Infinite dimensional spaces in quantum mechanics 

Video molto interessante, non avevo mai approfondito la questione nei miei studi ma questi dubbi effettivamente erano sempre un po' presenti, alcuni concetti come la "pseudo-ortogonalità" dove si usano vettori di stato che non sono in realtà appartenenti ad L^2 ma per i quali si usa, in sistemi continui, ugualmente il prodotto interno di questo spazio sono sempre stati grandi punti interrogativi. Vederli ritornare fa riaffiorare molta curiosità!

In any case, as Jacob Barandes often talks about, we do not live in a Hilbert space, the Hilbert space formalism for QM is gauge redundant so cannot be physical. In other words, it's just a model. One of many for QM. The algebraic GNS framework is a nice alternative, but still only a model, so best not take it too literally, hence be cautious about demanding that is is "physical". Models do not have to be physical, we want them to be mostly physical in correspondences --- but we do have to know, or try to know, when they aren't.

great video ! came here as a mathematician trying to figure out the point of braket notation when the only place ive seen it is in quantum computing papers that almost exclusively use finite-dimensional spaces only to come away with a much better understanding of physical principles ! pleasantly surprised by the amount of analysis involved in physics :)

@16:00 you can make the sequence of gaussians Cauchy by taking the variances to zero so that their ratio approaches 1.

Whelp in this case you're right... this did make me more confused. The things I did notice: 1. Can't the normalization condition set the criteria for which infinite sums are allowed? 2. A bit of confusion on the cauchy sequence and distance function: if our sequence is a sequence of orthogonal basis vectors ψ1,ψ2,ψ3,..., then the only distance we'll get is either 1 or 0. How can this be reconciled with the distance after index n getting smaller and smaller? 3. On 7:31 to 9:15 wouldn't it make perfect sense for the expectation value of particle number to be infinite since for the number state |i>, i itself is taken to the limit of infinity? 4. Physically, can't the previous problem be remedied by saying that the theory is valid only to a finite extent? like if expectation value of momentum is infinite then there is a momentum eigenstate (plane wave) with infinitesimally short wavelength... but this can only work if space can be divided to infinitely small parts. I think the same thing apply to harmonic oscillator eigenstates (Hermite polynomials) if we take the angular frequency (thus the quantum number) to infinity. 5. Near the end the term "quantum state tomography" is mentioned. If we can construct the wavefunction from the set of measured operator eigenvalues (experiment repeated many times) would that imply the wavefunction actually being physical? 6. If the Dirac delta is not included in Schwartz (and Hilbert) space, what is the worst implication for physics? I imagine Green's function method to construct wavefunctions could be problematic since it needs a sharp impulse. 7. I got a feeling somehow this will connect to renormalization, and the "field-ness" of QFT Pretty sure I got some assumptions wrong, but not sure which ones.

Thanks ! Do you know if someone already built a mathematically rigorous quantum theory on tempered distributions space ? To be honest I do not care as you seem to care about having a theory where every mathematical object is mapped on a physical object. It is like phase velocity VS group velocity. Phase velocity does exist mathematically in the theories but does not really map onto something physical. I appreciated your video (and subscribed).

Great ！👍👍👍