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Of course, you could try to perform instead a "big" direct sum / integral of all these disjoint Hilbert spaces and use that as your state space receptacle, but this space (if it can be constructed at all in this fashion) is bound to be also technically extremely unwieldy to use. If you need to work with disjoint states in the above sense, is technically and conceptually better to work instead with the algebraic concept of state: as a certain kind of linear functional on the algebra of local observables of the theory and perform the GNS construction with respect to a convenient reference state (e.g. the vacuum or more generally a thermal equilibrium state with respect to some Lorentz frame) when a concrete Hilbert space is needed. All the information you need for that is already in the algebra of local observables. In other words, you no longer treat all states of the theory as belonging to a single Hilbert space but only look at subspaces of "mutually locally accessible" states, so to speak. These subspaces are usually called sectors of the theory. Under fairly general assumptions such as the ones in the Wightman-Garding formalism, sectors are always separable.

Of course, you could try to perform instead a "big" direct sum / integral of all these disjoint Hilbert spaces and use that as your state space receptacle, but this space (if it can be constructed at all in this fashion) is bound to be also technically extremely unwieldy to use. If you need to work with disjoint states in the above sense, is technically and conceptually better to work instead with the algebraic concept of state: as a certain kind of linear functional on the algebra of local observables of the theory and perform the GNS construction with respect to a convenient reference state (e.g. the vacuum or more generally a thermal equilibrium state with respect to some Lorentz frame) when a concrete Hilbert space is needed. All the information you need for that is already in the algebra of local observables. In other words, you no longer treat all states of the theory as belonging to a single Hilbert space but only look at subspaces of "mutually locally accessible" states, so to speak. These subspaces are usually called sectors of the theory. Under fairly general assumptions such as the ones in the Wightman-Garding formalism, sectors are always separable. Another advantage of the algebraic notion of state is that it incorporates both pure and mixed states into a single concept.

All seems well and good, but does the reconstruction theorem recover all physical states? The answer is unfortunately no, not even within finite but arbitrary accuracy. A typical example of a state not accessible from the vacuum state by means of local operations within arbitrary precision is a thermal equilibrium state of finite, nonzero temperature with respect to some Lorentz frame. The same can be said about any state reached from the latter state by means of local operations - they live in "disjoint" Hilbert spaces. Moreover, there are theories where the vacuum state is not unique, and with distinct vacua not locally accessible from each other in the above fashion. It is by no means clear whether all these states fit into an infinite tensor product of Hilbert spaces such as the the one in the OP or not.

All seems well and good, but does the reconstruction theorem recover all physical states? The answer is unfortunately no, not even within finite but arbitrary accuracy. A typical example of a state not accessible from the vacuum state by means of local operations within arbitrary precision is a thermal equilibrium state of finite, nonzero temperature with respect to some Lorentz frame. The same can be said about any state reached from the latter state by means of local operations - they live in "disjoint" Hilbert spaces. Moreover, there are theories where the vacuum state is not unique, and with distinct vacua not locally accessible from each other in the above fashion. It is by no means obvious whether all these states fit into an infinite tensor product of Hilbert spaces such as the the one in the OP or not. If the theory is interacting, this is even less clear since (as mentioned in the previous paragraph) it may not be representable at all as a continuum of oscillators.

The "infinite tensor product" picture may be useful as a sort of concrete image of the state space of a quantum field theory, but in practice is rarely used because of the technical difficulties it brings along. Moreover, the paragraph from Streater and Wightman you quoted could be read as saying that infinite tensor products of Hilbert spaces could be thought of as a bigger mathematical "receptacle" for the actual "natural state space" of the theory, but even that may not be actually tenable.

The "infinite tensor product" picture may be useful as a sort of concrete image of the state space of a quantum field theory, but in practice is rarely used because of the technical difficulties it brings along. Moreover, the paragraph from Streater and Wightman you quoted could be read as saying that infinite tensor products of Hilbert spaces could be thought of as a bigger mathematical receptacle for the actual "natural state space" of the theory, but even that may not be tenable.