We review the steps needed to reduce quantum field theory to quantum mechanics as its limiting case. While processes involving a changing number of particles require a quantum field theoretical description, some works used quantum mechanical data based on the single particle Dirac equation to predict the outcome. These non-quantum field theoretical approaches have led to conceptual controversies that are often related to the negative energy states, which are characteristic of the Dirac equation. As a result, the issues about the physical reality of the relativistic localisation phenomenon, the Zitterbewegung and the Klein paradox are widely discussed in many textbooks. In order to contribute to this discussion, we review recent computer simulations of oversimplified model systems based on quantum field theory. The direct visualisation of intrinsically quantum field theoretical process with full spatial and temporal resolutions can provide us with some first insight into the spontaneous electron–positron pair creation from the vacuum triggered by a strong external (supercritical) force field and with the resolution of the Klein paradox.
We review the steps needed to reduce quantum field theory to quantum mechanics as its limiting case. While processes involving a changing number of particles require a quantum field theoretical description, some works used quantum mechanical data based on the single particle Dirac equation to predict the outcome. These non-quantum field theoretical approaches have led to conceptual controversies that are often related to the negative energy states, which are characteristic of the Dirac equation. As a result, the issues about the physical reality of the relativistic localisation phenomenon, the Zitterbewegung and the Klein paradox are widely discussed in many textbooks. In order to contribute to this discussion, we review recent computer simulations of oversimplified model systems based on quantum field theory. The direct visualisation of intrinsically quantum field theoretical process with full spatial and temporal resolutions can provide us with some first insight into the spontaneous electron–positron pair creation from the vacuum triggered by a strong external (supercritical) force field and with the resolution of the Klein paradox.
Using space-time resolved solutions in relativistic quantum field theory we analyze the electron-positron pair creation process from vacuum and resolve the Klein-paradox. We show that there are in principle no limitations to the localization length of an electron and demonstrate that its spatial probability density can be much narrower than the Compton wavelength. We also argue that quantum field theory prohibits the occurrence of Zitterbewegung for an electron in contrast to claims found in textbooks.
As in non-relativistic quantum theory, the simplest problem to solve in relativistic quantum theory is that of describing a free particle. Much can be learned from this case which will be of use in interpreting the topics covered in later chapters. Furthermore, some of the most profound features of relativistic quantum theory are well illustrated by the free particle, so it is a very instructive problem to consider in detail. Another advantage of the free-particle problem is that the mathematics involved in solving it is not nearly as involved as that necessary for solving problems involving particles under the influence of potentials. Firstly we shall look briefly at the formulae for the current and probability density, then we shall go on to examine the solutions of the Dirac equation and investigate their behaviour. This leads us to a discussion of spin, the Pauli limit, and the relativistic spin operator. Next we consider the negative energy solutions and show how relativistic quantum theory predicts the existence of antiparticles. Some of the dilemmas this concept introduces and their resolution are discussed. At the end of the chapter we go back to the Klein paradox, and examine it for an incident spin-1/2 particle. We find that the Klein paradox exists for Dirac particles in exactly the same way as it existed for Klein—Gordon particles and has the same resolution and interpretation.
Using space-time resolved solutions to relativistic quantum field theory we analyze the electron-positron pair creation process from vacuum. For early times the entangled electron-positron wave function can be obtained analytically. We show that there are, in principle, no limitations to the localization length of an electron and demonstrate that its spatial probability density can be much narrower than the Compton wavelength. We also find that quantum field theory prohibits the occurrence of Zitterbewegung for an electron.
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.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
Relativistic quantum mechanics was developed to reconcile the principles of quantum mechanics with Einstein’s theory of relativity. Despite its success in describing high-energy particles, the theory continues to face unresolved conceptual and mathematical difficulties, particularly in relation to the nature of time, causality, and relativistic consistency. In recent years, the idea of extending space-time into the complex domain has emerged as a useful and potentially meaningful approach to these problems. In this work, a theoretical synthesis is presented to examine the role of complex space-time in relativistic quantum mechanics. The study discusses how complex time arises naturally in relativistic wave equations, quantum field theory, and path integral formulations. Established methods such as Wick rotation, complex coordinate transformations, and non-Hermitian but parity–time (PT) symmetric Hamiltonians are reviewed to illustrate how complex space-time contributes to mathematical stability and clearer physical interpretation (Dirac, 1928; Wick, 1954; Bender & Boettcher, 1998). The relevance of complex space-time is further explored in the context of relativistic particle dynamics and quantum cosmology, where it helps address, issues related to vacuum structure, time evolution, and causality (Feynman & Hibbs, 1965; Hartle & Hawking, 1983). The paper suggests that complex space-time should be viewed not only as a computational technique, but as a framework that may reflect deeper features of relativistic quantum phenomena.