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  Is there a physical significance to non-normal states of the algebra of observables?

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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?

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user ACuriousMind

asked Nov 2, 2015 in Theoretical Physics by ACuriousMind (910 points) [ revision history ]
edited Mar 3, 2016 by Dilaton
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...

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user Valter Moretti
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...

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user Valter Moretti
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.

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user Valter Moretti
@ValterMoretti: Maybe that different notion of normal state is part of why I seem to have difficulty in nailing down their meaning - people are talking about two different things! If I'm understanding you correctly, are you saying in the curved spacetime case that we would not be able render both vacua as density operators on the same Hilbert space, thus making the generalization to non-normal states of the algebra necessary?

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user ACuriousMind
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.

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user Valter Moretti
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.

This post imported from StackExchange Physics at 2016-03-03 10:23 (UTC), posted by SE-user Urgje

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