Alberto Rimini

On the Number of Bound States of a Given Interaction

A method for deriving bounds to the number of bound states of a given interaction is built up. The class of interactions for which the method works includes the nonlocal interactions besides the local ones. Problems with many channels and spin‐dependent interactions can also be treated. Some bounds are explicitly given.

The puzzling entanglement of Schrödinger's wave function

A brief review of the conceptual difficulties met by the quantum formalism is presented. The main attempts to overcome these difficulties are considered and their limitations are pointed out. A recent proposal based on the assumption of the occurrence of a specific type of wave function collapse is discussed and its consequences for the above-mentioned problems are analyzed.

Old and New Ideas in the Theory of Quantum Measurement

The conceptually simplest interpretation of the quantum-mechanical wave function is that adopted by most textbooks. In the celebrated book by Dirac1 one reads — each state of a dynamical system at a particular time corresponds to a ket vector... if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state (pages 16, 17). And later — a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured (page 36). Similar sentences can be found, e. g., in the book by Messiah2 (pages 249 and 251). In the above statements the term state refers to a single system, not to a statistical ensemble of systems. This emerges clearly from the second statement, concerning reduction, which is quite incomprehensible in this form if the ket vector is not referred to a single system. In most textbooks the wave function up to a factor is interpreted just in this way — as the state of the single considered system.

Relativistic Spontaneous Localization: A Proposal

A new proposal for a Lorentz-invariant spontaneous localization process in the framework of relativistic quantum field theory is presented. As in all dynamical reduction models, a stochastic process is introduced, which drives the state vector towards the eigenspaces of a set of operators representing suitably chosen physical quantities. Such operators constitute a Lorentz scalar field and are built as time averages and space integrals of a local field-theoretic operator in such a way that the quantities they represent acquire a macroscopic character. As always in dynamical reduction theories, the action of the process on microscopic systems takes place via the micro-macro correlations which arise, e.g., as a consequence of measurements.

Quantum measurement in a family of hidden-variable theories

The measurement process for hidden-configuration formulations of quantum mechanics is analysed. It is shown how a satisfactory description of quantum measurement can be given in this framework. The unified treatment of hidden-configuration theories, including Bohmian mechanics and Nelsons stochastic mechanics, helps in understanding the true reasons why the problem of quantum measurement can succesfully be solved within such theories.

Dissipation and reduction effects of spontaneous localization on superconducting states

The problem of quantum measurement is briefly recalled with particular reference to theories of reduction as a real physical process. The spontaneous localization models for systems containing identical particles are described and the choice of the model when several kinds of particles are present is discussed. Dissipation and reduction effects induced by spontaneous localization in superconducting systems are analyzed. It is found that dissipation effects are negligibly small; on the other hand, reduction effects are present, but their characteristic time is long, say of the order of one hour. It follows that testing reduction experimentally is far beyond present day technology.

On the relationship between continuous and discontinuous stochastic processes in Hilbert space

Two different kinds of stochastic processes in Hilbert space used to introduce spontaneous localization into the quantum evolution are investigated. In the processes of the first type, finite changes of the wave function take place instantaneously with a given mean frequency. The processes of the second type are continuous. It is shown that under a suitable infinite frequency limit the discontinuous process transforms itself into the continuous one.

An Attempt at a Unified Description of Microscopic and Macroscopic Systems

When quantum mechanics is applied to a macroscopic particle, most features of the behaviour of such an object are accounted for: mean values of position and momentum evolve according to the classical laws and quantum effects such as the spread of wave packets and the tunnel effect are negligible. However, in the behaviour of a macroscopic particle, we do not find any trace of a stable existence of superpositions of macroscopically distinguishable (e.g. localized in far-away spatial regions) states. One can say that the superposition principle breaks down or, at least, that it suffers serious limitations. The discrepancy in this respect between quantum mechanics and the actual behaviour of a macroscopic particle would be dramatic if we would admit (as it is usual for microobjects) that any selfadjoint operator corresponds to an observable. On the other hand, if we are willing to accept serious limitations to the observability of macroobjects the discrepancy can be cured. We do not want to discuss here this attitude. We only note that it implies accepting a partition of all objects into two classes, those for which standard quantum mechanics is fully valid and those for which a limitation of the superposition principle is introduced which hardly can be considered as consistent with the conceptual framework of the theory.

Analytic Properties of Resolvents and Completeness

The main purpose of this paper is to prove, by using a simple formal procedure, the completeness of the set of eigenstates of a non‐Hermitian operator whose resolvent satisfies certain physically plausible analytic properties. It is also shown how the obtained completeness relation can be related to the scattering solutions of the eigenvalue equation with extension to the multichannel case. All proofs are heuristic only.

Spontaneous Localization and Superconductivity

The Schrodinger equation does not include the stochastic features which emerge in quantum mechanics during measurements and which are incorporated, in the ordinary presentation, into the reduction principle. In other words, the Schrodinger equa- tion does not describe the particularity of the world we experience. If one is not satisfied with the ordinary presentation because of the dualism between the Schrodinger equation and the reduction principle and of the related ambiguous dualism between quantum systems and measuring apparatuses, a possible reaction is to declare that the Schrodinger wave function describes statistical ensembles rather than individual systems, and to resort to the practical impossibility of detecting interference among the different macroscopically distinguishable terms appearing in the wave function (this impossibility is often referred to as decoherence). This interpretative attitude works to some extent, but it cannot avoid certain typical inconsistencies1,2 arising from the pretension to renounce in principle to any descriptive tool related to the result of an individual measurement. In this situation, one is induced to believe that any fully consistent theory of measurement must necessarily be based on a formulation of quantum mechanics such that the outcome of an individual measurement has a counterpart in the description of the system after the measurement (by system we mean here the quantum system plus the apparatus plus whatever else may be relevant). There are at least three types of formulations of quantum mechanics which satisfy this requirement.