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arXiv:2407.07160

Relativistic Quantum Field Theory Approach to Wavepacket Tunneling: Lack of Superluminal Transmission

M. Alkhateeb, X. Gutiérrez de la Cal, M. Pons, D. Sokolovski, A. Matzkin

Published Jul 9, 2024• 1 citation
arXivHTMLPDF

Abstract

We investigate relativistic wavepacket dynamics for an electron tunneling through a potential barrier employing space-time resolved solutions to relativistic quantum field theory (QFT) equations. We prove by linking the QFT property of micro-causality to the wavepacket behavior that the tunneling dynamics is fully causal, precluding instantaneous or superluminal effects that have recently been reported in the literature. We illustrate these results by performing numerical computations for an electron tunneling through a potential barrier for standard tunneling as well for Klein tunneling. In all cases (Klein tunneling \ or regular tunneling across a standard or a supercritical potential) the transmitted wavepacket remains in the causal envelope of the propagator, even when its average position lies ahead of the average position of the corresponding freely propagated wavepacket.

DOI
10.48550/arxiv.2407.07160

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We investigate relativistic wavepacket dynamics for an electron tunneling through a potential barrier employing space-time resolved solutions to relativistic quantum field theory (QFT) equations. We prove by linking the QFT property of micro-causality to the wavepacket behavior that the tunneling dynamics is fully causal, precluding instantaneous or superluminal effects that have recently been reported in the literature. We illustrate these results by performing numerical computations for an electron tunneling through a potential barrier for standard tunneling as well for Klein tunneling. In all cases (Klein tunneling \ or regular tunneling across a standard or a supercritical potential) the transmitted wavepacket remains in the causal envelope of the propagator, even when its average position lies ahead of the average position of the corresponding freely propagated wavepacket.
Recent results on electron tunneling across a potential barrier, inferred from observations or obtained from theoretical models, have suggested superluminal or instantaneous barrier traversal times. In this work we investigate relativistic wave-packet dynamics for an electron tunneling through a potential barrier employing space-time resolved solutions to relativistic quantum field theory (QFT) equations. We prove by linking the QFT property of microcausality to the wave-packet behavior that the tunneling dynamics is fully causal, precluding instantaneous or superluminal effects. We illustrate these results by performing numerical computations for an electron tunneling through a potential barrier for standard tunneling as well for Klein tunneling. In all cases (Klein tunneling or regular tunneling across a standard or a supercritical potential) the transmitted wave packet remains in the causal envelope of the propagator, even when its average position lies ahead of the average position of the corresponding freely propagated wave packet.
We show, in the framework of a space-time resolved relativistic quantum field theory approach to tunneling, that microcausality precludes superluminal tunneling dynamics. More specifically in this work dealing with Dirac and Klein-Gordon fields, we first prove that microcausality holds for such fields in the presence of a background potential. We then use this result to show that an intervention performed on a localized region of an initial wave packet subsequently scattering on a potential barrier does not result in any effect outside the light cone emanating from that region. We illustrate these results with numerical computations for Dirac fermions and Klein-Gordon bosons.
Abstract We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schrödinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein–Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.
We solve numerically the time-dependent Dirac equation for a quantum wave packet tunneling through a potential barrier. We analyze the spatial probability distribution of the transmitted wave packet in the context of the possibility of effectively superluminal peak and front velocities of the electron during tunneling. Both the Dirac and Schr\"odinger theories predict superluminal tunneling speeds. However, in contrast to the Dirac theory the Schr\"odinger equation allows a possible violation of causality. Based on an analysis of the tunneling process in full temporal and spatial resolution, we introduce an instantaneous tunneling speed that can be computed inside the potential barrier.
Abstract Wavepacket tunneling, in the relativistic limit, is studied via solutions to the Dirac equation for a square barrier potential. Specifically, the arrival time distribution (the time-dependent flux) is computed for wavepackets initiated far away from the barrier, and whose momentum is well below the threshold for above-barrier transmission. The resulting distributions exhibit peaks at shorter times than those of photons with the same initial wavepacket transmitting through a vacuum. However, this apparent superluminality in time is accompanied by very low transmission probabilities. We discuss these observations, and related observations by other authors, in the context of published objections to the notion that tunneling can be superluminal in time. We find that many of these objections are not consistent with our observations, and conclude that post-selected (for transmission) distributions of arrival times can be superluminal. However, the low probability of tunneling means a photon will most likely be seen first and therefore the superluminality does not imply superluminal signaling.
Previous 1+1-dimensional Dirac wavepacket calculations showed that the tunneling component of a relativistic electron wavepacket can generate an arrival-time distribution whose peak occurs earlier than the corresponding free-photon peak. However, adapting superluminal tunneling to signaling leads to subluminal signaling due to the low tunneling probability. In the present work we note that the barriers used in those calculations are supercritical with respect to the Sauter-Schwinger effect. Consequently, the single-electron evolution must be accompanied by spontaneous electron-positron production from the vacuum. We derive compact formulas for the electron and positron densities when one additional electron is present, showing that the evolved wavepacket contribution adds to the vacuum-produced electron density, while Pauli blocking reduces the positron density by the negative-energy component of the propagated electron. We then apply these formulas to a fourth-order super-Gaussian barrier which produces superluminal tunneling of an electron. The resulting densities are shown explicitly at several times, and are compared with a semiclassical resonance model for the pair number. The semiclassical description reproduces the numerical growth of the pair yield and clarifies the role of Klein-zone resonance energies and widths. Finally, we outline the extension from 1+1 to 1+3 dimensions by integrating over transverse momenta, using scaling properties of the 1+1-dimensional pair number.