/sci/ - Science & Math

>>10340609
The Cartan model was actually from Henri, Elie's son.
>>10349821
Fredholm operators are homotopically classified by K-theory via Atiyah-Janisch, so it's an alg-top object. KK-theory is a generalization of K-theory
through this perspective.
To answer your question, in symplectic geometry the Hamiltonian [math]H\in C^\infty(M)[/math] including the potential is first given on a symplectic manifold [math](M,\omega)[/math], then from which a Hamiltonian vector field [math]X_H\in TM[/math] satisfying [math]dH + \omega(X_H,\cdot) = 0[/math] is constructed. This vector field generates a flow [math]\gamma:\mathbb{R}\rightarrow M[/math] satisfying [math]\dot{\gamma} = X_H(\gamma)[/math] as physical trajectories.

>>10296282
Collapse is by definition the process by which a state thermalizes with a macroscopic system. It has been proven that, given any [math]\epsilon >0[/math] and a Hamiltonian [math]H = H_A\otimes I + I\otimes H_B + \epsilon H_\text{int}[/math] describing the dynamics of states in a Hilbert space [math]\mathcal{H}_A\otimes\mathcal{H}_B[/math], the density matrix [math]\rho_A[/math] for the subsystem time evolves into a classical state (i.e. one with only diagonal elements) at some time [math]0<T<\infty[/math].
>I know that some larger (relative) objects behave like waves/non collapsed objects
Macroscopic states are formed via condensation, from e.g. the BEC or the BCS mechanism. There's a reason they're called "condensed": even some very small thermal fluctuations are enough to destroy it.
>This basically happens with entanglement right?
No. Please stop posting until after you've read quantum stat mech textbook.

>>10296282
A non-interacting state cannot collapse, and collapse inevitably happens when the state is thermalized with a macroscopic system. It has been proven that, for any [math]\epsilon >0[/math] and a Hamiltonian [math]H = H_A\otimes I + I\iotimes H_B + \epsilon H_\text{int}[/math] describing the energies of states in the Hilbert space [math]\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B[/math], there exists [math]T>0[/math] such that the subsystem's density matrix [math]\rho_A[/math] time-evolves into a classical state, namely it only has diagonal elements. Any interaction will causetime
>I know that some larger (relative) objects behave like waves/non collapsed objects
Macroscopic states are formed from condensations, such as the order parameters of BEC state or the BCS superconducting state. There's a reason why it's called "condensation", small thermal fluctuations will destroy it, we don't even need a heat bath [math]\mathcal{H}_B[/math]. The fact that the state has macroscopic occupancy has nothing to do with its quantum coherence.
>This basically happens with entanglement right?
No. Please stop posting until you've read textbooks on quantum stat mech.

>>10263180
>singular/simplicial homology as counting the number of "holes" in each dimension
No, the ranks of the homology groups do that, not the groups themselves, and it's one-half of the rank that tells you the number of holes. If you think about CW complexes, using Hurewicz's homomorphism you can consider [math]H_1[/math] as an Abelianization of [math]\pi_1[/math].
For the projective plane we know [math]\mathbb{R}P^2 = S^2/\mathbb{Z}_2[/math] where [math]\mathbb{Z}_2[/math] is the group acting on the sphere by antipodal inversion [math]x\mapsto -x[/math]. Even though all loops on [math]S^2[/math] are contractible, the path from one point to its antipode will descend to a loop on [math]\mathbb{R}P^2[/math]. This is why you only have two loop classes in the projective plane, which lift to a trivial loop and an antipode path, respectively, on the sphere. Also since [math]S^2[/math] is simply connected and [math]\mathbb{Z}_2[/math] acts freely and properly discontinuously on it, we know [math]\pi_1(S^2/\mathbb{Z}_2) \cong \mathbb{Z}_2[/math].
>The whole thing seems incredibly unmotivated to me
Not at all. De Rham cohomology is extremely well motivated in physics, specifically classical EM. Not to mention topological quantum anomalies are classified by homology groups on spin [math]G[/math]-bundles via the Atiyah-Singer index theorem.

>>10232629
I doubt it will. It's an ever-developing field after all, and one that's getting more and more attention what with all the applications of gauge theory to pure topology.

>>10229644
>At one point I thought I wanted to do mathematical physics
So do you wish to study actual physics now? Try reading Nakahara (GR), Wen (condmat) and Bertlmann (TQFT), they're quite friendly to math-oriented people.

>>10200429
You're almost there.
Define projectors. The fact that [math]G[/math] acts "properly discontinuously" (for a lack of a better term) on [math]V[/math] means that you can mod out the action of [math]G[/math] on [math]V[/math] to get two orbits, one is [math]K(0)[/math] and the other is [math]G/K(0)[/math]. Then it suffices to show that this projection [math]\pi: V\rightarrow V/G[/math] is orthogonal.

>>10185754

>>10147915
Geometric topology