Journal of Physics A Mathematical and General

A generalization of Berry's connection

G. Papadopoulos

Published Apr 7, 1992• 1 citation

Abstract

A generalization of Berry's connection is proposed for quantum mechanical models whose wavefunctions are sections of a vector bundle over a bundle space with fibre, the configuration space of the model and base space a space of parameters. The transformation properties of the generalized Berry's connection under diffeomorphisms of the parameter space are also studied.

DOI: 10.1088/0305-4470/25/7/042

Related papers

Fibre bundle formulation of relativistic quantum mechanics

Bozhidar Zakhariev Iliev

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

arXiv:quant-ph/0201085Jan 19, 20022 citations

Fibre bundle formulation of relativistic quantum mechanics. II. Covariant approach

Bozhidar Zakhariev Iliev

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

arXiv:quant-ph/0107002Jun 30, 20010 citations

Fibre Bundle Formulation of Relativistic Quantum Mechanics: I. Time-dependent Approach

Bozhidar Zakhariev Iliev

We propose a new fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present first part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced by (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations.

arXiv:quant-ph/0105056Physica ScriptaJan 1, 20030 citations

Classical and quantum mechanics with time-dependent parameters

Gennadi Aleksandrovich Sardanashvily

We show that composite bundles Q→Σ→R, where Σ→R is the parameter bundle, provide the adequate mathematical description of mechanical systems with time-dependent parameters both in classical and quantum mechanics. In particular, the Berry phase phenomenon is described in terms of connections on composite Hilbert space bundles.

arXiv:quant-ph/0004005Journal of Mathematical PhysicsAug 1, 200028 citations

Generalized Bundle Quantum Mechanics

D. D. Ferrante

This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way. It starts off with a review of the mathematical tools needed and then moves on to build the objects necessary.

arXiv:math-ph/0308004Aug 4, 20030 citations

Fibre bundle formulation of time-dependent mechanics

Gennadi Aleksandrovich Sardanashvily

We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis. Connections on these bundles describe reference frames. Quantum time-dependent mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is leafwise geometric quantization.

arXiv:1303.1735Mar 7, 20130 citations