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Physical Review Letters

Induced gauge fields in a nongauged quantum system

Huazhong Li

Published Feb 9, 1987• 39 citations
Publisher

Abstract

We show that non-Abelian gauge fields arise in a nongauged quantum system in the adiabatic approximation by working out a model of N-dimensional rotational symmetry. The induced gauge fields are symmetric under N-dimensional rotations accompanied by compensating gauge transformations of the group SO(N).
DOI
10.1103/physrevlett.58.539

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We start by reviewing the concept of gauge invariance in quantum mechanics, for Abelian and Non-Ableian cases. Then we idescribe how the various gauge potential and field can be associated with the geometrical phase acquired by a quantum mechanical wave function while adiabatically evolving in a parameter space. Subsequently we show how this concept is exploited to generate light induced gauge field for neutral ultra cold bosonic atoms. As an example of such light induced Abelian and Non Abelian gauge field for ultra cold atoms we disucss ultra cold atoms in a rotating trap and creation of synthetic spin orbit coupling for ultra cold atomic systems using Raman lasers.
For a molecular system with an odd number of electrons, the electronic Hamiltonian possesses a twofold Kramers degeneracy for all nuclear configurations. It is shown that this leads to a non-Abelian gauge-field term in the effective nuclear Hamiltonian, with permissible gauge transformations belonging to SU(2). This should lead to observable effects analogous to the Berry phase for the nondegenerate case. In particular, an atom in a slowly rotating electric field exhibits changes in its component of angular momentum along the field because of the non-Abelian phase factor.
In this paper high-order adiabatic approximate method is proposed to study the evolution of degenerate quantum systums with slowly-changing parameters and varying symmetry. We not only prove the quantum adiabatic theorem for degenerate cases and discuss the corresponding non-abelian induced gauge structure, but also point out the non-adiabatic corrections resulting from higher-order approximations. With nuclear quadrupole resonance as an explicit example, we use this method to analyse observable effects of non-abelian Berry's phases in adiabatic and non-adiabatic experimental process.
This study shows that the usual time evolution operator can act as a U(1) gauge transformation on the initial wavefuntion. When considering the spin freedom in the system, the time evolution operator corresponds to a non-Abelian SU(2) gauge transformation for spin-1/2 particles and a SU(3) gauge transformation for spin-1 particles. If we adopt the adiabatic approximation in the magnetization dynamics of a ferromagnetic system driven by a spin-polarized current, a temperature-dependent non-Abelian thermal gauge potential will appear. We can employ a temperature-dependent Landau-Lifshitz-Gilbert (LLG) equation to describe the magnetization dynamics in the semi-classical limit, a linear approximated solution for this equation is given analytically.
It is commonly understood that the Yang-Mills non-Abelian gauge fields is the natural generalization of the well known Abelian gauge group symmetry U(1)U(1)U(1) in the electrodynamics. Taking into account that the problems of the localization and divergences in QFT are not solved in the framework of the Standard Model (SM), I proposed a different approach to the quantum theory of the single self-interacting electron. In connection with this theory, I would like attract the attention to the state-dependent gauge transformations U(1)×U(N−1)U(1) \times U(N-1)U(1)×U(N−1) associated with the Jacobi vector fields of the geodesic variations in the complex projective Hilbert space CP(N−1)CP(N-1)CP(N−1) of the unlocated quantum states (UQS's).