Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2208.09742 · Physical Review A

Subluminality of relativistic quantum tunneling

Lorenzo Gavassino, Marcelo M. Disconzi

Published Mar 13, 2023• 7 citations
arXivHTMLPDF

Abstract

We prove that the classical Dirac equation in the presence of an external (nondynamical) electromagnetic field is a relativistically causal theory. As a corollary, we show that it is impossible to use quantum tunneling to transmit particles or information faster than light. When an electron tunnels through a barrier, it is bound to remain within its future light cone. In conclusion, the relativistic quantum tunneling (if modeled using the Dirac equation) is an entirely subluminal process, and it is not instantaneous.

DOI
10.1103/physreva.107.032209

Related papers

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

M. Alkhateeb, X. Gutiérrez de la Cal, M. Pons +2 more

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.

arXiv:2407.07160•Jul 9, 2024•1 citation

Relativistic quantum field theory approach to electron wave-packet tunneling: A fully causal process

M. Alkhateeb, X. Gutiérrez de la Cal, M. Ll. Pons +2 more

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.

Physical Review A•Jan 29, 2025•6 citations

Microcausality and tunneling times in relativistic quantum field theory

M. Alkhateeb, A. Matzkin

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.

arXiv:2507.09066•Physical review. D/Physical review. D.•Sep 16, 2025•1 citation

Effects of relativity on the time-resolved tunneling of electron wave packets

Piotr Krekora, Qichang C. Su, R. Grobe

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.

Physical Review A•Feb 14, 2001•38 citations

On Times and Speeds of Time-Dependent Quantum and Electromagnetic Tunneling

М. В. Давидович

Abstract This is a review of studies of quantum tunneling, which is described by the one-dimensional Schrödinger equation, and electromagnetic tunneling, where “superluminal” velocities and times of tunneling are considered. Integral and integrodifferential equations have been presented to describe tunneling. According to these equations, superluminal motion is impossible. The paradoxical Hartman effect has been discussed and explained. It has been shown that the velocity of passage of a particle in a beam through a barrier in the case of steady-state and time-dependent quantum tunneling is equal to the velocity of its incidence on the barrier and quasiphotons inside any layer of matter carry the energy always at a subluminal velocity. However, the tunneling time of a single particle or a photon is meaningless.

Journal of Experimental and Theoretical Physics•Jan 1, 2020•12 citations

A subluminous Schroedinger equation

Philip Rosenau, Z. Schuss

The standard derivation of Schroedinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of Schroedinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit, which produces an intermediate model between non-relativistic and relativistic quantum mechanics: it produces Schroedinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. These boundary conditions relieve the Schroedinger equation of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), the EPR paradox, and so on. The practical implications of the present formulation are yet to be seen.

arXiv:1007.1780•Jul 11, 2010•0 citations
We prove that the classical Dirac equation in the presence of an external (nondynamical) electromagnetic field is a relativistically causal theory. As a corollary, we show that it is impossible to use quantum tunneling to transmit particles or information faster than light. When an electron tunnels through a barrier, it is bound to remain within its future light cone. In conclusion, the relativistic quantum tunneling (if modeled using the Dirac equation) is an entirely subluminal process, and it is not instantaneous.
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.
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 This is a review of studies of quantum tunneling, which is described by the one-dimensional Schrödinger equation, and electromagnetic tunneling, where “superluminal” velocities and times of tunneling are considered. Integral and integrodifferential equations have been presented to describe tunneling. According to these equations, superluminal motion is impossible. The paradoxical Hartman effect has been discussed and explained. It has been shown that the velocity of passage of a particle in a beam through a barrier in the case of steady-state and time-dependent quantum tunneling is equal to the velocity of its incidence on the barrier and quasiphotons inside any layer of matter carry the energy always at a subluminal velocity. However, the tunneling time of a single particle or a photon is meaningless.
The standard derivation of Schroedinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of Schroedinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit, which produces an intermediate model between non-relativistic and relativistic quantum mechanics: it produces Schroedinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. These boundary conditions relieve the Schroedinger equation of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), the EPR paradox, and so on. The practical implications of the present formulation are yet to be seen.