/sci/ - Science & Math

>>11439526
>Someone ask something, I'm bored.
Alright. Define [math]\mathcal{S}[/math] as the smooth subalgebra of trace-class operators in [math]\mathcal{B}(\mathscr{H}) \cap \mathcal{A}[/math] where [math]\mathcal{A}[/math] is the algebra of many-body observables, I'm trying to prove that the inclusion [math]\mathcal{S}\hookrightarrow \mathcal{A}[/math] is a quasi-isomorphism in operator [math]K[/math]-theory, meaning that [math]K_0(\mathcal{S}) \cong K_0(\mathcal{A})[/math]. Reason being I need to be able to take the von Neumann trace for an index formula but I also don't want to lose any topological info. I know in the one-body case I can do this if [math]\mathcal{S}=\mathcal{R}[/math] is a smooth subalgebra and [math]\mathcal{A}=C^*_r(G)[/math] is the reduced [math]C^*[/math]-algebra because of the Haagerupp inequality, but [math]C^*_r(G)[/math] is finitely generated for finite [math]G[/math] while the many-body observables [math]\mathcal{A}[/math] is literally the image of the Sobolev [math]H^1[/math] under the second quantization functor. Haagerupp inequality doesn't help here. Anyone knows of any ideas?
>>11440690
>Suppose I want to minimize a squared sum of fractional linear functions
Against what? What are the minimization parameters?
>linear fractional function
What? The Mobius transforms are projective, not linear.
Anyways suppose you want to minimize the sum with respect to the coefficients, i.e. [math]f(z) = \int_A g^2 d\mu(g) |g(z)|^2[/math] for some collection [math]A\subset PSL_2(\mathbb{C})[/math], then by the left-invariance of the Haar measure we have [math]d\mu(g) = -g^2d\mu(g^{-1})[/math] hence [math]f(z) = -|z|^2\int_{A^{-1}}d\mu(g) g^2[/math], then it's just an issue of maximizing the average [math]\mu(\cdot^2)[/math] of the squaring function [math]g\mapsto g^2[/math] as [math]g[/math] moves through [math]A^{-1}[/math].

>>11403755
I am both. A physical mathematician if you will.
>>11406409
Optics in general have an encompassing description y symplectic geometry; it is in fact the historical motivation. For quantum optics, you will need a lot of scattering theory, which also intersects spectral theory, applied PDEs and Riemannian geometry. Most people do things with the spectral density since that's all the optics people know how to measure.