Abstract This book describes one approach to resolving the ‘measurement problem’ of standard quantum theory (SQT), by changing the Schrodinger equation’s evolution of the wave function. The wave function of SQT does not describe reality. For example, the wave function corresponding to a completed experiment is the sum of terms describing the different experimental outcomes whereas one of these outcomes is what appears in Nature, The rules of SQT respond to this deficiency by imposing an ill-defined ‘collapse postulate.’ The prior smooth evolution of the wave function according to Schrodinger’s equation is replaced by an abrupt ‘jump’ to one of the outcomes. When, why, how are unanswered questions. I have added a term to Schrodinger’s equation, obtaining a theory (which I named the ‘Continuous Spontaneous Localization’ theory, or CSL) that answers these questions. The term has a random field in it: each field makes the sum of terms evolve to one of the terms. The collection of all possible fields produces the (Born rule) probabilistic predictions of SQT. Thus, CSL does something SQT does not: it describes the occurrence of events. But, it is a different theory than SQT, so it predicts different results for some carefully crafted experiments. CSL has now existed for over three decades. While no experiment as yet has proven CSL true, no experiment has conflicted with it either. In this book I have aimed to give a clear exposition of CSL and its consequences, accessible to anyone who has studied SQT.
Promotion of quantum theory from a theory of measurement to a theory of reality requires an unambiguous specification of the ensemble of realizable states (and each state's probability of realization). Although not yet achieved within the framework of standard quantum theory, it has been achieved within the framework of the continuous spontaneous localization (CSL) wave-function collapse model. In CSL, a classical random field
w(x,t) interacts with quantum particles. The state vector corresponding to each
w(x,t) is a realizable state. In this paper, I consider a previously presented model, which is predictively equivalent to CSL. In this completely quantized collapse (CQC) model, the classical random field is quantized. It is represented by the operator
W(x,t) which satisfies $[W(\mathbf{x},t),W({\mathbf{x}}^{\ensuremath{'}},{t}^{\ensuremath{'}})]=0$. The ensemble of realizable states is described by a single state vector, the ``ensemble vector.'' Each superposed state which comprises the ensemble vector at time
t is the direct product of an eigenstate of $W(\mathbf{x},{t}^{\ensuremath{'}})$, for all
x and for $0\ensuremath{\leqslant}{t}^{\ensuremath{'}}\ensuremath{\leqslant}t$, and the CSL state corresponding to that eigenvalue. These states never interfere (they satisfy a superselection rule at any time), they only branch, so the ensemble vector may be considered to be, as Schr\"odinger put it, a ``catalog'' of the realizable states. In this context, many different interpretations (e.g., many worlds, environmental decoherence, consistent histories, modal interpretation) may be satisfactorily applied. Using this description, a long-standing problem is resolved, where the energy comes from the particles gain due to the narrowing of their wave packets by the collapse mechanism. It is shown how to define the energy of the random field and its energy of interaction with particles so that total energy is conserved for the ensemble of realizable states. As a by-product, since the random-field energy spectrum is unbounded, its canonical conjugate, a self-adjoint time operator, can be discussed. Finally, CSL is a phenomenological description, whose connection to, or derivation from, more conventional physics has not yet appeared. We suggest that, because CQC is fully quantized, it is a natural framework for replacement of the classical field
w(x,t) of CSL by a quantized physical entity. Two illustrative examples are given.
Introduction In spite of its wonderful agreement with every experiment performed so far, standard quantum theory (SQT) fails to describe events and to define the circumstances under which such events occur [171]. It is the linearity of Schrödinger evolution that lies, as we have seen, at the root of the problem. A state vector always evolves to become a linear superposition of the states corresponding to several possible outcomes of a measurement, and it is only when an experiment is actually done that one of these possible outcomes is realized at a particular instant. Thereupon the state has to be changed to the one corresponding to the particular outcome in order to follow its subsequent evolution. This additional information is not contained in the theory and has to be obtained from outside . This means that the theory is unable to predict when an event will occur. All it can predict is that if an event occurs, the possible outcomes and their probabilities are such and such. Since events do occur in every experiment, there is something missing from the theory. Since a state vector can be written as the linear sum of a complete set of basis states and these basis states can be chosen in a number of ways, each of which corresponds to a different set of outcomes, the theory also fails to tell us how to choose the preferred basis .
Abstract This chapter introduces the problem (that the wave function of standard quantum theory (SQT) cannot correspond to reality in nature) and its resolution (change Schrodinger’s equation). The chapter begins by citing Schrodinger’s argument as to how the collapse postulate of SQT leads to the cited problem. It then presents a resolution embodied in the continuous spontaneous localization (CSL) theory‘s altered Schrodinger equation. The chapter shows how the “Gambler’s Ruin” game, where two gamblers bet on a coin toss until one wins all the money, is a precise and intuitively appealing analogy to the needed collapse behavior for a superposition of two states evolving under random influence to one or another of those states. Following this, the chapter ends by presenting the structure of the book. The first seven chapters show how CSL works, discusses experimentally testable and already tested predictions, and covers interpretive features of CSL. The purpose of the remaining eleven chapters, called “Supplementary,” is summarized as well.
We discuss a model of spontaneous collapse of the quantum state that does not require adding any stochastic processes to the standard dynamics. The additional ingredient with respect to the wave function is a position in the configuration space which drives the collapse in a completely deterministic way. This new variable is equivalent to a set of positions of all the particles, i.e., a set of Bohmian positions, which obey the usual guiding equation of Bohmian theory. Any superposition of quantum states of a macroscopic object occupying different regions of space is projected by a localization process onto the region occupied by the positions. Since the Bohmian positions are well defined in a single realization of the experiment, a space localization into one region is produced. The mechanism is based on the correlations between these positions arising from the cohesive forces inside macroscopic objects. The model introduces two collapse parameters which play a role very similar to those of the Ghirardi-Rimini-Weber (GRW) [Ghirardi et al., Phys. Rev. D 34, 470 (1986)] and continuous spontaneous localization (CSL) [Pearle, Phys. Rev. A 39, 2277 (1989); Ghirardi et al., Phys. Rev. A 42, 78 (1990)] theories. With appropriate values of these parameters, we check that the corresponding dynamics rapidly projects superpositions of macroscopic states localized in different regions of space into a single region, while it maintains a negligible effect in all situations where the predictions of standard quantum dynamics are known to be correct. We briefly speculate about the possible relations with gravity. We then study the evolution of the density operator and a mean-field approximation of the dynamical equations of this model, as well as the change of the evolution of the momentum introduced by the localization process. Possible theoretical interpretations are discussed. Generally speaking, this model introduces a sharper border between the quantum and classical worlds than the GRW and CSL theories and leaves a broader range of acceptable values for the parameters.