Pensieve header: The $sl_2^\epsilon$ dequantizator.

Code borrowed from Phi2CR-Classical.nb.

Monoblog on Oct 18, 2017:

{rhs=Simplify[((1-T (A^2))/ℏ) /. {T->(E^(ℏ(ϵ a -b))), A->(E^(-ℏ ϵ a))} /. b->ϵ a-t], Limit[rhs, ℏ->0]}

Representing (𝔤^ϵ).

ME=MatrixExp;
Simp[sol_] := Flatten[sol]/. C[_]->0/. (var_->val_) :> (var -> Simplify[PowerExpand@val]);

ρt=(({{1, 0}, {0, 1}}));  ρy=(({{0, 0}, {ϵ, 0}}));  ρa=(({{(1+1/ϵ)/2, 0}, {0, -(1-1/ϵ)/2}})); ρx=(({{0, 1}, {0, 0}}));
Simplify@{ρa.ρx-ρx.ρa==ρx, ρa.ρy-ρy.ρa==-ρy, ρx.ρy-ρy.ρx==2ϵ ρa-ρt}

The 2-Stitch of Exponentials

ϕ={(Subscript[τ, 0])->-((Log[1+ϵ (Subscript[η, 2]) (Subscript[ξ, 1])])/ϵ)+(Subscript[τ, 1])+(Subscript[τ, 2]),(Subscript[η, 0])->(Subscript[η, 1])+(((E^(-(Subscript[α, 1]))) (Subscript[η, 2]))/(1+ϵ (Subscript[η, 2]) (Subscript[ξ, 1]))),(Subscript[α, 0])->2 Log[1+ϵ (Subscript[η, 2]) (Subscript[ξ, 1])]+(Subscript[α, 1])+(Subscript[α, 2]),(Subscript[ξ, 0])->(((E^(-(Subscript[α, 2]))) (Subscript[ξ, 1]))/(1+ϵ (Subscript[η, 2]) (Subscript[ξ, 1])))+(Subscript[ξ, 2])};

Simplify[
ME[(Subscript[τ, 1])ρt].ME[(Subscript[η, 1])ρy].ME[(Subscript[α, 1])ρa].ME[(Subscript[ξ, 1])ρx].ME[(Subscript[τ, 2]) ρt].ME[(Subscript[η, 2])ρy].ME[(Subscript[α, 2])ρa].ME[(Subscript[ξ, 2])ρx]==ME[(Subscript[τ, 0]) ρt].ME[(Subscript[η, 0])ρy].ME[(Subscript[α, 0])ρa].ME[(Subscript[ξ, 0])ρx] /. ϕ]

Φ=Simplify[(Subscript[t, 0]) (Subscript[τ, 0]) + (Subscript[y, 0]) (Subscript[η, 0]) + (Subscript[a, 0]) (Subscript[α, 0]) + (Subscript[x, 0]) (Subscript[ξ, 0]) /. ϕ]

Collect[Limit[Φ, ϵ->0], {(Subscript[t, 0]), (Subscript[y, 0]), (Subscript[a, 0]),(Subscript[x, 0])}]

Finding the Dequantizator

Should satisfy x (OverHat[y])-q (OverHat[y])x=((1-(e^(ℏ(t-2ϵ a))))/ℏ) where q=(e^(ℏ γ ϵ)) (temporarily at γ=1).

The x(OverHat[y]) side

Simplify[(∂_(Subscript[ξ, 1]))(E^Φ)] /. {(Subscript[(τ|η|α|ξ), 1])->0, (Subscript[(v:(τ|η|α|ξ)), 2]):>v, (Subscript[(v:(t|y|a|x)), 0]):>v}

Makes x f[y,a-1]-(t-2 a ϵ+y ϵ (∂_y))(∂_y)f[y,a].

The q(OverHat[y])x side

Simplify[(∂_(Subscript[ξ, 2]))(E^Φ)]/. {(Subscript[(v:(τ|η|α|ξ)), 1]):>v, (Subscript[(τ|η|α|ξ), 2])->0, (Subscript[(v:(t|y|a|x)), 0]):>v}

Makes x (e^(ℏ ϵ))f[y,a].

The overall equation

x (f[y,a-1]-(e^(ℏ ϵ))f[y,a])-(t-2 a ϵ+y ϵ (∂_y))(∂_y)f[y,a]=((1-(e^(ℏ(t-2ϵ a))))/ℏ).

At ϵ=0 this is x (f[y,a-1]-f[y,a])-t(∂_y)f[y,a]=((1-(e^(ℏ t)))/ℏ), which is solved by f[y,a]=y(((e^(ℏ t))-1)/(ℏ t)).

eqn[f_]:=Simplify[x((f/.a->a-1)-(E^(ℏ ϵ))f)-(t-2ϵ a)(∂_y)f+y ϵ (∂_({y,2}))f - ((1-(E^(ℏ(t-2ϵ a))))/ℏ)]

eqn[f[y,a]]/. ϵ -> 0

Simplify[eqn[y (((E^(ℏ t))-1)/(ℏ t))]/. ϵ -> 0]

SeriesCoefficient[eqn[y (((E^(ℏ t))-1)/(ℏ t))+ϵ (Subscript[f, 1])[y,a]], {ϵ,0,1}]

SeriesCoefficient[eqn[y (((E^(ℏ t))-1)/(ℏ t))+ϵ (((y a)/t)-(((E^(t ℏ)) y a)/t)+(Subscript[f, 1])[y,a])], {ϵ,0,1}]

Simplify[eqn[y g[x y,a]] //. x y -> z]

eq[g_] := Simplify[(1/ℏ)(-1+(E^((t-2 a ϵ) ℏ))+z ℏ ((g/.a->a-1)-(E^(ϵ ℏ)) g)-(t-2 a ϵ) ℏ (g+z (∂_z)g)+z ϵ ℏ (2 (∂_z)g+z (∂_({z,2}))g))]

eq[g[z,a]]==Simplify[eqn[y g[x y,a]] //. x y -> z]

Simplify[eq[g[z,a]] /. ϵ->0]

SeriesCoefficient[eq[g[z,a]], {ϵ,0,0}]

SeriesCoefficient[eq[(((E^(ℏ t))-1)/(ℏ t))+ϵ (Subscript[g, 1])[z,a]], {ϵ,0,1}]

Simplify[eqn[(x^(-1)) g[(t^(-1)) x y,a]] //. (t^(-1))x y -> z]

Simplify[eqn[(x^(-1)) g[x y,a]] //.x y -> z]

eq[g_] := Simplify[(((E^((t-2 a ϵ) ℏ))-1)/ℏ)+(g/.a->a-1)-(E^(ϵ ℏ)) g-(t-2 a ϵ)(∂_z)g+z ϵ(∂_({z,2}))g]

Simplify[eq[g[z,a]]==eqn[(x^(-1)) g[x y,a]] //. x y -> z]

Changing the dependent variable:

Collect[((((z^(1-a+(t/(2 ϵ)))) ))^(-1))eq[(z^((t-2 a ϵ)/(2 ϵ)))g[z,a]], { g[z,a], g[z,-1+a], Derivative[2,0][g][z,a]}, Simplify]

Changing the independent variable:

eq[g[(z^((t+ϵ-2 a ϵ)/ϵ)),a]]

Changing the dependent and the independent variable:

Collect[eq[(z^(Subscript[s, 1]))g[(z^(Subscript[s, 2])),a]], Derivative[1,0][g][(z^(Subscript[s, 2])),a], Simplify]

Expanding in ϵ:

Simplify[eq[g[z,a]] /. ϵ->0]

SeriesCoefficient[eq[g[z,a]], {ϵ,0,0}]

Series[-(1/ℏ)+((E^(t ℏ))/ℏ), {ℏ,0,3}]

SeriesCoefficient[eq[(((E^(ℏ t))-1)/(ℏ t))z+ϵ (Subscript[g, 1])[z,a]], {ϵ,0,1}]

Series[-2 a (E^(t ℏ))+(z/t)-(((E^(t ℏ)) z)/t)-((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)), {ℏ,0,2}]

SeriesCoefficient[eq[(((E^(ℏ t))-1)/(ℏ t))z+ϵ((z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ)))+(Subscript[g, 1])[z,a])], {ϵ,0,1}]

Series[(z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ))), {t,0,1}]

FullSimplify[(z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ)))]

Expand@SeriesCoefficient[eq[(((E^(ℏ t))-1)/(ℏ t))z+ϵ((z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ)))+ϵ (Subscript[g, 2])[z,a])], {ϵ,0,2}]

Expand@SeriesCoefficient[eq[(((E^(ℏ t))-1)/(ℏ t))z+ϵ((z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ)))+ϵ(((z (a^2))/t)(-((4 (E^(t ℏ)))/t)-(4/((t^2) ℏ))+((4 (E^(t ℏ)))/((t^2) ℏ))+2 (E^(t ℏ)) ℏ)+a((z^2)/(2t))((4/(t^2))+((8 (E^(t ℏ)))/(t^2))+(12/((t^3) ℏ))-((12 (E^(t ℏ)))/((t^3) ℏ))-((2 (E^(t ℏ)) ℏ)/t))+(1/(12 (t^5) ℏ))(z^2) (-12 t+12 (E^(t ℏ)) t-24 z+24 (E^(t ℏ)) z+6 (t^2) ℏ-18 (E^(t ℏ)) (t^2) ℏ-12 t z ℏ-12 (E^(t ℏ)) t z ℏ+3 (t^3) (ℏ^2)+9 (E^(t ℏ)) (t^3) (ℏ^2)-2 (t^2) z (ℏ^2)+2 (E^(t ℏ)) (t^2) z (ℏ^2))+(Subscript[g, 2])[z,a]))], {ϵ,0,2}]

Expand@SeriesCoefficient[eq[(((E^(ℏ t))-1)/(ℏ t))z+ϵ((z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ)))+ϵ(((z (a^2))/t)(-((4 (E^(t ℏ)))/t)-(4/((t^2) ℏ))+((4 (E^(t ℏ)))/((t^2) ℏ))+2 (E^(t ℏ)) ℏ)+a((z^2)/(2t))((4/(t^2))+((8 (E^(t ℏ)))/(t^2))+(12/((t^3) ℏ))-((12 (E^(t ℏ)))/((t^3) ℏ))-((2 (E^(t ℏ)) ℏ)/t))+(1/(12 (t^5) ℏ))(z^2) (-12 t+12 (E^(t ℏ)) t-24 z+24 (E^(t ℏ)) z+6 (t^2) ℏ-18 (E^(t ℏ)) (t^2) ℏ-12 t z ℏ-12 (E^(t ℏ)) t z ℏ+3 (t^3) (ℏ^2)+9 (E^(t ℏ)) (t^3) (ℏ^2)-2 (t^2) z (ℏ^2)+2 (E^(t ℏ)) (t^2) z (ℏ^2))+ϵ (Subscript[g, 3])[z,a]))], {ϵ,0,3}]

Series[(((E^(ℏ t))-1)/(ℏ t))z+ϵ((z/t)(-2 a (E^(t ℏ)) -((2 a)/(t ℏ))+((2 a (E^(t ℏ)))/(t ℏ)))+(z/(2t))((z/t)+(((E^(t ℏ)) z)/t)+((2 z)/((t^2) ℏ))-((2 (E^(t ℏ)) z)/((t^2) ℏ)))+ϵ(((z (a^2))/t)(-((4 (E^(t ℏ)))/t)-(4/((t^2) ℏ))+((4 (E^(t ℏ)))/((t^2) ℏ))+2 (E^(t ℏ)) ℏ)+a((z^2)/(2t))((4/(t^2))+((8 (E^(t ℏ)))/(t^2))+(12/((t^3) ℏ))-((12 (E^(t ℏ)))/((t^3) ℏ))-((2 (E^(t ℏ)) ℏ)/t))+(1/(12 (t^5) ℏ))(z^2) (-12 t+12 (E^(t ℏ)) t-24 z+24 (E^(t ℏ)) z+6 (t^2) ℏ-18 (E^(t ℏ)) (t^2) ℏ-12 t z ℏ-12 (E^(t ℏ)) t z ℏ+3 (t^3) (ℏ^2)+9 (E^(t ℏ)) (t^3) (ℏ^2)-2 (t^2) z (ℏ^2)+2 (E^(t ℏ)) (t^2) z (ℏ^2))+ϵ (Subscript[g, 3])[z,a])), {t,0,2}]

eqng = FullSimplify[ℏ(x((y g[x y,a-1])-(E^(ℏ ϵ))(y g[x y,a]))-(t-2ϵ a)(∂_y)(y g[x y,a])+y ϵ (∂_({y,2}))(y g[x y,a]) - ((1-(E^(ℏ(t-2ϵ a))))/ℏ))] //. x y -> z

eqng = FullSimplify@PowerExpand[
Simplify[
ℏ(x((y g[Log[x y],a-1])-(E^(ℏ ϵ))(y g[Log[x y],a]))-(t-2ϵ a)(∂_y)(y g[Log[x y],a])+y ϵ (∂_({y,2}))(y g[Log[x y],a])) -( 1-(E^(ℏ(t-2ϵ a))))
] //. x y -> (E^z)
]

FullSimplify[eqng /. ϵ->0]