GRW Theory (Ghirardi, Rimini, Weber Model of Quantum Mechanics)

Consider a toy system consisting of a marble and box. The marble has two states, |Ψ_in⟩ and | Ψ_out⟩, corresponding to the marble being inside or outside the box. These states are eigenvectors of the operator B̂, measuring whether the marble is inside or outside the box. The formalism of quantum mechanics (QM) has it that not only |Ψ_in⟩ and | Ψ_out⟩ themselves, but any ► superposition |Ψ_m⟩ = a |Ψ_in + b |Ψ_out⟩ where a and b are complex numbers such that |a|² + |b|² = 1, can be the state of the marble. What are the properties of the marble in such a state? This question is commonly answered by appeal to the so-called Eigenstate-Eigenvalue Rule (EER): An observable Ô has a well-defined value for a quantum a system S in state |Ψ⟩ if, and only if, |Ψ⟩ is an eigenstate of Ô. Since |Ψ_in⟩ and |Ψ_out⟩ are eigen-states of B̂, EER yields that the marble is either inside (or outside) the box if its state is |Ψ_in⟩ (or |Ψ_out⟩). However, states like |Ψ_m⟩ defy interpretation on the basis of EER and we have to conclude that if the marble is in such a state then it is neither inside nor outside the box. This is unacceptable because we know from experience that marbles are always either inside or outside boxes. Reconciling this fact of everyday experience with the quantum formalism is the infamous measurement problem. See also ► Bohmian mechanics; Measurement theory; Metaphysics in Quantum Mechanics; Modal Interpretation; Objectification; Projection Postulate.

Standard quantum mechanics solves this problem, following a suggestion of von Neumann's, by postulating that upon measurement the system's state is instantaneously reduced to one of the eigenstates of the measured observable, which leaves the system in a state that can be interpreted on the basis of EER (► Measurement Theory). However, it is generally accepted that this proposal is ultimately unacceptable. What defines a measurement? At what stage of the measurement process does the ► wave function collapse take place (trigger problem)? And why should the properties of a system depend on actions of observers?