Spectral Breakdown Theory

We present the theory of spectral breakdown via renormalization and decoherence. In our concept, renormalization reduces the multiplicity of a function by dividing it by its or its multiple roots (multiplicity -1) and decoherence removes the imaginary parts of complex roots by projecting them onto the real axis which reduces the complexity of the function.Here the spectral break is a transformation of a complex function P into a simple function q and the spectral distance measures the proximity between the roots of P and q (the spectral distance between the roots of P and q), the spectral distance is in a spectral plane and not Euclidean so the spectral distance can be complex. In our article we introduce a new concept of minimal rupture, it is the minimum of P and q. Spectral breakdown is a nonlinear and irreversible transformation because of the decoherence which removes the majority of the identity of the function and the renormalization which reduces the multiplicity of function. NB: Renormalization is applied before decoherence.我们提出了基于重整化（renormalization）与退相干（decoherence）的谱破裂理论。在我们的理论框架中，重整化通过将函数除以其（或重根的）（重数减1）次幂来降低函数的重数；而退相干则通过将复根投影至实轴，消去复根的虚部，从而简化函数的复杂度。此处的谱破裂指将复函数P变换为简化函数q的过程，而谱距离（spectral distance）用于衡量P与q的根之间的贴近程度，即二者根集的谱距离；由于谱距离定义于谱平面而非欧几里得（Euclidean）空间，因此其取值可以为复数。在本文中，我们引入了最小破裂这一新概念，其为P与q间的极小值。谱破裂属于非线性不可逆变换，这是因为退相干会消去函数的绝大多数本征属性，而重整化会降低函数的重数。注：重整化需先于退相干执行。