Berry’s Phase and Fine Structure

Irrational numbers can be assigned to physical entities based on iterative processes of geometric objects. It is likely that iterative round trips of vector signals include a geometric phase component. If so, this component will couple back to the round trip frequency or path length generating an non-linear feedback loop (i.e. induced by precession). In this paper such a quantum feedback mechanism is defined including generalized fine structure constants in accordance with the fundamental gravitomagnetic relation of spin-orbit coupling. Supported by measurements, the general relativistic and topological background allows to propose, that the deviation of the fine structure constant from 1/137 could be assigned to Berry’s phase. The interpretation is straightforward: spacetime curvature effects can be greatly amplified by non-linear phase-locked feedback-loops adjusted to single-valued phase relationships in the quantum regime. Introduction A quantum mechanics of spin can not be complete without considering the phase evolution of a wave function including interference phenomena and geometric spin precession. Berry’s phase [1] can appear in purely classical situations such as round trip excursions on curved surfaces. Since spatial phases appear in any kind of wave propagation, different manifestations of this extremely general phenomenon have been found in several branches of physics from the high energy regime to the low. In addition to a Hamiltonian-induced dynamic phase, a quantum state evolving in parameter space on a trajectory that returns to the initial state acquires an extra phase termed geometric phase. This additional phase or angle depends only on the geometry of the Hamiltonian’s trajectory through parameter space and not on its time evolution. Various manifestations of geometric phases exist and are connected with names like Aharonov, Anandan, Berry, Bohm, Pancharatnam, Simon, Thomas, Wilczek and many others, see e.g. [2]. Although there are no widely recognized practical applications of the nonabelian gauge theory, its experimental observations have been reported in many fields of science. But quantum electrodynamics (QED) was established long before Berry’s phase was discovered. The successful concept of QED is perturbative and based on powers of the coupling constant α. QED handles (hyper)fine structure, Lamb shift, an spin anomalies at the most accurate level while ignoring Berry’s phase. A quick search for Berry in this context over the last 10 years in a physics archive returns almost now hits. The situation is probably unbalanced regarding the connections of Dirac’s theory and Berry’s phase. Dirac’s equation and Dirac’s theory of monopoles [3] are very important for the foundations of QED especially in the atomic range. A magnetic monopole as a logical consequence of the Dirac theory is necessary to quantize charge. Berry’s phase is intimately related to Dirac magnetic monopoles and arises naturally in the field of a monopole [1]. But magnetic monopoles have a similar status like Berry’s phase: an abstract geometrical and topological feature, where the topological structure of this abstract manifold is under special circumstances observable and physical. But no monopole has been found, at least been identified. This fact and the rather exotic status seems to imply for many scientists that Berry’s phase could have no big impact on the foundations of quantum mechanics manifested, email: [email protected], Weildorferstr.22, 88682 Salem-Neufrach, Germany c ©2002