We prove a uniqueness theorem showing that, subject to certain natural constraints, all 'no collapse' interpretations of quantum mechanics can be uniquely characterized and reduced to the choice of a particular preferred observable as determine (definite, sharp). We show how certain versions of the modal interpretation, Bohm's 'causal' interpretation, Bohr's complementarity interpretation, and the orthodox (Dirac-von Neumann) interpretation without the projection postulate can be recovered from the theorem. Bohr's complementarity and Einstein's realism appear as two quite different proposals for selecting (...) the preferred determinate observable--either settled pragmatically by what we choose to observe, or fixed once and for all, as the Einsteinian realist would require, in which case the preferred observable is a 'beable' in Bell's sense, as in Bohm's interpretation (where the preferred observable is position in configuration space). (shrink)

We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous-spectrum and continuous-in-time observations. The “collapsed state-vector” after the “objectification” is simply treated as a random vector of the a posterioristate given by the quantum filtering, i.e., the conditioning of the a prioriinduced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization” has been derived (...) from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. (shrink)

It is generally believed that the uncertainty relation Δq Δp≥1/2ħ, where Δq and Δp are standard deviations, is the precise mathematical expression of the uncertainty principle for position and momentum in quantum mechanics. We show that actually it is not possible to derive from this relation two central claims of the uncertainty principle, namely, the impossibility of an arbitrarily sharp specification of both position and momentum (as in the single-slit diffraction experiment), and the impossibility of the determination of the path (...) of a particle in an interference experiment (such as the double-slit experiment).The failure of the uncertainty relation to produce these results is not a question of the interpretation of the formalism; it is a mathematical fact which follows from general considerations about the widths of wave functions.To express the uncertainty principle, one must distinguish two aspects of the spread of a wave function: its extent and its fine structure. We define the overall widthW Ψ and the mean peak width wψ of a general wave function ψ and show that the productW Ψ w φ is bounded from below if φ is the Fourier transform of ψ. It is shown that this relation expresses the uncertainty principle as it is used in the single- and double-slit experiments. (shrink)

The problem of joint measurement of incompatible observables is investigated. Measurements are represented by positive operator-valued measures. A quantitative notion of inaccuracy is defined. It is shown that within this framework joint inaccurate measurements are possible for arbitrary maximal projection-valued measures on finite-dimensional spaces. The accuracy of such measurements is limited, as is shown by an inaccuracy inequality we derive. This new type of uncertainty relation can be unambiguously interpreted as referring to measurement precision rather than preparative quality. Several recent (...) experiments are seen to be realizations of such joint measurements. (shrink)

A partial ordering in the class of observables (∼ positive operator-valued measures, introduced by Davies and by Ludwig) is explored. The ordering is interpreted as a form of nonideality, and it allows one to compare ideal and nonideal versions of the same observable. Optimality is defined as maximality in the sense of the ordering. The framework gives a generalization of the usual (implicit) definition of self-adjoint operators as optimal observables (von Neumann), but it can, in contrast to this latter definition, (...) be justified operationally. The nonideality notion is compared to other quantum estimation theoretic methods. Measures for the amount of nonideality are derived from information theory. (shrink)

We attempt to clarify the main conceptual issues in approaches to ‘objectification’ or ‘measurement’ in quantum mechanics which are based on superselection rules. Such approaches venture to derive the emergence of classical ‘reality’ relative to a class of observers; those believing that the classical world exists intrinsically and absolutely are advised against reading this paper. The prototype approach (K. Hepp, Helv. Phys. Acta45 (1972), 237–248) where superselection sectors are assumed in the state space of the apparatus is shown to be (...) untenable. Instead, one should couple system and apparatus to an environment, and postulate superselection rules for the latter. These are motivated by the locality of any observer or other (actual or virtual) monitoring system. In this way ‘environmental’ solutions to the measurement problem (H.D. Zeh, Found. Phys.1 (1970), 69–76; W. H. Zurek, Phys. Rev.D26 (1982), 1862–1880 and Progr. Theor. Phys.89 (1993), 281–312) become consistent and acceptable, too. Points of contact with the modal interpretation are briefly discussed. We propose a minimal value attribution to observables in theories with superselection rules, in which only central observables have properties. In particular, the eigenvector-eigenvalue link is dropped. This is mainly motivated by Ockham's razor. (shrink)

The current status of localization and related concepts, especially localized statevectors and position operators, within Lorentz-invariant Quantum Theory (LIQT) is ambiguous and controversial.1 Ever since the early work of Newton & Wigner (1949), and the subsequent extensions of their work, particularly by Hegerfeldt (1974, 1985), it has seemed impossible to identify localized statevectors or position operators in LIQT that were not counterintuitive—strange—in one way or another; the most striking strange property being the superluminal propagation of the localized states. The ambiguous (...) and controversial status of these concepts arises from the varied reactions that workers have to the strange properties. Some regard them, particularly the superluminal propagation, as utterly unacceptable, and conclude that no precise concepts of localized statevectors and position operators exist in LIQT (Wigner 1973, p. 325-327; 1983, pp. 310-313; Malament 1996). Others downplay the whole issue, on the grounds that current theoretical and experimental practice has no need of sharply formulated concepts of localization, localized states etc. (Birrell & Davies 1984, pp. 48-59; Haag 1992, p. 34). Still others, including ourselves, believe that the superluminal propagation does not lead to causal contradictions, and is not in conflict with available empirical data: so that it should not be ruled out. We also believe that the other strange properties are merely unfamiliar novelties of LIQT which we must simply learn to accept. (Like the superluminal propagation, they do not lead to causal contradictions, nor conflict with available data.) So in this paper, we aim to help remove the ambiguous and controversial status of localization concepts in LIQT. Overall, our strategy will be to assess a number of localization concepts, and thereby clarify the often complex relationships among them. (shrink)