On the CSL Scalar Field Relativistic Collapse Model

Abstract

The CSL dynamical collapse structure, adapted to the relativistically invariant model where the collapse-generating operator is a one-dimensional scalar field $\hat\phi(x,t)$ (mass $m$) is discussed. A complete solution for the density matrix is given, for an initial state $|\psi,0\rangle=\frac{1}{\sqrt{2}}[|L\rangle+|R\rangle]$ when the Hamiltonian $\hat H$ is set equal to 0, and when $\hat H$ is the free field Hamiltonian. Here $|L\rangle, |R\rangle$ are coherent states which represent clumps of particles, with mean particle number density $N\chi_{i}^{2}(x)$, where $\chi_{1}(x),\chi_{1}(x) $ are gaussians of width $\sigma>>m^{-1}$ with mean positions separated by distance $>>\sigma$. It is shown that, with high probability, the solution for $\hat H=0$ (identical to the short time solution for $\hat H\neq 0$) favors collapse toward eigenstates of the scalar field whose eigenvalues are close to $\sim\chi_{i}(x)$. Thus, this collapse dynamics results in essentially one clump of particles. However, eventually particle production dominates the density matrix since, as is well known, the collapse generates energy/sec-volume of every particle momentum in equal amounts. Because of the particle production, this is not an experimentally viable physical theory but, as is emphasized by the discussion, it is a sound relativistic collapse model, with sensible collapse behavior.