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.
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.
In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.
Systems of interest in physics are usually composed by a very large number of interacting particles. At equilibrium, these systems are described by stationary states of the many-body Hamiltonian (at zero temperature, by the ground state). The reaction to perturbations, for example to a change of the external fields, is governed by the time-dependent many-body Schroedinger equation. Since it is typically very difficult to extract useful information from the Schroedinger equation, one of the main goals of non-equilibrium statistical mechanics is the derivation of effective evolution equations which can be used to predict the macroscopic behavior of the system. In these notes, we are going to consider systems of interacting bosons in the so called Gross-Pitaevskii regime, and we are going to show how coherent states and Bogoliubov transformations can be used to approximate the many body dynamics.
We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.
In this paper, we obtain the quantum dynamics in the framework of the general theory of relativity, where a quantum partide is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spațial coordinates and of the momentum, called wave functions. For a free partide, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time dependent phases proporțional to the relativistic lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the partide wave functions, we obtain lorentz’s force and the maxwell equations. For a quantum partide in electromagnetic field, we obtain dynamic equations in the coordinate and momentum spaces, and the partide and antiparticle wave functions. We obtain the scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
The recent experimental progress in parity-time (PT) symmetry has attracted great interest. However, compared with PT symmetry, there are only a few reported results on its counterpart, anti-PT symmetry, which would lead to new insights and applications. Experimentally simulating and demonstrating the properties of anti-PT symmetry is of particular interest. Here, we present experimental research for simulating the dynamics of bosonic Bogoliubov quasi-particles with anti-PT symmetry based on single photons generated from a point defect in a gallium nitride film. The dynamical evolution under a non-unitary operator is a continuous complex Lorentz transformation in a complex Minkowski space. The evolved states are located on hyperbolic curves, depending on the values of the new defined inner product, which remain invariant during the evolution. Three types of state, space-like, light-like, and time-like, which are analogous to those predicted by the special relativity, are demonstrated. The results of our work could be helpful to investigate the dynamics of quasi-particles in diverse systems and promote deep understanding of non-Hermitian quantum mechanics in open systems.