Collective excitations in the superfluid $A$ and $B$ phases of $\operatorname{He}^3$ in an electric field

The influence of an electric field on Bose excitations in the superfluid $A$ and $B$ phases of a system of $\operatorname{He}^3$ type is investigated by the functional integration method. To find the Bose spectrum of the system in an electric field, a functional of the hydrodynamic (effective) action is constructed. In the first approximation, the electric field influences only the acoustic modes in both the $A$ and the $B$ phase. In both phases, the acoustic spectrum has the same form and becomes anisotropic in the electric field. The velocity of sound is maximal in the direction of the electric field.