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Wigner Function for Spin-1/2 Fermions in Electromagnetic Fields

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Wigner Function for Spin-1/2 Fermions in Electromagnetic Fields Johann Wolfgang Goethe University Institute for Theoretical Physics University of Science and Technology of China Department of Modern Physics and Interdisciplinary Center for Theoretical Study Ph. D. Thesis Wigner Function for Spin-1/2 Fermions in Electromagnetic Fields Xin-Li Sheng October, 2019 arXiv:1912.01169v1 [nucl-th] 3 Dec 2019 Supervisors: Prof. Dr. Dirk H. Rischke, Institute for Theoretical Physics, Johann Wolfgang Goethe University. Prof. Dr. Qun Wang, Department of Modern Physics and Interdisciplinary Center for Theoretical Study, University of Science and Technology of China. Wigner function for spin-1/2 fermions in electromagnetic fields Xin-li Sheng1, 2 1Institute for Theoretical Physics, Goethe University, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: 07/24/19) I Abstract We study the Wigner function for massive spin-1/2 fermions in electromagnetic fields. The covariant Wigner function is a four by four matrix function in 8-dimensional phase space fxµ; pµg, whose compo- nents give various physical quantities such as the particle distribution, the current density, and the spin distribution, etc.. The kinetic equations for the Wigner function are obtained from the Dirac equation. We derive the Dirac form equations with first order differential operators, as well as the Klein-Gordon form equations with second order differential operators, both are matrix equations in Dirac space. We prove that some component equations are automatically satisfied if the rest are fulfilled, which means both the Dirac form and the Klein-Gordon form equations have redundancy. In this thesis two methods are proposed for calculating the Wigner function, which are proved to be equivalent. In addition to the covariant Wigner function, the equal-time Wigner function will also be introduced. The equal-time one is a function of time and 6-dimensional phase space variables fx; pg, which can be derived from the covariant one by taking an integration over energy p0. The equal-time Wigner function is not Lorentz-covariant but it is a powerful tool to deal with dynamical problems. In this thesis, it is used to study the Schwinger pair-production in the presence of an electric field. The Wigner function can be analytically calculated following the standard second-quantization procedure. We consider three cases: free fermions with or without chiral imbalance, and fermions in constant magnetic field with chiral imbalance. The computations are achieved via firstly deriving a set of orthonormal single- particle wavefunctions from the Dirac equation, then constructing the quantized field operator, and finally inserting the field operator into the Wigner function and determine the expectation values of operators under the wave-packet description. The Wigner functions are computed to leading order in spatial gradients. In Strong electric field the vacuum can decay into a pair of particle and anti-particle. The pair-production process is studied using the equal-time Wigner function. General solutions are obtained for pure constant electric fields and for constant parallel electromagnetic fields. We also solve the case of a Sauter-type electric field numerically. For an arbitrary space-time dependent electromagnetic field, the Dirac equation does not have an analytical solution and neither has the Wigner function. A semi-classical expansion with respect to the reduced Planck’s constant ~ are performed for the Wigner function as well as the kinetic equations. We calculate the Wigner function (and all of its components equivalently) to leading order in ~, in which order the spin component start playing a role. Up to this order, the Wigner function contains four independent degrees of freedoms, three of which describe the polarization density and the remaining one describes the net particle number density. A generalized Bargmann-Michel-Telegdi (BMT) equation and a generalized Boltzmann equation are obtained for these undetermined parts, which can be used to construct spin-hydrodynamics in the future. Using analytical results and semi-classical solutions, we compute physical quantities in thermal equilib- rium. In semi-classical expansion, we introduce the chiral chemical potential µ5 in the thermal distribution. This naive treatment is straightforward extension of the massless case but provides a good estimate of phys- ical quantities when µ5 is comparable or smaller than the typical energy scale, i.e., the temperature in a II thermal system. Meanwhile, by making comparison of the results of the semi-classical expansion and the ones in a constant magnetic field, we find that the semi-classical method works well for the chiral effects, including the Chiral Magnetic Effect, the Chiral Separation Effect, as well as the energy flux along the direction of the magnetic field. But when the mass and chemical potentials are much larger than the tem- perature, the semi-classical results over estimate these chiral effects. The magnetic field strength dependence of physical quantities is discussed. If we fix the thermodynamical variables, the net fermion number density, energy density, and the longitudinal pressure are proportional to the field strength, while the axial-charge density and the transverse pressure are inversely proportional to it. Schwinger pair-production rates in a thermal background are computed for a Sauter-type electric field and a constant parallel electromagnetic field, respectively. For the Sauter-type field, the total number of newly generated pairs is proportional to the field strength and the life time of the field. On the other hand, a parallel magnetic field will enhance the pair-production rate. Due to Pauli’s exclusion principle, the creation of pairs is forbidden for particles already exsiting in the same quantum state. Thus in both cases, the pair-production rate is proved to be inversely proportional to the chemical potential and temperature. Keywords: Wigner function, electromagnetic fields, chiral effect, pair-production. III Zusammenfassung In dieser Arbeit haben wir uns mit dem Wignerfunktionsansatz für Spin-1/2-Teilchen beschäftigt und diese Herangehensweise verwendet, um die chiralen Effekte und die Paarbildung in Gegenwart eines elektromagnetischen Feldes zu untersuchen. Die Wignerfunktion ist als Quasiverteilungs- funktion im Phasenraum definiert. Die Wignerfunktion ist eine komplexe 4 × 4-Matrix, die in die Generatoren der Clifford-Algebra Γi zerlegt werden kann. Die Zerlegungskoeffizienten werden gemäß ihrer Transformationseigenschaften unter Lorentz-Transformationen und Paritätsinversion jeweils als Skalar, Pseudoskalar, Vektor, Axialvektor und Tensor identifiziert. Sie können nach Integration über den Impuls mit verschiedenen Arten von makroskopischen physikalischen Größen wie dem Fermionstrom, der Spinpolarisation und dem magnetischen Dipolmoment in Beziehung gesetzt werden. Sie werden also als die Dichten im Phasenraum interpretiert. Da die Wignerfunktion mit Hilfe des Dirac-Feldes konstruiert wird, haben wir die kinetischen Gleichungen für die Wignerfunktion aus der Dirac-Gleichung erhalten. In dieser Arbeit haben wir die Dirac-Form-Gleichung abgeleitet, die im Differentialoperator linear ist. Daneben haben wir auch die Klein-Gordon-Form-Gleichung erhalten, die die Operatoren zur zweiten Ordnung enthält. Diese Gleichungen werden dann wie auch die Wignerfunktion selbst in Γi zerlegt, sodass sie ein Sys- tem mehrerer partieller Differentialgleichungen (PDG) liefern. Glücklicherweise sind die zerlegten Gleichungen nicht unabhängig voneinander. Durch Eliminieren der redundanten Gleichungen er- hielten wir zwei Möglichkeiten, die Lösung für die Wignerfunktion im massiven Fall zu bestimmen. Diese Redundanz beruht auf der Tatsache, dass die Vektor- und Axialvektorkomponenten Vµ, Aµ der Wignerfunktion in Form der Skalar-, Pseudoskalar- und Tensorkomponenten F, P, Sµν ausge- drückt werden können oder umgekehrt. Ein Ansatz zur Lösung des PDG-Systems besteht daher darin, Vµ, Aµ als Basisfunktionen zu verwenden und sich auf ihre Massenschalenbedingungen zu konzentrieren. Daneben besteht der andere Ansatz darin, F, P, Sµν als Basisfunktionen zu verwen- den. Durch eine Entwicklung in ~, die als semiklassische Entwicklung bezeichnet wird, haben wir die allgemeine Lösung der Wignerfunktion bis zur ersten Ordnung in ~ erhalten. Es hat sich gezeigt, dass die beiden oben genannten Ansätze zu dem gleichen Ergebnis führen und somit äquivalent sind. Die endgültige Lösung hat nur vier unabhängige Freiheitsgrade, was durch eine Eigenwer- 2 2 tanalyse bewiesen wird. Zur Ordnung ~ wird die übliche Massenschale p − m = 0 durch die spinmagnetische Kopplung verschoben. Wir haben weiterhin die Wignerfunktion für den masselosen Fall durch eine semiklassische En- twicklung reproduziert. Im masselosen Fall können die Fermionen nach ihrer Chiralität in zwei IV Gruppen eingeteilt werden. Unter Verwendung von Vµ und Aµ konstruierten wir die linkshändigen und rechtshändigen Ströme, die zur Ordnung ~ bestimmt werden. Die übrigen Komponenten F, P, Sµν sind proportional zur Teilchenmasse und verschwinden somit im masselosen Limes. In dieser Arbeit haben wir eine direkte Beziehung zwischen den masselosen und massiven Strömen Vµ, Aµ gefunden. Dies könnte darauf hinweisen, dass unsere massiven Ergebnisse allgemeiner sind als die chirale kinetische Theorie. In dieser Arbeit haben wir mehrere analytisch lösbare Fälle diskutiert. In
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