Friday, December 9, 2016

Relativistic Open Quantum Systems I: Foundations

I recently published a paper about Relativistic Open Quantum Systems

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 completely and utterly absent from 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 !!!!

One of the figures of the paper is now featured in the Kaleidoscope section of PRA:


The CUDA source code for my paper can be found at

The python notebook files that run specific examples can be found at

This link also contains many figures and many simulations not part of the paper.

Note that the file "" must be updated with the information of your own system.

It requires PyCUDA along with other standard python libraries and of course an NVIDIA graphics card.

Please ask me questions if you cannot run it.

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