Adiabatic Geometrical Phase for Scalar Fields in a Curved Spacetime

A convenient framework is developed to generalize Berry’s investigation of the adiabatic geometrical phase for a classical relativistic charged scalar field in a curved background spacetime which is minimally coupled to electromagnetism and an arbitrary (non-electromagnetic) scalar potential. The general formalism which is applicable for any scalar field equation which involves time derivatives of orders one and two leads to the fact that the geometric phase is independent of the choice of an inner product on the space of solutions of the field equations. It only depends on the inner product of the Hilbert space of the square integrable functions defined on the spatial hypersurfaces. Furtheremore, it is shown that unlike the non-relativistic case, the requirement of the adiabaticity also restricts the eigenvalues of the induced two-component Hamiltonian. This two-component formalism is applied in the investigation of the adiabatic geometric phases for several specific examples, including the systems of a rotating magnetic field in Minkowski space, a rotating cosmic string, and a homogeneous cosmological background. It is shown that the two-component formalism reproduces the known results for the first two examples. It also leads to several interesting results for the case of homogeneous cosmological models. In particular, it is shown that the geometric phase angles vanish for Bianchi type I models, where as Bianchi type IX models give rise to nontrivial non-Abelian geometrical phases. The analogy between the geometric phases induced by the Bianchi type IX backgrounds and the nuclear quadrupole Hamiltonians is also pointed out. ∗E-mail: [email protected]