Ghirardi-Rimini-Weber theory

The Ghirardi--Rimini--Weber theory modifies unitary time evolution by random Gaussian position localizations. Each constituent collapses at a small fixed rate, while the rate for an entangled macroscopic body grows with its number of constituents.

For localization length $r_C$, the GRW localization operator acting on particle $i$ isA collapse centred at $\mathbf{X}$ sends $|\psi ⟩$ to the normalized state $L_i(\mathbf{X})|\psi ⟩$ with probability density $\Vert L_i(\mathbf{X})|\psi ⟩{\Vert }^2$.

For equal particle rates in the GRW model, the next centre falls in $B$ with probability $N^{-1}{\sum }_i{\int }_B\Vert L_i(c)\Psi {\Vert }^{2}dc$. With zero Hamiltonian operator, two successive centres in $B$ have joint probability $N^{-2}{\sum }_{ij}{\int }_{B}dc{\int }_{B}dd\Vert L_j(d)L_i(c)\Psi {\Vert }^2$. Dividing by the first-event probability gives the conditional second event, including repeated particle labels. Spatial probabilities depend on actual packet mass, not solely the nominal packet centres.

For $N$ entangled constituents, independent localization rates add to approximately $N\lambda$. One constituent's collapse localizes the correlated macroscopic configuration, rapidly suppressing macroscopically separated branches.

A first localization near one macroscopically separated branch suppresses the other branch through the Gaussian GRW localization operator. This selects a correlated pointer alternative. However, if a region contains only packet centres and excludes substantial packet mass, the next centre need not lie in that region's small enlargement. Probabilities $|\alpha {|}^2$ for the first event and approximately one for the next require regions containing essentially all corresponding branch mass. The general GRW collapse-centre event probabilities retain the geometry exactly.

GRW position localization increases the ensemble mean momentum variance. In three dimensions one collapse adds $3{\hslash }^2/(2r_C^2)$ to $⟨{\mathbf{p}}^2⟩$, producing mean kinetic-energy increase $3{\hslash }^2/(4m{r}_C^2)$.