Lorentz quantum mechanics

TL;DR

This paper introduces Lorentz quantum mechanics, a new theoretical framework where system dynamics are described by complex Lorentz transformations in Minkowski space, contrasting with traditional unitary quantum evolution.

Contribution

It develops the foundational aspects of Lorentz quantum mechanics, including state classification, matrix representations, and adiabatic evolution, and links it to physical systems like fermion gases and Bose-Einstein condensates.

Findings

Defines three types of states: space-like, light-like, and time-like.

Establishes matrix form of Lorentz transformations and Pauli-like matrices.

Identifies physical systems where Lorentz quantum dynamics can occur.

Abstract

We present a theoretical framework called Lorentz quantum mechanics, where the dynamics of a system is a 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, there exist three types of states, space-like, light-like, and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as matrix form of a Lorentz transformation, construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in this mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where this Lorentz quantum dynamics can arise, are presented. They are one dimensional fermion gas, Bose-Einstein condensate (or superfluid), and one dimensional antiferromagnet.