Microcausality and Tunneling Times in Relativistic Quantum Field Theory

TL;DR

The paper investigates whether tunneling times in relativistic quantum field theory can be superluminal. It develops a space-time resolved QFT framework for Dirac and Klein-Gordon fields in the presence of a background potential and introduces an intervention protocol on a localized region of an initial wave packet to probe causality. The authors prove that microcausality holds for density observables even with a background field and show that modifying the initial density within a compact region cannot affect densities outside the corresponding light cone; they support this with numerical simulations for both Dirac and KG fields demonstrating subluminal propagation through barriers. The work provides a rigorous relativistic QFT basis that rules out superluminal tunneling in barrier scattering and clarifies how causal information propagates in such tunneling scenarios, with implications for related approaches to quantum tunneling.

Abstract

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.

Figures (3)

• Figure 1: The initial wave packet at $t_0=0$, with average position $x_0$ and momentum $p_0$ is shown along with the intervention region $\mathcal{D}$ defined on a compact support. The causally disconnected region at time $t^\prime = \Delta t - t_0$ on the other side of the potential barrier, displayed with purple dots, is defined as the region lying outside the light-cone emanating from the right edge $x_+$ of $\mathcal{D}$.
• Figure 2: The left panel shows, in dotted blue, an initial Dirac Gaussian wave packet of width $\sigma = 1 \lambda$ ($\lambda = \frac{\hbar}{m_e c}$ is the Compton wavelength of the electron), initial position $x_0 = -15 \lambda$, and average momentum $p_0 = \sqrt{V_0^2 - m^2 c^4}/c$, where $V_0$ is the potential barrier height. The mutilated wave packet is shown in solid red, with the intervention function defined by Eq. (\ref{['mutif']}) and $D = 1 \lambda$. The vertical black lines mark the edges of the intervention area in the left panel and the light cone emanating from them in the middle and right panels. The potential barrier is given by Eq. (\ref{['poti']}) with $V_0 = 2.5 mc^2$, $\kappa = 0.2 \lambda$, and $d = 3 \lambda$. The middle panel shows the reflected and tunneled wave packets at $t = 0.137 \lambda/c$. In the right panel, we zoom in on the region around the light cone, showing the tunneled part of the Gaussian wave packet in dotted blue and the difference between the Gaussian and the mutilated wave packets in dashed black. The calculations are performed on a lattice of width $100 \lambda$, with $2^{11}$ sites and time steps of $\delta t = 137 \times 10^{-6} \lambda/c$.
• Figure 3: Initial and tunneled KG wave packets, depicted using the same color scheme as in the Dirac case. The wave packet, barrier, and lattice parameters are identical to those used in the Dirac analysis, with $\lambda$ now denoting the Compton wavelength of the boson.