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Kluwer Academic Publishers eBooks

The Relativistic Many Body Problem in Quantum Mechanics

M. Moshińsky

Published Dec 17, 2005• 2 citations
Publisher

Abstract

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.

DOI
10.1007/1-4020-2634-x_4

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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.
The present work deals with the idea of a multi-time wave function, i.e. a wave function with N space-time arguments for N particles. Firstly, a careful derivation of the necessity of multi-time wave functions in relativistic quantum mechanics is given and a general formalism is developed. Secondly, the physical meaning of multi-time wave functions is discussed in connection with realistic relativistic quantum theories, in particular the "Hypersurface Bohm-Dirac" model. Thirdly, a first interacting model for multi-time wave functions of two Dirac particles in 1+1 space-time dimensions is constructed. Interaction is achieved by means of boundary conditions on configuration space-time, a mechanism closely related to zero-range physics. This is remarkable, as a restrictive consistency condition rules out various types of interaction and consequently no rigorous interacting model was known before. Fourthly, the model is extended to more general types of interaction and to the N-particle case. Higher dimensions are also discussed. Finally, the "Two-Body Dirac equations" of constraint theory are placed within the context of the multi-time formalism. In particular, the question of probability conservation is critically discussed, leading to further implications both for fundamental and applied questions.
We construct a non-Gaussian field and its associated creation and annihilation operators, describing non-interacting bound states with no momentum in a Coulomb potential. The system is shown to behave quantum mechanically as a collection of coupled harmonic oscillators, allowing us to construct the field by injection into the quantum mechanical space of a free Gaussian field, and to study its time development. 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.
There is no relativistic Hamiltonian for many particles systems except for free particles and this has been accepted since the 1960s from the work of Currie, Jordan and Sudarshan, Cannon and Jordan, and Leutwyler. This is the problem we will address in this paper by coupling relativistic particles with a scalar field. The Hamiltonian we derive is explicitly relativistic and in the non-relativistic limit gives a Hamiltonian with two body interaction. However, we find that the Poincare algebra is still not satisfied because of the inherent time delayed interaction in relativistic particle systems.
The prototype of the simplest periodic deterministic systems is a model whose states move on a circle. The prototype of the quantum mechanical periodic system is the harmonic oscillator. In line with the philosophy on which this book it based, these two models can be mapped one onto the other. However, we need an intermediate case: the harmonic rotator . We illustrate the use of the group SU(2)\mathit{SU}(2)SU(2) for this model, and we explain how to take the ‘continuum limit’. 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.