Continuous spontaneous localization model

The continuous spontaneous localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle. and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini.
Introduction.
The most widely studied among the dynamical reduction (also known as collapse) models is the CSL model. Building on the Ghirardi-Rimini-Weber model, the CSL model works as a paradigm of collapse models. In particular, it describes the collapse as occurring continuously in time, in contrast to the Ghirardi-Rimini-Weber model.
The main features of the model are:
Dynamical equation.
The CSL dynamical equation for the wavefunction is stochastic and non-linear:formula_3where formula_4 is the Hamiltonian describing the quantum mechanical dynamics, formula_5 is a reference mass taken equal to that of a nucleon, formula_6, and the noise field formula_7 has zero average and correlation equal toformula_8where formula_9 denotes the stochastic average over the noise. Finally, we introducedformula_10where formula_11 is the mass density operator, which readsformula_12where formula_13 and formula_14 are, respectively, the second quantized creation and annihilation operators of a particle of type formula_15 with spin formula_16 at the point formula_17 of mass formula_18. The use of these operators satisfies the conservation of the symmetry properties of identical particles. Moreover, the mass proportionality implements automatically the amplification mechanism. The choice of the form of formula_11 ensures the collapse in the position basis.
The action of the CSL model is quantified by the values of the two phenomenological parameters formula_1 and formula_2. Originally, the Ghirardi-Rimini-Weber model proposed formula_22sformula_23 at formula_24m, while later Adler considered larger values: formula_25sformula_23 for formula_24m, and formula_28sformula_23 for formula_30m. Eventually, these values have to be bounded by experiments.
From the dynamics of the wavefunction one can obtain the corresponding master equation for the statistical operator formula_31:formula_32Once the master equation is represented in the position basis, it becomes clear that its direct action is to diagonalize the density matrix in position. For a single point-like particle of mass formula_33, it readsformula_34where the off-diagonal terms, which have formula_35, decay exponentially. Conversely, the diagonal terms, characterized by formula_36, are preserved. For a composite system, the single-particle collapse rate formula_1 should be replaced with that of the composite systemformula_38where formula_39 is the Fourier transform of the mass density of the system.
Experimental tests.
Contrary to other solutions of the measurement problem, collapse models are experimentally testable. The experiments testing the CSL model can be divided in two classes: interferometric and non-interferometric experiments, which respectively probe direct and indirect effects of the collapse mechanism.
Interferometric experiments.
Interferometric experiments can detect the direct action of the collapse, which is to localize the wavefunction in space. They include all experiments where a superposition is generated and, after some time, its interference pattern is probed. The action of CSL is a reduction of the interference contrast, which is quantified by the reduction of the off-diagonal terms of the statistical operatorformula_40where formula_41 denotes the statistical operator described by quantum mechanics, and we defineformula_42Experiments testing such a reduction of the interference contrast are carried out with cold-atoms, molecules and entangled diamonds.
Similarly, one can also quantify the minimum collapse strength to actually solve the measurement problem at the macroscopic level. Specifically, an estimate can be obtained by requiring that a superposition of a single-layered graphene disk of radius formula_43m collapses in less than formula_44s.
Non-interferometric experiments.
Non-interferometric experiments consist in CSL tests, which are not based on the preparation of a superposition. They exploit an indirect effect of the collapse, which consists in a Brownian motion induced by the interaction with the collapse noise. The effect of this noise amounts to an effective stochastic force acting on the system, and several experiments can be designed to quantify such a force. They include:
formula_47where formula_48 is the vacuum dielectric constant and formula_49 is the light speed. This prediction of CSL can be tested by analyzing the X-ray emission spectrum from a bulk Germanium test mass.
Dissipative and colored extensions.
The CSL model describes consistently the collapse mechanism as a dynamical process. It has, however, two weak points.