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AIP conference proceedings

Quantum geometric phases in deformable extended objects

Chryssomalis Chryssomalakos, Héctor H. Hernández, Elias Okon, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov

Published Jan 1, 2008• 1 citation
Publisher
DOI
10.1063/1.3043875

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Quantum Phase via Geometry

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We consider particles constrained to move in the close vicinity of a space curve through a steep quadratic potential in the plane normal to the curve. As is known, the effective 1D Hamiltonian that governs the motion along the curve involves both its curvature and torsion. Thus, the adiabatic cyclic deformations of the curve might give rise to geometric phases being accumulated by the particle. We report that this is indeed, in general, the case, and analyze in detail the origin of the phases, as described in both an inertial Cartesian frame and a noninertial one, adapted to the curve. We extend previous work to include non-locally arclength preserving deformations, and derive general formulas for Berry’s curvature in this case.
On the basis of the results of some experiments dealing with the violation of Local Lorentz Invariance (LLI) and on the formalism of the Deformed Special Relativity (DSR), we examine the connections between the local geometrical structure of space-time and the foundation of Quantum Mechanics. We show that Quantum Mechanics, beside being an axiomatic theory, can be considered also a deductive physical theory, deducted from the primary physical principle of Relativistic Correlation. This principle is synonym of LLI and of a rigid and at minkowskian space-time. The results of the experiments mentioned above show the breakdown of LLI and hence the violation of the principle of Relativistic Correlation. The formalism of DSR allows to highlight the deep meaning of LLI breakdown in terms of the geometrical structure of local space-time which, far from being rigid and at, is deformed by the energy of the physical phenomena that take place and in this sense it has an active part in the dynamics of the whole physical process. This perspective has a far reaching physical meaning that extends its consequences to the foundations of Quantum Mechanics according to the interpretation of Copenhagen. It provides a 'real' explanation and description of quantum phenomena enriching, by the concept of deformed space-time, the realistic interpretation in terms of pilot wave and hence it uncovers the reality hidden below the probabilistic interpretation and dualistic nature of quantum objects.