Abstract It is shown that the interpretation of quantum mechanics as a theory describing systems for which their propositions are always valid or not leads to a contradiction within the theory. The proof does not depend on any specific property of measurements, but only the usual description of ensembles of quantum objects by statistical operators is used.
Abstract It is shown that the interpretation of quantum mechanics as a theory describing systems for which their propositions are always valid or not leads to a contradiction within the theory. The proof does not depend on any specific property of measurements, but only the usual description of ensembles of quantum objects by statistical operators is used.
In recent notes by Einstein, Podolsky and Rosen and by Bohr, attention has been called to the fact that certain results of quantum mechanics are not to be reconciled with the assumption that a system has independently real properties as soon as it is free from mechanical interference. We here investigate in general, and in abstract terms, the extent of this disagreement. When suitably formulated, such an assumption gives to certain types of questions the same answers as does quantum mechanics; this is true of the formulas usually given in discussions of the theory of measurement. There exists, however, a general class of cases in which contradictions occur. That such contradictions are not restricted to the abstract mathematical theory, but can be realized in the commonest physical terms, is shown by the working out of an example..
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.
Within the last decade, significant progress has been made towards a consistent and complete reformulation of the Copenhagen interpretation (an interpretation consisting in a formulation of the experimental aspects of physics in terms of the basic formalism; it is consistent if free from internal contradiction and complete if it provides precise predictions for all experiments). The main steps involved decoherence (the transition from linear superpositions of macroscopic states to a mixing), Griffiths histories describing the evolution of quantum properties, a convenient logical structure for dealing with histories, and also some progress in semiclassical physics, which was made possible by new methods. The main outcome is a theory of phenomena, viz., the classically meaningful properties of a macroscopic system. It shows in particular how and when determinism is valid. This theory can be used to give a deductive form to measurement theory, which now covers some cases that were initially devised as counterexamples against the Copenhagen interpretation. These theories are described, together with their applications to some key experiments and some of their consequences concerning epistemology.
The main theme of this book is the idea that quantum mechanics is valid not only for microscopic objects but also for the macroscopic apparatus used for quantum mechanical measurements. The author demonstrates the intimate relations that exist between quantum mechanics and its interpretation which are induced by the quantum mechanical measurement process. Consequently, the book is concerned both with the philosophical, metatheoretical problems of interpretations and with the more formal problems of quantum object theory. The consequences of this approach turn out to be partly very promising and partly rather disappointing. On the one hand, it is possible to give a rigorous justification of some important parts of interpretation, such as probability, by means of object theory. On the other hand, the problem of the objectification of measurement results leads to inconsistencies which cannot be resolved in an obvious way. This open problem has far-reaching consequences for the possibility of recognising an objective reality in physics.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.