标题： 洛伦兹变换 显示英文标题

标题： Lorentz Transformations

摘要： 本文描述了洛伦兹群基本性质的一种特别具有教育意义且透明的推导方法。 对于任意方向的旋转和速度增加，生成器以及它们的对易关系被写成定义旋转轴或速度方向的单位向量的函数（这种方法可以与电动力学中的方法进行比较，在电动力学中使用电场和磁场而不是麦克斯韦应力张量）。 对于旋转角度或速度的有限值，统称为V，坐标变换矩阵M的指数展开式（以GV的形式，其中G是生成器）的存在要求矩阵相对于V的导数等于GM。 只有当变换是可加时，才能满足这个条件，这确实是旋转的情况，但不是速度的情况。 然而，如果假设对于速度增加存在这样的展开式，其中V = V(v)，v表示速度，并且如果上述条件应用于速度矩阵，则可以恢复以双曲余弦cosh(V)和双曲正弦sinh(V)表示的表达式，并确定V(= arc tanh(v))的表达式。 一个一般的洛伦兹变换可以写成包含旋转和速度增加的指数之和的形式，到一阶近似时等于速度增加与旋转的乘积。 计算二阶和三阶项表明，本文中使用的生成器方程允许可靠地推导出高阶生成器的表达式，而无需借助对易关系。 研究了Weyl旋子的变换矩阵，针对旋转和速度的有限值，以及导致角动量算符表达式的场表示进行了探讨。 显示英文摘要

摘要： This paper describes a particularly didactic and transparent derivation of basic properties of the Lorentz group. The generators for rotations and boosts along an arbitrary direction, as well as their commutation relations, are written as functions of the unit vectors that define the axis of rotation or the direction of the boost (an approach that can be compared with the one that in electrodynamics, works with the electric and magnetic fields instead of the Maxwell stress tensor). For finite values of the angle of rotation or the boost's velocity, collectively denoted by V, the existence of an exponential expansion for the coordinate transformation's matrix, M (in terms of GV where G is the generator) requires that the matrix's derivative with respect to V, be equal to GM. This condition can only be satisfied if the transformation is additive as it is indeed the case for rotations, but not for velocities. If it is assumed, however, that for boosts such an expansion exists, with V = V(v), v being the velocity, and if the above condition is imposed on the boost's matrix then its expression in terms of hyperbolic cosh(V) and sinh(V} is recovered, and the expression for V(= arc tanh(v)) is determined. A general Lorentz transformation can be written as an exponential containing the sum of a rotation and a boost, which to first order is equal to the product of a boost with a rotation. The calculations of the second and third order terms show that the equations for the generators used in this paper, allow to reliably infer the expressions for the higher order generators, without having recourse to the commutation relations. The transformationmatrices for Weyl spinors are derived for finite values of the rotation and velocity, and field representations, leading to the expression for the angular momentum operator, are studied.