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Repeating the question on the title: Are Hilbert spaces uniquely defined for a given system?

I started to think about this when I was reading about Schrödinger and Heisenberg pictures/formulations. From my understanding, you can describe a system analyzing the time dependent state defined by [tex]\left\vert \psi \left( t\right) \right\rangle =U\left( t,t_{0}\right)\left\vert \psi \left( t_{0}\right) \right\rangle[/tex], while keeping the observables time-independent (wich would be the Schrödinger picture). And this is totally equivalent to description of the system by a time-independent state, and observables defined by [tex] A^{\prime }\left( t\right) =U^{\dagger }\left( t,t_{0}\right) AU\left(t,t_{0}\right) [/tex]. (Heisenberg picture) However, I don't think it's the same Hilbert space! In the first one, the elements of the space are time dependent, while in the other they're not.

Am I making any sense? :uhh:

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# Are Hilbert spaces uniquely defined for a given system?

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