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Harmonic analysis on the Lorentz group as dynamical relativistic fourier analysis

Mavrodiev, S. Cht.

Published Jan 1, 1977• 0 citations
OpenAlex

Abstract

A dynamical relativistic Fourier analysis connecting both the relative momentum and the relativistic coordinate has been constructed in the framework of the theory of Galilei and Poincare group unitary representations and the concept of „geometrization“ of the relativistic two-body problem. The usual three-dimensional quantum-mechanical language of the formalism enables one to obtain easily some consequences for relativistic quantum mechanics.

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A dynamical relativistic Fourier analysis connecting both the relative momentum and the relativistic coordinate has been constructed in the framework of the theory of Galilei and Poincare group unitary representations and the concept of „geometrization“ of the relativistic two-body problem. The usual three-dimensional quantum-mechanical language of the formalism enables one to obtain easily some consequences for relativistic quantum mechanics.
It is well known that the analysis of a relativistic n -body problem invariant under the transformations of the Poincaré group and involving one time has only been done for n =1. For n >1 one uses the second quantization formalism of field theory. In this paper we state it in the ordinary space time coordinates associated with the n-bodies as Dirac did it in the one body case. We apply the formalism first to the two body problem if the interaction is of an harmonic oscillator form and then extend it to n particles. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincare sphere on polarization optics.
In this paper, a Unitary Quantum Relativistic Theory was developed. The wavy nature of a quantum particle involves the relativistic dynamics of the wave packet of this particle, and a field of interaction described by Lorentz’s force and three of the four Maxwell equations: the electromagnetic induction law, and the flow laws of the electric and magnetic fields. These equations arise from the group velocity, which is of the form of the Lagrange equation, and from a relativistic principle for wave functions, which asserts that in any system of reference, a wave function has a bounded spectrum with a velocity limit c. When a magnetic circuit law is considered for a field interacting with a quantum particle, this is an electromagnetic field, propagating with the velocity c. This means that the limit velocity c of a quantum particle according to the theory of relativity is also obtained as a necessary condition of quantum dynamics. The relativistic particle wavefunctions considered in this framework, describing finite distributions of densities in the two conjugated spaces of the coordinates and momentum, provide a better description of a collisional phenomenon, in a finite volume occupied by these particles and the photon mediating this collision according to Quantum Electrodynamics.
This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincaré sphere on polarization optics.