Relativistic Quantum Mechanics of Single Particles

Zoltán Papp

Τ. Padmanabhan

Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.

Sol Wieder

Owen L. de Lange, R. E. Raab

Abstract In this chapter we give a brief introduction to the relativistic equations for spin 0 and spin ½ particles (the Klein-Gordon and Dirac equations), and we solve these equations for two simple problems. Throughout we consider only single-particle theory: thus we do not discuss ‘second quantization’ of the relativistic equations (i.e. the use of quantized fields that create and annihilate particles and antiparticles).

Walter Greiner

Stephen L. Adler

Abstract We turn in this chapter and the next to a discussion of topics relating to relativistic single-particle equations in quaternionic quantum mechanics. In Secs. 11.1 through 11.3 we develop the theory of the quaternionic Klein-Gordon equation, and then proceed in Secs. 11.4 and 11.5 to develop the theory of the quaternionic Dirac equation.