Following the previous post about a relativistic classical particle in a laser field, here we have the quantum version where the center of the wavepacket still follows the classical trajectory.

This movie corresponds to a weak relativistic electron interacting with a laser field. The laser wavelength is 800 nm with an intensity
of 10^20 W/m^2. The propagation follows the z direction. The
electric field
is polarized along the x direction where the oscillation is observed.

As we can clearly see, the wavepacket spreading is significant even in a single oscillation! Of course, this information is absent in the classical model.

A particle in a laser field with fixed direction and infinite wavefront has analytic solutions in both the quantum and classical realms.
These solutions allow for the possibility of electromagnectic wavepackets modulated in the direction of motion.

The quantum solution of the Dirac equation was found by Volkov in the early days of quantum mechanics!

Volkov D M 1935 Z. Physik 94, 250

Contrary to expectations, the classical solutions appeared much later. The following book contains a very complete analysis of the classical case

Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) by William E. Baylis, BirkhĂ¤user; Corrected edition (January 12, 2004)

The trajectories for linear and circular polarization can be seen in the following movies:

I am preparing these notes in Mathematica, so, there are actual functions that perform symbolic and numerical calculations. If you do not have Mathematica, you can download the companion pdf documents or download the Mathematica CFD player for free at:

These lecture notes employ the language of Clifford algebras in two flavors: The Algebra of the Physical Space (APS) and the Space Time Algebra (STA).

Contrary to the literature about Clifford algebras in physics, I decided to heavily rely on matrix representations. Clifford algebras can be developed from elegant axiomatic principles where no matrix is necessary at all. Nevertheless, I personally saw that most people claim to be too busy for that. My hope is that a direct exposure of the matrix representation will give them a more familiar environment based on simple standard linear algebra.

In this post I will explain some of the main ideas furnished in the theoretical framework we refer to as Operational Dynamical Modeling (ODM). The formal publication can be found at:

[1] Operational Dynamic Modeling Transcending Quantum and Classical Mechanics, Denys I. Bondar, Renan Cabrera, Robert R. Lompay, Misha Yu. Ivanov, and Herschel A. Rabitz, Phys. Rev. Lett. 109, 190403, 2012

The early success of Lagrangian and Hamiltonian classical mechanics established the variational principle as the main tool of theoretical physics. Since then, the variational principle solidified its reputation in virtually all branches of fundamental physics and beyond. Considering such triumph, one may think that this technique could well be employed to deal with all the new challenges of theoretical physics. Nevertheless, there are important physical phenomena such as quantum decoherence and quantum dissipation that are inherently outside of the range of applicability of traditional Lagrangian and Hamiltonian treatments. The reason of this limitation is that the latter are only suitable to describe conservative systems that also maintain the quantum/classical information invariant. Therefore, systems undergoing energy dissipation and/or loss of information require an alternative approach. One such possibility is the application of stochastic processes that naturally addresses the loss of information.

In [1] we propose an alternative approach based on the crucial observation that the Ehrenfest equations can be used to model a very wide range of physical systems that can be quantum/classical and/or conservative/dissipative.

A first look at the Ehrenfest theorem

may give us the wrong impression that these equations can be easily reduced to Newton's equations; thus to classical mechanics. However, this is only true for quadratic potentials. In this case the Ehrenfest equations become a closed set of ordinary differential equations that exactly obey Newton's equation. Otherwise, there are higher order statistical moments of the position operator that prevent to turn the Ehrenfest equations into a consistent system of ordinary differential equations.

Much lesser known, the Ehrenfest equations can be written for classical mechanics in almost exactly the same form with one single critical difference: the position and momentum operators commute. Yes, classical mechanics can be expressed in the Hilbert space according to the Koopman-von Neumann mechanics, where the observables x and p commute.

Therefore, we conclude that the Ehrenfest equations shown above are compatible with both quantum and classical mechanics. In this sense, these equations transcend both quantum and classical mechanics implying that we have in hands something much more fundamental. From this perspective, the Ehrenfest equations coalesce to either quantum or classical mechanics only after we provide the algebra of the observable operators.

All this becomes really interesting when we engage with modifications of the Ehrenfest equations. For example, we could have a dissipative dynamics according to

where gamma is the dissipation constant. There is plenty of stuff in the literature and sometimes names such as quantum Brownian motion appear in this context. Nevertheless, no satisfactory quantum solution existed until we published the following paper

Wigner–Lindblad Equations for Quantum Friction, Denys I. Bondar, Renan Cabrera, Andre Campos, Shaul Mukamel, and Herschel A. Rabitz, J. Phys. Chem. Lett., 2016, 7 (9), pp 1632–1637

This solution overcomes all the shortcomings of previous proposals that appeared since the birth of quantum mechanics. In particular, the evolution is Lindbladian. This means that the quantum states maintain full quantum consistency without violating the uncertainty principle; no matter what the initial condition are and what the temperature is. In second place, the equations of motion are numerically very stable and easy to solve with the described methods in the paper.

In the near future we will present related work in the context of relativistic quantum mechanics, but there are many other opportunities that certainly go beyond physics.

Klein's paradox is beautifully visualized in the phase space through the relativistic Wigner function.

In introductory quantum mechanics we learn about quantum tunneling and how it allows the transmission of particles through potentials that are otherwise insurmountable in mechanics.

In relativity we have another twist, the transmission is not only present but it can be the dominant effect for strong potentials. In the following animation we illustrate this effect where we clearly see that the transmitted wave-packet is composed of antiparticles (negative momentum but moving to the positive direction).

This type of effect can be seen in effective Dirac materials in terms of an enhanced tunneling. The animation of such process is

These effects are well know in the literature. My contribution in [1] was to show that the presence of quantum decoherence does not affect them in a significant measure.

[1] Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz,
Dirac open-quantum-system dynamics: Formulations and simulations , Phys. Rev. A 94, 052111 (2016)
https://doi.org/10.1103/PhysRevA.94.052111

Relativistic states are much richer in features than those in non-relativity. One of the most important relativistic hallmarks is the ability to describe of antiparticles. We already know this from books of quantum mechanics, but it is in the phase space where we can visualize them in full glory.

Let us first watch the time-evolution of a free relativistic cat-state:

This movie might as well describe a non-relativistic cat state, so, no surprise (the dynamics is undergoing quantum coherence).

Now, let us observe the evolution of the corresponding particle-antiparticle coherent superposition:

In this case we observe that the antiparticle at the bottom advances to the positive direction despite of having a negative momentum. This type of dynamics is characteristic of antiparticles!

One critical piece of information is that both states are undergoing the same degree of quantum decoherence due to the contact with an environment. Nevertheless, the particle-antiparticle superposition maintains the interference robust. We just found a state that belongs to a free-decoherence space!

Al should also mention that the relativistic state happens to be a Majorana state.

[1] Dirac open-quantum-system dynamics: Formulations and simulations,

Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz Phys. Rev. A 94, 052111 – Published 14 November 2016