为什么量子动力学是线性的
Why quantum dynamics is linear
摘要：
量子动力学是线性的。我们怎麽知道？理论还是实验？回顾了这一问题的历史。量子力学的非线性推广已经被提出。他们预测了小但明显的非线性效应，这是非常精确的实验所没有看到的。原则上是否存在未发现非线性的原因？这不可能吗？量子动力学必须是线性的吗？试图证明这一点的努力并不是决定性的，因为要么他们的假设不令人信服，要么他们的论点不是决定性的。这个问题一直悬而未决。有一个简单的答案，基于一个简单的假设。它是在相隔44年的两个步骤中发现的。它们是回到更简单、更有说服力的假设的步骤。维格纳-巴格曼证明的假设的证明自1962年以来就已为人所知。它假定密度矩阵在时间上的映射是线性的。对于该步骤，还假设密度矩阵被一对一地映射到密度矩阵上。另一种方法是假设纯态是一对一映射到纯态上的，并且熵不会减少。在2006年采取的步骤中，证明了密度矩阵在时间上的映射是线性的。和前面的步骤一样，我们假设在每一时刻，物理量和状态都是由量子力学通常的线性结构描述的，所以问题只是关于事物如何随时间变化。除此之外，证明仅假设动力学不依赖于系统之外的任何东西，但必须允许系统被描述为更大系统的一部分。
英文摘要：
Quantum dynamics is linear. How do we know? From theory or experiment? The history of this question is reviewed. Nonlinear generalizations of quantum mechanics have been proposed. They predict small but clear nonlinear effects, which very accurate experiments have not seen. Is there a reason in principle why nonlinearity is not found? Is it impossible? Does quantum dynamics have to be linear? Attempts to prove this have not been decisive, because either their assumptions are not compelling or their arguments are not conclusive. The question has been left unsettled. There is a simple answer, based on a simple assumption. It was found in two steps separated by 44 years. They are steps back to simpler and more compelling assumptions. A proof of the assumptions of the Wigner-Bargmann proof has been known since 1962. It assumes that the maps of density matrices in time are linear. For this step, it is also assumed that density matrices are mapped one-to-one onto density matrices. An alternative is to assume that pure states are mapped one-to-one onto pure states and that entropy does not decrease. In a step taken in 2006, it is proved that the maps of density matrices in time are linear. It is assumed, as in the earlier step, that at each time the physical quantities and states are described by the usual linear structures of quantum mechanics, so the question is only about how things change in time. Beyond that, the proof assumes only that the dynamics does not depend on anything outside the system, but must allow the system to be described as part of a larger system.
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