It is known that relativistic wavefunctions formally propagate beyond the light cone when the propagator is limited to the positive energy sector. By construction, this is the case for solutions of the Salpeter (or relativistic Schrödinger) equation or for Klein–Gordon and Dirac wavefunctions defined in the Foldy–Wouthuysen representation. In this work, we quantitatively investigate the degree of non-causality for free propagation for different types of wavepackets that all initially have a compact spatial support. In the studied examples, we find that non-causality appears as a small transient effect that can in most cases be neglected. We display several numerical results and discuss the fundamental and practical consequences of our findings concerning this peculiar dynamical feature.
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.
Relativistic diffraction in time wave functions can be used as a basis for causal scattering waves. We derive such exact wave function for a beam of Dirac and Klein-Gordon particles. The transient Dirac spinors are expressed in terms of integral defined functions which are the relativistic equivalent of the Fresnel integrals. When plotted versus time the exact relativistic densities show transient oscillations which resemble a diffraction pattern. The Dirac and Klein-Gordon time oscillations look different, hence relativistic diffraction in time depends strongly on the particle spin.
In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [ Phys. Rev. X , 4 , 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.
The principal differences between non-relativistic and relativistic quantum mechanics are: the existence of negative energy states, and the natural emergence of spin within a relativistic framework. We consider both these effects in the creation and evolution of superoscillating wavepackets. Superoscillations are a phenomena in which a function can oscillate faster than it’s fastest Fourier component. As a trade-off they suffer by being exponentially smaller than conventional oscillations. Previous studies of superoscillations within the construct of the Schrödinger equation have identified two key features: disappearance after a time,
td, and an asymmetrical evolution either side of a line in space (the wall effect). In this thesis, we expand this work into a relativistic framework by applying it to the Klein-Gordon and Dirac equations, to do this we require a thorough examination of 1+1 dimensional relativistic propagators. Although initially derived interms of Bessel functions, these have two key limits which allow for simpler calculation: the light-cone limit
(x→x0+ct) and the WKB limit
(h→0). Positive and negative energy superoscillations are best described in the WKBlimit. Through application of asymptotic integration, we find that both positiveand negative energy superoscillations possess a disappearance time and wall effect. For both Klein Gordon and Dirac equations,
td is equal. This implies, in terms of disappearance spin has no effect on the evolution of a superoscillatory, relativistic wavefunction of positive or negative energy. However, relativistic superoscillations disappear faster than non-relativistic superoscillations, and in each case, the disappearance time tends to a finite value as a superoscillatory parameter is increased. For non-relativistic superoscillations
td→1/2 compared to
td→0.3715 for the relativistic case. The wall effect is also noticeably different in relativistic and non-relativistic contexts. The walls appear at different points in space and, despite the relativistic wall effect being evident from
t=0, within the Schrödinger equation, it doesn’t appear until a time of
t=1/32. This contrast in the wall effects of both cases is the leading cause of the inequality of the disappearance times. The asymmetries between the wall effects are caused by differences in the behaviour of saddles (complex momenta) found when evaluating the wavefunction as an integral over the propagator. Again, there is no disparity between the walls of the Klein-Gordon or Dirac equation; spin does not affect the evolution of positive ornegative energy superoscillations. Where positive and negative energy wavefunctions appear as
h→0, mixed energy superoscillations appear at the light-cone. Mixed energy superoscillations do not exhibit the wall effect and neither do they exist in a non-relativistic description. However, they do have a disappearance time. In contrast to the positive and negative energy states,
td→0 as a superoscillatory parameter is increased. It is within a mixed energy construct that the effect of spin on the evolution of relativistic superoscillations appears; one of the components of the Dirac equation does not superoscillate. This is caused by this term existing at the WKB limit as opposed to the light-cone.
We obtain the solution of a relativistic wave equation and compare it with the solution of the Schr\"odinger equation for a source with a sharp onset and excitation frequencies below cutoff. A scaling of position and time reduces to a single case all the (below cutoff) nonrelativistic solutions, but no such simplification holds for the relativistic equation, so that qualitatively different ``shallow'' and ``deep'' tunneling regimes may be identified relativistically. The nonrelativistic forerunner at a position beyond the penetration length of the asymptotic stationary wave does not tunnel; nevertheless, it arrives at the traversal (semiclassical or B\"uttiker-Landauer) time $\ensuremath{\tau}.$ The corresponding relativistic forerunner is more complex: it oscillates due to the interference between two saddle-point contributions and may be characterized by two times for the arrival of the maxima of lower and upper envelopes. There is in addition an earlier relativistic forerunner, right after the causal front, which does tunnel. Within the penetration length, tunneling is more robust for the precursors of the relativistic equation.