Abstract
In this study, we introduce a relativistic quantum mechanical wave equation of the spin-1 particle as an excited state of the zitterbewegung and show that it is consistent with the 2 + 1 dimensional Proca theory. At the same time, we see that in the rest frame, this equation has two eigenstates; particle and antiparticle states or negative and positive energy eigenstates, respectively, and satisfy SO(2,1) spin algebra. As practical applications, we derive the exact solutions of the equation in the presence of a constant magnetic field and a curved spacetime background. From these solutions, we find Noether charge by integrating the zeroth component of the spin-1 particle current, called charge density, on hyper surface and discuss pair production from this charge. And, we see that the discussion on the Noether charge is a useful tool for understanding the pair production phenomenon because it is derived from a probabilistic particle current.
Highlights
Physics in 2 + 1 dimensional spacetime presents many interesting and surprising results, both experimentally and theoretically
We find the Noether charge derived from the zeroth component of the current, called charge density, and discuss pair production in terms of this charge
Taking a complex vector potential, Aμ, and fields, F μν, in terms of three components, we show that the equation is consistent with the 2 + 1 dimensional Proca theory
Summary
Physics in 2 + 1 dimensional spacetime presents many interesting and surprising results, both experimentally and theoretically. Discussions still continue on such an equation in the 2 +1 dimensional spacetime structure, physically and mathematically [55,56,57,58] From all these points of view, a relativistic quantum mechanical wave equation for a spin-1 particle in the 2 + 1 dimensional spacetime should be expected to satisfy the Proca equation and the relation between vector potentials, Aμ , and tensor fields, F μν , in classical limit. The outline of this study is as follows: In Section 2, in the 2 + 1 dimensional spacetime, we introduce a relativistic quantum mechanical wave equation for a spin-1 particle as an excited state of the classical zitterbewegung model and construct its free particle solutions.
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