Vacuum polarization of gauge-independent spin-1 field

Abstract

In a recent paper (2) we have investigated the propagation of the massive spin-1 totally antisymmetric rank-two field given by TAKAHASttI and PALMER (~). Hereafter we refer to it as the T.P. field. We have found that when this field is coupled minimally to the electromagnetic field, it propagates acausally, that is some of the characteristic surfaces are spacelike. Furthermore, we have shown that by adding an appropriate dipole moment interaction to the Lagrangian, causal propagation will be recovered provided the strength of the coupling constant is equal to e, any other value will always lead to acausal propagation. This clearly indicates that the T.P. spin-1 field possesses a fixed dipole moment and does not admit an anomalous arbi trary dipole moment if causality is to be met. The fixed dipole moment interaction that we add to the T.P. minimally coupled field can be introduced in a more natural way namely, by adding a fixed four divergence to the free T.P. Lagrangian. This new Lagrangian yields causal equations of motion when minimally coupled to the electromagnetic field. We now compare the T.P. field with the well-known spin-1 Proca field. I t is known (3) that the Proea field is causal when minimally coupled to the electromagnetic field and remains so even when an arbi t rary dipole moment interaction is added to it. The Proca field, on the other hand, is not invar iant under a gauge transformation of the second kind, while the T.P. field is gauge independent (s). Thus the advantage of gauge independence restricts the T.P. field to a fixed dipole moment, while the disadvantage of noninvarianee under a gauge transformation of the second kind anows the Proca field to have an arbi t rary dipole moment without disturbing its propagation characteristics. I t is interesting to see whether the minimally coupled Proea field and the minimally coupled T.P. field with the added dipole moment which makes it causal, will yieId the same physical quantities or not. In the free case the two fields are physically indis-