Distinguishing NPCs, free geodesics, and constrained geodesics via Energy Fluctuations AND Lorentz-Boost effects on superluminal probability of relativistic waves under weak measurement

Som, Abhishek (2018) Distinguishing NPCs, free geodesics, and constrained geodesics via Energy Fluctuations AND Lorentz-Boost effects on superluminal probability of relativistic waves under weak measurement. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

The entire work consists of two topics. The abstract for each of them is discussed in the following couple paragraphs. 1. There are two independent ways of looking at the ground state of a quantum system, the metric tensor and the geometric phase. The association of a metric tensor to the ground state manifold very naturally leads to creation of geodesics. They have a nice property of associating non-trivial complex invariant quantities to the geometric phase. Later, it was shown that this property can be generalized to curves known as null phase curves(NPC), of which geodesics are a subset. Geometric phases yield same results for all NPCs and thus, handicaps one in regard to knowing whether a curve is a geodesic or merely a NPC. We, therefore, ask the question- Can we distinguish geodesics from NPCs that are not geodesics? We show that mean energy variance, originating from the other independent way, the metric tensor, does the job for us. This quantity reaches its global minima for the free geodesic and a local minima for the 'constrained geodesic', which is a geodesic constrained to a subspace. Picking up a specific example of the quadrupolar subspace, it has also been shown that free geodesics can not be slowly transformed to a constrained geodesic and that these two kind of 'geodesics' should be treated as very different entities in general despite apparent similarity in their nomenclature. 2. For Klein-Gordon and Dirac waves for massive particles, the local group velocity, defined as the weak value of the velocity operator can take values greater than c. For waves that are written as a superposition of several plane waves, each with subluminal group velocities,this superluminal probability depends on the speed of the frame of the observer. This dependence is nonmonotonic in case of Klein-Gordon and monotonically increasing in case of Dirac. Curiously, the probability increases if we take contributions from both positive and negative energy sectors instead of choosing only positive energy solutions.

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