On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

Abstract

We consider the relation between so called continuous localization models -i.e. non-linear stochastic Schrödinger evolutions -and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula , which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.

Bounds on quantum collapse models from matter-wave interferometry: calculational details

“…λ gives the frequency of the localization events for a reference object of mass m 0 = 1 amu, while r C describes how well an object is localized. The QMUPL model has only the parameter η, which can be related to the GRW/CSL parameters [20]. A common open question of both the GRW and CSL models, is to explain the origin of the noise in the dynamical equations.…”

Adiabatic theorem for a class of stochastic differential equations on a Hilbert space

“…Remark 3 In a Banach space version of the proposition, the condition iL 2 is self-adjoint should be replaced by L 2 is a generator of isometries. The proof is technically more involved and requires a version of Trotter-Kato formula that does not seem to be available in the literature ( [12] assumes compact state space, while [8,6] assume the Hilbert space structure). In particular that if dL j , j = 1, 2 is the generator of a propagator U j (s, s ′ ), then the propagator U(s, s ′ ) generated by a sum dL 1 + dL 2 can be expressed as…”

“…Writing γ = (β + ||X|| 2 ∞ )/(1 − 2α||X|| 2 ∞ ) and choosing the optimal α = 1/(4||X|| 2 ∞ ) we get Bound (6).…”

Bohmian mechanics, collapse models and the emergence of classicality

“…In order to do that, we will refer to another collapse model, the Quantum Mechanics with Universal Position Localizations (QMUPL) model [26,27]. The connection between the GRW and the QMUPL model has been studied in [28],…”

“…Collapse regime: In a very short time an arbitrary initial wave function localizes in space according to the Born rule. The collapse time, from an arbitrary initial wave function to a wavefunction of spread l, can be quantified as t C = 3 2l 2 Λ QMUPL , where Λ QMUPL = Λ/r 2 C in an appropriate limit as discussed in [28]. For example, let us consider a sphere of radius R = 1mm in the Earth's atmosphere as described in section IV: (1/ √ 2)r C = λ th = 3 · 10 −12 m and Λ = η = 3, 6 × 10 22 s −1 .…”