYqRiojRiZGLA==

Z1:=-x1/omega/t;Z2:=-x2/omega/t;Z3:=-x3/omega/t;

LCQqKEkjeDFHNiIiIiJJJm9tZWdhR0YlISIiSSJ0R0YlRihGKA==

LCQqKEkjeDJHNiIiIiJJJm9tZWdhR0YlISIiSSJ0R0YlRihGKA==

LCQqKEkjeDNHNiIiIiJJJm9tZWdhR0YlISIiSSJ0R0YlRihGKA==

Zd:=array(1..5,[Z0,-Z1,-Z2,-Z3,-Z5]): Zd:=array(1..5,[Z0,-Z1,-Z2,-Z3,-Z5]): for i from 1 to 5 do dZu[i]:=add(diff(Zu[i],Xu[k])*dXu[k],k=1..4);end do;

LCoqJiwmKihJInRHNiIhIiNJJm9tZWdhR0YnRigsJiIiIkYrKiZGKSIiIywqKiRGJkYtRisqJEkjeDFHRidGLSEiIiokSSN4MkdGJ0YtRjIqJEkjeDNHRidGLUYyRitGMkYrI0YrRi1GK0YrRitJI2R0R0YnRitGKyooRiZGMkYxRitJJGR4MUdGJ0YrRjIqKEYmRjJGNEYrSSRkeDJHRidGK0YyKihGJkYyRjZGK0kkZHgzR0YnRitGMg==

LCYqKkkjeDFHNiIiIiJJJm9tZWdhR0YlISIiSSJ0R0YlISIjSSNkdEdGJUYmRiYqKEYnRihGKUYoSSRkeDFHRiVGJkYo

LCYqKkkjeDJHNiIiIiJJJm9tZWdhR0YlISIiSSJ0R0YlISIjSSNkdEdGJUYmRiYqKEYnRihGKUYoSSRkeDJHRiVGJkYo

LCYqKkkjeDNHNiIiIiJJJm9tZWdhR0YlISIiSSJ0R0YlISIjSSNkdEdGJUYmRiYqKEYnRihGKUYoSSRkeDNHRiVGJkYo

LCoqJiwmKihJInRHNiIhIiNJJm9tZWdhR0YnRigsJiIiIkYrKiZGKSIiIywqKiRGJkYtRisqJEkjeDFHRidGLSEiIiokSSN4MkdGJ0YtRjIqJEkjeDNHRidGLUYyRitGK0YrI0YrRi1GMkYrRitJI2R0R0YnRitGKyooRiZGMkYxRitJJGR4MUdGJ0YrRisqKEYmRjJGNEYrSSRkeDJHRidGK0YrKihGJkYyRjZGK0kkZHgzR0YnRitGKw==

dZd:=array(1..5,[dZu[1],-dZu[2],-dZu[3],-dZu[4],-dZu[5]]): Dp:=proc(cucu,k) local t; global Xu; map(diff,cucu,Xu[k+1]); end proc: Dpp:=proc(cucu,k,j) local t; global Xu; map(diff,cucu,Xu[k+1],Xu[j+1]); end proc: Metric tensor line:=simplify(dZu[1]^2-add(dZu[j]^2,j=2..5)): Gdd:=array(symmetric,1..4,1..4): for i from 1 to 4 do for j from 1 to 4 do der:=(1/2)*diff(line,dXu[i],dXu[j]);Gdd[i,j]:=simplify(der);end do;end do: print(Gdd);

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g:=simplify(sqrt(-det(Gdd)));

KiQqJkkidEc2IiEiKUkmb21lZ2FHRiVGJiMiIiIiIiM=

Guu:=array(symmetric,1..4,1..4): Guu:=simplify(evalm(Gdd^(-1)));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmKiZJInRHRiNGLkkmb21lZ2FHRiNGLjYkRi5GLiwkRjQhIiI2JEYnRidGODYkRipGKkY4NiRGJ0YmRis2JEYuRipGKzYkRipGLkYrNiRGJkYnRis2JEYqRiZGKw==

1-forms: form:=array(1..4): xdx:=x1*dx1+x2*dx2+x3*dx3: form[1]:=-dt/omega/t;

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form[2]:=-dx1/omega/t;

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form[3]:=-dx2/omega/t;

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form[4]:=-dx3/omega/t;

LCQqKEkmb21lZ2FHNiIhIiJJInRHRiVGJkkkZHgzR0YlIiIiRiY=

verific_line:=simplify(add(eta[k,k]*form[k]^2,k=1..4)-line);

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Tetrades H:=array(1..4,1..4):H1:=array(1..4,1..4): E:=array(1..4,1..4): for i from 1 to 4 do for j from 1 to 4 do; H1[i,j]:=simplify(diff(form[i],dXu[j])); end do; end do; print(H1);

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QyQ+SSZUcmFuc0c2IkkjSWRHRiUhIiI= H:=evalm(Trans.H1);

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmLCQqJkkmb21lZ2FHRiMhIiJJInRHRiNGN0Y3NiRGLkYuRjQ2JEYnRidGNDYkRipGKkY0NiRGJ0YmRis2JEYuRipGKzYkRipGLkYrNiRGJkYnRis2JEYqRiZGKw==

E:=simplify(inverse(H));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmLCQqJkkmb21lZ2FHRiNGJkkidEdGI0YmISIiNiRGLkYuRjQ2JEYnRidGNDYkRipGKkY0NiRGJ0YmRis2JEYuRipGKzYkRipGLkYrNiRGJkYnRis2JEYqRiZGKw==

simplify(multiply(E,H)): Connection in local frames Cartan coefficients with lower indices Cart:=array(1..4,1..4,1..4): aux1:=array(1..4,1..4,1..4): aux2:=array(1..4,1..4,1..4): for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do cucu:=simplify((diff(H[k,i],Xu[j])-diff(H[k,j],Xu[i]))*eta[k,k]);aux1[i,j,k]:=cucu; end do;end do; end do; Next i:=0:j:=0:k:=0: for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do cucu1:=simplify(sum('aux1[s,k,i]*E[s,j]','s'=1..4));aux2[j,k,i]:=cucu1 end do; end do; end do; aux3[2,1,2]: Next for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do cucu2:=simplify(sum('aux2[j,p,i]*E[p,k]','p'=1..4)):Cart[j,k,i]:=simplify(cucu2) end do;end do; end do: Cart[2,1,2]: Connection Conex:=array(1..4,1..4,1..4): for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do cucu:=simplify(Cart[i,j,k]-Cart[i,k,j]+Cart[k,j,i]):Conex[i,j,k]:=cucu/2 end do;end do; end do: Dirac matrices Dir:=proc(s,t,x,y,z) local matr; matr:=<<s|0|t-z|-x+I*y>,<0|s|-x-I*y|t+z>,<t+z|x-I*y|-s|0>,<x+I*y|t-z|0|-s>>; matr; end proc: Id:=<<1|0|0|0>,<0|1|0|0>,<0|0|1|0>,<0|0|0|1>>: gam1:=Dir(1,0,0,0,0);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNT0iXCRvVjJXblc9

gam2:=Dir(0,0,-1,0,0): gam3:=Dir(0,0,0,-1,0): gam4:=Dir(0,0,0,0,-1): g0:=evalm(gam1):g1:=evalm(gam2):g2:=evalm(gam3):g3:=evalm(gam4): g5:=evalm(I*g0.g1.g2.g3);

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGJjYkIiIjRidGJjYkRi5GJkYrNiRGJkYuRis2JEYnRi5GJjYkRipGJ0YrNiRGJkYmRis2JEYuRi5GKzYkRidGJ0YrNiRGKkYqRis2JEYnRiZGKzYkRi5GKkYrNiRGKkYuRis2JEYmRidGKzYkRipGJkYm

Gam1:=evalm(E[1,1]*g0+E[1,2]*g1+E[1,3]*g2+E[1,4]*g3): Gam2:=evalm(E[2,1]*g0+E[2,2]*g1+E[2,3]*g2+E[2,4]*g3): Gam3:=evalm(E[3,1]*g0+E[3,2]*g1+E[3,3]*g2+E[3,4]*g3): Gam4:=evalm(E[4,1]*g0+E[4,2]*g1+E[4,3]*g2+E[4,4]*g3): G0:=Gam1:G1:=Gam2:G2:=Gam3:G3:=Gam4: SL(2,C) generators with upper indices S12:=evalm((I/2)*(gam1.gam2));Sp01:=S12;

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmRis2JEYuRi5GKzYkRidGJ0YrNiRGKkYqRis2JEYnRiZeIyNGJkYuNiRGLkYqRjg2JEYqRi5GODYkRiZGJ0Y4NiRGKkYmRis=

SSRTMTJHNiI=

S13:=evalm((I/2)*(gam1.gam3)):Sp02:=S13: S14:=evalm((I/2)*(gam1.gam4)):Sp03:=S14: S23:=evalm((I/2)*(gam2.gam3)):Sp12:=S23: S24:=evalm((I/2)*(gam2.gam4)):Sp13:=S24: S34:=evalm((I/2)*(gam3.gam4)):Sp23:=S34: Sigma1:=evalm(2*S34): Sigma2:=evalm(-2*S24): Sigma3:=evalm(2*S23): Spin Connection and Dirac operator SCon1:=I*simplify(evalm(Conex[1,1,2]*S12+Conex[1,1,3]*S13+Conex[1,1,4]*S14 +Conex[1,2,3]*S23+Conex[1,2,4]*S24+Conex[1,3,4]*S34));

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SCon2:=I*simplify(evalm(Conex[2,1,2]*S12+Conex[2,1,3]*S13+Conex[2,1,4]*S14 +Conex[2,2,3]*S23+Conex[2,2,4]*S24+Conex[2,3,4]*S34));

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SCon3:=I*simplify(evalm(Conex[3,1,2]*S12+Conex[3,1,3]*S13+Conex[3,1,4]*S14 +Conex[3,2,3]*S23+Conex[3,2,4]*S24+Conex[3,3,4]*S34)): SCon4:=I*simplify(evalm(Conex[4,1,2]*S12+Conex[4,1,3]*S13+Conex[4,1,4]*S14 +Conex[4,2,3]*S23+Conex[4,2,4]*S24+Conex[4,3,4]*S34)): Momentum operators-chiral version PG0:=evalm(I*(E[1,1]*D0+E[2,1]*D1+E[3,1]*D2+E[4,1]*D3)*Id+I*SCon1): PG1:=evalm(I*(E[1,2]*D0+E[2,2]*D1+E[3,2]*D2+E[4,2]*D3)*Id+I*SCon2);

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmKipeIyEiIkYmSSNEMUdGI0YmSSZvbWVnYUdGI0YmSSJ0R0YjRiY2JEYuRi5GNDYkRidGJ0Y0NiRGKkYqRjQ2JEYnRiYqJl4jI0YmRi5GJkY4RiY2JEYuRipGPjYkRipGLkY+NiRGJkYnRj42JEYqRiZGKw==

PG2:=evalm(I*(E[1,3]*D0+E[2,3]*D1+E[3,3]*D2+E[4,3]*D3)*Id+I*SCon3): PG3:=evalm(I*(E[1,4]*D0+E[2,4]*D1+E[3,4]*D2+E[4,4]*D3)*Id+I*SCon4): Dirac operator OpD:=simplify(evalm(g0.PG0+g1.PG1+g2.PG2+g3.PG3));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRioqKl4jISIiRiZJI0QzR0YjRiZJJm9tZWdhR0YjRiZJInRHRiNGJjYkIiIjRicqKl4jRiZGJkYwRiZGMUYmRjJGJjYkRjRGJkYrNiRGJkY0Ris2JEYnRjRGLTYkRipGJ0YrNiRGJkYmLCYqKkYuRiZJI0QwR0YjRiZGMUYmRjJGJkYmKiZeIyNGKkY0RiZGMUYmRiY2JEY0RjRGPDYkRidGJywmKipGNkYmRj5GJkYxRiZGMkYmRiYqJl4jIyEiJEY0RiZGMUYmRiY2JEYqRipGRDYkRidGJiwmKipGNkYmSSNEMUdGI0YmRjFGJkYyRiZGJiooSSNEMkdGI0YmRjFGJkYyRiZGLzYkRjRGKiwmKipGLkYmRk5GJkYxRiZGMkYmRiZGT0YmNiRGKkY0LCZGTUYmRk9GJjYkRiZGJywmRlNGJkZPRi82JEYqRiZGNQ==

ED:=evalm(OpD-m*Id): Operator algebra Products of scalar operators Prod:=proc(X,Y) local AA,AX,ProdAB,A0,A1,A2,A3,A11,A00,A01,A02,A03,A22,A33,A12,A23,A13; global Xu,Dd; A00:=diff(X,Dd[1],Dd[1])/2;A01:=diff(X,Dd[1],Dd[2]);A02:=diff(X,Dd[1],Dd[3]);A03:=diff(X,Dd[1],Dd[4]);A11:=diff(X,Dd[2],Dd[2])/2;A22:=diff(X,Dd[3],Dd[3])/2;A33:=diff(X,Dd[4],Dd[4])/2;A12:=diff(X,Dd[3],Dd[2]);A23:=diff(X,Dd[4],Dd[3]);A13:=diff(X,Dd[2],Dd[4]);AA:=A00*Dd[1]^2+A01*Dd[1]*Dd[2]+A02*Dd[1]*Dd[3]+A03*Dd[1]*Dd[4]+A11*Dd[2]^2+A22*Dd[3]^2+A33*Dd[4]^2+A12*Dd[3]*Dd[2]+A13*Dd[2]*Dd[4]+A23*Dd[3]*Dd[4];AX:=simplify(X-AA);A0:=diff(AX,Dd[1]);A1:=diff(AX,Dd[2]);A2:=diff(AX,Dd[3]);A3:=diff(AX,Dd[4]); ProdAB:=X*Y+A0*diff(Y,Xu[1])+A1*diff(Y,Xu[2])+A2*diff(Y,Xu[3])+A3*diff(Y,Xu[4])+A00*(2*Dd[1]*diff(Y,Xu[1])+diff(Y,Xu[1],Xu[1]))+A11*(2*Dd[2]*diff(Y,Xu[2])+diff(Y,Xu[2],Xu[2]))+A22*(2*Dd[3]*diff(Y,Xu[3])+diff(Y,Xu[3],Xu[3]))+A33*(2*Dd[4]*diff(Y,Xu[4])+diff(Y,Xu[4],Xu[4]))+A12*(Dd[2]*diff(Y,Xu[3])+Dd[3]*diff(Y,Xu[2])+diff(Y,Xu[2],Xu[3]))+A23*(Dd[3]*diff(Y,Xu[4])+Dd[4]*diff(Y,Xu[3])+diff(Y,Xu[3],Xu[4]))+A13*(Dd[4]*diff(Y,Xu[2])+Dd[2]*diff(Y,Xu[4])+diff(Y,Xu[2],Xu[4]))+A01*(Dd[2]*diff(Y,Xu[1])+Dd[1]*diff(Y,Xu[2])+diff(Y,Xu[1],Xu[2]))+A02*(Dd[3]*diff(Y,Xu[1])+Dd[1]*diff(Y,Xu[3])+diff(Y,Xu[3],Xu[1]))+A03*(Dd[4]*diff(Y,Xu[1])+Dd[1]*diff(Y,Xu[4])+diff(Y,Xu[4],Xu[1]));simplify(ProdAB);end proc: Commutation Com:=proc(X,Y) local cucu;cucu:=simplify(Prod(X,Y)-Prod(Y,X));cucu;end proc: Standard Form StandardForm :=proc(X) local AA,AX,A0,A1,A2,A3,A11,A00,A01,A02,A03,A22,A33,A12,A23,A13,stf; global Xu,Dd; A00:=simplify(diff(X,Dd[1],Dd[1])/2);A01:=simplify(diff(X,Dd[1],Dd[2]));A02:=simplify(diff(X,Dd[1],Dd[3]));A03:=simplify(diff(X,Dd[1],Dd[4]));A11:=simplify(diff(X,Dd[2],Dd[2])/2);A22:=simplify(diff(X,Dd[3],Dd[3])/2);A33:=simplify(diff(X,Dd[4],Dd[4])/2);A12:=simplify(diff(X,Dd[3],Dd[2]));A23:=simplify(diff(X,Dd[4],Dd[3]));A13:=simplify(diff(X,Dd[2],Dd[4]));AA:=A00*Dd[1]^2+A01*Dd[1]*Dd[2]+A02*Dd[1]*Dd[3]+A03*Dd[1]*Dd[4]+A11*Dd[2]^2+A22*Dd[3]^2+A33*Dd[4]^2+A12*Dd[3]*Dd[2]+A13*Dd[2]*Dd[4]+A23*Dd[3]*Dd[4];AX:=simplify(X-AA);A0:=simplify(diff(AX,Dd[1]));A1:=simplify(diff(AX,Dd[2]));A2:=simplify(diff(AX,Dd[3]));A3:=simplify(diff(AX,Dd[4])); stf:=AA+A0*Dd[1]+A1*Dd[2]+A2*Dd[3]+A3*Dd[4]+subs(Dd[1]=0,Dd[2]=0,Dd[3]=0,Dd[4]=0,X);stf;end proc: Products of Dirac operators ProdD:=proc(X::matrix(4,4),Y::matrix(4,4))::matrix(4,4); local i, j, mat::array(1..4,1..4); global Prod, StandardForm; mat:=array(1..4,1..4);for i from 1 to 4 do;for j from 1 to 4 do; mat[i,j]:=simplify(add(Prod(X[i,k],Y[k,j]),k=1..4)); end do; end do; return map(StandardForm,evalm(mat)); end proc: Commutation of Dirac operators ComD:=proc(X::matrix(4,4),Y::matrix(4,4))::matrix(4,4);global ProdD; simplify(evalm(ProdD(X,Y)-ProdD(Y,X)));end proc: Anticommutation of Dirac operators AComD:=proc(X::matrix(4,4),Y::matrix(4,4))::matrix(4,4); global ProdD;simplify(evalm(ProdD(X,Y)+ProdD(Y,X)));end proc: Trace Tr:=proc(Y::matrix(4,4));simplify(Y[1,1]+Y[2,2]+Y[3,3]+Y[4,4]);end proc: The Dirac Equation s:=Xu[1],Xu[2],Xu[3],Xu[4];

NiZJInRHNiJJI3gxR0YkSSN4MkdGJEkjeDNHRiQ=

psi:=matrix(4,1,[X1(s),X2(s),Y1(s),Y2(s)]);

PTYiNiQ7IiIiIiIlO0YmRiZFXFtsJTYkIiIjRiYtSSNYMkdGIzYmSSJ0R0YjSSN4MUdGI0kjeDJHRiNJI3gzR0YjNiRGJkYmLUkjWDFHRiNGLjYkRidGJi1JI1kyR0YjRi42JCIiJEYmLUkjWTFHRiNGLg==

ED1:=expand(subs(D0=0,D1=0,D2=0,D3=0,simplify(add(Prod(OpD[1,k],psi[k,1]),k=1..4))))-m*psi[1,1]: ED2:=expand(subs(D0=0,D1=0,D2=0,D3=0,simplify(add(Prod(OpD[2,k],psi[k,1]),k=1..4))))-m*psi[2,1]: ED3:=expand(subs(D0=0,D1=0,D2=0,D3=0,simplify(add(Prod(OpD[3,k],psi[k,1]),k=1..4))))-m*psi[3,1]: ED4:=expand(subs(D0=0,D1=0,D2=0,D3=0,simplify(add(Prod(OpD[4,k],psi[k,1]),k=1..4))))-m*psi[4,1]: Conserved operators The Killing vectors Kd:=array(1..5,1..5,1..4);

PTYiNiU7IiIiIiImRiU7RiYiIiVFXFtsIQ==

Ku:=array(1..5,1..5,1..4);

PTYiNiU7IiIiIiImRiU7RiYiIiVFXFtsIQ==

RezKu:=array(1..4);

PTYiNiM7IiIiIiIlRVxbbCE=

Cons:=array(antisymmetric,1..5,1..5);

PUkuYW50aXN5bW1ldHJpY0c2IjYkOyIiIiIiJkYmRVxbbCE=

for j from 1 to 5 do for k from 1 to j do Cons[j,k]:=simplify(Zd[k]*dZd[j]-Zd[j]*dZd[k]);end do;end do: for i from 1 to 5 do for j from 1 to 5 do for k from 1 to 4 do Kd[i,j,k]:=diff(Cons[i,j],dXu[k]);end do;end do;end do; for i from 1 to 5 do for j from 1 to 5 do for k from 1 to 4 do cucu:=add(Kd[i,j,n]*Guu[n,k],n=1..4);Ku[i,j,k]:=simplify(cucu);end do;end do;end do; Killing:=proc(a,b)::array(1..4); global Ku,RezKu; RezKu[1]:=Ku[a,b,1];RezKu[2]:=Ku[a,b,2];RezKu[3]:=Ku[a,b,3];RezKu[4]:=Ku[a,b,4];return(evalm(RezKu));end proc: Ku01:=Killing(1,2);

PTYiNiM7IiIiIiIlRVxbbCVGJiooSSN4MUdGI0YmSSZvbWVnYUdGI0YmSSJ0R0YjRiYiIiMsJComLCwqJkYrRi1GLEYtRiYqJkYrRi1GKkYtRiYqJkYrRi1JI3gyR0YjRi0hIiIqJkYrRi1JI3gzR0YjRi1GNUY1RiZGJkYrRjUjRiZGLSIiJCooRitGJkYqRiZGNEYmRicqKEYrRiZGKkYmRjdGJg==

Ku02:=Killing(1,3);

PTYiNiM7IiIiIiIlRVxbbCVGJiooSSN4MkdGI0YmSSZvbWVnYUdGI0YmSSJ0R0YjRiYiIiMqKEYrRiZJI3gxR0YjRiZGKkYmIiIkLCQqJiwsKiZGK0YtRixGLUYmKiZGK0YtRi9GLSEiIiomRitGLUYqRi1GJiomRitGLUkjeDNHRiNGLUY2RjZGJkYmRitGNiNGJkYtRicqKEYrRiZGKkYmRjlGJg==

Ku03:=Killing(1,4);

PTYiNiM7IiIiIiIlRVxbbCVGJiooSSN4M0dGI0YmSSZvbWVnYUdGI0YmSSJ0R0YjRiYiIiMqKEYrRiZJI3gxR0YjRiZGKkYmIiIkKihGK0YmSSN4MkdGI0YmRipGJkYnLCQqJiwsKiZGK0YtRixGLUYmKiZGK0YtRi9GLSEiIiomRitGLUYyRi1GOComRitGLUYqRi1GJkY4RiZGJkYrRjgjRiZGLQ==

Ku05:=Killing(1,5);

PTYiNiM7IiIiIiIlRVxbbCVGJkkidEdGIyIiI0kjeDFHRiMiIiRJI3gyR0YjRidJI3gzR0Yj

Ku12:=Killing(2,3);

PTYiNiM7IiIiIiIlRVxbbCVGJiIiISIiIywkSSN4MkdGIyEiIiIiJEkjeDFHRiNGJ0Yp

Ku13:=Killing(2,4);

PTYiNiM7IiIiIiIlRVxbbCVGJiIiISIiIywkSSN4M0dGIyEiIiIiJEYpRidJI3gxR0Yj

Ku23:=Killing(3,4);

PTYiNiM7IiIiIiIlRVxbbCVGJiIiISIiI0YpIiIkLCRJI3gzR0YjISIiRidJI3gyR0Yj

Ku15:=Killing(2,5);

PTYiNiM7IiIiIiIlRVxbbCVGJiwkKihJI3gxR0YjRiZJJm9tZWdhR0YjRiZJInRHRiNGJiEiIiIiIywkKiYsLComRixGL0YtRi9GJiomRixGL0YrRi9GJiomRixGL0kjeDJHRiNGL0YuKiZGLEYvSSN4M0dGI0YvRi5GJkYmRiZGLEYuI0YuRi8iIiQsJCooRixGJkYrRiZGNkYmRi5GJywkKihGLEYmRitGJkY4RiZGLg==

Ku25:=Killing(3,5);

PTYiNiM7IiIiIiIlRVxbbCVGJiwkKihJI3gyR0YjRiZJJm9tZWdhR0YjRiZJInRHRiNGJiEiIiIiIywkKihGLEYmSSN4MUdGI0YmRitGJkYuIiIkLCQqJiwsKiZGLEYvRi1GL0YmKiZGLEYvRjJGL0YuKiZGLEYvRitGL0YmKiZGLEYvSSN4M0dGI0YvRi5GJkYmRiZGLEYuI0YuRi9GJywkKihGLEYmRitGJkY7RiZGLg==

Ku35:=Killing(4,5);

PTYiNiM7IiIiIiIlRVxbbCVGJiwkKihJI3gzR0YjRiZJJm9tZWdhR0YjRiZJInRHRiNGJiEiIiIiIywkKihGLEYmSSN4MUdGI0YmRitGJkYuIiIkLCQqKEYsRiZJI3gyR0YjRiZGK0YmRi5GJywkKiYsLComRixGL0YtRi9GJiomRixGL0YyRi9GLiomRixGL0Y2Ri9GLiomRixGL0YrRi9GJkYmRiZGJkYsRi4jRi5GLw==

Scalar generators related to Killing vectors K01:=add(Ku01[i]*Dd[i],i=1..4);

LCoqKkkjeDFHNiIiIiJJJm9tZWdhR0YlRiZJInRHRiVGJkkjRDBHRiVGJkYmKigsLComRiciIiNGKEYtRiYqJkYnRi1GJEYtRiYqJkYnRi1JI3gyR0YlRi0hIiIqJkYnRi1JI3gzR0YlRi1GMUYxRiZGJkYnRjFJI0QxR0YlRiYjRiZGLSoqRidGJkYkRiZGMEYmSSNEMkdGJUYmRiYqKkYnRiZGJEYmRjNGJkkjRDNHRiVGJkYm

K02:=add(Ku02[i]*Dd[i],i=1..4);

LCoqKkkjeDJHNiIiIiJJJm9tZWdhR0YlRiZJInRHRiVGJkkjRDBHRiVGJkYmKipGJ0YmSSN4MUdGJUYmRiRGJkkjRDFHRiVGJkYmKigsLComRiciIiNGKEYwRiYqJkYnRjBGK0YwISIiKiZGJ0YwRiRGMEYmKiZGJ0YwSSN4M0dGJUYwRjJGMkYmRiZGJ0YySSNEMkdGJUYmI0YmRjAqKkYnRiZGJEYmRjVGJkkjRDNHRiVGJkYm

K03:=add(Ku03[i]*Dd[i],i=1..4);

LCoqKkkjeDNHNiIiIiJJJm9tZWdhR0YlRiZJInRHRiVGJkkjRDBHRiVGJkYmKipGJ0YmSSN4MUdGJUYmRiRGJkkjRDFHRiVGJkYmKipGJ0YmSSN4MkdGJUYmRiRGJkkjRDJHRiVGJkYmKigsLComRiciIiNGKEYzRiYqJkYnRjNGK0YzISIiKiZGJ0YzRi5GM0Y1KiZGJ0YzRiRGM0YmRjVGJkYmRidGNUkjRDNHRiVGJiNGJkYz

K05:=add(Ku05[i]*Dd[i],i=1..4);

LCoqJkkjRDBHNiIiIiJJInRHRiVGJkYmKiZJI0QxR0YlRiZJI3gxR0YlRiZGJiomSSNEMkdGJUYmSSN4MkdGJUYmRiYqJkkjRDNHRiVGJkkjeDNHRiVGJkYm

K12:=add(Ku12[i]*Dd[i],i=1..4);

LCYqJkkjRDFHNiIiIiJJI3gyR0YlRiYhIiIqJkkjRDJHRiVGJkkjeDFHRiVGJkYm

K13:=add(Ku13[i]*Dd[i],i=1..4);

LCYqJkkjRDFHNiIiIiJJI3gzR0YlRiYhIiIqJkkjRDNHRiVGJkkjeDFHRiVGJkYm

K23:=add(Ku23[i]*Dd[i],i=1..4);

LCYqJkkjRDJHNiIiIiJJI3gzR0YlRiYhIiIqJkkjRDNHRiVGJkkjeDJHRiVGJkYm

K15:=add(Ku15[i]*Dd[i],i=1..4);

LCoqKkkjeDFHNiIiIiJJJm9tZWdhR0YlRiZJInRHRiVGJkkjRDBHRiVGJiEiIiooLCwqJkYnIiIjRihGLkYmKiZGJ0YuRiRGLkYmKiZGJ0YuSSN4MkdGJUYuRioqJkYnRi5JI3gzR0YlRi5GKkYmRiZGJkYnRipJI0QxR0YlRiYjRipGLioqRidGJkYkRiZGMUYmSSNEMkdGJUYmRioqKkYnRiZGJEYmRjNGJkkjRDNHRiVGJkYq

K25:=add(Ku25[i]*Dd[i],i=1..4);

LCoqKkkjeDJHNiIiIiJJJm9tZWdhR0YlRiZJInRHRiVGJkkjRDBHRiVGJiEiIioqRidGJkkjeDFHRiVGJkYkRiZJI0QxR0YlRiZGKiooLCwqJkYnIiIjRihGMUYmKiZGJ0YxRixGMUYqKiZGJ0YxRiRGMUYmKiZGJ0YxSSN4M0dGJUYxRipGJkYmRiZGJ0YqSSNEMkdGJUYmI0YqRjEqKkYnRiZGJEYmRjVGJkkjRDNHRiVGJkYq

K35:=add(Ku35[i]*Dd[i],i=1..4);

LCoqKkkjeDNHNiIiIiJJJm9tZWdhR0YlRiZJInRHRiVGJkkjRDBHRiVGJiEiIioqRidGJkkjeDFHRiVGJkYkRiZJI0QxR0YlRiZGKioqRidGJkkjeDJHRiVGJkYkRiZJI0QyR0YlRiZGKiooLCwqJkYnIiIjRihGNEYmKiZGJ0Y0RixGNEYqKiZGJ0Y0Ri9GNEYqKiZGJ0Y0RiRGNEYmRiZGJkYmRidGKkkjRDNHRiVGJiNGKkY0

Omega matrix and spin terms Oaux:=array(1..4,1..4);

PTYiNiQ7IiIiIiIlRiVFXFtsIQ==

Omega:=proc(ku::array(1..4))::array(1..4,1..4); local cucu1, cucu2, cucu,i,j; global E,H,Xu,Oaux; for i from 1 to 4 do; for j from 1 to 4 do; cucu1:=H[i,1]*diff(ku[1],Xu[j])+H[i,2]*diff(ku[2],Xu[j])+H[i,3]*diff(ku[3],Xu[j])+H[i,4]*diff(ku[4],Xu[j]);cucu2:= diff(H[i,j],Xu[1])*ku[1]+diff(H[i,j],Xu[2])*ku[2]+diff(H[i,j],Xu[3])*ku[3]+diff(H[i,j],Xu[4])*ku[4]; cucu:=simplify(cucu1+cucu2);Oaux[i,j]:=cucu; end do;end do;evalm(simplify(multiply(multiply(Oaux,E),eta)));end proc: SpinTerm:=proc(ku::array(1..4))::array(1..4,1..4);global Oaux,Omega;evalm(-Omega(ku)[1,2]*S12-Omega(ku)[1,3]*S13-Omega(ku)[1,4]*S14+Omega(ku)[2,3]*S23+Omega(ku)[2,4]*S24+Omega(ku)[3,4]*S34);end proc: Conserved Dirac operators The Hamiltonian operator Ham:=simplify(evalm(evalm(-I*omega*K05*Id)+omega*SpinTerm(Ku05)));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmKiheIyEiIkYmSSZvbWVnYUdGI0YmLCoqJkkjRDBHRiNGJkkidEdGI0YmRiYqJkkjRDFHRiNGJkkjeDFHRiNGJkYmKiZJI0QyR0YjRiZJI3gyR0YjRiZGJiomSSNEM0dGI0YmSSN4M0dGI0YmRiZGJjYkRi5GLkY0NiRGJ0YnRjQ2JEYqRipGNDYkRidGJkYrNiRGLkYqRis2JEYqRi5GKzYkRiZGJ0YrNiRGKkYmRis=

HH:=map(StandardForm,simplify(ProdD(Ham,Ham)));

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

simplify(ComD(OpD,Ham));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmRis2JEYuRi5GKzYkRidGJ0YrNiRGKkYqRis2JEYnRiZGKzYkRi5GKkYrNiRGKkYuRis2JEYmRidGKzYkRipGJkYr

Total angular momentum J1:=evalm(evalm(-I*K23*Id)+SpinTerm(Ku23));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQjRiYiIiM2JEYmRioiIiE2JEYsRidGLjYkRixGJkYrNiRGJkYsRis2JEYnRixGLjYkRipGJ0YrNiRGJkYmKiZeIyEiIkYmLCYqJkkjRDJHRiNGJkkjeDNHRiNGJkY3KiZJI0QzR0YjRiZJI3gyR0YjRiZGJkYmNiRGLEYsRjU2JEYnRidGNTYkRipGKkY1NiRGJ0YmRi42JEYsRipGLjYkRipGLEYuNiRGJkYnRi42JEYqRiZGLg==

J2:=evalm(evalm(I*K13*Id)-SpinTerm(Ku13)): J3:=evalm(evalm(-I*K12*Id)+SpinTerm(Ku12)): JJ:=simplify(evalm(ProdD(J1,J1)+ProdD(J2,J2)+ProdD(J3,J3))): Lorentz boost K1:=evalm(evalm(-I*K01*Id)+SpinTerm(Ku01));

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

K2:=evalm(evalm(-I*K02*Id)+SpinTerm(Ku02)): K3:=evalm(evalm(-I*K03*Id)+SpinTerm(Ku03)): KK:=simplify(evalm(ProdD(K1,K1)+ProdD(K2,K2)+ProdD(K3,K3))): Runge-Lenz operator R1:=evalm(evalm(-I*K15*Id)+SpinTerm(Ku15));

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

R2:=evalm(evalm(-I*K25*Id)+SpinTerm(Ku25)): R3:=evalm(evalm(-I*K35*Id)+SpinTerm(Ku35)): RR:=simplify(evalm(ProdD(R1,R1)+ProdD(R2,R2)+ProdD(R3,R3))): Momentum P1:=simplify(evalm(-omega*(R1+K1)));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiciIiQiIiE2JEYmRipGKzYkIiIjRidGKzYkRi5GJkYrNiRGJkYuRis2JEYnRi5GKzYkRipGJ0YrNiRGJkYmKiZeIyEiIkYmSSNEMUdGI0YmNiRGLkYuRjQ2JEYnRidGNDYkRipGKkY0NiRGJ0YmRis2JEYuRipGKzYkRipGLkYrNiRGJkYnRis2JEYqRiZGKw==

P2:=simplify(evalm(-omega*(R2+K2))): P3:=simplify(evalm(-omega*(R3+K3))): PP:=simplify(evalm(ProdD(P1,P1)+ProdD(P2,P2)+ProdD(P3,P3))): Q1:=simplify(evalm(-omega*(R1-K1)));

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

Q2:=simplify(evalm(-omega*(R2-K2))): Q3:=simplify(evalm(-omega*(R3-K3))): QQ:=simplify(evalm(ProdD(Q1,Q1)+ProdD(Q2,Q2)+ProdD(Q3,Q3))): QP:=map(StandardForm,simplify(evalm(ProdD(Q1,P1)+ProdD(Q2,P2)+ProdD(Q3,P3)))): Algebra #simplify(evalm(ComD(J1,J2)-I*J3)); #simplify(evalm(ComD(R1,R2)-I*J3)); #simplify(evalm(ComD(K1,K2)+I*J3)); #simplify(evalm(ComD(K1,R1)+I*Ham/omega)); #ComD(evalm(AA+JJ),R1); Casimir operators Cas1:=map(StandardForm,simplify(evalm((-JJ-RR+KK)*omega^2+HH))): simplify(evalm(HH+3*I*omega*Ham-QP-omega^2*JJ-Cas1));

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DD:=ProdD(OpD,OpD): Test:=simplify(evalm(Cas1-DD));

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CasSO4:=simplify(evalm(JJ+RR)): StandardForm(CasSO4[1,1]): Wu0:=simplify(evalm(ProdD(J1,R1)+ProdD(J2,R2)+ProdD(J3,R3))): Wu1:=simplify(evalm(ProdD(K3,R2)-ProdD(K2,R3)-ProdD(Ham,J1)/omega)): Wu2:=simplify(evalm(ProdD(K1,R3)-ProdD(K3,R1)-ProdD(Ham,J2)/omega)): Wu3:=simplify(evalm(ProdD(K2,R1)-ProdD(K1,R2)-ProdD(Ham,J3)/omega)): Wu5:=simplify(evalm(ProdD(J1,K1)+ProdD(J2,K2)+ProdD(J3,K3))): WW:=simplify(evalm(ProdD(Wu0,Wu0)-ProdD(Wu1,Wu1)-ProdD(Wu2,Wu2)-ProdD(Wu3,Wu3)-ProdD(Wu5,Wu5))): Cas2:=evalm(-omega^2*WW): simplify(evalm(-Cas2+3/4*DD+3/16*omega^2*Id));

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ComD(WW,R1);

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#ComD(WW,J1): #ComD(WW,K1): #ComD(WW,Ham): #simplify(evalm(ComD(J1,R2)-I*R3)): #simplify(evalm(ComD(J1,K2)-I*K3)): #implify(evalm(ComD(R1,K2))): TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzI4NjA1MjcwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjOiImIiYiIiIiIiFGJ0YnRidGJyEiIkYnRidGJ0YnRidGKEYnRidGJ0YnRidGKEYnRidGJ0YnRidGKEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzI4NjA1OTkwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjMSIlIiUiIiIiIiFGJ0YnRichIiJGJ0YnRidGJ0YoRidGJ0YnRidGKEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY4MzQ5MTE4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjMSIlIiUiIiIiIiFGJ0YnRidGJkYnRidGJ0YnISIiRidGJ0YnRidGKEYl