On a formal difference between the individual and statistical interpretation of quantum theory

Dipankar Home, M. A. B. Whitaker

Miguel Ferrero, Víctor Gómez Pin, J. L. Sánchez‐Gómez

The preparation procedure, an undefined notion in quantum theory, has not had the relevance that it deserves in the interpretation of quantum mechanical formalism.Here we utilize the concepts of identical and similar preparation procedures to show the conceptual differences between the statistical and the conventional interpretation of quantum formalism.Although the statistical understanding, and its final logical consequence, hidden variables theories (this connexion being explained in the text), have a great intuitive appeal due to its fewer ontological difficulties, both recent experimental results and theoretical developments seem to support an epistemic alternative closer to the conventional one.Nevertheless, we must not rule out the possibility that new theorems or new explanatory principles may modify the reigning supremacy of this interpretation. 1.-Introduction.Although the conventional interpretation of the quantum mechanical formalism (CIQM), very close to the Copenhagen interpretation, is still dominant (is used in almost all textbooks), there are physical situations that ask for an interpretation of the quantum mechanical formalism (QMF) in terms of the Statistical Interpretation of Quantum Mechanics (SIQM) or even for hidden variables theories (HVT).The SIQM and HVT are different research programs, but they are closely related, as we will see further below.To introduce properly the distinctions between the three previous acronyms we will ask an old and apparently simple question.Is the wave function associated to an individual system or to a statistical ensemble?A first tentative answer could be formulated as follows.QMF introduces probabilities, and probabilities are related to statistical ensembles.Otherwise, the same concept of probability would lose its meaning.In QM, if it is said that you would get some concrete result with 0,63 probability it means that if the experiment is carried out 100 times, that concrete result would appear approximately 63 times.Hence, the obvious answer will be that the wave function should be related with statistical ensembles.However, despite its intuitiveness this idea does not capture completely what we want to express.The real problem to analyze could be best introduced with the following example.Imagine that we prepare identically two radioactive atoms with half-life of 30 minutes.By preparation we understand all those theoretical methods and physical procedures necessary to fully characterize the state of the system.It is, thus, a macroscopic reproducible act that lefts the system, without destroying it, in a concrete state after the procedure.In quantum theory, this preparation is represented by a vector state (or a density matrix).Identically in this context will mean identity of procedures, that is that the experimenter has repeated all previous operations, so that there is no difference whatsoever between them.Note, however, that identical preparations are not necessarily followed by identical results, so there is

V. F. Lenzen

Christian DE RONDE, Graciela Domenech, Federico Holik +1 more

The concept of individuality in quantum mechanics shows radical differences from the one used in classical physics. In particular, it is not possible to consider the fundamental particles described by quantum theory as individual distinguishable objects. In this paper we present arguments in favor of quantum non-individuality, which -in addition to those based on quantum statistics- relate to the Kochen-Specker theorem and the principle of superposition. Then, we analyze the possibility of referring to 'possible individuals' instead of 'actual individuals', and show that the Modal-Kochen-Specker theorem precludes this interpretational move.

Peter Mittelstaedt

Peter Mittelstaedt

Historical remarks The quantum mechanical formalism discovered by Heisenberg [Heis 25] an Schrödinger [Schrö 26] in 1925 was first interpreted in a statistical sense by Born [Born 26]. The formal expressions p (φ, a i ) = |(φ,φ ai )| 2 , i ∈ N, were interpreted as the probabilities that a quantum system S with preparation φ possesses the value a i that belongs to the state φ ai . This original Born interpretation, which was formulated for scattering processes, was, however, not tenable in the general case. The probabilities must not be related to the system S in state φ, since in the preparation φ the value a i of an observable A is in general not subjectively unknown but objectively undecided. Instead, one has to interpret the formal expressions p ( a i , a i ) as the probabilities of finding the value a i after measurement of the observable A of the system S with preparation φ. In this improved version, the statistical or Born interpretation is used in the present-day literature. On the one hand, the statistical (Born) interpretation of quantum mechanics is usually taken for granted, and the formalism of quantum mechanics is considered as a theory that provides statistical predictions referring to a sufficiently large ensemble of identically prepared systems S (φ) after the measurement of the observable in question. On the other hand, the meaning of the same formal terms p (φ, a i ) for an individual system is highly problematic.