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A theoretical physicist who, with Giancarlo Ghirardi and Tullio Weber, proposed a common dynamical description of microscopic systems and macroscopic objects. Their Ghirardi-Rimini-Weber theory modifies ordinary quantum evolution so that widely separated macroscopic alternatives do not persist indefinitely. The GRW amplification mechanism connects a weak microscopic modification to strong macroscopic localization.^[ref]

A theoretical physicist who coauthored Ghirardi-Rimini-Weber theory, a proposed solution to the measurement problem based on modified dynamics. The 1986 model treats localization as a physical process, with a GRW amplification mechanism that makes its consequences depend strongly on system size.^[ref]

His review with Angelo Bassi explains how such models aim to obtain definite macroscopic outcomes from the same law used for microscopic objects.^[ref]

The GRW model takes a normalized wave function in $L^2({\mathbb{R}}^{3N})$. Between jumps it obeys the Schrödinger equation, $i\hslash {\partial }_t\Psi =H\Psi$, with the usual many-particle Hamiltonian operator, for example $H={\sum }_i[-{\hslash }^2{\nabla }_i^2/(2m_i)]+V({\mathbf{x}}_1,\dots ,{\mathbf{x}}_N)$. Each particle has an independent Poisson process of collapse times with rate $1/\tau$. Thus the total jump rate is $N/\tau$, the mean wait is $\tau /N$, and the jumping label is uniform on $\{1,\dots ,N\}$.

At a jump of particle $i$, use the GRW localization operatorIts centre has probability density function $p_i(\mathbf{c})=\Vert L_i(\mathbf{c})\Psi {\Vert }^2$. The post-jump quantum state is $L_i(\mathbf{c})\Psi /\Vert L_i(\mathbf{c})\Psi \Vert$. The normalization is consistent because $\int d^3\mathbf{c}{L}_i(\mathbf{c}{)}^2=I$. The collapse times and labels have these state-independent rates; the centres depend on the wave function through the Born rule-like squared norm.

For precision about the stated spatial event, let $B=Y_{10a}=\{\mathbf{c}:\mathrm{dist}(\mathbf{c},Y)\le 10a\}$. With $H=0$, the GRW collapse-centre event probabilities are exactlyThe last expression is defined only for $p_1(B)>0$. It includes the possibility that the next jump hits the same particle.

Write $f_i^Y(\mathbf{x})=|\varphi (\mathbf{x}-{\mathbf{y}}_i){|}^2$, and similarly $f_i^Z$. Putwith the same definitions for $Z$. The macroscopic separation and negligible packet tails make the two branches essentially orthogonal, so their interference contribution can be neglected. The first-event probability is thereforeThe $Z$ contribution is extremely small: its actual positions are within $r$ of $Z$, whereas centres in $B$ are within $10a$ of $Y$, leaving a distance much larger than $a$ in the Gaussian. Packet-tail errors add to this Gaussian-tail error.

For the next-event calculation, two distinct labels have independent position densities within either product branch, giving factors ${\eta }_i{\eta }_j$. A repeated label instead gives $R_i$, since both localization Gaussians multiply the same position variable. ConsequentlyThese formulas, with $p_{2\mid 1}=p_{12}/p_1$, answer the spatial questions without an extra containment assumption. If the first event is dominated by the $Y$ branch, they simplify toNeglecting the $Z$ term in this conditional expression requires it to be small relative to the retained first-event probability, not merely small in absolute value. The exact operator formulas above cover rare-event exceptions as well.

The intended GRW branch persistence versus geometric containment approximation is obtained if $Y$ and $Z$ contain essentially all the position mass of their corresponding branches. Then positions in $Y$ have a Gaussian-centre margin of $10a$ inside $B$, so ${\eta }_i^Y\simeq R_i^Y\simeq 1$, while ${\eta }_i^Z,R_i^Z\simeq 0$. This givesFor the Gaussian convention used here, the three-dimensional probability of a displacement exceeding $10a$ is $\mathrm{erfc}(10)+20e^{-100}/\sqrt{\pi }$, an extremely small number. A first centre near $Y$ also suppresses the $Z$ branch relative to the $Y$ branch by the very small ratio $|\beta {|}^2{q}_i^Z(\mathbf{c})/(|\alpha {|}^2{q}_i^Y(\mathbf{c}))$ when the latter branch has appreciable weight. All particles are correlated with the same pointer alternative, so one localization selects the macroscopic branch even when the next particle label differs.

The literal PDF, however, only places the packet centres ${\mathbf{y}}_i$ in $Y$; it does not say that the packets themselves fit inside $Y$. Since it permits $r>a$, the two numerical approximations just displayed do not follow for every allowed packet and region. For example, take all ${\mathbf{y}}_i=0$, let $Y$ be a very small ball, and let $|\varphi {|}^2$ be uniform on a ball of radius $r\gg 10a$. Translate the other branch to a similarly small region $Z$ at distance much larger than $100r$. ThenIf these equal packet probabilities are denoted by $\eta$ and their repeated-label integral by $R$, the next probability iswhich approaches $\eta$, not one, for large $N$. Thus packet containment is needed for the stated near-region event to have the usual branch probabilities. The literal general answer is given by the integrals above.

The physical GRW amplification mechanism remains intact: with properly chosen macroscopic branch regions, spontaneous localization selects the alternatives with approximately the squared branch amplitudes and stabilizes the selected record, on a time scale $\tau /N$. Gaussian tails are not exactly eliminated, so the persistence is approximate. The containment qualification concerns how the region is defined, rather than whether the dynamics selects a macroscopically separated branch.

Let $q=({\mathbf{q}}_1,\dots ,{\mathbf{q}}_N)$ denote the particle positions. In the Ghirardi--Rimini--Weber model, the wavefunction obeys the ordinary Schrödinger equationbetween random collapses. Each particle $i$ has an independent Poisson process of collapse times with rate $\lambda$. At such a time the state jumps according towhere the random centre has probability density $\Vert L_i(\mathbf{X})\psi {\Vert }^2$. The normalization is arranged so that $\int d^{3}X{L}_i(\mathbf{X}{)}^2=I$.

The original GRW scales are approximatelyAn isolated microscopic particle is therefore exceedingly unlikely to collapse during a laboratory experiment. A macroscopic pointer containing about $N\sim 10^{23}$ relevant particles has total collapse rate $N\lambda \sim 10^7{\mathrm{s}}^{-1}$ and collapse time about $10^{-7}\mathrm{s}$. After a measurement interaction correlates different microscopic outcomes with pointer positions separated by much more than $r_C$, one constituent's localization suppresses all incompatible pointer branches. This GRW amplification mechanism produces one definite macroscopic outcome with probabilities given by the Born rule, while leaving ordinary microscopic unitary time evolution almost unchanged.

For one spatial coordinate, average over the random centre of one collapse. The resulting density operator has position-space kernelIts diagonal is unchanged, so $⟨x⟩$ and $⟨x^2⟩$ are unchanged. The first derivative of the Gaussian factor vanishes at $x=x^{\prime }$, so $⟨p⟩$ is also unchanged. Its second derivative does not vanish, and the position representation of the momentum operator givesIn three dimensions the increase in total ${\mathbf{p}}^2$ is $3{\hslash }^2/(2r_C^2)$, corresponding to kinetic-energy increase $3{\hslash }^2/(4m{r}_C^2)$ per collapse. These are ensemble statements: conditioning on one specified collapse centre can shift the position moments. Repetition at rate $\lambda$ predicts GRW spontaneous heating, so precision searches for anomalous bulk heating, spontaneous radiation, momentum diffusion, and loss of matter-wave interference test the model.

The GRW model supports robust macroscopic locations, but not universal exact point locations supplied by a list of particle coordinates. It modifies the wave function rather than adding the actual particle configuration of Bohmian mechanics. A claim that objects are located must therefore specify both the collapse dynamics and the proposed ontology. “Localized” is not synonymous with “has compact support” or “is a position eigenstate”.

Using one standard convention, a collapse on constituent $i$ with centre $X$ acts by the GRW localization operatorIts centre has probability density $\Vert L_i(X)\Psi {\Vert }^2$. Independent constituent collapse rates $\lambda$ give a body with $N$ relevant constituents an approximately $N\lambda$ localization rate. Correlations within a macroscopic superposition allow one constituent's localization to suppress an entire alternative configuration: the GRW amplification mechanism.^[ref]

For two narrow packets separated by $d\gg r_C$, a hit near one multiplies the other's relative amplitude approximately by $e^{-d^2/(2r_C^2)}$. This explains a sharply concentrated pointer reading. Yet the Gaussian function is never zero at finite separation. A nonzero distant tail therefore remains, and a finite-width packet is not an exact position eigenstate. This is the tails problem in collapse theories.^[ref] The assertion that each hit produces exact bounded support is false; an initially everywhere nonzero Gaussian wave packet is a concrete counterexample.

A GRW matter-density ontology specifies matter in ordinary SPACE byA macroscopic body's density is concentrated near one region, making its approximate location physically meaningful. The formula is the proposed matter field itself, not a probability distribution over a hidden set of exact point positions. Residual tails correspond to very low density elsewhere, motivating the further question of why such tails do not constitute ordinary rival objects. Concentration provides a practical account of records; a full ontology must also address that structured-tail objection.

A GRW flash ontology instead identifies the primitive events with collapse centres in spacetime. Individual flashes have definite locations and times, while a persisting macroscopic object is a dense pattern of flashes. There is no continuous particle trajectory between them. An isolated microscopic system may flash rarely, so the model must not be read as assigning it an ordinary point location at every instant. Matter-density and flash accounts express distinct ontologies with related empirical predictions.^[ref]

An approximate-location rule, such as requiring almost all matter density inside a suitably sized region, is meaningful if its tolerance and scale are stated. It does not turn the small tail into zero, and arbitrary sharp property thresholds can create artificial counting puzzles. The defensible result is emergent, highly stable macroscopic localization, or exact locations of flashes, depending on the ontology. Exact Bohmian positions are a different theoretical commitment and cannot be attributed to the collapse equation alone.