/sci/ - Science & Math

>>8755642
Interesting.
A wavefunction [math]\psi[/math] in quantum theory can be considered as a Lie algebra-valued section on the Hermitian G-line principal bundle [math]G \rightarrow P \rightarrow M[/math], and therefore [math]\psi \in \Psi^{k}(X,\operatorname{Lie}G)[/math] if we can write [math]M = X^k[/math], such as the case for the [math]k[/math]-body Euclidean space [math]M = (\mathbb{R}^n)^k[/math] or the torus [math]M = (S^1 \times S^1)^k[/math]. The diagonals in these manifold can be considered as the points at which each of the [math]k[/math] particles have identical positions, which are forbidden by Fermi statistics if we're interested in electrons. This means that either we have to exclude the diagonals [math]\Delta[/math] from [math]M[/math] or identify any (Fermi) wavefunctions that agree on the diagonal. Is there a way to extend Alexander-Spanier cohomology to global equivalence, where I define my equivalence classes of [math]k[/math]-functions if the representative coincide on the entire [math]\Delta[/math] instead of just an open subset? How would that change the cohomology structure?
If we are able to do this then we may be able to extrapolate [math](k+1)[/math]-body quantum properties from what we know about [math]k[/math]-body systems with exact sequences.