Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a density operator formalism where a quantum state is given by a density operator $\rho : \mathcal{H}\to\mathcal{H}$ on some Hilbert space $\mathcal{H}$ where $\rho$ is self-adjoint, positive semi-definite and is trace-class with unit trace.

However, in an alternative approach, one starts with the abstract $C^*$-algebra $\mathcal{A}$ of observables and calls any positive linear functional on $\mathcal{A}$ a state. A normal state is then one that can be represented by a density operator on some Hilbert space on which the algebra is represented (there are several equivalent definitions, but this one is the one which makes it obvious that the normal states are the usual pure+mixed states of quantum mechanics).

But, in general, there exist non-normal states at least on some admissible algebras of observables. One can find several vague statements about what those may or may not mean, but I have not yet found a unique satisfactory answer to the question:

Do the non-normal states of the algebra of observables have physical significance, that is, is there an actual quantum system where only considering the normal states leads you to different physical predictions than taking all states?

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    $\begingroup$ Consider a pair of unitarily inequivalent representations of a given $C^*$-algebra induced by a pair of pure algebraic states by means of the GNS construction. Each one of these states is not a normal state for the other GNS representation. (Otherwise it would be represented by a cyclic unit vector in the Hilbert space of the other rep. and the two reps would be unitarily equivalent via the identity operator). Unitarily inequivalent reps arise in several contexts in QFT...Spontaneous breaking of symmetry, for instance, leads to unitarily inequivalent reps... $\endgroup$ – Valter Moretti Nov 2 '15 at 15:49
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    $\begingroup$ In curved spacetime, consider a spacetime which is flat both in the future and in the past of a given Cauchy surface, but including a curved region. The $*$-algebra of the fields is fixed but there are two natural pure states on that algebra: the Minkowski vacuum of the future and the Minkowski vacuum of the past. In general, depending on the curvatures, they are not unitarily equivalent, each one is not normal for the representation of the other of the same algebra of quantum fields... $\endgroup$ – Valter Moretti Nov 2 '15 at 16:04
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    $\begingroup$ Actually there is another possiblity, different from the two cases I mentioned above, referring to non-normal states with respect to a von Neumann algebra. For a given C*-algebra, fix a state, represent the C*-algebra in the GNS Hilbert space and take the generated von Neumann algebra therein. There exist algebraic states for that C* algebra (the von Neumann algebra interpreted as a C* algebra), which are not normal. Maybe you were referring to this other notion of non-normal state. $\endgroup$ – Valter Moretti Nov 2 '15 at 16:12
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    $\begingroup$ Yes, in curved spacetime the picture is the one you described. There are states which cannot be represented in the same Hilbert space in the standard way. However, there is a class of states, called Hadamard states, which are locally unitarily equivalent i.e. referring to the subalgebras of observables restricted to bounded spacetime regions. $\endgroup$ – Valter Moretti Nov 2 '15 at 16:30
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    $\begingroup$ Non-normal states can occur if a thermodynamic limit is taken for instance for a system of particles in a box. Another case is that of scattering of Schródinger particles in the presence of bound states. $\endgroup$ – Urgje Nov 2 '15 at 20:15

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