Causality and relativistic localization in one-dimensional Hamiltonians

We compare the relativistic time evolution of an initially localized quantum particle obtained from the relativistic Schrödinger, the Klein-Gordon and the Dirac equations. By computing the amount of the spatial probability density that evolves outside the light cone we quantify the amount of causality violation for the relativistic Schrödinger Hamiltonian. We comment on the relationship between quantum field theoretical transition amplitudes, commutators of the fields and their bilinear combinations outside the light cone as indicators of a possible causality violation. We point out the relevance of the relativistic localization problem to this discussion and comment on ideas about the supposed role of quantum field theory as a vehicle of making a theory causal by introducing antiparticles.

Figure 1

The final charge densities for Dirac (dot-dash), Klein-Gordon (dot), and relativistic Schrödinger Hamiltonians. While the Dirac and Klein-Gordon densities stay within the light cone, the relativistic Schrödinger density has a portion of weight $1.2\%$ outside of the light cone. (δ$=$0.1/$c$, $T=$ 0.001 a.u., numerical box length $L=$ 0.4 a.u. with NDIM$=$200 000 grid points.)

Figure 2

The amount of probability outside of the light cone (δ$+$$\mathit{ct}$ < $z$) as a function of time for various initial wave function widths δ for the relativistic Schrödinger Hamiltonian. ($L=$ 0.4 a.u., NDIM$=$800 000.)

Figure 3

The quantum mechanical charge densities associated with the quantum field theoretical state $|a,0\rangle \equiv {\mathrm{\Psi ̂}}_a^{†}(0,t=0)|\mathrm{vac}\rangle$ for the Dirac theory.