Stone's theorem & construction of observables - L12 - Frederic Schuller

This is from a series of lectures - "Lectures on Quantum Theory" delivered by Dr.Frederic P Schuller

[@Bignic2008]

1:32:30 I live for the moment where his writing fits perfectly on the pre-drawn underline.

[@mashedpotatoez99]

At 1:15:09 to be fully explicit, it should be noted that \psi(t)=0 only implies U(t)\psi = exp(-itA** )\psi. i.e thus far this equality is only known for \psi in the domain D_A (and for all t). But of course since D_A is dense in the Hilbert space and we have two *bounded* operators (U(t) and exp(-itA** )) which agree on this dense subset, it follows they are equal on the entire Hilbert space.

[@active285]

I love his subtle humour!

[@madarauchiha9805]

Danke Sir,

[@abrlim5597]

I was expecting the explanation of why the operators look like how they look like. But then it is the Stone-von Neumann theorem gives the explanation, not Stone's theorem. A bit disappointed. Anyone knows where I can find the proof of the Stone-von Neumann theorem? Thank you

[@productivelb]

At 34:26 won't Psi(t)/t be in neighbourhood N but not necessarily Psi(t). In particular Psi(t0)/t0 will be in N but not Psi(t0) itself as it is off by a factor of 1/t0. How did we conclude Psi(t0) is in N?

[@rifathkhan9995]

Yes. I think he made a error. But it is easily fixable. Psi(t0)/t0 is what lies in N and in D_A. Just replace everything he says about Psi(t0) to Psi(to)/to

[@alexandrestehlick4929]

Sir, I think there is a problem with the the “path ψ_τ” created at 32:45 . ψ_τ is not close to ψ, but close to 0. In fact, next, when you extract the consequence that ψ_τ/τ goes to ψ as τ goes 0, it implies that ψ_τ is actually going to 0. If you want to use ψ_τ/τ as the element close to ψ, then should compute if the derivative exists at ψ_τ/τ, not at ψ_τ. I checked the book of Teschl, and indeed he writes the same step. However, Brian Hall proves in a different way.

[@carlesv1488]

Hi Sith Sobolev. You are right. What is arbitrarily close to \phi is \phi_tau/tau. But this does not invalidate the proof nor the reasoning. After all \phi_tau/tau is simply \phi_tau multiplied (I should say re-scaled) by the real number 1/ tau.

[@mathjitsuteacher]

I don't even understand the integral he builts. He integrates U(t)(psi) from 0 to tau. How does one define the integral of a function from IR into H?

[@rifathkhan9995]

Yes. I think he made a error. But it is easily fixable. Psi(t0)/t0 is what lies in N and in D_A. Just replace everything he says about Psi(t0) to Psi(to)/to

[@wdlang06]

1:09

[@tomassuleiman9986]

17:40 isn't U an element of the group and not the group itself??

[@tomassuleiman9986]

I think as t is not fixed, U(.) may be the entire group. If someone know a little more about it i will be grateful :)

[@wdlang06]

which uni is this?

[@danideboe]

Erlangen

[@StephenCrowley-dx1ej]

The chalk no longer works