本文是对量子力学与量子场论核心概念的非标准阐述，涵盖了部分最新研究成果。其理论基础分为两种：一是以星代数为出发点的代数方法，二是以凸态集为出发点的几何方法。\n量子概率的标准公式由退相干推导而来，这一推导使得我们能够在几何方法的框架下超越量子理论的范畴。粒子被定义为基态的元激发（而准粒子则被定义为任意平移不变态的元激发）。\n传统散射矩阵不适用于准粒子（若理论不具备粒子诠释，其甚至不适用于粒子）。针对这类情形的散射分析基于包容性散射矩阵（inclusive scattering matrix）的概念，该矩阵与包容性截面密切相关。研究证明，传统散射矩阵可通过格林函数表示（即LSZ公式），而包容性散射矩阵则可通过非平衡统计物理Keldysh形式体系中出现的广义格林函数表示。\n演化算符及其他物理量的泛函积分表达式推导基于算符符号（symbol of an operator）的概念，这些推导思路同样适用于几何方法。包容性散射矩阵的概念在几何方法中具有意义（但目前看来，在该情形下无法给出传统散射矩阵的定义）。\n借助几何方法可以证明，量子力学及其推广形式可被视为经典理论，只是我们的测量设备仅能测量部分可观测量。
英文摘要：
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory including some recent results.
It is based on the algebraic approach where the starting point is a star-algebra and on the geometric approach where the starting point is a convex set of states.
Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state).
The conventional scattering matrix does not work for quasiparticles (and even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the
notion of inclusive scattering matrix, closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics.
The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied also in the geometric approach.
The notion of inclusive scattering matrix makes sense in the geometric approach (but it seems that one cannot give a definition of the conventional scattering matrix in this situation).
The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices can measure only a part of observables.
A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance
我回顾了量子力学的各种代数基础。从这一理论诞生到去年，一直有人提出这些问题。它们是：Heisenberg-Born-Jordan（1925），Weyl（1928），Dirac（1930），von Neumann（1936），Segal（1947），T.F. Jordan（1986）、Morchio and Strocchi（2009）以及Buchholz and Fregenstrom（2019）。强调三种情况：本文主要讨论了三个方面的问题：1）狄拉克基础的误解; 2）冯·诺依曼从希尔伯特空间的解析方法到算子环的代数方法的转换; 3）量子力学最近建立在微扰拉格朗日代数上。此外，历史上的考虑去和停止路径上执行的代数方法在QM的历史。形式主义的水平已经从仅仅引入矩阵增加到群论和C*-代数。但是，在接近物理学基础方面没有取得进展;因此，发现QM的代数公式作为一种基于问题的理论并利用不超过建设性数学的问题是开放的。
Step-by-step derivation of the algebraic structure of quantum mechanics (or from nondisturbing to quantum correlations by connecting incompatible observables)
