Abstract
We present a theoretical framework for the dynamics of bosonic Bogoliubov quasiparticles. We call it Lorentz quantum mechanics because the dynamics is a continuous complex Lorentz transformation in complex Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the unitary transformation in Hilbert space. In our Lorentz quantum mechanics, three types of state exist: space-like, light-like and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as a matrix form of a Lorentz transformation, and the construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in these mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where bosonic Bogoliubov quasi-particles and their Lorentz quantum dynamics can arise, are presented. They are a one-dimensional fermion gas, Bose–Einstein condensate (or superfluid), and one-dimensional antiferromagnet.
Highlights
At the core of theoretical physics, two forms of vector transformations are of fundamental importance: the unitary transformation and the Lorentz transformation
While Eq (10) formally resembles the conventional Lorentz evolution in special relativity, there are delicate differences: (i) in contrast to the conventional Lorentz transformation where only real numbers are involved, here we are dealing with a complex vector specified by complex numbers, the interval of which requires the notion of modulus; (ii) unlike the real Minkowski space where x(t) must be a space-like component, here a freedom is left as we define the space-like axis and time-like one, i.e., we can either define a(b) as the space-like component or time-like component
We have studied the dynamics of bosonic quasiparticles based on Bogoliubov-de Gennes (BdG) equation for the (1, 1)-spinor
Summary
At the core of theoretical physics, two forms of vector transformations are of fundamental importance: the unitary transformation and the Lorentz transformation. The former, usually representing rotation of a real vector in space, preserves the modulus of the vector. In the context of quantum mechanics based on the Schrodinger equation, unitarity is an essential requirement for transformations of space, time, and spin, such that the modulus of a state vector in the Hilbert space – representing the total probability of finding the particle – is ensured invariant under these transformations. As we show below, such Lorentz quantum mechanics describes, and allow new insights into, the dynamical behavior of bosonic Bogoliubov quasiparticles. The bosonic Bogoliubov operator studied here stands for a class of generalized PT symmetric Hamiltonian [12], or more precisely, the anti-PT Hamiltonian [13], which can be realized experimentally by making use of refractive indices in optical settings [13, 14]
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