Abstract This chapter introduces the two equations that completely define CSL, for the simplest case, that is, a state vector in a two-dimensional Hilbert space. One equation is the modified evolution equation for the state vector which depends upon a Brownian motion function of time, B(t), a Hermitian operator A called the “collapse-generating operator,” and a collapse rate parameter λ. The state vector evolution is linear, but not unitary, so its norm changes with time. The second equation is called the probability rule. It says the probability a particular B(t) occurs in Nature is proportional to the squared norm of the state vector, that is, high norm state vectors are most likely to occur. The chapter shows how an initial state vector that is a superposition of the eigenstates of A evolves to one eigenstate, with the Born probability, as predicted by standard quantum theory. Finally, it discusses the density matrix description of the ensemble of evolutions under all possible B(t).
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.
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.
A linear equation of motion for the state vector is presented, in which an anti-Hermitian Hamiltonian that fluctuates randomly is added to the usual Hamiltonian of the Schr\"odinger equation. It is shown how the resulting theory describes the continuous evolution of a state vector to an ensemble of reduced state vectors while retaining important physical features of the Ghirardi, Rimini, and Weber [Phys. Rev. D 34, 470 (1986)] theory of spontaneous localization, in which the state vector reduction occurs discontinuously. A novel aspect, compared with ordinary quantum theory, is that the state-vector norm changes with time. The squared norm of each state vector is interpreted as being proportional to the probability possessed by that state vector in the ensemble of state vectors. This interpretation is shown to be consistent with the independent Markovian evolution of each state vector.
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.
Abstract Since the Hilbert space for state vectors describing Nature is, of course, of higher dimension than two, generalization of the state vector evolution equation given in the previous chapter is needed. It is explained why the Brownian motion function needs to be replaced by its time derivative, that is, white noise w(t). The collapse-generating operator A is allowed to be arbitrary. As an example, the case of a free particle moving in one dimension, where A=X (the particle’s position operator), is completely solved. Then, further generalizing, the single collapse-generating operator A is replaced by a family Ai, a complete commuting set of operators, one of whose joint eigenstates is the end result of collapse. Each operator is linked to its own white noise function wi(t). The chapter then discussed time-ordering, which comes into play in the generalized evolution equation. The chapter also covers the associated density matrix. The chapter concludes with a final generalization where the discrete index i is replaced by the continuous index x, so the operators A(x) (x is a point in space) and the white noise field w(x,t) appear in the evolution equation.