/sci/ - Science & Math

whatever appears in yukarifag's last post on /mg/

>>11352521
Let [math]H = H_E \otimes I + I \otimes H_B + \epsilon H_\text{int}[/math] be the Hamiltonian describing an interaction with range [math]R<\infty[/math] between the quantum system [math]B[/math] and the environment [math]E[/math] where [math]N = \operatorname{dim}E \gg \operatorname{dim}B[/math]. Suppose [math]\rho = \rho_E\otimes \rho_B[/math] is the initial density matrix of the coupled system with [math]\rho_E[/math] pure, then the thermalization hypothesis implies that there exists [math]T\in\mathbb{R}_{>0}[/math] and a pure state [math]\rho_0[/math] on [math]B[/math] such that the Heisenberg dynamics [math]\tau_\ast^H[/math] drives [eqn]||\operatorname{tr}_E(\tau_T^H(\rho)) - \rho_0|| \leq \epsilon O(N^{-\infty}).[/eqn]
What this means is that, given a macroscopic classical environment [math]E[/math], its massive number of degrees of freedom will eventually "drown out" the quantum properties of states on [math]B[/math]. In other words, putting the cat in a box will decohere all the quantum particles in the box. After 1 min all particles will behave as classically as the cat initially did.
>deactivate the chamber
I don't know what that means.

>>11248917
Read Simon & Reed's texts on methods of mathematical physics back to back.

>>10709648
Well yeah he called me a retard once but that doesn't make him a hack

>>10376771
I'll do that once I find art that makes reference to Clifford modules bundles.

>>10193027
A section [math]f\in \mathcal{F}(U)[/math] over a stalk [math]\mathcal{F}_x[/math] for [math]x\in U[/math] is by definition a collection of sections over stalks [math]f: U\rightarrow \coprod_{x\in U}\mathcal{F}_x[/math]. The intuition for the sheaf of sections over a manifold/topological space is useful here: the section [math]s: U \rightarrow E[/math] of a fibre bundle over [math]U[/math] is by definition a collection of sections over fibres [math]s: U\rightarrow \coprod_{x\in U} F = U \times F[/math] by a local trivialization (in the case of Seifert fibre bundles, for instance, the equality [math]\coprod_{x\in U}F = U \times F[/math] no longer holds for all [math]U[/math]).
Evaluating the section [math]f[/math] at a point [math]x\in U[/math] results in an element of the stalk [math]\mathcal{F}_x[/math], and the compatibility condition tells you that the sections are now continuous.
For instance, suppose we have a cover [math]\{U\}[/math] of the manifold [math]M[/math], then the compatibility condition states that a refinement [math]\{V\}[/math] exists such that sections [math]s_U[/math] on the cover [math]\{U\}[/math] restricts to sections [math]s_V[/math] on the cover [math]\{V\}[/math]; in particular they restrict to the intersection [math]U\cap U'[/math] of two open sets.

>>10160569
That's wrong though. Are you conflating a smooth structure with an embedding? Because I don't think that's a good idea.