二阶Boltzmann方程：规范依赖性和规范不变性
Second order Boltzmann equation : gauge dependence and gauge invariance
摘要：
在宇宙学扰动理论的背景下，我们导出了描述辐射分布函数演化的二阶Boltzmann方程，其中没有特定的规范选择。给出了一个更一般的框架，重新讨论并扩展了导出玻尔兹曼方程的基本步骤：（1）通过使用张量值分布函数，将光的偏振引入到这一形式中;（2）强调了在描述分布函数时，选择四元组场来定义局部惯性系的重要性; iii）我们第一次对强度和偏振进行温度和光谱失真之间的分离; iv）导出进入玻尔兹曼方程的所有微扰量的规范依赖性，这使我们能够通过明确地显示其强度和偏振的规范不变性来检查微扰玻尔兹曼方程的正确性。最后，我们讨论了几个影响的规范依赖所观察到的温度。
英文摘要：
In the context of cosmological perturbation theory, we derive the second order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: i) the polarisation of light is incorporated in this formalism by using a tensor-valued distribution function; ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; iii) we perform a separation between temperature and spectral distortion, both for the intensity and for polarisation for the first time; iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature.
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