Helical Vacuum Currents for a Scalar Field in Models with Nontrivial Spatial Topology

The vacuum expectation value of the current density for a massive scalar field in a (D + 1)-dimensional flat space-time with compact dimension is investigated. Along a compact dimension, the field operator obeys the helical quasiperiodicity condition with a constant phase. The results previously studied in the literature are obtained in the special case of a zero helicity parameter. In contrast to this case, the vacuum current, in addition to the component along the compact dimension, has a non-zero component in the direction of the non-compact dimension along which the field operator is shifted. A simple relation between these components is obtained. They are odd periodic functions of magnetic flux penetrating the compact dimension, with the period equal to the magnetic flux quantum. For a given non-zero value of the helicity parameter, the component of the vacuum current along the compact dimension tends to zero in the limit when the length of that dimension tends to zero. In the problem considered earlier with the usual quasiperiodic condition, the vacuum current diverges in that limit.