洛仑兹和指数的关系：费米近似，傅立叶变换和因果关系
Relating the Lorentzian and exponential: Fermi's approximation,the Fourier transform and causality
摘要：
傅里叶变换通常用于将共振散射的洛伦兹能量分布与衰减态的指数时间依赖性联系起来。然而，为了应用傅里叶变换，人们必须弯曲标准量子力学的规则;洛伦兹能量分布必须扩展到完整的实轴$-\infty<E<\infty$，而不是从$0\leq E <\infty$下面有界（"费米近似“）。然后，扩展的洛伦兹函数的傅里叶变换变成指数，但只对时间$t\geq 0$，时间不对称，这与标准量子力学的幺正群时间演化相冲突。将傅里叶变换从分布扩展到广义向量，我们得到了伽莫夫钟，它具有与$-\infty<E<\infty$的洛伦兹能量分布，并且仅对$t\geq t_0 =0$具有指数时间演化。这导致不违反因果关系的概率预测。
英文摘要：
The Fourier transform is often used to connect the Lorentzian energy distribution for resonance scattering to the exponential time dependence for decaying states. However, to apply the Fourier transform, one has to bend the rules of standard quantum mechanics; the Lorentzian energy distribution must be extended to the full real axis $-\infty<E<\infty$ instead of being bounded from below $0\leq E <\infty$ (``Fermi's approximation''). Then the Fourier transform of the extended Lorentzian becomes the exponential, but only for times $t\geq 0$, a time asymmetry which is in conflict with the unitary group time evolution of standard quantum mechanics. Extending the Fourier transform from distributions to generalized vectors, we are led to Gamow kets, which possess a Lorentzian energy distribution with $-\infty<E<\infty$ and have exponential time evolution for $t\geq t_0 =0$ only. This leads to probability predictions that do not violate causality.
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