Relativistic Quantum Mechanics and Quantum Fields

Sol Wieder

R. F. Álvarez-Estrada, José L. Sánchez-gómez

We study the time evolution of decaying particles in renormalizable models of Relativistic Quantum Field Theory. Significant differences between the latter and Non Relativistic Quantum Mechanics are found —in particular, the Zeno effect seems to be absent in such RQFT models. Conventional renormalization yields finite time behaviour in some cases but fails to produce finite survival probabilities in others. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at short times in non relativistic quantum mechanics (QM henceforth); see, for instance, [1] for a comprehensive review.

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.

Walter Greiner

Hartmut M. Pilkuhn

S. Rupp

In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schr\"odinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can be derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.