Several theoretical and experimental studies in the past decades have examined phenomena involving superluminal waves and objects because of their implication in quantum and cosmological physics [1, 2, 3, 4, 5, 6, 7, 8, 9]. Among them, the study of the tunneling time problem is one of the topics that has most attracted the interest of quantum physicists [10, 11, 12, 13, 14, 15]. Researchers have approached this issue both from the perspective of non-relativistic  and relativistic  quantum theory. In both cases the tunneling time does not depend on the barrier width (at least for large enough barriers), thus proving superluminal behaviour of the quantum object (wave or particle). However, the tunneling time problem remains a controversial one in quantum physics. A comprehensive and clear theory to explain how long does it take a particle to tunnel through a barrier still does not exist . As is well known, classical quantum mechanics does not treat time as an Hermitian operator but rather as a parameter . Time does not appear in the commutation relationships typical of the Hermitian operators, even if it appears in one of the forms of the Heisenberg uncertainty principle, being a physical variable conjugated to the energy. For this reason we have to give up directly knowing the tunneling time. We may bypass the obstacle by assuming that the wave packet inside the barrier is stationary, with an imaginary wave vector. We can then interpret the tunneling time as the phase variation of the evanescent stationary wave that crossing the potential barrier induces. This is the definition of tunneling phase time [20, 21] and is the most widely used quantity in the studies of tunneling phenomena.
In this paper we investigate, by the study of the phase time, the one-dimensional scattering process of a relativistic half-integer spin free particle through a rectangular barrier. Because the motion takes place in one dimension, particle and antiparticle states are bi-spinor (there is no spin-flip in one-dimensional motion). Thus, we investigate the tunneling time for each of the spinor components. When the barrier height is greater than the particle energy, for the first component of particle and antiparticle states, we prove that the tunneling process always occurs at superluminal velocity (negative tunneling times). This transpires regardless of the width of the barrier and the energy gap between the barrier and the relativistic particle. Second spinor components of particle and antiparticle states, on the contrary, behave in a different way. In these, the scattering through the barrier may be subluminal or superluminal depending on the barrier width (positive tunneling time). In total, the tunneling time is almost always positive for particle states, while it becomes negative for antiparticle ones. Therefore, the Hartman effect always occurs for both particles and antiparticles. Furthermore, the study shows that when the energy gap increases, the tunneling time tends to zero for both particle and antiparticle states, and the crossing velocity of the barrier diverges to infinite values. Finally, we prove that superluminality is more accentuated for antiparticle states than particle states. This peculiarity disappears as the particle velocity increases, i.e., when the second spin component becomes more and more important (its magnitude increases as the relativistic factor β = u/c increases).
2 Quantum tunneling from the perspective of the Heisenberg uncertainty principle
In 1928 George Gamow gave a simple but elegant interpretation of quantum tunneling : the particle with energy E borrows from the vacuum an amount of energy (U – E) to surmount the barrier potential U > E. Recalling the uncertainty principle in the form: (1)
provides the time needed to repay the borrowed energy as: (2)
Eq. (2) gives the tunneling time, i.e., the time the quantum particle needs to cross the potential barrier. It is interesting to note that this time does not depend on the geometry of the barrier, providing simple evidence of the Hartman effect . If a is the one-dimensional barrier width, the tunneling velocity is: (3)
Since the barrier width and the energy gap can assume whatever value, from Eq. (3) we conclude that there are geometrical and energetic conditions under which the tunneling is a superluminal process. In particular, for large values of a(high confinement) and (U – E) (high-energy systems), the tunneling velocity inside the barrier is higher than the speed of light, according to a previous work by the present author .
3 The scattering model
Let us consider a half-integer spin relativistic free particle of rest mass m0, energy ± E (depending on its particle or antiparticle state) and velocity u, scattering through a potential barrier of width a and potential energy U. The potential energy is always greater than the particle energy (U > ± E):
In regions I and II, where the potential energy is zero, the particle moves freely and behaves like a relativistic monochromatic de Broglie wave plane , which spreads in space-time. To simplify the study we suppose a two-dimensional space-time, so that the plane wavefront impinging on the barrier has the two-component spinor form (there is no spin-flip): (4)
In Eq. (4), β is the relativistic factor (u/c), while |Ψ〉+ and |Ψ〉− are respectively the particle and antiparticle spinors. N is the normalization constant that, for a given finite volume, provides, where γ is the Lorentz factor. Wave vectors k0 and kx are: (5)
At the beginning of the barrier (x = x0), the de Broglie wave may be reflected or transmitted. In the latter case, the wave behaves like an evanescent stationary wave . Since the wave is confined within the barrier (evanescent waves are not propagating waves), its wave vector has only a spatial component: (6)
At the end of the barrier, the evanescent wave is transmitted in region III, where it returns to be a free relativistic particle with half-integer spin. Since the wave function (4) is a two-component vector, we must study the scattering process for both of them. At the barrier edges, each component of the Dirac spinor has to match the continuity conditions that quantum theory imposes.
We will investigate the tunneling time by the phase time (which is asymptotic and non-local) defined as : (7)
Therefore, the study object of this research preliminarily requires the calculation of the transmission amplitude CT for the spinor component of the particle and antiparticle states.
4 Transmission and reflection amplitudes
Let us calculate the transmission and reflection amplitudes of the scattering process that we will use for tunneling time investigation. Considering initially the first component of the particle spinor (4), the continuity conditions are: (8)
cR and cT are respectively the reflection and the transmission coefficients of the particle wave, while α and δ are the coefficients of the evanescent wave (see Appendix A). Substituting functions (9) in (8) and solving the system by the Cramer algorithm, we get: (10) (11)
where Δk = (k0 − k1) = γm0 (c−u)/ℏ. For antiparticle states, Eq. (10) provides the transmission amplitude, replacing Δkwith −Δkand using the evanescent wave vector χ−. Through some algebraic manipulations, we use the explicit forms (5) and (6) of the Dirac wave vectors and set m0 ≡ c ≡ ℏ = 1. Eq. (10) thus provides the probability transmission of the particle: (12)
When increasing the barrier height, i.e., increasing the potential energy U, the probability that the particle is transmitted through the barrier rapidly goes to zero, while if U = E this probability tends to one: (13)
The limits in (13) also hold for the antiparticle state, but for U → ∞ the probability tends to zero more rapidly. The reflection probability has an opposite trend to the transmitted one, which the quantum relation = 1 has to satisfy.
5 Tunneling time for particle states
As mentioned earlier, the phase time (7) helps perform the investigation of the tunneling time. The first step for its calculation is to determine the argument of the complex number CT exp[i(k0cta − kxxa)]. To simplify the formalism, from now on the Greek letter θ will represent this. Through some algebraic calculations (see Appendix B), we get: (14)
Assuming that the barrier width is large enough, we may set the term (e2χxa − 1/e2χxa + 1) equal to one, so that the argument function becomes: (15)
where the superscript (1∘) means that the phase time refers the first component of the Dirac spinor. Eq. (16) is independent from the barrier width a, and as expected, the Hartman effect  occurs. Moreover, the value of the phase time is always negative whatever the particle energy and the barrier height. According to references [25, 26], this means the tunneling always occurs at superluminal velocities. Figure 2 shows the phase time trends for U = 10 and U = 100 vs the relativistic factor β (as usual we set m0 ≡ c ≡ ℏ = 1):
The trends show a minimum tunneling time that shifts towards low values of the relativistic factor as the barrier height increases. Moreover, when increasing the barrier height, the phase time goes rapidly to zero. This result is in agreement with the superluminality that the Heisenberg uncertainty principle predicts. The ratio between the barrier width and the tunneling time provides the barrier-crossing speed, at least in principle. From the obtained results, we thus conclude that the higher the barrier is, the faster the scattering through it is. This phenomenon occurs as if the potential barrier boosts the particle towards higher velocities, even if its occurrence probability (that the squared transmission amplitude provides) goes to zero as the barrier height decreases.
We may also display the superluminal tunneling of the first spinor component by the refraction index, calculated as the ratio between the velocity of the free particle before impinging on the barrier and its velocity during the tunneling:
As expected, the refraction index is always lower than one and increases with the relativistic factor. This means that as the velocity of the free particle approaches the speed of light, the superluminality of the tunneling tends to increase, and the particle velocity inside the barrier becomes infinite. This result is explainable by the fact that by keeping the potential barrier constant and increasing the particle velocity, the energy gap rapidly tends to zero.
This shows that the two components of the spinor emerge from the far side of the barrier with a phase time difference given by: (18)
This quantity, which from now we will call phase time delay, is always positive. Figure 4 shows the trend of the second spinor component for the particle state:
This delay is considerable at non-relativistic regimes (u≪c), at which the magnitude of the secondary component of spinor (4) is very low, and tends to zero when the particle velocity approaches the speed of light. For this reason, whatever the barrier width, the secondary spinor component always emerges later than the principal one. This phenomenon leads to an anomalous distortion of the Dirac wave function that disappears when the relativistic behaviour of the particle becomes relevant. Overall, the phase time (17) is always positive, except when the particle velocity approaches the speed of light, and follows the trend shown in Figure 5:
As expected, the total tunneling time is asymptotic, and the Hartman effect may occur depending on the barrier width and on the particle energy.
6 Tunneling time for antiparticle states
Let us calculate the tunneling time for the antiparticle states following the same approach used in the previous section. The negative values of energy E and the relativistic wave number Δk represent the only difference. Eq. (16) can then provide the phase time by replacing E with −E, Δk with −Δk and using the second variant of Eq. (6): (19)
The negative sign of Δk is due to the fact that the sign of the argument of the exponential function for the antiparticle state is opposite to that of the particle state. Eq. (19) does not depend on the barrier width, and the Hartman effect occurs. Figure 6 shows the trend for U = 10:
Unlike the first spinor component of the particle state, the phase time is always positive, and the tunneling through the barrier could be subluminal or superluminal depending on the barrier width. This is expected behaviour considering that the first spinor component of the antiparticle is the opposite of the second spinor component of the particle. After setting the barrier height, the phase time tends to zero as the velocity approaches the speed of light, just like what occurs for the particle state.
Once again unexpectedly, the secondary component of the antiparticle spinor emerges from the far side of the barrier before the principal one. This always leads to an anomalous distortion of the Dirac antiparticle spinor that, however, is opposite to the one that occurs for the particle state. This distortion disappears as the antiparticle velocity approaches the speed of light but is slower than what occurs for the particle spinor.
In total, the antiparticle phase time given by the contribution of the two spinor components is always negative and turns out to be positive only when the particle velocity approaches the speed of light. Figure 8 shows the asymptotic trend:
This study proves that relativistic free particles and antiparticles behave differently when a potential barrier scatters them, even at velocities close to the speed of light. For large barrier widths, these behaviours tend to converge and become uniform. Although the Hartman effect occurs for both particles and antiparticles, the tunneling is always superluminal for antiparticle states (negative phase time) except when the particle velocity approaches the speed of light (positive phase time tending to zero). However, it can be subluminal or superluminal for particle states depending on the barrier width, as the phase time is always positive except when the particle velocity approaches the speed of light (negative phase time tending to zero). The interaction of the relativistic wave packet with the potential barrier generates superluminality. Inside this, forward and backward evanescent waves describe the particle or antiparticle. As references [16, 27] suggest, this interaction leads to a complex set of interferences between the incident and reflected waves that cause an acceleration of the incoming wave. However, the Heisenberg uncertainty principle may also explain superluminal tunneling. The energy confinement of the particle within the barrier reduces the time uncertainty, and consequently increases the one affecting the energy. Such high uncertainty leads to the transmitted wave exceeding the speed of light. We think this interpretation of the phenomenon we are investigating is more understandable and closer to quantum theory.
We must still explain the reason why an antiparticle behaves differently than a particle, i.e., why it almost always tunnels in superluminal mode. The negative energy of the antiparticle decreases the transmission probability that Eq. (12) provides. This is especially due to the strong increases of the hyperbolic function at the denominator and changes the interference mechanism between forward and backward waves. In other words, the barrier tends to reflect an antiparticle more than a particle. An antiparticle tunneling through the potential barrier U is equivalent to a particle that tunnels through a potential barrier (U + E). This provides a more physical explanation of why an antiparticle has a lower probability of penetrating the barrier. However, once the antiparticle overcomes the barrier edge, it becomes more energy confined than the particle and accelerates more strongly. In fact, comparing the trend of Figure 5 with that of Figure 8 shows that the tunneling time of the antiparticle state is much closer to zero than the particle one, whatever its energy is. This means the superluminality of the antiparticle state is greater than the superluminality of the particle state.
If the high-energy confinement (very high barriers) facilitates acceleration towards superluminal velocity, why does increasing β tend to decrease the phase time? Increasing the relativistic factor decreases the energy gap between the barrier and the particle/antiparticle. This reduces the energy confinement and, consequently, the uncertainty affecting the energy E. But because E is so high, it is small enough of an uncertainty to boost the particle towards ever higher superluminal velocities. In fact, the trends show that accelerating the particle/antiparticle within the barrier is a more important kinetic factor than the energy confinement.
Finally, the different mechanism of interference between the transmitted and reflected waves may also explain spinor distortion. This provides different accelerations for the principal and secondary spinor components . We can also state that the potential barrier contributes to spreading the wave packet. An earlier occurrence or delay of the secondary spinor component is equivalent to frequency changes, i.e., energy changes between the plane waves forming the wave packet.
Summarizing, we have considered tunneling of a relativistic free particle with half-integer spin through a one-dimensional potential barrier. The analysis suggests that the total transmission is almost always superluminal for antiparticle states and may be subluminal or superluminal for particle states. Within the potential barrier, the two components of the spinor behave as evanescent modes, and the high energy confinement accelerates the particle/antiparticle further towards the speed of light. Considering the interference between forward and backward evanescent waves may explain the phenomenon, but an easier explanation holds if we account for the Heisenberg uncertainty principle. As the particle/antiparticle velocity approaches the speed of light, the tunneling time tends to zero, making the acceleration divergent towards infinite values. The study also proves that the two spinor-transmitted components emerge from the potential barrier at different times, providing an anomalous distortion of the Dirac wave function. This distortion occurs in an opposite way for particles and antiparticles. More precisely, the secondary spinor component is delayed compared with the first one for particle states and is anticipated for antiparticle states. All of these results are innovative with respect to those available in the literature. They suggest a way to obtain a source of superluminal massive elementary particles: ultra-relativistic particles scattering through huge potential barriers. As discussed above, the probability of total transmission of the particle/antiparticle through the barrier strongly decreases as its incoming velocity approaches the speed of light. However, we cannot exclude the occurrence of the model used in this study a priori in a cosmological scenario.
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The Cramer rule can solve linear system (8). To simplify the mathematical formalism, we set: (A1)
The determinant of the coefficient matrix A is: (A2)
The four unknown quantities, i.e., the coefficients of the linear combination are: (A3)
The calculation of phase time requires knowledge of the complex number argument: (B1)
Using (A1) and considering that (χ − iΔk)2/i = (iΔk − χ)2 and (Δk − iχ)2/i = (iχ − Δk)2, we get: (B2)
Solving the squares of the two binomials in (B2), we obtain: (B3)
Denoting by z′ the denominator of (B3) and recalling that its inverse is z′−1 = z′/|z′|2 and that arctg(β /α) provides the argument of a complex number z = α + iβ, we obtain: (B4)
Some simple algebraic manipulations allow us to rewrite (B4) as: (B5)
If the barrier width a is large enough, then the fraction and (B5) becomes: (B6)
which is independent of the barrier width.
In this section we explain the steps to get the derivative of the argument function (B6). Using the chain rule shows: (C1)
The last derivative of (C1) is: (C2)
from which we obtain the particle phase time (16).
The argument function of the secondary component of spinor (4) is: (D1)
Before calculating its derivative, we have to explain the relativistic factor βas the function of the particle energy. We proceed as follows: (D2)
Now we calculate the derivative of function (D1) using the chain rule: (D3)
We have already calculated the first derivative in the square brackets in Appendix C. We calculate the second derivative using expression (D2): (D4)
from which we obtain the phase time (17) of the secondary spinor component.
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Published Online: 2017-11-10