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).
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).
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.
Abstract CSL so far is a general framework to describe dynamical collapse of a statevector to one or another joint eigenstate of a set of mutually commuting collapse-generating operators. This chapter makes the final selection of the collapse generating operators. First, the choice of mass-density operators, A(x)= D(x), is discussed. This leads to an infinite rate of energy increase for particles, which is ameliorated by a slightly different choice, “smeared” mass-density operators, A(x)=∫dx’f(x-x’)D(x’), where f is a Gaussian of width a, f(z) exp-z2/2a2. This introduces a second parameter to the theory (in addition to the collapse rate λ), the width parameter a. This completes construction of non-relativistic CSL, the theory proposed to describe Nature. The chapter goes on to discuss the density matrix, followed by two important calculations, the first consequences of the theory. First, for an initial state that is an object in a superposition of two different places, is the rate to collapse to one or another place, and second, the rate of energy increase of an object due to the wave function narrowing inherent in collapse.
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.
The continuous-spontaneous-localization (CSL) description of state-vector reduction allows one to describe reality by the state vector. The mathematical structure of the theory, based upon a c-number white-noise source interacting with the quantum system, is reviewed. Then this description is generalized in two ways. First, a generalization is made to a wide class of c-number stochastic processes: using the square root of the probability density functional of the process, it is shown how to construct an evolution equation for the state vector which results in state-vector reduction. Second, within this wider framework the c-number source is completely eliminated and replaced by quantum operators, dubbed ``index'' operators, representing new physical variables which interact with the quantum system. In this case the state vector cannot represent reality as in the previous case, but it is shown how to write it unambiguously as a sum of ``preferred basis'' states, any one of which can represent reality. Therefore this quantum theory is open to the ensemble interpretation, where ``ensemble'' refers to the collection of possible real states. Three requirements sufficient for a viable ensemble interpretation are specified. It is shown how to satisfy these requirements, obtaining a purely quantum model that is equivalent to CSL and capable of describing reality.
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 Before examining experimental consequences of CSL, this chapter discusses the spontaneous localization (SL) theory of Ghirardi, Rimini and Weber (GRW). As for CSL, the chapter presents the structure of the theory, and then specializes it to a non-relativistic theory. SL does not add a term to Schrodinger’s equation. Instead, the state vector is suddenly altered by a “hit” (GRW’s ‘spontaneous localization’), that is, multiplication by an operator, followed by renormalization. The hits occur at random times. The chapter shows the specific choice of operator made by GRW for their non-relativistic theory. Emphasized are two aspects of SL that helped the journey to CSL. One is that the collapse is always acting (one need not “turn it on.” The other is the “smeared” character of the GRW operator. The chapter then shows how the density matrix is calculated. Finally, it discusses the numerical values, chosen by GRW, of the two parameters of SL theory, λ and a, that are provisionally adopted for CSL.