  We present a method of simulating the Dirac equation in 3+1 dimensions for a
free spin-1/2 particle in a single trapped ion. The Dirac bispinor is
represented by four ionic internal states, and position and momentum of the
Dirac particle are associated with the respective ionic variables. We show also
how to simulate the simplified 1+1 case, requiring the manipulation of only two
internal levels and one motional degree of freedom. Moreover, we study relevant
quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the
transition from massless to massive fermions, with the relativistic and
nonrelativistic limits, via the tuning of controllable experimental parameters.

Quantum Physics;Mesoscale and Nanoscale Physics;High Energy Physics - Phenomenology