/sci/ - Science & Math

>>11558493
Arnol'd or Sternberg-Guillemin
>>11558735
Bring the [math]-\hbar\omega a^\dagger[/math] term in Eq. (4.3.13) to the other side. >>11558760 is wrong as it that misses the [math]\frac{1}{2}[/math] factor.
>>11558747
Recall that harmonic maps [math]u[/math] minimizes the energy functional [math]-\frac{1}{2}\int_D |\nabla u|^2[/math] and satisfies the min/max principle. The condition here is "shrunk" to make [math]u[/math] satisfy min/max only in tangent circles [math]C[/math] and hence minimizes the energy functional [math]-\frac{1}{2}\int_{C(z)}|\nabla u|^2[/math] for each [math]z \in \operatorname{int}D[/math]. However, note that as [math]z[/math] is in the interior so you can find [math]z \neq z' \in \operatorname{int}D[/math] such that [math]C(z) \cap C(z') \neq \emptyset[/math], allowing you to analytically extend [math]u[/math] to [math]C(z)\cup C(z')[/math]. Since [math]\bigcup_{z\in D}C(z)[/math] is an open cover of [math]D[/math] and since [math]D[/math] is compact, you're done.