The Ghirardi–Rimini–Weber (GRW) theory is a spontaneous collapse theory in quantum mechanics, proposed in 1986 by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber.
Measurement problem and spontaneous collapses Quantum mechanics has two fundamentally different dynamical principles: the linear and deterministic Schrödinger equation, and the nonlinear and stochastic wave packet reduction postulate. The orthodox interpretation, or Copenhagen interpretation of quantum mechanics, posits a wave function collapse every time an observer performs a measurement. One thus faces the problem of defining what an "observer" and a "measurement" are. Another issue of quantum mechanics is that it forecasts superpositions of macroscopic objects, which are not observed in nature (see Schrödinger's cat paradox). The theory does not tell where the threshold between the microscopic and macroscopic worlds is, that is when quantum mechanics should leave space to classical mechanics. The aforementioned issues constitute the measurement problem in quantum mechanics. Collapse theories avoid the measurement problem by merging the two dynamical principles of quantum mechanics in a unique dynamical description. The physical idea that underlies collapse theories is that particles undergo spontaneous wave-function collapses, which occur randomly both in time (at a given average rate), and in space (according to the Born rule). The imprecise "observer" and "measurement" that plague the orthodox interpretation are thus avoided because the wave function collapses spontaneously. Furthermore, thanks to a so-called "amplification mechanism" (later discussed), collapse theories recover both quantum mechanics for microscopic objects, and classical mechanics for macroscopic ones. The GRW is the first spontaneous collapse theory that was devised. In the following years several different models were proposed. Among these are
the continuous spontaneous localization model (CSL model), which is formulated in terms of identical particles; the Diósi–Penrose model, which relates the spontaneous collapse to gravity; the quantum mechanics with universal position localization (QMUPL) model, which proves important mathematical results on collapse theories; and the coloured QMUPL model, which is the only collapse model involving coloured stochastic processes for which the exact solution is known.
Description The first assumption of the GRW theory is that the wave function (or state vector) represents the most accurate possible specification of the state of a physical system. This is a feature that the GRW theory shares with the standard Interpretations of quantum mechanics, and distinguishes it from hidden variable theories, like the de Broglie–Bohm theory, according to which the wave function does not give a complete description of a physical system. The GRW theory differs from standard quantum mechanics for the dynamical principles according to which the wave function evolves. More philosophical issues related to the GRW theory and to collapse theories in general one have been discussed by Ghirardi and Bassi.
Working principles Each particle of a system described by the multi-particle wave function | ψ ⟩ {\displaystyle |\psi \rangle } independently undergoes a spontaneous localization process (or jump):
| ψ ⟩ → | ψ x i ⟩ ⟨ ψ x i | ψ x i ⟩ , {\displaystyle |\psi \rangle \rightarrow {\frac {|\psi _{x}^{i}\rangle }{\sqrt {\langle \psi _{x}^{i}|\psi _{x}^{i}\rangle }}},}
where | ψ x i ⟩ = L ^ x i | ψ ⟩ {\displaystyle |\psi _{x}^{i}\rangle ={\hat {L}}_{x}^{i}|\psi \rangle } is the state after the operator L ^ x i {\displaystyle {\hat {L}}_{x}^{i}} has localized the i {\displaystyle i} -th particle around the position x {\displaystyle x} .
The localization process is random both in space and time. The jumps are Poisson distributed in time, with mean rate λ {\displaystyle \lambda } ; the probability density for a jump to occur at position x {\displaystyle x} is P i ( x ) = ⟨ ψ x i | ψ x i ⟩ {\displaystyle P_{i}(x)=\langle \psi _{x}^{i}|\psi _{x}^{i}\rangle } . The localization operator has a Gaussian form:
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