Objective Collapse Theories

Objective collapse theories replace exact universal unitary evolution by dynamics that spontaneously suppress macroscopic superpositions. GRW uses random localization events; continuous spontaneous localization uses continuous stochastic dynamics. These are modified physical theories with parameter-dependent predictions, not merely alternative vocabulary for textbook projection.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

A physical localization scale

In the GRW model, each constituent undergoes Poisson-distributed localization events at rate $\lambda$, with Gaussian width $r_C$. The localization operators and their probability law are defined in probability in collapse theories.

For two narrow position packets separated by $\Delta x$, the ensemble coherence is multiplied at rate

$$\frac{d\rho_{LR}}{dt} \approx-\lambda\!\left[1-e^{-(\Delta x)^2/(4r_C^2)}\right]\rho_{LR},$$

apart from Hamiltonian evolution. At small separation the effect scales as $\lambda(\Delta x)^2/(4r_C^2)$; at large separation it saturates at $\lambda$. This convention matches a localization amplitude proportional to $e^{-(q-x)^2/(2r_C^2)}$.

Amplification and outcomes

If a rigid object’s $N$ constituents all occupy distinguishably displaced branches, independent GRW events can suppress the collective superposition at a rate of order $N\lambda$. This amplification allows weak microscopic effects and rapid suppression of macroscopically separated records.

The precise scaling changes in mass-proportional CSL and depends on geometry and correlation length. A universal $N^2$ claim is therefore inappropriate.

Collapse does not by itself specify what matter is. A mass-density ontology and a flash ontology interpret the same kind of stochastic state history differently. Gaussian tails also remain: “one outcome” cannot mean that every competing component becomes mathematically zero after one Gaussian hit.

Heating as a consistency check

For the nondissipative one-particle GRW convention and a state with finite kinetic energy, localization contributes

$$\left.\frac{d\langle H_{\rm kin}\rangle}{dt}\right|_{\rm loc} =\frac{3\lambda\hbar^2}{4mr_C^2}.$$

Each Cartesian momentum component gains $\hbar^2/(2r_C^2)$ in its ensemble second moment per localization event. Summing the three components, dividing by $2m$, and multiplying by the event rate $\lambda$ gives the result. This is the full kinetic-energy rate for a free particle. In a potential, ordinary Hamiltonian exchange between kinetic and potential energy must also be included.

Localization narrows position and injects momentum spread. The units are energy per time, and the effect increases as the localization width decreases. Dissipative variants modify this behavior, so a bound derived for one model must not be applied to every collapse proposal.

This concrete prediction shows why altered dynamics can be tested even if ordinary measurement frequencies remain close to Born probabilities.

Locality, classicality, and empirical status

Collapse models are designed to produce definite macroscopic records and approximate quantum behavior for microscopic systems. Consistency with no-signaling constrains their ensemble evolution; relativistic formulations require additional structure and are not automatic consequences of nonrelativistic GRW.

Environmental decoherence and fundamental collapse can both suppress observed interference, but they differ in reversibility and underlying dynamics. Collapse-model tests separate model parameters, environmental backgrounds, and measured observables. This page gives stable model structure, not a claim about a universal current exclusion bound.

Exercises

Compare the large- and small-separation decoherence rates for $\Delta x=r_C/10$ and $\Delta x=10r_C$.

Solution

For the first, the bracket is approximately $0.01/4=0.0025$. For the second it is $1-e^{-25}$, essentially one. The rates are therefore about $0.0025\lambda$ and $\lambda$. Increasing separation indefinitely does not make the single-particle GRW rate grow indefinitely.

References

• A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave-function collapse, underlying theories, and experimental tests,” Reviews of Modern Physics 85, 471–527 (2013), author manuscript.
• G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic systems,” Physical Review D 34, 470–491 (1986), doi:10.1103/PhysRevD.34.470.