/sci/ - Science & Math

>>11551599
>No counter-arguments, just anime pic
Thanks, I've won this debate.

>>11504084
If I understood you correctly you wish to optimize, for each [math]t[/math], [math]f_t = f(\cdot, t)[/math] on the zero locus [math]\Sigma_t = g^{-1}_t(0) \subset E[/math], right? Assuming time-independence, Rockefeller allows you to use variational methods to optimize [math]f[/math] on a dense subset, which necessarily intersects the zero locus transversally. This then allows you to study optimizers as the joint kernel of a bilinear form [math]q_f = \langle \partial f(x) \cdot,\cdot\rangle[/math] on the tangent space [math]T_x \Sigma_0[/math]. The idea is that optimizers satisfy [math]v\in \bigcap_u \operatorname{ker}q_f(u,\cdot)[/math].
Now for the time-dependent case, my first thought is this: we wish to find the regularity conditions on [math]f,g[/math] necessary for the existence of a bounded linear operator [math]\alpha_t[/math] (the time-evolution operator) on the Banach algebra [math]\mathcal{A}[/math] of convex functions such that [math]\alpha_t(g_0(x)) = g(x,t)[/math], then you may apply this operator to [math]\partial f[/math] to find the bilinear form [math]q_{f,t} = \langle \alpha_t(\partial f(x))\cdot,\cdot\rangle[/math] on [math]T_x\Sigma_t[/math], since [math]\Sigma_t = g^{-1}_t(0) = (\alpha_t g_0)^{-1}(0)[/math]. This then allows you to apply Rockefeller at each [math]t[/math] and your solutions will evolve along with [math]\alpha_t[/math]. In particular if we have a C^*-structure on [math]\mathcal{A}[/math] then it'd be even better if we can find unitaries [math]U_t[/math] such that [math]\alpha_t(g)= U_t^\dagger g U_t[/math], since in this case we can just evolve optimizers [math]v_0 \mapsto v_t = (dU_t) v_0[/math] like a propagator.
This is just a formal idea ofc, it's an interesting problem.

>>11470867
Just through Coulomb? No. However with electron-phonon coupling, [math]q \sim \omega[/math] where [math]\omega[/math] is the effective phonon frequency, and this can be tuned such that the Coulomb potential becomes attractive in a certain region. These then allow electrons to form Cooper pairs [math]c_k^\dagger c_{-k}^\dagger[/math] at which point superconductivity occurs.

>>11432079
Good effort, but what you've said doesn't illustrate why category theory would be useful in QM; you've just modeled the observables with von Neumann algebras in which, over a fixed system, no reference to any categorical structure is required.
When the system starts to evolve in (space)time, we can define an operator net as a functor [math](\mathcal{R},\coprod)\rightarrow ({\bf vNA},\bigotimes)[/math] from the category of open sets on spacetime with inclusions into the category of von Neumann algebras. Locality is then the condition that only spacelike inclusions leads to an algebra embedding, otherwise the the algebra in the image splits into direct sums. Now this [math]still[/math] doesn't illustrate why category theory is useful (though it does organize the data of an operator net neatly); people had been working with local operator nets of observables as foundations of QFT for decades without reference to category theory. This is Haag-Kastler's non-perturbative QFT.
What [math]does[/math] demonstrate the usefulness of categories, however, is when we consider [math]{\bf vNA} = {\bf vNA}^2[/math] as a 2-category, with bimodules [math]M: A\rightarrow B[/math] as 2-morphisms. The reason for this is that QM is modeled by [math]representations[/math] of von Neumann algebras and not the algebras themselves, and the representation spaces, as algebra modules, induces a [math]{\bf unitary ~Morita~ equivalence}[/math] between the algebras aside from the usual algebra isomorphisms. Theories defined on Morita equivalent local nets are unitarily equivalent, hence we must consider something like [math]\pi_0 {\bf Rep}_{\mathbb{C}}({\bf vNA}^2)[/math] if we want to properly characterize QFT/QM theories

>>11323392
It will shift complexity space around a little, but certain problems will remain NP. The boldest claim I've heard is from Nayak who says that the traveling salesman will become P with a quantum computer. Other than that it'll just be the issue of teaching people how to design and implement quantum algorithms.
>>11323884
I doubt that, people have already been developing quantum encryption algorithms since the 80's.
>>11323904
In BCS theory, Cooper pairs form between electron states lying at diameterically opposite ends of the Fermi surface, so their states are distinct and aren't subject to Pauli exclusion. Even non-BCS (e.g. FFLO) SC have Cooper pairs labeled by different bands; no violation of Pauli exclusion here at all, and no violation of that will ever occur.
>tremendous energy
Quite the opposite my dear, it takes extremely low energy and temperatures; BCS theory is a weak-coupling theory between electrons due to an effective attraction, typically an electron-phonon interaction. This requires a very "still" tight-binding configuration of the lattice which is easily destroyed by thermal fluxtuations. Once Coulomb takes over the electrons are never getting back.

>>11270142
Regularizing [math]\mu(S) = \sum_k \mu(V)[/math] amounts to regularizing infinite sums of constants, which comes down to evaluating [math]\mu(V)\zeta(0) = -\frac{1}{2}\mu(V)[/math]. Now since the analytic continuation of the generalized [math]\zeta[/math] is regular, other regularization schemes such as Borel would give the same answer.

>>11047636
>What do you mean by "implicit" integration?
I meant that no actual integration has been performed, but only argued that it has the effect of "smearing out" contributions from non-saddle points so we just write down the saddle-point contribution. For equivariant fibre integration on oriented vector bundles, this is the essence of Atiyah-Bott.
>can you say something general about expectations
If you can find sequences of measures [math]d\mu_n[/math] that converges to the Gibbs measure [math]d\mu = D[\phi]e^{S}[/math] then it's definitely possible to regularize expressions like [math]\langle A(\phi)\rangle = \lim_n \frac{\int d\mu_n A(\phi)}{\int d\mu_n}[/math]. However, typically these measures are just better behaved (regularity-wise) and are still infinite-dimensional. You can try to section out finite-dimensional subspaces in the measure, but then convergence is extremely ill-behaved.
In certain special cases such as in CFT, we can again circumvent these difficulties by defining correlations geometrically by inserting primaries [math]\phi_i(z)[/math] into marked points of a Riemannian surface [math]\Sigma[/math]. By expressing the partition function [math]Z(\overline{q},q)[/math] (as a section of a V-bundle on moduli space of Riemann surfaces) in terms of the nome [math]q[/math] of the punctures you open up at the marked points, you can directly evaluate [math]\langle \prod_i \phi_i(z)\rangle[/math] by just "counting Virasoro weights" on [math]\overline{q},q[/math] using the factorization algebra. This is also an "implicit" evaluation similar to the Atiyah-Bott localization.

>>11026231
Completely untrue, especially in infinite-dimensional locally convex spaces.
>>11027197
Two hints:
1. What group rotates rectilinear states into circular, spinning states?
2. What is the double-cover and representation theory of that group?