/sci/ - Science & Math

>>11552397
No... the Bloch sphere is literally just a sphere, and is not the main ingredient in TQC. One may use the Hopf fibration to describe circle bundles on the sphere, the sections of which describe (compactified) quantum states. Just a stationary qubit is given by sections of a line bundle over a single point, there's nothing topological that can happen here, you actually need your qubit to move on a sphere in spacetime
Besides, quantum states are described via the trivial fibration (i.e. not Hopf) unless the Pontrjagyn invariant [math]\mu[/math] is non-zero.Non-trivial topology occurs only when the wavefunction is multi-valued across a great circle, which occurs when there's a monopole enclosed.

>>11283612
How is "the universe just [math]happens[/math] to settle in one ground state out of hundreds of other degenerate states" satisfactory to you, sweetie?
Besides, the topological LSM theorem states that - unless some LSM anomaly is non-trivial - degenerate ground states in the SRE phase is either gapless or hosts fractional excitations. Now as string theory (as a CFT) have discrete spectra by virtue of the Viasoro reps of the underlying affine Lie algebra, the ground state is automatically gapped; coupled with the fact that fractionalized anyons don't exist in dimensions higher than 2, this implies that the LSM anomaly is non-trivial. Topological anomalies are typically more energetically costly to produce, so you better have a good explanation for why the universe settled down in such a topological false vacuum if you want to advocate for string theory as a ToE. Else you're literally blowing smoke up your own bum.

>>11013193
>the spectrification functor does not commute with the CW-approximation functor

>>11007523
What? Who's talking about girls?

>>10164470
Literally just use the universal property

>>10134838
>[math]j^2 + k^2[/math]
That's wrong, unless you meant to compute [math]\operatorname{dim}(L(W_1) \oplus L(W_2))[/math]. In general the two bases can overlap and you'll need to compute [math]\operatorname{dim}\operatorname{ker}P_qP_2[/math] where [math]P_{1,2}[/math] are projectors onto [math]W_{1,2}[/math].

>>10111362
Mountainpass theorem
>he ACTUALLY has never done any function analysis
Jesus

>>9789606
>Well then you are clearly retarded.
Coming from a complete idiot like yourself? That's a compliment.
>Well, there is no absolute definition of a configuration manifold
Yes there is. A configuration space is the space on which particles move without the diagonals (i.e. coinciding positions), if the particles are indistinguishable. This is NOT the symplectic manifold by the way, so you using retarded terminology that is literally irrelevant to the convo just makes you look worse, if such a thing is even possible.
>"any general submanifold in 3N dimensional coordinate space".
THIS is what you think a configuration space is? LMFAO holy shit.
>I don't know how someones who claims comprehension cannot even see a simple concept.
I cannot see the retarded terminology you're using, not the concept. If you actually learn the material instead of skimming shit you'd know the right words to use. Until then you can sit in your tard corner.
>the derivative of this map defines a function to the fucking tangent bundle
And you call this a pairing? LMFAO you're literally just evaluating the tangent vector at some point, this is by no means a pairing. Again, learn the right words, dilettante.
>Who the falk talked about the hamiltonian?
You did. The Hamiltonian formalism is built FROM a Hamiltonian [math]H\in C^\infty(M)[/math] as an a priori given. Fuck's sake it's like talking with a neckbeard undergrad who wears a Schrodinger's cat shirt.
>The lagrangian is defined however the fuck you want
LMFAO. The entire point of symplectic geometry is to put analytical mech on rigorous grounds. If the Lagrangian isn't defined in those terms then it is NOT analytical mechanics. Are you actually this fucking dense?
>So you see exactly
I see exactly that the reason I gave completely breaks what OP was trying to do, since he HAS to change the Lagrangian in order for it to admit what he drew as a extremizer. Are you actually this stupid?

>be literally the funniest guy in the dept
>still no bf