DarMM said:
Sorry, that should read the Weyl algebra. The generators of the Weyl algebra might have no representation on a non-separable Hilbert Space. In other words, there are no strongly continuous representations of the Weyl algebra on a non-separable Hilbert Space.
I realized that this explanation might be seen as somewhat vague, so I thought I would expand on it. The Weyl Algebra can be understood in many ways, but for quantum mechanics it is essentially any combination of unitary operators representing translations in position or momentum space.
The generators of the Weyl algebra are the position and momentum operators themselves, or in quantum field theory they are the field and its conjugate momentum field.
So, on a non-seperable Hilbert space it is possible for the field/position operator to be ill-defined, which would make calculations extremely difficult. Of course we "know" that position is a well-defined observable, so we can ignore non-seperable Hilbert spaces when it comes to quantum mechanics.
In fact if you do quantum mechanics from the abstract algebraic point of view you can actually prove that most temporal automorphisms (jargon word in algebraic quantum theory for time evolution) have GNS representations* as self-adjoint operators on a seperable Hilbert space.
In field theory non-seperable representations can occur for thermal states.
*Algebraic field theory views the operators as the primary objects and states are viewed simply as linear maps from the operators to their expectation values. The GNS theorem is a fundamental result which states that if you take some algebra of operators:
\mathcal{A}
and a state on that algebra:
\rho
defined as a map from elements of the algebra to their expectation values:
\rho(A), \quad A \in \mathcal{A}
Then the action of this state on this algebra can always be replicated by a ket in some Hilbert space and bounded operators on that Hilbert space. Specifically we can always find:
\Psi_{\rho}, \quad \mathcal{H}_{\rho}, \qquad \Psi_{\rho} \in \mathcal{H}_{\rho}
that is a Hilbert space and a vector element of that Hilbert space,
and we can also construct a map:
\pi : \mathcal{A} \rightarrow \mathcal{B}(\mathcal{H}_{\rho})
mapping the algebra into the bounded operators over that Hilbert space, such that:
\rho(A) = \left(\Psi_{\rho}, \pi(A) \Psi_{\rho}\right)_{\mathcal{H}_{\rho}}
This is the GNS theorem.
In other words, using Hilbert spaces becomes a theorem in the Algebraic approach. The GNS theorem proves that Hilbert space, their elements and their operators, can be used as tools in computing maps on the algebra of observables.
Now of course often several different states result in the same \mathcal{H}_{\rho} You say that such states are in the same folium. Time evolution can only move you around inside a given folium, which is why in "normal" quantum mechanics and field theory we only work with Hilbert spaces, time evolution always keeps you inside one folium.
However for deeper issues in field theory, for instance if you want to look at exactly what is the relationship between a free and interacting theory then you need to look at different folia at the same time, which the standard formalism is very bad at doing.
