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On the Concept of State in Quantum Mechanics: Another Way to Decoherence?


According to a formulation of the quantum theory presented by Rudolf Haag and Daniel Kastler more than half a century ago, the observables of a system, rather than the states, are the primary objects. In this paper we assume that only those observables that do really exist as measurement instruments determine the properties of the states. If this is the case, states that would be pure if all the self-adjoint operators corresponded to existing instruments may turn out to be statistical mixtures. This proposal, although not free from disconcerting implications on the very concept of state, may entail a new way of looking both at some of the most popular “paradoxes” and, more significantly, at the measuring process.

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  1. The ideas expressed in this note are already present in [2].

  2. Due to the probabilistic nature of quantum mechanics, it must always be assumed the possibility of preparing many copies of the system in the same way: if all the copies are identical we say that the system is in a pure state, say |a〉; if we have ni copies of the system in the state |ai〉, we say that the system is in the statistical mixture \(\big \{|{a_{1}}\rangle ,p_{1}=\frac {n_{1}}{n};\ |{a_{2}}\rangle ,p_{2}=\frac {n_{2}}{n};\cdots \big \}\), \(n={\sum }_{i} n_{i} \).

  3. The ‘laboratory physics’ assumption differs from the algebraic approach of Haag-Kastler: there, the local algebras \(\mathfrak {A}(\text {B)}\) are the set of all observable in the bounded space-time region B, instead the ‘laboratory physics’ assumption has to do only with the observables (instruments) actually existing in a laboratory and these do not necessarily constitute an algebra.

  4. In the example considered no observable has nonvanishing matrix elements between |r〉 and |l〉 and, as a consequence, all states α|r〉 + β|l〉 are statistical mixtures. If instead both T1 and T3 (but not T2) were observables, not all such states are mixture: states with α and β real are pure, while α|r〉 + iβ|l〉 are statistical mixtures.

  5. The correct formulation of Schrödinger’s cat paradox should include not only the state of the cat, but also that of the ampoule of cyanide (|broken〉,|not broken〉) or, equivalently, the decaying radioactive atom. This was pointed out in [2]. See also [7].

  6. The concept expressed by the term decoherence finds an ante litteram application, long before the term itself was invented, see [10].


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Correspondence to Luigi E. Picasso.

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Picasso, L.E. On the Concept of State in Quantum Mechanics: Another Way to Decoherence?. Int J Theor Phys 62, 46 (2023).

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