Pensieve header: Trying to understand the U(1)-indeterminacy of KV solutions.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2015-12"];
Get["../Projects/WKO4/FreeLie.m"];
Get["../Projects/WKO4/AwCalculus.m"];

l=LW["l"]; r=LW["r"]; x=LW["x"]; y=LW["y"]; z=LW["z"];
A=LS[{l,r}, as];
V = El[
A // LieMorphism[{l->⟨x->LS[y], y->LS[0]⟩, r->⟨x->LS[0],y->LS[x]⟩}, ⟨x,y⟩, tb],
CWS[0]
] // Γ;
V@{4}

Rl[a_,b_]:= El[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
iRl[a_,b_]:= El[⟨a->LS[0], b->-LS[LW@a]⟩, CWS[0]];
Rs[a_,b_]:= Es[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
iRs[a_,b_]:= Es[⟨a->LS[0], b->-LS[LW@a]⟩, CWS[0]];
Θl[x_,y_, s_] := El[⟨x->LS[s LW@y], y->LS[s LW@x]⟩, CWS[0]];
Θs[x_,y_,s_] := Θl[x,y,s] // Γ;
R4Eqn[V_] := V**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**V;
UnitarityEqn[V_] := V**(V//dA) ≡ Es[⟨x->LS[0], y->LS[0]⟩,CWS[0]];
CapEqn[V_,Cap_] := (V**(Cap//dΔ[x,x,y])//dc[x]//dc[y]) ≡ (Cap*(Cap//dσ[x,y])//dc[x]//dc[y]);

as["l"]=-1/4;
msgs=SeriesSolve[A, (ℏ^(-1))R4Eqn[V]];
A[1]

A[1]

A[2]

κs[xs___] := κs[xs] = SeriesCoefficient[(1/4)Log[((ℏ/2)/(Sinh[ℏ/2]))], {ℏ,0,Length@{xs}}];
κ=CWS[{x}, κs];
(Subscript[Cap, 0])=Es[⟨x->LS[0]⟩,κ];
κ@{16}

SymmetryFactor[l_List] := Module[{ll=l, sf=1},
While[(ll=RotateLeft[ll]) =!= l, ++sf];
sf];
SymmetryFactor[{1,1,1,2,1,1,1,2}]

γs[xs___] := γs[xs]= If[{xs}===RotateLeft@{xs}, 0, SymmetryFactor[{xs}]]*κs[xs];
γ=CWS[{x,y}, γs];
γ@{6}

βs[t_]["x"]:= t; βs[t_]["y"]:=0;
α[t_] := α[t]=LS[{x,y}, αs[t]]; β[t_] := β[t]=LS[{x,y}, βs[t]];
V[t_] := (
V[t] = Es[⟨x->α[t], y->β[t]⟩, γ];
SeriesSolve[{α[t], β[t]}, ((ℏ^(-1))R4Eqn[V[t]]) ∧ UnitarityEqn[V[t]] ∧ CapEqn[V[t],(Subscript[Cap, 0])]];
V[t]
);
V[1/2]@{6}

τ[V_]=Rs[x,y]**(V//dσ[{x,y}->{y,x}])**Θs[x,y,-1/2];
(V[#]≡τ[V[#]])@{6}& /@ {1/4,1/3,1/2,1}

Relations

(V[0]**dA[V[1]])@{6}

(V[0]**dS[V[1]])@{6}

(V[1/4]≡(V[1/4] // dσ[{x,y} -> {y,x}] // (((-1))^deg)))@{8}

((V[1/4]//dA)**(V[1/4] // dσ[{x,y} -> {y,x}] // (((-1))^deg)))@{6}

(V[1/2]**(V[1/2]//dS))@{6}

((V[1/4]**Θs[x,y,1/12])≡V[1/3])@{7}

Play

{V[0], V[1/4], V[1/2], V[3/4], V[1]} // ColumnForm

V[1/2]@{4}

(dA[V[0]]**V[1])@{5}

Series[(E^(ℏ/2)),{ℏ,0,8}]

(dS[V[0]]**V[1])@{5}

(V[0]**dS[V[1/2]])@{6}

V[1/4]

V[1/4] // dσ[{x,y} -> {y,x}] // (((-1))^deg)

V[1/2]

V[1/2] // dσ[{x,y} -> {y,x}] // (((-1))^deg)

((V[0]//dA)**(V[0] // dσ[{x,y} -> {y,x}] // (((-1))^deg)))@{6}

(V[1/4]**(V[1/4]//dS))@{6}

Θs[x,y,1/2]

V[s]

V[1/2]

V[1/4] ** Θs[x,y,1/4]

V[1/3]

((V[1/4]**Θs[x,y,1/12])≡V[1/3])@{7}

((Θs[x,y,1/12]**V[1/4])≡V[1/3])@{4}