Relativistic collapse model with tachyonic features

A finite relativistic model for free particles, which describes the collapse of the state vector, is presented. The interaction of particles with a classical fluctuating field $w\left(x\right)\mathrm{}$ causes collapse, as in the (so far) physically satisfactory nonrelativistic continuous spontaneous localization (CSL) model. In previous relativistic models, where $w\left(x\right)\mathrm{}$ has a white-noise spectrum, collapse is accompanied by spontaneous creation of particles with all energies out of the vacuum [amounting to infinite energy/(sec-volume)]. In this paper we explore one way of eliminating such vacuum excitation. We present a formalism which allows an arbitrary spectrum for $w\left(x\right).$ We point out, in lowest order of perturbation theory, that it is timelike components of this spectrum which are responsible for vacuum excitation but (appropriate, it seems, to a fundamentally nonlocal phenomenon) it is spacelike components which are responsible for collapse. We restrict the spectrum of $w\left(x\right)\mathrm{}$ to that of a tachyon of mass $\mu =ħ/ac\approx 1\hspace{0.5em}\mathrm{eV},$ where $a\approx {10}^{-5}\hspace{0.5em}\mathrm{cm}\mathrm{}$ is the Ghirardi-Rimini-Weber (GRW) collapse model scale parameter used in CSL. However, in higher orders than the first, there is still vacuum excitation due to the energy-momentum supplied by particle propagators. We opt to explore a simple means of eliminating this vacuum excitation: by removing the time-ordering operation from the state vector evolution so that particle propagators are on the mass shell. The result is a finite theory: there is no vacuum excitation (and no need for renormalization) as each vertex describes a finite physical process, the spontaneous emission or absorption of a real tachyon by a particle (pair creation or annihilation is not permitted by energy-momentum conservation). This process may be thought of as analogous to the GRW “hitting” description of collapse. The cost of no time ordering is that at each vertex a mass M particle’s wave function can spread by as much as $\approx cT(\mu /M)$ in time T. This has the result that, beginning at perturbation order $2M/\mu ,$ there is a nonvanishing probability for the particle to be found outside its light cone: for electrons and nucleons, the probability of this spacelike transport is negligibly small. Apart from this effect, the density matrix of particles in one region evolves independently of particles in another spacelike separated region. We thus obtain a finite, reasonably sensible, relativistic collapse model for free particles.