Subluminality of relativistic quantum tunneling

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.

Figure 1

Minkowski diagrams of the thought experiment outlined in the introduction. Left panel: Alice's viewpoint. The light bulbs (black dashed lines) are at rest, and they are all turned on simultaneously at $t_A=1$ (blue dashed line). In Alice's coordinates, the space-time event at which the $k\mathrm{th}$ bulb is turned on is $(1,k)$, and it is marked with a yellow star. Right panel: Bob's viewpoint. The event at which the $k\mathrm{th}$ bulb is turned on is now $(\gamma +\gamma vk,\gamma v+\gamma k)$. These events are no longer simultaneous for different values of $k$, and it looks as if there was a signal traveling along the blue dashed line $x_B^1=t_B/v-{\left(\gamma v\right)}^{-1}$.

Figure 2

Relativistic principle of causality. Let $\mathrm{\Sigma }$ (blue plane) be the initial-data hypersurface, e.g., the hyperplane $\{t=0\}$. Pick an event $x$ in the future of $\mathrm{\Sigma }$. Such event can be influenced only by that portion of $\mathrm{\Sigma }$ that can be reached by a nonspacelike world line emitted from $x$ (yellow line). In other words, the value of $\mathrm{\Psi }\left(x\right)$ depends only on the initial data prescribed inside the past light cone of $x$ (in red). Changes of initial data outside the past light cone of $x$ cannot affect the value of $\mathrm{\Psi }\left(x\right)$, when we solve the Dirac equation. Furthermore, we cannot change the value of $\mathrm{\Psi }\left(x\right)$ even by altering the external potential $A_{\mu }$ outside the past light cone of $x$.

Figure 3

Visual representations of our arguments (i)–(iii), respectively, which are direct consequences of Theorem t1. The shades of red represent the electron density ${\mathrm{\Psi }}^{†}\mathrm{\Psi }$ (red large, white small). Upper panel: The field $\mathrm{\Psi }$ is bound to propagate within the lightcone. Hence, electrons cannot travel faster than light. Quantum tunneling through a barrier is no exception. Middle panel: Alice can generate a disturbance in $\mathrm{\Psi }$ by altering the value of $A_{\mu }$ at her location. However, such disturbance travels slower than light, and it cannot reach Bob, who is causally disconnected from Alice (no superluminal signaling). Lower panel: The only way for a tunneled wave packet to exit at a time $t<L$ is that $\mathrm{\Psi }\ne 0$ on the right of $Q$ already at $t=0$.

Figure 4

Visualization of the Gauss-theorem argument discussed in Sec. 4. The yellow region represents the space-time volume where we integrate the divergence of the Dirac current (which vanishes). The hypersurfaces ${\mathrm{\Sigma }}_I$ and ${\mathrm{\Sigma }}_F$ have constant time, and thus they are spacelike. The hypersurface ${\mathrm{\Sigma }}_L$ is lightlike. Since the integral of ${\mathrm{\Psi }}^{†}\mathrm{\Psi }$ across all space is normalized to 1, we know that $\mathrm{\Psi }$ decays to zero at space-like infinity, and thus we can extend the integration region up to $z=+\infty$. Note that we have located the down-left corner of the trapezium in the origin only for convenience. This argument still holds if we translate the trapezium anywhere else.

Figure 5

A forbidden process. A wave function has support over the pink region, which expands at the speed of light. Most of the probability density $j^0$ is initially located to the left of the origin (darker region). Can there be a large transfer of probability (dark red beam) from the left to the right of the yellow lightlike path? According to our inequality (9), no. In fact, the probability stored in the segment $BC$ can never exceed the probability stored in its nonzero causal past (the segment $0A$).

Figure 6

Location of point $Q$ in the numerical experiment of Dumont et al. [8]. The mathematical procedure for computing it is the following. First step: Identify the position of the tunneled wave packet in the Minkowski diagram (red “beam” in the upper right corner). Second step: Draw a right-travelling lightlike path (in blue) located immediately on the left of the tunelled wavepacket. Third step: find the location $Q$ where this lightlike path intersects the line $t=0$. The region $z>Q$ defines the “tail” of the incoming wave function in the causal past of the tunneled wavepacket. Equation (10) states that the tunneling probability cannot exceed the probability associated to such tail (because the tunneled wave packet is the causal evolution of such tail). In other words, the probability flows subluminally from the tail to the tunneled wave packet. We note that, since the incoming wave packet is very far from the barrier at $t=0$, the point $Q$ that marks the beginning of the tail lies outside the barrier ($Q<0$).