In quantum mechanics, the evolution of systems is described by unitary transformations in Hilbert spaces. When the number of particles of a system is large, the dimensionality of the corresponding Hilbert space turns uncountable, so the Hilbert space in non-separable. It is usually assumed that this makes no difference as compared to the case where the Hilbert space has countably many dimensions. But is this true? Actually the case of unitary transformations in uncountable dimensions might be trickier than what might be extrapolated from the countable case. Are there known results on such unitary transformations in uncountable dimension/non-separable Hilbert spaces?

(Added below for motivation and claity)

The question relates to quantum measurement. As you might know, the description of quantum systems distinguishes two different kinds of evolutions:

  1. when the system is not observed, time evolution of its state is a continuous unitary transform in its Hilbert space of states

  2. when the system is observed through a measurement, in which case the evolution is a discontinuous projection of the state of the system into a eigensubspace of the observed quantity that is chosen randomly.

What happens during a measurement seems to be that the relevant space of states changes from a well behaved Hilbert space to a much bigger, non-separable one through interaction with the measurement device's own space of states. Yet we physicist would like to keep something that includes unitary evolution even for large systems such as measurement devices (Okham...). So the question could be reworded

"Is there in non-separable spaces a notion of transformation that could have either a unitary or a projection aspect, and on what would this aspect depend?"

  • 2
    $\begingroup$ It's hard to answer this specifically. Unitary operators on inseparable Hilbert Spaces will indeed share some properties with unitary operators on separable Hilbert Spaces, but there will be some differences. Without knowing specifically which properties you're interested in, it's hard to comment on how appropriate this assumption of separability actually is. $\endgroup$
    – user754697
    Mar 3, 2020 at 3:13
  • $\begingroup$ See (Added below for motivation and claity) to specify slightly what I meant. $\endgroup$
    – Mathias
    Jul 15, 2020 at 2:46
  • $\begingroup$ Well - the basic need is to see whether there would be non unitary solutions (esp. protective ones) to Schrödinger equation in a Hilbert space built as the complete tensor product ( à la von Neumann) of an infinite number of separable Hilbert spaces. $\endgroup$
    – Mathias
    Aug 2, 2020 at 7:17


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