tag:blogger.com,1999:blog-8391924215274794862024-02-18T21:02:08.074-08:00Renan Cabrera L. PhDDiverse topics in physics, mathematics and scientific computing.Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.comBlogger108125tag:blogger.com,1999:blog-839192421527479486.post-10801904639425542552017-10-13T18:07:00.000-07:002017-10-16T08:08:22.857-07:00Particles and Antiparticles, a mathematical difference.The discovery of antiparticles was a revolutionary event in physics and now we are capable to produce and analyze antimatter.<br />
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We are usually told that the world of antiparticles is like an exact copy of ours but with some peculiarities. The lack of evidence of large quantities of antimatter in the observable universe is an indication that the symmetry between particles and antiparticles is not perfect. However, from a mathematical perspective it is possible to argue about a significant asymmetry between particles and antiparticles.<br />
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Group theory tell us that the Dirac spinor is a double cover representation of the Special Lorentz Group denoted by SO(1,3). This group has a subgroup characterized for maintaining the direction of time. As a result, the operations in this group can be visualized as either spatial rotations or boosts of velocity. In other words, this subgroup has a simple and intuitive interpretation and can be associated with particles. Therefore, antiparticles are the complement of the total group with the subgroup of particles. This mathematical construction is called coset. Consequently the world of pure particles has the structure of a group while the world of pure antiparticles has the structure of a coset!<br />
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This argument can be illustrated in the following diagram where O_{+}(1,3) is the group that maintains the direction of time.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggono_t26DAjbBXH1RFqqxdpEtSJ_ktX-NXGr2bcV9By3nXLatiZn-m2xssLHpncAOUfP4x98WApr1Xu0uHFeV1mq62pg3mihqttPe6fHKx63cTnY3Ujg5dJrDs7vf2Mu_hz3VNKmTfPo/s1600/LorentzGroup.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="354" data-original-width="500" height="226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggono_t26DAjbBXH1RFqqxdpEtSJ_ktX-NXGr2bcV9By3nXLatiZn-m2xssLHpncAOUfP4x98WApr1Xu0uHFeV1mq62pg3mihqttPe6fHKx63cTnY3Ujg5dJrDs7vf2Mu_hz3VNKmTfPo/s320/LorentzGroup.jpg" width="320" /></a></div>
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The book that contains a detailed description of the group structure of spinors is<br />
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* Pertti Lounesto, Clifford Algebras and Spinors, Cambridge University Press, May 3, 2001 <br />
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-84753504842591704762017-10-07T16:21:00.000-07:002017-10-07T16:25:46.249-07:00Bounded states in a linear potentialThe Dirac equation with interaction in the mass is<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9_c-0y0_UwhrvDpieIXI1wtSijh-szNXE2CXOImulk5AEkX4SfurYYL3iFN9cG6Ns-OxPjc7B8KPt4ZTxS3a06WXIQ-jDfFtrKVn3ORr1dJoc7WY1yMKHFiomk80qtlwqix2q8aQm4Wg/s1600/Screenshot-2017-10-7+HostMath+-+Online+LaTeX+formula+editor+and+browser-based+math+equation+editor%25281%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="60" data-original-width="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9_c-0y0_UwhrvDpieIXI1wtSijh-szNXE2CXOImulk5AEkX4SfurYYL3iFN9cG6Ns-OxPjc7B8KPt4ZTxS3a06WXIQ-jDfFtrKVn3ORr1dJoc7WY1yMKHFiomk80qtlwqix2q8aQm4Wg/s1600/Screenshot-2017-10-7+HostMath+-+Online+LaTeX+formula+editor+and+browser-based+math+equation+editor%25281%2529.png" /></a></div>
<br />In this case the potential does not correspond to an electric potential, nevertheless, it is often used in the literature.
Something that may initially look surprising is the fact that this type of interaction supports bounded states for monotonically increasing potentials as shown in the figure below<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnoZwlK18QKqXrnik_S2eqyjiQ7gLDEGlnK1PpUJwDtqk9t2GGdPv8T72uP7rSfGAUTi7J3BTVTgHdI_yj1Tri-64c3a9HmH_lbHKQXfu1RUyvj1BZDE8h-kBAgTp-brrqn9cwalOBEPs/s1600/PseudoScalar.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="301" data-original-width="400" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnoZwlK18QKqXrnik_S2eqyjiQ7gLDEGlnK1PpUJwDtqk9t2GGdPv8T72uP7rSfGAUTi7J3BTVTgHdI_yj1Tri-64c3a9HmH_lbHKQXfu1RUyvj1BZDE8h-kBAgTp-brrqn9cwalOBEPs/s320/PseudoScalar.jpg" width="320" /></a></div>
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This has been reported in the literature for sometime but without much of an explanation.
The deeper insight came from the study of the Dirac equation in solid state physics where this type of system was recognized as the result of the contact of two topologically distinct phases: one with positive mass and the other one with negative mass. This contact gives origin to the so called <b>zero mode states</b> that have many interesting features and are the subject of intensive research today.<br />
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I published a generalization of this bounded state with additional nonlinear interactions in my recent paper at PRL<br />
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<a href="https://journals.aps.org/prl/accepted/14070Y2cTa61f86ba97d95a1f5d388e103887cbff">https://journals.aps.org/prl/accepted/14070Y2cTa61f86ba97d95a1f5d388e103887cbff</a><br />
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-3652883637396878002017-09-20T23:08:00.000-07:002017-09-26T12:43:54.469-07:00Relativistic electron in a strong laser field: Quantum spreading <br />
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.<br />
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/V1HpJeX5Wu0/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/V1HpJeX5Wu0?feature=player_embedded" width="320"></iframe></div>
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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.<br />
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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.<br />
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A second animation is shown below for a stronger laser field<br />
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<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/vY-cWWA5BQ0/0.jpg" src="https://www.youtube.com/embed/vY-cWWA5BQ0?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-87437799597154092462017-08-31T19:33:00.001-07:002017-10-07T15:34:08.260-07:00Relativistic classical electron in a laser field<br />
A particle in a laser field with fixed direction and infinite wavefront has analytic solutions in both the quantum and classical realms.<br />
These solutions allow for the possibility of electromagnectic wavepackets modulated in the direction of motion.<br />
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The quantum solution of the Dirac equation was found by Volkov in the early days of quantum mechanics!<br />
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Volkov D M 1935 Z. Physik 94, 250<br />
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Contrary to expectations, the classical solutions appeared much later. The following book contains a very complete analysis of the classical case<br />
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Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) by William E. Baylis, Birkhäuser; Corrected edition (January 12, 2004)<br />
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The trajectories for linear and circular polarization can be seen in the following movies:<br />
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Linear polarization<br />
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/G2lzFJHiApU/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/G2lzFJHiApU?feature=player_embedded" width="320"></iframe></div>
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Circular polarization<br />
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/iYzyMgsUyY8/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/iYzyMgsUyY8?feature=player_embedded" width="320"></iframe></div>
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An interactive Mathematica cdf document can be downloaded from<br />
<a href="http://www.princeton.edu/~rcabrera/Volkov_LinearPolarization.cdf">http://www.princeton.edu/~rcabrera/Volkov_LinearPolarization.cdf</a><br />
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The general theory is part of the lectures notes at<br />
<a href="https://github.com/cabrer7/Lectures-On-Relativity">https://github.com/cabrer7/Lectures-On-Relativity </a><br />
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News: The case with linear polarization was published at the Wolfram Demonstrations Project:<br />
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<a class="relatedLink" href="http://demonstrations.wolfram.com/author.html?author=Renan%20Cabrera" target="top">Renan Cabrera</a>"<a class="relatedSiteLink" href="http://demonstrations.wolfram.com/ClassicalRelativisticParticleInALinearlyPolarizedLaserField/" target="top">Classical Relativistic Particle in a Linearly Polarized Laser Field</a>"<br /> <a class="relatedSiteLink" href="http://demonstrations.wolfram.com/ClassicalRelativisticParticleInALinearlyPolarizedLaserField/" target="top">http://demonstrations.wolfram.com/ClassicalRelativisticParticleInALinearlyPolarizedLaserField/</a><br /> <a class="relatedSiteLink" href="http://demonstrations.wolfram.com/" target="top">Wolfram Demonstrations Project</a><br /> Published: September 15, 2017<br />
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-70822780288799701552017-05-10T16:38:00.001-07:002017-09-27T14:03:47.553-07:00Relativistic Dynamical Inversion RDI<h2 class="title mathjax">
Analytic solutions to coherent control of the Dirac equation and beyond </h2>
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My latest arxiv paper is online </div>
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<a href="https://arxiv.org/abs/1705.02001">https://arxiv.org/abs/1705.02001</a></div>
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This work introduces Relativistic Dynamical Inversion (RDI) as a technique to find analytic solutions to the Dirac equation.<br />
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Update: This paper was just accepted to appear at Phys. Rev. Lett.<br />
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<a href="https://journals.aps.org/prl/accepted/14070Y2cTa61f86ba97d95a1f5d388e103887cbff">https://journals.aps.org/prl/accepted/14070Y2cTa61f86ba97d95a1f5d388e103887cbff</a><br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqOO5VrKoLyKx9d1I4ptE43KCGI8tJbPeEb9jp6-VL7Sas9xZJ-AM0qzHYnuLdqUYCbczt8WkhkAht66ZCUUIaqRLPETT2VFsuUTpPPadUK7U2C1aaf9jpJy8MAp842FtXlCqrndfI82Q/s1600/SoftCoulomb-Ground-3D.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="365" data-original-width="361" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqOO5VrKoLyKx9d1I4ptE43KCGI8tJbPeEb9jp6-VL7Sas9xZJ-AM0qzHYnuLdqUYCbczt8WkhkAht66ZCUUIaqRLPETT2VFsuUTpPPadUK7U2C1aaf9jpJy8MAp842FtXlCqrndfI82Q/s320/SoftCoulomb-Ground-3D.jpg" width="316" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Ground state for a Dirac system confined by a combination of magnetic and electric fields.</td></tr>
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Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-8440394540520915302017-03-30T15:58:00.000-07:002017-04-05T15:25:03.309-07:00A course on Relativity using Clifford (geometric) algebras in Mathematica I am posting my lectures notes on <i><b>Classical and Quantum Relativistic Mechanics </b></i>at<br />
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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:<br />
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<a href="https://www.wolfram.com/cdf-player/">https://www.wolfram.com/cdf-player/</a><br />
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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). <br />
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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. <br />
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Two snapshots of randoms pages <br />
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-38262626704381704872016-12-26T23:06:00.003-08:002016-12-30T20:38:17.837-08:00Operational Dynamical Modelling<div style="text-align: justify;">
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: </div>
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[1] <i>Operational Dynamic Modeling Transcending Quantum and Classical Mechanics</i>, Denys I. Bondar, Renan Cabrera, Robert R. Lompay, Misha Yu. Ivanov, and Herschel A. Rabitz, Phys. Rev. Lett. 109, 190403, 2012</div>
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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. </div>
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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.<br />
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A first look at the Ehrenfest theorem </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKWZbMGwt3HqQMXMRAQISDD6UJ-pMnALO86KzgEO8ffz485twhSORGGZNkwARSXWo2AmkVdNnQUh05YQy3P-xa3l1ajlPtBD95X3WqYxv2T-io8u55Ww5srqBwuP2pLR_1V8MnrqZoMOo/s1600/Ehrenfest.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKWZbMGwt3HqQMXMRAQISDD6UJ-pMnALO86KzgEO8ffz485twhSORGGZNkwARSXWo2AmkVdNnQUh05YQy3P-xa3l1ajlPtBD95X3WqYxv2T-io8u55Ww5srqBwuP2pLR_1V8MnrqZoMOo/s320/Ehrenfest.png" width="320" /></a></div>
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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. </div>
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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.<br />
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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. <br />
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All this becomes really interesting when we engage with modifications of the Ehrenfest equations. For example, we could have a dissipative dynamics according to<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiT-3_5boAYdsPUqwhVdQikycG9Y1-rf7hxxANwt8cN3lRD4Q6AowRbXstz9DRSLoD6yMVHsJsy8EsAehmLl3SNsaghDxkDYS-QdF_dgtPpgB_ybyOqKlGShoCyUNpia2JaiaI1B6yNng/s1600/Ehrenfest-Dissipation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiT-3_5boAYdsPUqwhVdQikycG9Y1-rf7hxxANwt8cN3lRD4Q6AowRbXstz9DRSLoD6yMVHsJsy8EsAehmLl3SNsaghDxkDYS-QdF_dgtPpgB_ybyOqKlGShoCyUNpia2JaiaI1B6yNng/s320/Ehrenfest-Dissipation.png" width="320" /></a></div>
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<br />
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<i>Wigner–Lindblad Equations for Quantum Friction</i>, Denys I. Bondar, Renan Cabrera, Andre Campos, Shaul Mukamel, and Herschel A. Rabitz, J. Phys. Chem. Lett., 2016, 7 (9), pp 1632–1637<br />
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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. <br />
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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. <br />
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Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-32426627027782437282016-12-12T19:46:00.000-08:002016-12-12T19:48:42.887-08:00Relativistic Open Quantum Systems III: Klein's paradox<h2>
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Klein's paradox is beautifully visualized in the phase space through the relativistic Wigner function.<br />
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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. <br />
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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).<br />
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This type of effect can be seen in effective Dirac materials in terms of an enhanced tunneling. The animation of such process is<br />
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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.<br />
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[1] Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz,
<i> Dirac open-quantum-system dynamics: Formulations and simulations </i>, Phys. Rev. A 94, 052111 (2016)
<a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.052111">https://doi.org/10.1103/PhysRevA.94.052111 </a>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-3393285976445294822016-12-10T22:33:00.000-08:002016-12-12T19:46:51.778-08:00Relativistic Open Quantum Systems II: Majorana particlesRelativistic 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.<br />
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Let us first watch the time-evolution of a free relativistic cat-state:<br />
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This movie might as well describe a non-relativistic cat state, so, no surprise (the dynamics is undergoing quantum coherence).<br />
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Now, let us observe the evolution of the corresponding particle-antiparticle coherent superposition:<br />
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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!<br />
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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<i> free-decoherence</i> space!<br />
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Al should also mention that the relativistic state happens to be a <i><b>Majorana</b></i> state.<br />
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[1]<i> Dirac open-quantum-system dynamics: Formulations and simulations</i>, <br />
<div style="text-align: left;">
Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz Phys. Rev. A <b>94</b>, 052111 – Published 14 November 2016 </div>
<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-36808199550912000152016-12-09T00:37:00.001-08:002016-12-25T20:13:37.140-08:00Relativistic Open Quantum Systems I: Foundations<div style="text-align: left;">
I recently published a paper about <b>Relativistic Open Quantum Systems</b></div>
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<i>Dirac open-quantum-system dynamics: Formulations and simulations</i> </div>
<div style="text-align: left;">
Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz Phys. Rev. A <b>94</b>, 052111 – Published 14 November 2016 </div>
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<a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.052111">https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.052111</a> </div>
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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. </div>
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In this post I will highlight some items treated in the <b>introduction</b> of my paper describing the theoretical foundations of relativistic open quantum systems. More highlights will appear in future posts.</div>
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<li style="text-align: justify;">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<br /><div class="separator" style="clear: both;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgooMT_B2C6HX4uyurIEjfJXEtcxBxvfZ90PEOkvjJO5NYxsNDKmMeH9UfkA3hUbVDIdvYzFo8y8QJ2dSsNZWugmeIckNabz-wJ4zCwgiX0oVOnzLI0ar0lymabohQlh32gNJvBxOHiLXs/s1600/RelativisticVonNeumannEquation.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="31" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgooMT_B2C6HX4uyurIEjfJXEtcxBxvfZ90PEOkvjJO5NYxsNDKmMeH9UfkA3hUbVDIdvYzFo8y8QJ2dSsNZWugmeIckNabz-wJ4zCwgiX0oVOnzLI0ar0lymabohQlh32gNJvBxOHiLXs/s320/RelativisticVonNeumannEquation.jpg" width="320" /></a></div>
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 <span style="color: #073763;"><i><b>completely</b></i> </span><i><b><span style="color: #073763;">and utterly absent</span> </b></i>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 </li>
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<li style="text-align: justify;">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 <span style="color: #073763;"><i><b>two equations of motion</b></i></span> 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. </li>
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<li style="text-align: justify;">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 <span style="color: #073763;"><i><b>incorporates the shell mass condition</b></i></span>. 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. </li>
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<li> Final observation in this post: The equation of motion (6) is not made with Hermitian generators of motion. In fact <span style="color: #073763;"><i><b>the state P is not Hermitian</b></i></span> either !!!! </li>
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One of the figures of the paper is now featured in the Kaleidoscope section of PRA: <a href="http://journals.aps.org/pra/kaleidoscope/pra/94/5/052111">http://journals.aps.org/pra/kaleidoscope/pra/94/5/052111</a><br />
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<b>RUNNING THE CODE:</b><br />
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<span data-offset-key="s8bs-0-0" data-reactid="85"><span data-text="true">The CUDA source code for my paper can be found at </span></span></div>
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<span data-offset-key="1c0sl-0-0"><span data-text="true"><a href="https://github.com/cabrer7/PyWignerCUDA">https://github.com/cabrer7/PyWignerCUDA</a> </span></span></div>
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<span data-offset-key="a960r-0-0"><span data-text="true">The python notebook files that run specific examples can be found at </span></span></div>
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<a href="https://github.com/cabrer7/PyWignerCUDA/tree/master/instances/WignerDirac2D_4x4"><span data-offset-key="2blsu-0-0"><span data-text="true">https://github.com/cabrer7/PyWignerCUDA/tree/master/instances/WignerDirac2D_4x4</span></span></a></div>
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<span data-offset-key="2f9b4-0-0">This link also contains many figures and many simulations not part of the paper.</span></div>
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<span data-offset-key="8mkjj-0-0"><span data-text="true">Note that the file "pywignercuda_path.py" must be updated with the information of your own system.</span></span></div>
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<span data-offset-key="7kn1p-0-0"><span data-text="true">It requires PyCUDA along with other standard python libraries and of course an NVIDIA graphics card. </span></span></div>
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<span data-offset-key="4tssh-0-0"><span data-text="true">Please ask me questions if you cannot run it. </span></span></div>
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-40281742391658350682016-12-09T00:05:00.002-08:002016-12-11T00:22:34.686-08:00Relativistic quantum controlHere is a plot from my upcoming paper on<span style="color: #073763;"> <b>Relativistic Quantum Control</b></span>. In this particular example I show how to trap relativistic spin1/2 particles that obey the Dirac equation<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi96clBesJk4E1tQNXCCjrHfR75jolVY0DtHzHiVYAvK2SW2RrR3cd74KdBEnJnnDR7y_zyy1JECcwvulVBngFf4uiEEzdqB1wq2NfKbkYLJ_LRV5l7pSFSPr0roC-PcbdSHXPODixO6hE/s1600/Zeld512.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi96clBesJk4E1tQNXCCjrHfR75jolVY0DtHzHiVYAvK2SW2RrR3cd74KdBEnJnnDR7y_zyy1JECcwvulVBngFf4uiEEzdqB1wq2NfKbkYLJ_LRV5l7pSFSPr0roC-PcbdSHXPODixO6hE/s320/Zeld512.jpg" width="320" /></a></div>
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-70101253148806033782014-01-27T20:54:00.000-08:002016-12-13T13:50:49.625-08:00Quantum Mechanics in the phase space<div style="text-align: justify;">
The representation of quantum mechanics in the phase space (position-momentum) is carried out in terms of the Wigner function. This formulation has the advantage of providing an intuitive visualization of the state and allowing the introduction of interactions with the environment in the context of open quantum dynamics. However, the phase space has the disadvantage of requiring significantly more computational power. Fortunately, new algorithms and modern computational techniques such as GPU computing can be used to overcome this difficulty as I show in a sequence of papers [1,2]. These techniques can be even applied to relativistic systems [3]. </div>
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The following is a list of some sample simulations. Please click on the links to see the videos.</div>
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<ul>
<li><a href="http://www.youtube.com/watch?v=LOhdF6hKu9M"> Quantum tunnelling </a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlBkUnkkog_1KWmjKaFvrDVw16ut0KzEziLVEkfiopJOJKZyE-XhyphenhyphenmqqS9p2-wXQuJZyD038CLHVXdCKs0-6m69SF7o_GeWKzzY5T-N9hooUJCdMHOOZ_vtbrJcdaPhcQm0mThF66y1mY/s1600/tunneling.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlBkUnkkog_1KWmjKaFvrDVw16ut0KzEziLVEkfiopJOJKZyE-XhyphenhyphenmqqS9p2-wXQuJZyD038CLHVXdCKs0-6m69SF7o_GeWKzzY5T-N9hooUJCdMHOOZ_vtbrJcdaPhcQm0mThF66y1mY/s1600/tunneling.png" width="400" /></a><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/LOhdF6hKu9M/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/LOhdF6hKu9M?feature=player_embedded" width="320"></iframe></div>
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<li><a href="http://www.youtube.com/watch?v=_wrkDCnRBpg">Quartic potential</a></li>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhy2h8YQPMq7zvIVeynkNMhUKeoZ9WtETobVef4VphyyIpurI0aIBjDnjPWOaBxG7Dy0cU6DQVD3L-fhT3Ki8pAU_CLmitfbzYPwxkspSgfEpPpGJETIuBc7yY0rXPCmw8eBjBQENW068U/s1600/X4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhy2h8YQPMq7zvIVeynkNMhUKeoZ9WtETobVef4VphyyIpurI0aIBjDnjPWOaBxG7Dy0cU6DQVD3L-fhT3Ki8pAU_CLmitfbzYPwxkspSgfEpPpGJETIuBc7yY0rXPCmw8eBjBQENW068U/s1600/X4.png" width="400" /></a><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/_wrkDCnRBpg/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/_wrkDCnRBpg?feature=player_embedded" width="320"></iframe></div>
<ul>
<li><a href="http://www.youtube.com/watch?v=lvlIZyvQKws">Free particle evolution of a coherent superposition of two Gaussian states. </a></li>
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<li>Free particle evolution of an INCOHERENT superposition </li>
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[1] <i>Efficient method to generate time evolution of the Wigner function for open quantum systems</i>, Renan Cabrera, Denys I. Bondar, Kurt Jacobs, and Herschel A. Rabitz, Phys. Rev. A 92, 042122 – Published 28 October 2015<br />
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[2] <i>Efficient computations of quantum canonical Gibbs state in phase space,</i><br />
Denys I. Bondar, Andre G. Campos, Renan Cabrera, and Herschel A. Rabitz<br />
Phys. Rev. E 93, 063304 – Published 13 June 2016<br />
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[3] <i>Dirac open-quantum-system dynamics: Formulations and simulations,</i><br />
Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz<br />
Phys. Rev. A 94, 052111 – Published 14 November 2016<br />
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<br />Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-54937935668764019902014-01-27T20:04:00.000-08:002014-01-27T20:04:19.799-08:00Simulation of Relativistic quantum systems<br />
The effective and efficient simulation of relativistic quantum systems is now possible to accomplish with a common desktop computer thanks to the development of novel numerical algorithms and the emergence of modern computational techniques such as GPU computing.<br />
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I am in the process to post some animations of relativistic quantum systems in two and three dimensions<br />
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<ul>
<li><a href="http://www.youtube.com/watch?v=uJjOiNehZ3c">The Zitterbewegung in two dimensions</a> </li>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFJyKKPm5keoE2AGj8fnASrc35MP5sRTZIxrc3aW7pOLpEd0mS1_sjyBkPiS4Vqr9_oVE1OvZA0DZmQ96FPHgO2IkG__jd4kJ14tDT8Hk-5Y5iZEkEP7IMH99yKPhiSrdBv5UV3k3tEQ0/s1600/Zitterbewegung.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFJyKKPm5keoE2AGj8fnASrc35MP5sRTZIxrc3aW7pOLpEd0mS1_sjyBkPiS4Vqr9_oVE1OvZA0DZmQ96FPHgO2IkG__jd4kJ14tDT8Hk-5Y5iZEkEP7IMH99yKPhiSrdBv5UV3k3tEQ0/s1600/Zitterbewegung.png" height="106" width="320" /></a></div>
<br />
<br />
<ul>
<li><a href="http://www.youtube.com/watch?v=ph57kBezkpE">Klein paradox in two dimensions</a> </li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWnpQnyK8MK3OIGKcxcHKx0olqSOvfjkSZUrAZ__POtQVzMfjIWpFv7e2GeGkvKVtsMUSr1RLJzzOj7nhFW_IIzALLz1LNDIbcCBH7FCxTB67-WbH_6ZSEJF7JpBWMi-bqHsQ3vDT3gxc/s1600/Klein2D.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWnpQnyK8MK3OIGKcxcHKx0olqSOvfjkSZUrAZ__POtQVzMfjIWpFv7e2GeGkvKVtsMUSr1RLJzzOj7nhFW_IIzALLz1LNDIbcCBH7FCxTB67-WbH_6ZSEJF7JpBWMi-bqHsQ3vDT3gxc/s1600/Klein2D.png" height="160" width="320" /></a></div>
<br />
<ul>
<li><a href="http://www.youtube.com/watch?v=x3vH-yULs64">Klein paradox in three dimensions</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjno9YA9ShX_6v9fYxs3rKunv4lMWVK-L3x6sTVJ46V3ryzLQVFFJfiTG8-MwkkWYGg7y0F0rAuEMFGoVHeGh2VHrdyXbhJJi2Tu3qsbThb32TvYhXZZS4xwAvcIjiiK0Ns9O3xdwc5PjA/s1600/Klein3D.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjno9YA9ShX_6v9fYxs3rKunv4lMWVK-L3x6sTVJ46V3ryzLQVFFJfiTG8-MwkkWYGg7y0F0rAuEMFGoVHeGh2VHrdyXbhJJi2Tu3qsbThb32TvYhXZZS4xwAvcIjiiK0Ns9O3xdwc5PjA/s1600/Klein3D.png" height="320" width="313" /></a></div>
<br />
Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-15364644517969743932012-02-23T17:43:00.000-08:002012-02-23T17:43:37.387-08:00Split operator propagator<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br />
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This is a frameshot from the propagation of a 2D wavefunction<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCjxoQR3S6qCn0vUVP516dcoPUO8-cLPqs9yG33izE2Zh13p3YNEU3A82JG4cQg7I5HqJZc9KvrwhOD3Fy72E8eRFPeHO-AjOGwnp-FlIpHMx0W4YxdFbA6MVwTTEMvBPatSX5wcHQItk/s1600/splitoperator2D.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="310" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCjxoQR3S6qCn0vUVP516dcoPUO8-cLPqs9yG33izE2Zh13p3YNEU3A82JG4cQg7I5HqJZc9KvrwhOD3Fy72E8eRFPeHO-AjOGwnp-FlIpHMx0W4YxdFbA6MVwTTEMvBPatSX5wcHQItk/s320/splitoperator2D.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCjxoQR3S6qCn0vUVP516dcoPUO8-cLPqs9yG33izE2Zh13p3YNEU3A82JG4cQg7I5HqJZc9KvrwhOD3Fy72E8eRFPeHO-AjOGwnp-FlIpHMx0W4YxdFbA6MVwTTEMvBPatSX5wcHQItk/s1600/splitoperator2D.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br />
</a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCjxoQR3S6qCn0vUVP516dcoPUO8-cLPqs9yG33izE2Zh13p3YNEU3A82JG4cQg7I5HqJZc9KvrwhOD3Fy72E8eRFPeHO-AjOGwnp-FlIpHMx0W4YxdFbA6MVwTTEMvBPatSX5wcHQItk/s1600/splitoperator2D.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br />
</a></div>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-68486551584128092882012-02-12T00:02:00.000-08:002012-05-05T20:58:18.195-07:00Conservation of Shannon information in classical mechanicsThere are many places where to find the proof of the conservation of the quantum Shannon information under unitary evolution. However, I just could not find the equivalent proof that everybody talks in classical mechanics, so here it goes. <br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=S%20=%20-%20%5Cint%20dxdp%5C,%20%5Crho%5C,%20%5Cln%20%5Crho" target="_blank"><img src="http://latex.codecogs.com/gif.latex?S%20=%20-%20%5Cint%20dxdp%5C,%20%5Crho%5C,%20%5Cln%20%5Crho" title="S = - \int dxdp\, \rho\, \ln \rho" /></a><br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Cdot%7BS%7D%20=%20-%20%5Cint%20dxdp%5C,%20%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D%5C,%20%5Cln%20%5Crho%20@plus;%20%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Cdot%7BS%7D%20=%20-%20%5Cint%20dxdp%5C,%20%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D%5C,%20%5Cln%20%5Crho%20+%20%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D" title="\dot{S} = - \int dxdp\, \frac{\partial \rho}{\partial t}\, \ln \rho + \frac{\partial \rho}{\partial t}" /></a>
<br />
<br />
The Lioville equation can be expressed in terms of a self-adjoint operator as observed by Koopman and von Neumann, such that<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=i%20%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D%20=%20%5Cmathcal%7BL%7D%20%5Crho" target="_blank"><img src="http://latex.codecogs.com/gif.latex?i%20%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D%20=%20%5Cmathcal%7BL%7D%20%5Crho" title="i \frac{\partial \rho}{\partial t} = \mathcal{L} \rho" /></a>
<br />
<br />
Given that the Liouvillian follows the Leibniz rule, which is true for conservative systems<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Cmathcal%7BL%7D%20f%28%5Crho%29%20=%20f%27%20%5Cmathcal%7BL%7D%5Crho" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Cmathcal%7BL%7D%20f%28%5Crho%29%20=%20f%27%20%5Cmathcal%7BL%7D%5Crho" title="\mathcal{L} f(\rho) = f' \mathcal{L}\rho" /></a><br />
<br />
it is now easy to prove that<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Cdot%7BS%7D%20=%200" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Cdot%7BS%7D%20=%200" title="\dot{S} = 0" /></a><br />
<br />
This fact is in complete contrast with respect to the thermodynamical entropy, which can be fundamentally irreversible.<br />
<br />
<br />
-This entry was powered by http://www.codecogs.com/latex/eqneditor.phpRenan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-69904502557331412402012-02-09T13:31:00.000-08:002012-02-10T08:24:43.365-08:00DUSTDUST stands for Differential Unitary Space Time coding. This coding method is used in the transmission of information from multiple antennas to multiple antennas. <br />
<br />
The transmission of information in a block can be modelled with the following equation<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=X = \sqrt{\rho} S H @plus; W" target="_blank"><img src="http://latex.codecogs.com/gif.latex?X = \sqrt{\rho} S H + W" title="X = \sqrt{\rho} S H + W" /></a><br />
<br />
where S is the input matrix, H is the channel matrix, \rho is a scalar coefficient that modulates the channel and W is the noise.<br />
<br />
DUST encodes the information in a unitary matrix U_n, which is indirectly encoded through a sequence of two S unitary blocks as<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=S_n = U_n S_{n-1}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?S_n = U_n S_{n-1}" title="S_n = U_n S_{n-1}" /></a><br />
<br />
If H is full-rank and the channel does not change significantly in the time the two blocks are transmitted. The unitary matrix U can be recovered from the sequence of two received unitary blocks X without! the need to know the channel H as<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=X_n = U_n X_{n-1} @plus; W_n^{\prime}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?X_n = U_n X_{n-1} + W_n^{\prime}" title="X_n = U_n X_{n-1} + W_n^{\prime}" /></a><br />
<br />
where the new noise is defined as<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=W_n^{\prime} = W_n - U_n W_{n-1}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?W_n^{\prime} = W_n - U_n W_{n-1}" title="W_n^{\prime} = W_n - U_n W_{n-1}" /></a><br />
<br />
-This entry was powered by http://www.codecogs.com/latex/eqneditor.phpRenan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-2826545321280740122012-02-07T11:58:00.000-08:002012-02-11T20:07:23.911-08:00Random Unitary MatricesThe generation of random unitary matrices has many applications in many fields.<br />
The simplest method is based on the QR decomposition of a random complex matrix with independent elements following a Normal distribution.<br />
<br />
In Mathematica one can generate a random complex number with Normal distribution for both the real and imaginary parts with the following function <br />
<br />
<span style="font-family: "Courier New",Courier,monospace;">NormalRandomComplex[] := (#[[1]] + #[[2]]*I) &@</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> RandomReal[MultinormalDistribution[{0, 0}, {{1, 0}, {0, 1}}]]</span><br />
<br />
where <span style="font-family: "Courier New",Courier,monospace;">{0, 0}</span> specifies the origin and <span style="font-family: "Courier New",Courier,monospace;">{{1, 0}, {0, 1}}</span> the covariance matrix (standard deviation 1)<br />
<br />
The function that generates the random n x n unitary matrix is <br />
<br />
<div style="font-family: "Courier New",Courier,monospace;">RandomUnitaryMatrix[n_] := Module[{z, q, r},<br />
z = Table[NormalRandomComplex[], {n}, {n}];<br />
{q, r} = QRDecomposition[z];<br />
q.DiagonalMatrix[Sign[Tr[r, List]]]</div><span style="font-family: "Courier New",Courier,monospace;">] </span><br />
<br />
In this code, the QR decomposition of the complex random matrix with normal distribution outputs a unitary matrix "q" and a triangular matrix "r". The sought random unitary matrix is a correction of "q", which must be multiplied by a diagonal matrix filled with 1 and -1 according to the sign of the corresponding diagonal element of "r". <br />
<br />
<br />
<br />
<br />
<br />
<span style="font-family: "Courier New",Courier,monospace;"><br />
</span><br />
<span style="font-family: "Courier New",Courier,monospace;"><br />
</span>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-59200619477795387772012-01-19T12:42:00.000-08:002012-01-19T12:42:00.350-08:00New numerical algorithmsTwo recent developments on numerical analysis have called my attention<br />
<br />
A new very efficient sparse Fourier transform <br />
<a href="http://web.mit.edu/newsoffice/2012/faster-fourier-transforms-0118.html">http://web.mit.edu/newsoffice/2012/faster-fourier-transforms-0118.html</a><br />
<br />
A significantly faster algorithm for matrix multiplication<br />
<a href="http://www.newscientist.com/article/mg21228422.500-mathematical-matrix-multiplier-sees-first-advance-in-24-years.html">http://www.newscientist.com/article/mg21228422.500-mathematical-matrix-multiplier-sees-first-advance-in-24-years.html</a>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com1tag:blogger.com,1999:blog-839192421527479486.post-36726655614386499792011-12-08T20:31:00.000-08:002011-12-08T20:33:26.861-08:00De Broglie waves and relativityAnother extremely interesting connection between relativity and quantum mechanics is that the quantum De Broglie waves can be understood as an effect of the relativistic desynchronization of clocks given that the clocks rotate with a frequency proportional to the total energy of the particle. <br />
<br />
Imagine a train with a long chain of synchronized clocks from one extreme to the other with the synchronization established according to an observer moving along with the train. Another person standing still on the train station will conclude that the clocks are actually not synchronized (what he would see is a different thing). If the train is moving to the right, the clocks on the right will be ahead respect to the clocks on the left. The De Broglie wave length will be the length that stretches 12 hours of desynchronization. <br />
<br />
<div class="citationLine"><span class="italic">[1] W. Baylis, Canadian Journal of Physics</span><a href="http://www.nrcresearchpress.com/doi/abs/10.1139/p07-121">, 2007, 85:(12) 1317-1323, 10.1139/p07-121</a></div>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-16037182648352710272011-12-06T11:48:00.000-08:002012-02-10T13:31:10.409-08:00The ball is round!The first thing we learn in a course of relativity is the relativistic length contraction effect for moving objects relative to an observer. However, what one would actually see is a different thing ! because another important effect must be considered; the time required for the light to arrive from the different points of the observed object to our eyes. The combined effect of the Lorentz contraction along with the time delay was investigated reaching to curious results. For example, a relativistic ball remains with a round appearance!, as discussed by Penrose in the following paper <br />
<br />
<a href="http://adsabs.harvard.edu/abs/1959PCPS...55..137P">http://adsabs.harvard.edu/abs/1959PCPS...55..137P</a><br />
<br />
A nice visulaization can be found at<br />
<br />
<a href="http://www.spacetimetravel.org/ueberblick/ueberblick1.html">http://www.spacetimetravel.org/ueberblick/ueberblick1.html</a>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-29754757058433213142011-11-07T10:50:00.000-08:002012-02-22T13:06:59.213-08:00Thermodynamical entropy and Shannon informationThe moment I read about information theory I thought that it was evident that the thermodynamical entropy was completely equivalent; the same thing, just measured in different units with the Boltzmann constant in the middle. I was shocked when I read that this issue is still under debate today. There are two books that share the idea that the thermodynamical entropy is fundamentally a measure of information <br />
<ul><li>A Farewell To Entropy: Statistical Thermodynamics based on Information. By Arieh Ben-Naim</li>
</ul><ul><li>E.T. Jaynes: Papers on Probability, Statistics and Statistical Physics. By R.D. Rosenkrantz (Editor)<br />
</li>
</ul>The main issue is that the Shannon information is fundamentally conserved, while the thermodynamical entropy may describe irreversible systems where it is not conserved. More about this topic can be found at<br />
<ul><li><a href="http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory">http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory</a></li>
</ul><br />
<br />
<br />
<h1 class="parseasinTitle "><span id="btAsinTitle"></span></h1>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-2062155993366302652011-11-01T14:47:00.000-07:002011-11-10T12:25:28.828-08:00PyNewtonCUDADenys Bondar and I, just released to the public: PyNewtonCUDA<br />
<br />
<a href="https://code.google.com/p/py-newton-cuda/">https://code.google.com/p/py-newton-cuda/</a><br />
<br />
This program is able to propagate a large number of classical particles using GPU technology. The code is written in Python and PyCUDA, which allows to write high-level code for GPU programming while maintaining very high performance.<br />
<br />
<br />
You can download the code with the following command<br />
<br />
<tt>svn checkout https://py-newton-cuda.google.com/svn/trunk PyNewtonCUDA</tt>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-8408937216484048292011-06-24T23:31:00.000-07:002011-11-05T14:43:49.519-07:00Classical Relativistic Many-Body DynamicsI found a book on the interesting and very difficult topic of many-body classical relativistic theory <br />
<h1 class="headline" style="font-weight: normal;"><span style="font-size: small;"><b>Classical Relativistic Many-Body Dynamics</b></span></h1><b>Trump</b>, M.A., <b>Schieve</b>, W.C<br />
Springer, <a class="greyRaquoLink" href="http://www.springer.com/series/6001">Fundamental Theories of Physics</a>, Vol. 103<br />
<br />
This relativistic covariant many-body formulation overcomes the no go theorem by Jordan and Sudarshan [1] by giving up the locality of the interactions and the individual mass-shell conditions. These are not simply cosmetic upgrades but have serious consequences on how we should understand classical physics.<br />
<br />
"The single particle has no significance; it is the whole system that counts" --Lanczos<br />
<br />
<br />
[1] FROM CLASSICAL TO<br />
QUANTUM MECHANICS<br />
Giampiero Esposito, Giuseppe MarmoRenan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com0tag:blogger.com,1999:blog-839192421527479486.post-34533480169828980012011-06-03T05:33:00.000-07:002016-12-13T14:29:24.732-08:00Geo-coordinates for citations on scientific papersHere I draw a map for the places that either published a paper mentioning the word <i>Bures measure</i> or were referenced by somebody who used the word <i>Bures measure </i><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx_bK9xL8Aa7s3fDm9MkZZyk2_XAyfvxe6R8eeTztI-WEkj4NzpuMVFzz9qTiqQneCYUXS0vFMXvW8txG0rwArRA0Qt0S8wzNj06SLuHw__BJBRYpLxLJasPbYKXC4Xu8X1RJySUd2nGo/s1600/BuresGoeCoordinates.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx_bK9xL8Aa7s3fDm9MkZZyk2_XAyfvxe6R8eeTztI-WEkj4NzpuMVFzz9qTiqQneCYUXS0vFMXvW8txG0rwArRA0Qt0S8wzNj06SLuHw__BJBRYpLxLJasPbYKXC4Xu8X1RJySUd2nGo/s320/BuresGoeCoordinates.png" width="320" /></a></div>
The corresponding plot with the citations as links was created by my brother Ruben Cabrera<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbsDZthVgdMB1vKzH-dwwuX_bf6ojU4olhflD95lMizB2RQXpGKdIt_oqeVPAcX3rAl9f36W7dKaSgNr-G7gy1kVihHLwNg_l4pLfz0NU65pihVb4lUmBQ41l_99JeLKpmXmD9V0t-bDQ/s1600/citations_buresmeasure.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="133" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbsDZthVgdMB1vKzH-dwwuX_bf6ojU4olhflD95lMizB2RQXpGKdIt_oqeVPAcX3rAl9f36W7dKaSgNr-G7gy1kVihHLwNg_l4pLfz0NU65pihVb4lUmBQ41l_99JeLKpmXmD9V0t-bDQ/s320/citations_buresmeasure.png" width="320" /></a></div>
<br />
<br />
The total number of papers in consideration were 662. The extraction of coordinates from heterogeneous references is an expensive computational task.Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com1tag:blogger.com,1999:blog-839192421527479486.post-78240174874733407312011-05-19T17:07:00.000-07:002011-05-19T18:13:11.444-07:00Mathematica CDF documentsThis is my first Mathematica cdf document on the Internet.<br />
The most remarkable feature of a cdf document is that it can be posted on a web site and evaluated by anybody with a web browser and the freely available <a href="http://www.wolfram.com/cdf-player/">cdf player plugin</a>.<br />
<br />
<img alt="Graph demo" border="0" height="320" id="Graph_applet" src="http://dl.dropbox.com/u/24239307/graphs.png" width="290" /><br />
Note.- Unfortunately, it only works on Windows and Mac.<br />
<script src="http://www.wolfram.com/cdf-player/plugin/v1.0/cdfplugin.js" type="text/javascript">
</script><script type="text/javascript">
// <![CDATA[
var cdf = new cdf_plugin();
cdf.addCDFObject("Graph_applet", "http://dl.dropbox.com/u/24239307/graphs.cdf", 603,609);
// ]]>
</script>Renan Cabrera L. Ph.D.http://www.blogger.com/profile/01274204455427708304noreply@blogger.com4