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martinbn said:What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.

In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this representation is different (in the sense of non-isomorphic). The Lie algebra defines the kinematics of the system of interest. The semidirect product of ##(2dN+1)##-dimensional Heisenberg algebra with ##N## copies of ##so(d)## has a unique irreducible unitary representation, which defines the Hilbert space of ##N## particles in ##d##-dimensional Euclidean space.

Hilbert spaces don't have a classical limit. The latter is restricted to linear operators, which may have one. To have a classical limit, the above representation must depend on Planck's constant hbar in such a way that ##i[A,B]/\hbar## tends (at least for ##A## and ##B## in the Lie algebra of preferred observables) to a finite limit ##\{B,A\}##, which represents a Poisson bracket.

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