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Frontiers in Optics

Space-time resolved quantum field theory

Piotr Krekora, Qichang C. Su, R. Grobe

Published Jan 1, 2004• 0 citations
Publisher

Abstract

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.

DOI
10.1364/fio.2004.ftuh3

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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.
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.
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.
Based on space-time-resolved solutions to relativistic quantum field theory we illustrate interpretational difficulties in associating field-theoretical quantities with properties of particles. These difficulties are related to the fact that the definition of the spatial probability density of particles depends on the choice of the Hilbert subspace on which the field operator is projected. We illustrate these problems by analyzing pair-production probabilities and spatial densities for the electron-positron dynamics associated with a spatially localized subcritical potential that is turned on and off in time.
Considers the Dirac equation with external electromagnetic fields and gives a careful formulation of the Stuckelberg-Feynman interpretation. This electron-positron theory may be applied whenever electron and positron states can be separated, which is possible for a wide class of time-independent potentials vanishing at infinity. One then may introduce a system of observables with a position operator X(t) which is free of Zitterbewegung and yields the usual relativistic connection between momentum and velocity. The Stuckelberg-Feynman interpretation is usually associated with a scattering theory (propagator formalism), which differs from the one of a one-particle interpretation (formalism imitating non-relativistic techniques). For the above-mentioned potentials there is a simple relationship between these physically different formalisms. There are, however, more general potentials for which the Stuckelberg-Feynman interpretation may still be applied asymptotically. With the help of Klein's paradox the authors show that the Dirac equation may then have unphysical solutions not conserving the total charge.
We have calculated normalized wave function solutions to the time-dependent Dirac equation on a space-time lattice grid. Using these solutions we systematically explore phenomena that are consistent with the classical relativistic Liouville phase space approach and those that do not have a direct classical counterpart. The examples reviewed are the paradoxical Zitterbewegung, the time-resolved Klein-paradox, the dynamics of spin in an inhomogeneous magnetic field, the suppression of the spatial spreading rate and new dynamical features that occur close to relativistic resonances.