Probability in Collapse Theories

In an objective collapse theory, probability enters a physical stochastic law for the wavefunction or another specified ontology. A collapse event is not merely an observer’s update. The theory must specify the event rate, the distribution of collapse centers or noise, and how the resulting records reproduce Born statistics in an appropriate regime.

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

A normalized localization law

In a simple one-particle GRW model in three spatial dimensions, define

$$L_{\mathbf x}=(\pi r_C^2)^{-3/4} \exp\!\left[-\frac{(\hat{\mathbf q}-\mathbf x)^2}{2r_C^2}\right].$$

Here $r_C$ is the localization length, and the convention gives $\int d^3x\,L_{\mathbf x}^\dagger L_{\mathbf x}=I$. At a collapse time, the center is drawn with density

$$p(\mathbf x)=\|L_{\mathbf x}\psi\|^2, \qquad \psi\longmapsto\frac{L_{\mathbf x}\psi}{\|L_{\mathbf x}\psi\|}.$$

Poisson-distributed event times introduce a further rate parameter $\lambda$. Normalization of $p(\mathbf x)$ follows from the operator identity. The probability law is part of the modified dynamics, not a consequence of deterministic Schrödinger evolution alone.

Why branch weights matter

Let $\psi=c_L\psi_L+c_R\psi_R$, where normalized packets are separated by much more than both their widths and $r_C$. For centers near the left packet, $L_{\mathbf x}\psi_R$ is exponentially suppressed. Integrating $p(\mathbf x)$ over that region gives approximately $|c_L|^2$, with corrections governed by packet overlap and Gaussian tails.

Thus localization approximately selects macroscopically separated components with Born weights. The approximation matters: Gaussian localization does not delete every other component exactly. The residual tails and the choice of matter-density or flash ontology are distinct issues discussed in objective collapse theories.

Ensembles and individual histories

Individual normalized trajectories are stochastic and nonlinear. Averaging over collapse events gives a linear density-operator evolution of the form

$$\frac{d\rho}{dt}= -\frac{i}{\hbar}[H,\rho] +\lambda\!\left(\int d^3x\,L_{\mathbf x}\rho L_{\mathbf x}^\dagger-\rho\right).$$

The ensemble equation predicts unconditioned experimental statistics. A chosen stochastic unraveling alone does not prove that the individual paths are physically real: open-system theories can generate similar reduced equations without fundamental collapse. Ontology and dynamics together define the proposal.

What remains empirical

Collapse parameters produce departures from standard isolated-system quantum dynamics, including suppression of interference and, in common nondissipative models, energy growth. The numerical constraints depend on the precise model, environment, and apparatus. Collapse-model tests explain how to connect a parameterized law to measurements.

A model may recover approximate Born weights for well-separated records while changing other predictions. It is therefore inaccurate to label all collapse theories empirically equivalent to standard quantum mechanics.

Exercises

Why does multiplying every $L_{\mathbf x}$ by a constant without changing the probability prescription spoil the model?

Solution

The integrated probability becomes the squared constant instead of one. A localization operator, its center density, and its normalization convention must be specified together. Renormalizing each post-collapse state does not repair an incorrectly normalized distribution of which collapse occurs.

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.