Geometric Phase Locked in Fine Structure

Berry’s phase carries physical information coded as topological and geometrical objects that can be directly verified in measurements. In some cases the situation can be reduced to an irrational phase shift, that can be usually obtained by an iterative process. Take the Berry phase as the geometric object and let the iterative process be a non-linear phase-locked feedback mechanism defined by spin-orbit coupling and precession, a coupling of fast and slow rotating vectors. For spin-orbit coupling the realization is easy and fast generating irrational and rational numbers: generalized fine structure constants. As a result, this paper provides for additional evidence, that the Sommerfeld fine structure constant α carries a Berry phase component 2π(1− 137α). PACS 03.65.Bz, 03.65.Vf, 06.20.Jr, 12.20.-m, 31.30.Jv Due to a lack of knowledge, physics theories tend to define and postulated ‘second level’ constructions and abstractions that cannot directly be observed. This can lead to an inflation of redundant parameters and dimensions. It is the success of math to find short-cuts or ‘first level’ topological and geometrical constructions that can be directly verified and involve fewer dimensions, Berry’s phase is a good example. Generally, phase factors or phases representing the ‘holonomy’ provide for important boundary conditions while reducing the degree of redundancy in variables, including the phase shifts generated by Berry’s connection [1]. The non-adiabatic generalization of [2] defines a geometric phase factor for any cyclic evolution of a quantum system, for an introduction see i.e. [3]. This is one of the reasons why phases and gauge theories are not unimportant in quantum mechanics, despite of the central role of amplitude densities. The fine structure coupling in atomic and molecular dimensions is a candidate for a geometric phase shift: • it involves fast/slow vector couplings, • on round trips on a curved surface, • with exact frequency and phase relationships. If a geometric phase component generates spin precession, it can couple back to the dynamical phase evolution generating an non-linear feedback loop. In the previous paper [4] such a quantum feedback mechanism has been defined including generalized fine structure constants. The resonances of (atomic) clocks usually depend on phase-locked loops. Controlled by external fields, the geometric phase has a passive role, but in a phase-locked feedback loop the geometric phase has a double role: it is generated on the closed path and controls the closed path length and dynamic phase. Consequently, it is very interesting to consider round trips of vector signals additionally constrained by the precession dynamics induced by an emerging geometric phase. Round trips Based on phase-locked round trips of vector signals, a non-linear feedback situation controlled by Berry’s phase can be modelled that is compatible with any kind of spin-rotation coupling email: [email protected], c ©2002