一个婴儿马约拉纳量子形式主义
A baby Majorana quantum formalism
摘要：
本文的目的是通过一个简单的教学例子来介绍和讨论量子物理学的最基本的基本概念。其内容的相当一部分提出了原始的结果。我们从欧几里德平面开始，它当然是希尔伯特空间的一个典型例子。纯态形成单位圆（实际上是它的一半），混合态形成单位圆盘（实际上是它的一半），平面上的旋转通过封闭和开放系统中只涉及实数的马约拉纳式方程来控制时间演化。一组纯态或一组混合态解决了恒等式，它们用于理解单位圆上函数的积分量子化概念，并给出量子可观测量的半经典描述。有趣的概率方面的发展。由于两个平面的张量积，它们的直和，它们的卡耐基积，是同构的（2是x^x= x \times x = x+x的唯一解），它们也同构于C^2，并且同构于四元数场H（作为向量空间），我们描述了形成SU群$(2)$的实数纠缠、半自旋猫态和单位范数四元数之间的有趣关系。我们解释的最一般形式的哈密顿量在实平面考虑的积分量子化的磁类相互作用势视为一个经典的观察单位$2$ -球。最后，我们给出了一个指针态也位于欧几里德平面的量子测量的例子。
英文摘要：
The aim of the present paper is to introduce and to discuss the most basic fundamental concepts of quantum physics by means of a simple and pedagogical example. An appreciable part of its content presents original results. We start with the Euclidean plane which is certainly a paradigmatic example of a Hilbert space. The pure states form the unit circle (actually a half of it), the mixed states form the unit disk (actually a half of it), and rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. The set of pure states or a set of mixed states solve the identity and they are used for understanding the concept of integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU$(2)$. We explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit $2$-sphere. Finally, we present an example of quantum measurement with pointer states lying also in the Euclidean plane.
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