[Paper Review] Adiabatic Geometrical Phase for Scalar Fields in a Curved Spacetime

Alí Mostafazadeh|arXiv (Cornell University)|Aug 9, 1996

Cosmology and Gravitation Theories9 references3 citations

TL;DR

This paper generalizes Berry's adiabatic geometric phase to relativistic scalar fields in curved spacetime using a two-component formulation of the Klein-Gordon equation. It establishes a geometric phase framework independent of field equation inner products, relying only on spatial L² inner products; key results show vanishing phases for Bianchi I models and non-Abelian phases for Bianchi IX models, with analogies to nuclear quadrupole Hamiltonians.

ABSTRACT

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. It involves a two-component formulation of the corresponding Klein-Gordon equation. A precise definition of the adiabatic approximation is offered and conditions of its validity are discussed. It is shown that the adiabatic geometric phase can be computed without making a particular choice for an inner product on the space of solutions of the field equations. What is needed is just an inner product on the Hilbert space of the square integrable functions defined on the spatial hypersurfaces. The two-component formalism is applied in the investigation of the adiabatic geometric phases for several specific examples, namely, a rotating magnetic field in Minkowski space, a rotating cosmic string, and an arbitrary spatially 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 spatially homogeneous cosmological models. In particular, it is shown that the adiabatic geometric phase angles vanish for Bianchi type I models. The situation is completely different for Bianchi type IX models where a variety of nontrivial non-Abelian adiabatic geometrical phases can occur. The analogy between the adiabatic geometric phases induced by the Bianchi type IX backgrounds and those associated with the well-known time-dependent nuclear quadrupole Hamiltonians is also pointed out.

Motivation & Objective

• To extend the concept of adiabatic geometric phases—originally developed for quantum systems—to classical relativistic scalar fields in curved spacetime.
• To develop a framework that computes geometric phases without requiring a specific inner product on the space of field solutions.
• To analyze the geometric phase in specific physical scenarios: a rotating magnetic field in Minkowski space, a rotating cosmic string, and spatially homogeneous cosmological models.
• To clarify the conditions under which the adiabatic approximation holds in relativistic field theory.
• To explore the emergence of non-Abelian geometric phases in Bianchi type IX cosmological models.

Proposed method

• A two-component formulation of the Klein-Gordon equation is introduced to describe the dynamics of minimally coupled scalar fields in curved spacetime with electromagnetic and arbitrary scalar potentials.
• The adiabatic approximation is precisely defined using slow variation of background parameters, with validity conditions derived from the field equations.
• The geometric phase is computed using an inner product defined on spatial hypersurfaces, avoiding dependence on field solution space structure.
• The formalism is applied to three case studies: rotating magnetic fields, rotating cosmic strings, and spatially homogeneous cosmological models.
• For Bianchi models, the geometric phase is analyzed using the holonomy of the connection derived from the two-component evolution equation.
• The analogy between Bianchi IX backgrounds and time-dependent nuclear quadrupole Hamiltonians is established through isomorphism of the geometric phase structures.

Experimental results

Research questions

• RQ1 How can the adiabatic geometric phase be consistently generalized to relativistic scalar fields in curved spacetime?
• RQ2 What conditions ensure the validity of the adiabatic approximation in relativistic field theories?
• RQ3 Can geometric phases be computed without specifying an inner product on the space of field solutions?
• RQ4 What are the geometric phase structures in spatially homogeneous cosmological models, particularly for Bianchi types I and IX?
• RQ5 What is the nature of the geometric phase in a rotating cosmic string background?

Key findings

• The adiabatic geometric phase for a rotating magnetic field in Minkowski space is correctly reproduced by the two-component formalism.
• The rotating cosmic string model yields a geometric phase consistent with known results, validating the formalism.
• For Bianchi type I cosmological models, the adiabatic geometric phase angles vanish identically due to the Abelian nature of the holonomy.
• In Bianchi type IX models, nontrivial non-Abelian geometric phases emerge, indicating a rich structure of holonomy-induced phases.
• The geometric phase structure in Bianchi IX models is isomorphic to that of time-dependent nuclear quadrupole Hamiltonians, suggesting deep analogies in geometric phase physics.
• The formalism enables computation of geometric phases without fixing an inner product on the solution space, relying only on spatial L² inner products.