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.
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 .
We emphasize that standard quantum theory (SQT) is incomplete because it doesn't describe what is experimentally observed, namely events, nor does it satisfactorily define the circumstances under which events may occur. Simple models are given (all of which have the same density matrix evolution) to illustrate schemes which claim to complete SQT. It is shown how the model based upon the Continuous Spontaneous Localization (CSL) theory, in which an individual statevector undergoes "True Collapse," gives a satisfactory description of events. It is argued that various "decoherence" based approaches, illustrated by models in which a density matrix undergoes "False Collapse," do not satisfactorily resolve these problems of 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.
Different attempts to solve the measurement problem of the quantum me-chanics (QM) by denying the collapse principle, and replacing it with changes in the quantum formalism, failed because the changes in the formalism lead to contradictions with QM predictions. To the difference, Ghirardi, Rimini and Weber took the collapse as a real phenomenon, and proposed a calculus by which the wave-function should undergo a sudden localization. Later on, Ghirardi, Pearle and Rimini came with a change of this calculus into the CSL (continuous spontaneous localization) model of collapse. Both these proposals rely on the experimental fact that the reduction of the wave-function occurs when the microscopic system encounters a macroscopic object and involves a big amount of its particles. Both these proposals also change the quantum formalism by introducing in the Schr?dinger equation additional terms with noisy behavior. However, these terms have practically no influence as long as the studied system contains only one or a few components. Only when the amount of components is very big, these terms become significant and lead to the reduction of the wave-function to one of its components. The present work has two purposes: 1) proving that the collapse postulate is unavoidable; 2) ap-plying the CSL model to the process in a detector and showing step by step the modification of the wave-function, until reduction. As a side detail, it is argued here that the noise cannot originate in some classical field, contrary to the thought/hope of some physicists, because no classical field is tailored by the wave-functions of entanglements.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.
This is the first single volume about the collapse theories of quantum mechanics, which is becoming a very active field of research in both physics and philosophy. In standard quantum mechanics, it is postulated that when the wave function of a quantum system is measured, it no longer follows the Schrödinger equation, but instantaneously and randomly collapses to one of the wave functions that correspond to definite measurement results. However, why and how a definite measurement result appears is unknown. A promising solution to this problem are collapse theories in which the collapse of the wave function is spontaneous and dynamical. Chapters written by distinguished physicists and philosophers of physics discuss the origin and implications of wave-function collapse, the controversies around collapse models and their ontologies, and new arguments for the reality of wave function collapse. This is an invaluable resource for students and researchers interested in the philosophy of physics and foundations of quantum mechanics.