Pensieve header: Simplifying the messy functions in the symmetric dequantizator.

FS=FullSimplify;

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

FS[(Sqrt[(a^2)+ϵ b])==a+((ϵ b)/(a+(Sqrt[(a^2)+ϵ b])))]

FS[(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])==t+((ϵ(4ϖ+ϵ (γ^2)))/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))]

FS[t+((ϵ(4ϖ+ϵ (γ^2)))/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))]

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

FS[Cosh[x]-Cosh[y] == 2Sinh[((x+y)/2)] Sinh[((x-y)/2)]]

With[{x=(ℏ/2)(t+((ϵ(4ϖ+ϵ (γ^2)))/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))), y=(ℏ/2)(t-(2 a+γ) ϵ)},
Simplify/@{((x+y)/2),((x-y)/2)}]

FS[g==2(Sqrt[(( γ Sinh[(1/4) (2 t-(2 a+γ) ϵ+((ϵ ((γ^2) ϵ+4 ϖ))/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))) ℏ] Sinh[(1/4) ϵ (2 a+γ+(((γ^2) ϵ+4 ϖ)/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))) ℏ])/(Sinh[((γ ϵ ℏ)/2)](t (2 a+γ)-2 a (a+γ) ϵ+2 ϖ) ℏ))])]

FS[-(2 a+γ) ϵ+((ϵ ((γ^2) ϵ+4 ϖ))/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))]

FS[2 a+γ+(((γ^2) ϵ+4 ϖ)/(t+(Sqrt[(t^2)+(γ^2) (ϵ^2)+4 ϵ ϖ])))]