#005: Find a better mathematical characterization for quantum states

Category: Quantum mechanics - Tags: Vector spaces, Function spaces

Find a better physically motivated characterization for the state space of quantum mechanics.

Mathematical problem. The standard way to represent quantum states is using vectors in Hilbert spaces. The requirements for a Hilbert space can be broken up into the following components:

  • Vector space
  • + normed (normed vector spaces)
  • + complete under the norm (Banach space)
  • + every closed linear subspace is the range of a projection (Hilbert space) (see here) The completion under the norm seems the only non-physical requirement.

On the other hand, the Schwartz_space is a dense subset of \(L^2\), which seems better suited as:

  • It is closed under Fourier transform
  • It has finite expectations for all polynomials of position and momentum
  • It is dense over \(L^2\) In fact, requiring the second feature means recovering the Schwartz space.

Does the Schwartz space satisfy all other characteristics of a Hilbert space, except the closure under the norm? If we just drop the closure under the norm, what do we lose?

Physical significance. If we look at the list of defining properties of a Hilbert space, completeness is the only one that does not make physical sense. The linearity of the normed vector space can be understood as coming from linearity of probability space; the existence of projections is the requirement of being able to identify states (i.e. a measurement that outputs 1 if the state matches and 0 if it doesn’t). The completeness would mean that the limit of a sequence of state preparation always leads to a state, but this is not the case: we can (in principle) prepare states with narrower and narrower spatial distribution, but not zero spatial distribution (delta Dirac). The idea would be that the requirement of completeness makes the mathematical space nicer to work with, but not physically meaningful.

Notes. Even listing all problems with Hilbert spaces may be useful.

For example, a related problem with Hilbert spaces (\(L^2 specifically\) see Adrian Heathcote - Unbounded operators and the incompleteness of quantum mechanics) is that self-adjoint operators that are defined on the whole space must be bounded. This means that unbounded operators must not be defined on all states: there must be states for which the average position is not well-defined. Again, note that this is not true for the Schwartz space: the position is unbounded (there is no maximum value for position) but all states have a well defined average position.