Constraints on the noise in dynamical reduction models

Abstract

The dynamics of a quantum system with internal degrees of freedom undergoing spontaneous collapse in the position basis are analysed; e.g., neutral mesons or neutrinos. Surprisingly, the value of the Heaviside function θ(x) at x = 0 that can in general be chosen in the interval [0,1] leads to different physical predictions. For the QMUPL (Quantum Mechanics with Universal Position Localization) model only a single value leads to probabilities conserving the particle number. Herewith the physical properties of the noise field can be constrained. This opens a road to study the physical properties of the noise field essential for collapse models.PACS numbers: 03.65.-w, 03.65.Tu, 05.40.-a Dynamical reduction models have been shown to be proper frameworks to circumvent the measurement problem of standard quantum mechanics and allow for a clear definition of a microscopic and a macroscopic system, respectively. In their seminal work [1] Ghirardi, Rimini and Weber introduced a concise framework for microscopic and macroscopic physical systems by introducing a spontaneous collapse of the wave function. Mathematically, this can be achieved by adding specific nonlinear and stochastic terms to the Schrdinger equation. The stochasticity is required to avoid superluminal signaling in general. Up to date such modifications of the unitary evolution have not been found to be in conflict to any experiment. The testability of collapse models is one of its merits and experiments such as, e.g., the ones with X-rays [2, 3] are putting upper limits on the reduction rate parameter. In this letter we are not interested per se to single out an experimental testable observable, we focus on conceptual issues when considering systems at high energies. We will prove that the mathematical structure of the interaction with the noise field leads to constraints if one imposes particle number conservation.In general, the dynamics is described by the following non-linear stochastic differential equation [4]with = 1 and  i t := φ t | i |φ t . HereĤ is the standard Hamiltonian of the quantum system, i are a set of N operators introducing the collapse in a certain basis choice (position basis in most cases). W i,t present a set of independent standard Wiener processes and λ quantifies the strength of the collapse. Finding solutions to a non-linear stochastic equation is a cumbersome problem. A first step to simplify the issue is to replace the real noises introduced via W i,t by imaginary ones iW i,t . As shown in Ref.[5] this does not change the master equation of the corresponding density operator: the physics is left invariant. Via that "imaginary transformation", equation (1) becomes a Schrödinger-like equation with a random Hamiltonianwhere w i,t := dWi,t dt . In the following we focus on the QMUPL (Quantum Mechanics with Universal Position Localization) model [6,7] for one particle (N = 1), which is one of the conceptually simplest spontaneous reduction models, however, the results we draw are not limited to this model. Accordingly the opera...

Particle mixing and the emergence of classicality: A spontaneous-collapse-model view

“…Now, we turn to the collapse models acting in the Hilbert space L 2 (R d ) ⊗ H M with d = 1, 2, 3, which combines the position and flavor spaces, and the collapse is assumed to the spatial part of the state of the system. For the sake of concreteness, we consider the QMUPL and the mass-proportional CSL models, which were widely analyzed in the context of flavor oscillations [51][52][53][54][55][56], and generalize them to the models with the time-asymmetric noise field. In this way, we analyze the quantum state equation…”

“…It has been put under discussion whether the spontaneous collapse can be considered also as a source of the particle decay [54][55][56]. It has been shown that the decay dynamics can be recovered by the mass-proportional CSL model with an asymmetric correlation function of the noise field underlying this collapse model.…”

“…In this paper, we review the results of [54][55][56] in the context of the master equation associated with a collapse model and investigate how the noise field affects the dynamics of neutral mesons. We derive a class of collapse models with a free choice of time-asymmetry of the underlying white noise field and show that the decay property of a flavor oscillating system (or its absence) can be governed by the corresponding spontaneous collapse dynamics in accordance with [55].…”

Transient Dynamics from Quantum to Classical- From the Developed Coherent State via Extreme Squeezing -

“…gives the dominant interaction term λ X (t) x (t) 3 . The lowest contribution of this term yields the two-loop quantum effect for X (t)…”

“…There is no explicit observer nor detector in the early Universe and therefore quantum mechanics cannot yield any definite outcome of the quantum fluctuations. In order to solve this problem, some authors introduce detectors [1] [2]or modify the basic laws of quantum mechanics so that the wave function spontaneously collapses [3]. This situation is similar to the quantum-information-theory aspect of the black hole [4].…”