The Quantum Formalism and the GRW Formalism

Abstract

The Ghirardi-Rimini-Weber (GRW) theory of spontaneous wave function collapse is known to provide a quantum theory without observers, in fact two different ones by using either the matter density ontology (GRWm) or the flash ontology (GRWf). Both theories are known to make predictions different from those of quantum mechanics, but the difference is so small that no decisive experiment can as yet be performed. While some testable deviations from quantum mechanics have long been known, we provide here something that has until now been missing: a formalism that succinctly summarizes the empirical predictions of GRWm and GRWf. We call it the GRW formalism. Its structure is similar to that of the quantum formalism but involves different operators. In other words, we establish the validity of a general algorithm for directly computing the testable predictions of GRWm and GRWf. We further show that some well-defined quantities cannot be measured in a GRWm or GRWf world. PACS: 03.65.Ta. Key words: quantum theory without observers; Ghirardi-RiminiWeber (GRW) theory of spontaneous wave function collapse; empirical predictions; quantum measurement theory; predicted deviations from quantum mechanics; primitive ontology; limitations to knowledge; positive-operator-valued measure (POVM); completely positive superoperator. *

Epistemology of Wave Function Collapse in Quantum Physics

“…This theorem has been proven for Bohmian mechanics [18], GRWf [39,30], and GRWm [30]; a similar result for GRWm can be found in [6]. In orthodox quantum mechanics, the theorem is true as well, taking for granted that, after E , a quantum measurement of the position observable of the pointer of E 's apparatus will yield the result of E .…”

“…with ψ the initial wave function [39,30]. Of course, this fact does not imply that F can be measured-and we are claiming that it cannot.…”

“…According to the GRW theories, the wave function ψ of the universe evolves according to this stochastic law, starting from the initial time (say, the big bang). As a consequence [5,30], a subsystem of the universe (comprising M < N "particles") will have a wave function ϕ of its own that evolves according to the appropriate M -particle version of the GRW process during the time interval [t 1 , t 2 ], provided that ψ(t 1 ) = ϕ(t 1 ) ⊗ χ(t 1 ) and that the system is isolated from its environment during that interval.…”

“…More precisely, no experiment on the system can produce an outcome Z such that the conditional distribution P(C|Z) would be any different from the a priori distribution (p, 1 − p): Proposition 4. Consider the collapse process as in (29)- (30) and an arbitrary experiment E at time t 2 , possibly with more than two possible outcomes. Let ψ be uniformly distributed on S(H ).…”

Detecting wave function collapse without prior knowledge

“…A relevant fact is that for every conceivable experiment E that can be carried out on S (with random outcome Z from some value space Z), there is a positive-operator-valued measure (POVM) M (·) on Z acting on H such that the probability distribution of Z, when E is carried out on S with wave function ψ ∈ S, is given by P(Z ∈ ∆) = ψ|M (∆)|ψ (3) for all measurable sets ∆ ⊆ Z. The statement containing (3) was proved for GRW theory in [8] and for Bohmian mechanics in [4]. In orthodox quantum mechanics, which does not allow for an analysis of measurement processes, it cannot be proved rigorously, but it can be derived from the assumption that, after E, a quantum measurement of the position observable of the pointer of E's apparatus will yield the result of E. However, it is important to note that while every experiment E can be characterized in terms of a POVM, it is not necessarily true that every POVM is associated with a realizable experiment.…”