大腔体中振子辐射过程的精确解
An Exact Approach to the Oscillator Radiation Process in an Arbitrarily Large Cavity
摘要：
从一个机械振子耦合到半径为$R$的反射球内的标量场的问题的解决方案开始，我们研究了系统在自由空间中的行为，作为限制解中任意大半径的限制。从数学的角度来看，我们表明，这种方式面对的问题是不等价于考虑系统{ \it a} { \it先验}嵌入在无限空间。特别地，将系统转向主轴的变换的矩阵元素不倾向于分布在任意大的球体的极限中，因为如果两个过程在数学上是等价的，则应该是这种情况。此外，我们引入“打扮”的坐标，它允许一个精确的描述的振荡器辐射过程的任何值的耦合，强或弱。在弱耦合的情况下，我们恢复我们的精确表达式从微扰理论的众所周知的衰变公式。
英文摘要：
Starting from a solution of the problem of a mechanical oscillator coupled to a scalar field inside a reflecting sphere of radius $R$, we study the behaviour of the system in free space as the limit of an arbitrarily large radius in the confined solution. From a mathematical point of view we show that this way of facing the problem is not equivalent to consider the system {\it a} {\it priori} embedded in infinite space. In particular, the matrix elements of the transformation turning the system to principal axis, do not tend to distributions in the limit of an arbitrarily large sphere as it should be the case if the two procedures were mathematically equivalent. Also, we introduce "dressed" coordinates which allow an exact description of the oscillator radiation process for any value of the coupling, strong or weak. In the case of weak coupling, we recover from our exact expressions the well known decay formulas from perturbation theory.
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