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Quantum Mechanics with Spontaneous Localization

T. Weber

Published Jan 1, 1991• 0 citations

Abstract

One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.

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One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.

One of the main goals of no-collapse interpretations of quantum mechanics is to show how results of measurements are objectified, or, in a more poignant terminology, how in a measurement situation facts arise. One needs to show how in their dynamical evolution the measured system and, perhaps more importantly, the measuring apparatus acquire, or at least appear to acquire, the definite properties that are otherwise explained by the collapse of the state. This can be seen as a special case of the problem whether quantum mechanics alone can account for the classical behaviour of macroscopic systems; in particular, the permanent definiteness of at least those properties of macroscopic systems that are directly observable, such as the position of the pointer of a measuring apparatus. In other words, whether quantum mechanics can account for macrofacts of the familiar kind.

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Stochastic differential equations describing the Markovian evolution of state vectors in the quantum Hilbert space are studied as possible expressions of a universal dynamical principle. The general features of the considered class of equations as well as their dynamical consequences are investigated in detail. The stochastic evolution is proved to induce continuous dynamical reduction of the state vector onto mutually orthogonal subspaces. A specific choice, expressed in terms of creation and annihilation operators, of the operators defining the Markov process is then proved to be appropriate to describe continuous spontaneous localization of systems of identical particles. The dynamics obtained in such a way leaves practically unaffected the standard quantum evolution of microscopic systems and induces a very rapid suppression of coherence among macroscopically distinguishable states. The classical behavior of macroscopic objects as well as the reduction of the wave packet in a quantum measurement process can be consistently derived from the postulated universal dynamical principle.

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On the level of quantum phenomena, there is no precise possibility of separating an examined physical system from a measuring instrument. This fact lends support to the assertion that the evolution of a quantum system ceases to be rigorously causal from the moment when the system is subjected to investigation. A quantum measurement is special in that it always causes a jump in the system under study to an eigenstate of the considered variable. Possible interactions between the measuring and measured objects will be neglected, so that a given quantum system, which is free from any disturbances, e.g. measurements or other perturbations, will evolve in a rigorously predictable manner. Globally speaking, a quantum-mechanical scheme of description of physical phenomena can be summarized by introducing the so-called Schrödinger picture. A physical field is obtained when a given physical feature is attributed to each point of a space in which the process under study is developing.