相对论性等离子体输运性质的研究
摘要
在相对论重离子碰撞实验中测量输运系数对认识强作用物质即核物质与夸克物质的性质具有重要意义。其中一个重要的发现是剪切粘滞系数与熵密度之比与弱耦合理论计算的结果相去甚远,而与强耦合情形使用引力规范场对偶理论计算得到的下限非常接近,由此可以判断RHIC和LHC实验中产生的夸克物质类似于完美液体。本文利用相对论性流体力学和经典和量子动理学理论在几个方面研究夸克胶子等离子体的输运性质。
电磁性质是夸克胶子等离子体的一个重要特性。电磁信号与夸克胶子等离子体耦合较弱,一旦产生就几乎无作用地离开物质系统,所以它是探测夸克胶子的比较干净的信号。折射率是电磁波在等离子体里传播的一个重要物理量,它与介质的电磁响应性质相关。我们使用有限温度场论中的硬热圈重求和方法计算了夸克等离子体的介电常数和磁导率,由此得出可磁化和不可磁化等离子体的折射率。在可磁化等离子体里,磁导率和折射率存在频率极点ωmp,折射率对ω∈[k, ωmp]范围内的频率呈现负值,此处K为波数,但是在此范围不存在传播模式。在不可磁化等离子体中磁导率和折射率总是正的。与此对照,在强耦合等离子体系统存在负折射率,这显示了弱耦合和强耦合等离子体系统的主要差别。
理论上计算输运系数的通常方法是利用Kubo公式,它根据含时微扰论导出物理量的线性响应。利用Kubo公式和微扰理论可以逐级计算多体系统的输运系数。与Kubo公式等价的方法是利用玻尔兹曼方程。假设系统偏离平衡不远,可以把玻尔兹曼方程线性化,通过比较动力学项和碰撞项得到输运系数。手征磁效应是相对论重离子碰撞实验中重要的物理现象,同时也是解释重离子碰撞实验中观测到的电荷分离效应的理论之一。重离子实验中,两个带电重核相互对撞,在极短时间内可以产生高达1018高斯的强电磁场。带正(负)电的夸克的自旋会在强磁场下极化至磁场(磁场反)方向。如果忽略夸克的质量,则夸克的手征性以及螺旋度都是固定的,比如右手(左手)正粒子的螺旋度是+1(一1),即该粒子自旋沿着动量(动量反)方向,对于反粒子则相反。但由于量子三角反常(手征反常),在某个事例中,左右手粒子的数目会不同,从而会产生沿着磁场方向的电流。这种现象就是手征磁效应。与此效应类似,重离子碰撞中也存在手征电分离效应,即在电场方向会诱导出一个手征流,其比率即手征电导率σe。我们使用线性化玻尔兹曼方程计算了σe,发现它随μ和μA呈非平庸行为,但对二者的依赖具有对称性,并且当μ和μA趋于零时,σe对μ和μA呈线性依赖,与文献中的结果一致。
由Wigner函数可以导出量子输运理论,这是研究量子多体系统输运性质的基本方法。我们采用Wigner函数,系统地研究了(2+1)维(2维空间和1维时间)下含有宇称破缺的费米子系统的量子输运过程。为了简化计算,我们忽略了粒子间的相互作用,并且采用梯度展开的方法,得到了一阶无耗散流体力学的各个输运系数。首先,我们得到了含有温度和化学势依赖的Hall电导率,所得结果与有限温度QED计算结果一致。其次,我们还自然地得到了含有涡旋效应的输运过程,这些涡旋效应都是以往不曾发现的。所得的结果,将有助于人们更加深刻地认识(2+1)维量子电动力学系统,同时也对凝聚态物理中石墨烯的研究有所帮助和借鉴。
Transport properties of quark and nuclear matter in high energy heavy ion colli-sions provide important information for the nature of strong interaction. One surprising discovery of the experiment of relativistic heavy ion collisions at Brookhaven National Laboratory and at CERN-LHC is the experimental data for the ratio of shear viscosi-ty to entropy density are very close to the lower limit given by the gravity-gauge field duality. This means the quark gluon plasma produced at RHIC and LHC is a strongly coupled system like a perfect liquid. The work of this thesis is to study the transport properties of the quark gluon plasma in relativistic fluid dynamics and classical and quantum kinetic theory with focus on a few respects.
The electromagnetic probes such as photons are expected to provide clean sig-natures for the quark gluon plasma in heavy ion collisions due to their instant emis-sions once produced. These thermal photons contain undistorted information about the space-time trace of the new state of matter formed in such collisions. One of the most important electromagnetic properties of a plasma is the refractive index which measures the speed of electromagnetic wave in a medium relative to vacuum. We calculate the refractive index within the Hard-Thermal-Loop perturbation theory via the electric per-mittivity and the magnetic permeability. Since the quark gluon plasma is composed of electrically charged quarks instead of magnetic monopoles, the electric and magnetic sector do not play equal roles. We show that the physical definition and behavior of the magnetic permeability and then the refractive index can be very different due to specific magnetic response of the quark gluon plasma. Therefore a plasma can be classified into two types:magnetizable and non-magnetizable. In a magnetizable plasma the magne-tization is realistic, while in a non-magnetizable plasma it does not make physical sense any more. We calculate the refractive index and analyze their properties in these two types of plasmas. If the quark gluon plasma is magnetizable, we will show that there is a frequency pole ωmp in the magnetic permeability and then the refractive index, leading to the negative refractive index in the range ω∈[k, ωmp], where k is the wave number, but there are no propagating modes in the negative refractive index region. In a non-magnetizable plasma, and the magnetic permeability the refractive index are always positive.
There are two methods to calculate transport coefficients. One method is the Kubo formula, which is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Another method equivalent to the Kubo formula is linearized Boltzmann equation. Suppose the distribution function can be expanded near equilibrium with respect to space-time derivative. Then we can derive the integral equation for the deviation part of the distribution, from which we can obtain the transport coefficients. Using the linearized Boltzmann equation, we calculate the conductivity of the chiral current due to electric field. Similar to the chiral magnetic effect, there is also a chiral electric effect in heavy ion collisions. The chiral conductivity is just the ratio of the chiral current to the electric field. The chiral conductivity depends on the chemical potential and chiral chemical potential in a non-trivial way. Our results are consistent to those obtained in other methods in literature.
Quantum kinetic theory based on Wigner functions is a powerful tool to for trans-port phenomena. We investigate fluid-dynamics of a fermionic system in (2+1)-dimensions in a constant background electromagnetic field. We work in a quantum kinetic approach with gauge invariant Wigner function. A perturbative method is used to determine the Wigner function to the first order of the spatial derivative and electromagnetic field. The fermionic current induced by the magnetic field is obtained from the Wigner function which gives the Hall effect. The parity violating Hall electric conductivity is consistent to the previous result from quantum field theory. The vorticity term also emerges nat-urally. The entropy is proved to conserve indicating that this is a parity-violating and dissipationless system.
电磁性质是夸克胶子等离子体的一个重要特性。电磁信号与夸克胶子等离子体耦合较弱,一旦产生就几乎无作用地离开物质系统,所以它是探测夸克胶子的比较干净的信号。折射率是电磁波在等离子体里传播的一个重要物理量,它与介质的电磁响应性质相关。我们使用有限温度场论中的硬热圈重求和方法计算了夸克等离子体的介电常数和磁导率,由此得出可磁化和不可磁化等离子体的折射率。在可磁化等离子体里,磁导率和折射率存在频率极点ωmp,折射率对ω∈[k, ωmp]范围内的频率呈现负值,此处K为波数,但是在此范围不存在传播模式。在不可磁化等离子体中磁导率和折射率总是正的。与此对照,在强耦合等离子体系统存在负折射率,这显示了弱耦合和强耦合等离子体系统的主要差别。
理论上计算输运系数的通常方法是利用Kubo公式,它根据含时微扰论导出物理量的线性响应。利用Kubo公式和微扰理论可以逐级计算多体系统的输运系数。与Kubo公式等价的方法是利用玻尔兹曼方程。假设系统偏离平衡不远,可以把玻尔兹曼方程线性化,通过比较动力学项和碰撞项得到输运系数。手征磁效应是相对论重离子碰撞实验中重要的物理现象,同时也是解释重离子碰撞实验中观测到的电荷分离效应的理论之一。重离子实验中,两个带电重核相互对撞,在极短时间内可以产生高达1018高斯的强电磁场。带正(负)电的夸克的自旋会在强磁场下极化至磁场(磁场反)方向。如果忽略夸克的质量,则夸克的手征性以及螺旋度都是固定的,比如右手(左手)正粒子的螺旋度是+1(一1),即该粒子自旋沿着动量(动量反)方向,对于反粒子则相反。但由于量子三角反常(手征反常),在某个事例中,左右手粒子的数目会不同,从而会产生沿着磁场方向的电流。这种现象就是手征磁效应。与此效应类似,重离子碰撞中也存在手征电分离效应,即在电场方向会诱导出一个手征流,其比率即手征电导率σe。我们使用线性化玻尔兹曼方程计算了σe,发现它随μ和μA呈非平庸行为,但对二者的依赖具有对称性,并且当μ和μA趋于零时,σe对μ和μA呈线性依赖,与文献中的结果一致。
由Wigner函数可以导出量子输运理论,这是研究量子多体系统输运性质的基本方法。我们采用Wigner函数,系统地研究了(2+1)维(2维空间和1维时间)下含有宇称破缺的费米子系统的量子输运过程。为了简化计算,我们忽略了粒子间的相互作用,并且采用梯度展开的方法,得到了一阶无耗散流体力学的各个输运系数。首先,我们得到了含有温度和化学势依赖的Hall电导率,所得结果与有限温度QED计算结果一致。其次,我们还自然地得到了含有涡旋效应的输运过程,这些涡旋效应都是以往不曾发现的。所得的结果,将有助于人们更加深刻地认识(2+1)维量子电动力学系统,同时也对凝聚态物理中石墨烯的研究有所帮助和借鉴。
Transport properties of quark and nuclear matter in high energy heavy ion colli-sions provide important information for the nature of strong interaction. One surprising discovery of the experiment of relativistic heavy ion collisions at Brookhaven National Laboratory and at CERN-LHC is the experimental data for the ratio of shear viscosi-ty to entropy density are very close to the lower limit given by the gravity-gauge field duality. This means the quark gluon plasma produced at RHIC and LHC is a strongly coupled system like a perfect liquid. The work of this thesis is to study the transport properties of the quark gluon plasma in relativistic fluid dynamics and classical and quantum kinetic theory with focus on a few respects.
The electromagnetic probes such as photons are expected to provide clean sig-natures for the quark gluon plasma in heavy ion collisions due to their instant emis-sions once produced. These thermal photons contain undistorted information about the space-time trace of the new state of matter formed in such collisions. One of the most important electromagnetic properties of a plasma is the refractive index which measures the speed of electromagnetic wave in a medium relative to vacuum. We calculate the refractive index within the Hard-Thermal-Loop perturbation theory via the electric per-mittivity and the magnetic permeability. Since the quark gluon plasma is composed of electrically charged quarks instead of magnetic monopoles, the electric and magnetic sector do not play equal roles. We show that the physical definition and behavior of the magnetic permeability and then the refractive index can be very different due to specific magnetic response of the quark gluon plasma. Therefore a plasma can be classified into two types:magnetizable and non-magnetizable. In a magnetizable plasma the magne-tization is realistic, while in a non-magnetizable plasma it does not make physical sense any more. We calculate the refractive index and analyze their properties in these two types of plasmas. If the quark gluon plasma is magnetizable, we will show that there is a frequency pole ωmp in the magnetic permeability and then the refractive index, leading to the negative refractive index in the range ω∈[k, ωmp], where k is the wave number, but there are no propagating modes in the negative refractive index region. In a non-magnetizable plasma, and the magnetic permeability the refractive index are always positive.
There are two methods to calculate transport coefficients. One method is the Kubo formula, which is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Another method equivalent to the Kubo formula is linearized Boltzmann equation. Suppose the distribution function can be expanded near equilibrium with respect to space-time derivative. Then we can derive the integral equation for the deviation part of the distribution, from which we can obtain the transport coefficients. Using the linearized Boltzmann equation, we calculate the conductivity of the chiral current due to electric field. Similar to the chiral magnetic effect, there is also a chiral electric effect in heavy ion collisions. The chiral conductivity is just the ratio of the chiral current to the electric field. The chiral conductivity depends on the chemical potential and chiral chemical potential in a non-trivial way. Our results are consistent to those obtained in other methods in literature.
Quantum kinetic theory based on Wigner functions is a powerful tool to for trans-port phenomena. We investigate fluid-dynamics of a fermionic system in (2+1)-dimensions in a constant background electromagnetic field. We work in a quantum kinetic approach with gauge invariant Wigner function. A perturbative method is used to determine the Wigner function to the first order of the spatial derivative and electromagnetic field. The fermionic current induced by the magnetic field is obtained from the Wigner function which gives the Hall effect. The parity violating Hall electric conductivity is consistent to the previous result from quantum field theory. The vorticity term also emerges nat-urally. The entropy is proved to conserve indicating that this is a parity-violating and dissipationless system.
引文
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