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:
 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  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  was to show that the presence of quantum decoherence does not affect them in a significant measure.
 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)
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.
 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
The work on non-relativistic open quantum systems is vast but comparatively, very little can be found involving relativity. This situation is not changing significantly despite of the urgent need to advance in an ever growing number of fields beyond high energy physics, such as solid state, cold atoms, trapped ions, quantum optics and more. This paper reviews the diverse literature and consolidates the fundamental principles to present a unified formalism for Relativistic Open Quantum Systems. As a result, for the first time in the literature, we are able to simulate some relativistic open quantum systems and analyze the effect of quantum decoherence. This allowed us to gain new insights in one of the most fundamental problems of physics such as the quantum to classical transition and to recognize antiparticles as a new potential resource in quantum information.
In this post I will highlight some items treated in the introduction of my paper describing the theoretical foundations of relativistic open quantum systems. More highlights will appear in future posts.
The formulation of the Manifestly Relativistic Covariant von Neumann Equation for the Dirac equation is absent from the literature in the sense that it is not explicitly stated or treated as it is the case for the non-relativistic analog. Nevertheless, there are many works on the relativistic Wigner function that can be analyzed and traced back to finally obtain
where P is the relativistic density operator state and D is the Dirac generator. Both operators are 4x4 matrices with each entry as an operator dependent of x and p. This equation is important because it can be considered as the first step to formulate relativistic open quantum systems because P can be a pure or a mixed state in general. Moreover, the only hope to formulate covariant environments must rely on Eq (6) as the underlying generator or coherent dynamics. It must be emphasized that the the study of relativistic covariant environments is completelyand utterly absentfrom the literature. Considering that the vacuum is arguably an example of such environment, this could lead to the discovery of insights on some of the most fundamental problems of modern physics
An important observation of Eq. (6) is that the time and space are treated on the same footing. The time is itself an operator, that can be parametrized by two degrees of freedom (rows and columns)! In the paper we show how to disentangle those two time parameters and obtain two equations of motion that can be integrated independently. Surprise, surprise, one of those equations is the von Neumann equation that can be written directly using the Dirac Hamiltonian. The second equation of motion is strange and never even suspected in the literature. This additional dynamical equation is not necessary to describe the propagation of an initial state defined at a fixed point in time. So, what is the use of this equation? Answer: It is required to describe the propagation of the state in a different inertial frame of reference.
A third important observation of Eq (6) above, is that the mass appears in the generator of motion D. In fact, as we show in the paper, this equation already incorporates the shell mass condition. This is interesting when we compare the corresponding classical Hamiltonian formalism where the mass is not hard-coded and appears later as an integral of motion. Somehow, the mass is diluted in the quantum to classical process.
Final observation in this post: The equation of motion (6) is not made with Hermitian generators of motion. In fact the state P is not Hermitian either !!!!