Pensieve header: The dequantizators.

Follows code in Projects/SL2Portfolio/SL2PortfolioProgram.nb.

Startup

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\SL2Portfolio2"];
Once[<<"../Profile/Profile.m";];
<<"Engine-Speedy.m";
<<"Objects.m";
$k=3;
HL[ℰ_] := Style[ℰ, Background -> Green];

The Asymmetric Dequantizator

Following pensieve://People/VanDerVeen/Dequant1.pdf.

A𝔻$f=γ((Cosh[ℏ(a ϵ+((γ ϵ)/2)-(t/2))]-Cosh[ℏ(Sqrt[(((((t-γ ϵ)/2)))^2)+ϵ ω])])/(ℏ (E^(ℏ((a+γ)ϵ-t/2)))Sinh[((γ ϵ ℏ)/2)]((a^2)ϵ+a γ ϵ-a t-ω)));

A𝔻$ω=γ CU[y,x]+ϵ CU[a,a]-(t-γ ϵ)CU[a];

DeclareMorphism[A𝔻,QU->CU,{a->(Subscript[a, CU]),x->CU@x, y:>(Subscript[𝕊, CU])[SS[A𝔻$f],a->(Subscript[a, CU]),ω->A𝔻$ω]**(Subscript[y, CU])}]

The Symmetric Dequantizator

Following pensieve://People/VanDerVeen/Dequant1.pdf.

S𝔻$g=(Sqrt[((2γ(Cosh[(ℏ/2)(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])]-Cosh[((t-ϵ γ-2ϵ a)/(2/ℏ))]))/(Sinh[((γ ϵ ℏ)/2)](t (2 a+γ)-2 a (a+γ) ϵ+2 ϖ) ℏ))]);

S𝔻$f=Simplify[(E^(ℏ(t/2-ϵ a)))(S𝔻$g /. {a->-a, t->-t})];

S𝔻$ϖ=γ CU[y,x]+ϵ CU[a,a]-(t-γ ϵ)CU[a]-t γ (Subscript[1, CU])/2;

DeclareMorphism[S𝔻,QU->CU,{a->(Subscript[a, CU]),
x:>(Subscript[𝕊, CU])[SS[S𝔻$f],a->(Subscript[a, CU]),ϖ->S𝔻$ϖ]**(Subscript[x, CU]),
y:>(Subscript[𝕊, CU])[SS[S𝔻$g],a->(Subscript[a, CU]),ϖ->S𝔻$ϖ]**(Subscript[y, CU]) }]

A𝔻$f=γ((Cosh[ℏ(a ϵ+((γ ϵ)/2)-(t/2))]-Cosh[ℏ(Sqrt[(((((t-γ ϵ)/2)))^2)+ϵ ω])])/(ℏ (E^(ℏ((a+γ)ϵ-t/2)))Sinh[((γ ϵ ℏ)/2)]((a^2)ϵ+a γ ϵ-a t-ω)))

Series[A𝔻$f,{ℏ,0,4}]//Simplify

(Series[A𝔻$f,{ℏ,0,4}]/.ϵ->0)//Simplify

Collect[Series[A𝔻$f,{ℏ,0,4}]//Normal,ϵ]

Series[Normal@Series[A𝔻$f, {ℏ,0,5}], {ϵ,0,2}]

Series[A𝔻$f,{a,0,1}]//Simplify

Series[A𝔻$f,{ω,0,1}]//Simplify

Simplify[Series[A𝔻$f,{ϵ,0,2}],Assumptions->{t>0}]

Limit[A𝔻$f, ϵ->0]

1+(E^(t ℏ))-2 (E^((t ℏ)/2)) Cosh[((t ℏ)/2)]//TrigExpand

(1/(24 (t^5) (((a t+ω))^2) ℏ))(E^((t ℏ)/2)) (-2 (t^4) (γ^2) (((a t+ω))^2) (ℏ^2) Sinh[((t ℏ)/2)]+24 (t^4) (((a+γ))^2) ((ω^2) (ℏ^2)+2 a ω ℏ (-1+t ℏ)+(a^2) (2-2 t ℏ+(t^2) (ℏ^2))) Sinh[((t ℏ)/2)]+6 (t^2) (a+γ) (ω ℏ+a (-1+t ℏ)) (4 t (a t+t γ-ω) (a t+ω) ℏ Cosh[((t ℏ)/2)]+8 ω (-t γ+ω) Sinh[((t ℏ)/2)])-(a t+ω) (24 t (t γ-2 ω) (t γ-ω) ω ℏ Cosh[((t ℏ)/2)]-48 (t γ-2 ω) (t γ-ω) ω Sinh[((t ℏ)/2)]-(t^5) (((2 a+γ))^3) (ℏ^2) Sinh[((t ℏ)/2)]+(t^2) (((t γ-2 ω))^3) (ℏ^2) Sinh[((t ℏ)/2)])) (ϵ^2)//TrigExpand//Together

A𝔻$ω=γ CU[y,x]+ϵ CU[a,a]-(t-γ ϵ)CU[a];

(Subscript[Exp, cm,1,0])[γ (Subscript[y, 1])(Subscript[x, 1])+ϵ ((Subscript[a, 1])^2)-((Subscript[t, 1])-γ ϵ )(Subscript[a, 1])]

(Subscript[Exp, cm,1,1])[γ (Subscript[y, 1])(Subscript[x, 1])+ϵ ((Subscript[a, 1])^2)-((Subscript[t, 1])-γ ϵ )(Subscript[a, 1])]