Relativistic dynamical collapse model

A model is discussed where all operators are constructed from a quantum scalar field whose energy spectrum takes on all real values. The Schrödinger picture wave function depends upon space and time coordinates for each particle, as well as an inexorably increasing evolution parameter $s$ which labels a foliation of spacelike hypersurfaces. The model is constructed to be manifestly Lorentz invariant in the interaction picture. Free particle states and interactions are discussed in this framework. Then, the formalism of the continuous spontaneous localization (CSL) theory of dynamical collapse is applied. The collapse-generating operator is chosen to be the particle number space-time density. Unlike previous relativistically invariant models, the vacuum state is not excited. The collapse dynamics depends upon two parameters, a parameter $\mathrm{\Lambda }$ which represents the collapse rate/volume and a scale factor $\ell$. A common example of collapse dynamics, involving a clump of matter in a superposition of two locations, is analyzed. The collapse rate is shown to be identical to that of nonrelativistic CSL when the GRW-CSL choice of $\ell =a=10^{-5}\mathrm{cm}$, is made, along with $\mathrm{\Lambda }=\lambda /a^3$ (GRW-CSL choice $\lambda =10^{-16}{s}^{-1}$). The collapse rate is also satisfactory with the choice $\ell$ as the size of the Universe, with $\mathrm{\Lambda }=\lambda /\ell a^2$. Because the collapse narrows wave functions in space and time, it increases a particle’s momentum and energy, altering its mass. It is shown that, with $\ell =a$, the change of mass of a nucleon is unacceptably large but, when $\ell$ is the size of the Universe, the change of mass over the age of the Universe is acceptably small.

Figure 1

The region where $\mathcal{I}(\mathbf{z},t;{\mathbf{x}}_1,0;{\mathbf{x}}_2,0)=1$ is the diamond shaped region, with vertices $(z,t)=(\frac{x_1+x_2}{2},0),(\frac{x_1+x_2}{2}+\frac{x_2-x_1}{4},\frac{x_2-x_1}{4}),(\frac{x_1+x_2}{2},\frac{x_2-x_1}{2}),(\frac{x_1+x_2}{2}-\frac{x_2-x_1}{4},\frac{x_2-x_1}{4})$.