arXiv 1807.005192018-07-02quant-phmath-phmath.MP

A gentle introduction to Schwinger's formulation of quantum mechanics: The groupoid picture

Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo

备注：: 7 pages
发表信息：: Modern Physics Letters A, Volume 33, Issue 20, 2018
DOI：: 10.1142/S0217732318501225

英文摘要：

In this short letter we review Schwinger's formulation of Quantum Mechanics and we argue that the mathematical structure behind Schwinger's "Symbolism of Atomic Measurements" is that of a groupoid. In this framework, both the Hilbert space (Schrödinger picture) and the $C^{*}$-algebra (Heisenberg picture) of the system turn out to be derived concepts, that is, they arise from the underlying groupoid structure.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Schwinger's picture of quantum mechanics: 2-groupoids and symmetries

Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows Ω$ associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global".The microscopic point of view leads to the introduction of an additional layer over the grupoid $G$, giving rise to a suitable algebraic structure of 2-groupoid.On the other hand, taking advantage of the notion of group of bisections of a given groupoid…

2021-04-28math-phmath.MPquant-ph

Schwinger's Picture of Quantum Mechanics I: Groupoids

A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger's algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac-Schrödinger and Born-Jordan-Heisenberg pictures are succinctly discussed.

2019-05-29hep-thmath-phmath.MP

Quantum Tomography and Schwinger's Picture of Quantum Mechanics

In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger's picture of Quantum Mechanics in which a groupoid is associated with every quantum system. The attention is focused on spin tomography: In this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in pro…

2022-04-30math-phmath.MPquant-ph

Schwinger's Picture of Quantum Mechanics IV: Composition and independence

The groupoids description of Schwinger's picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger's groupoid of the system. The groupoids picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, …

2020-04-06hep-thmath-phmath.MP

Schwinger's picture of Quantum Mechanics

In this paper we will present tha main features of what can be called Schwinger's foundational approach to Quantum Mechanics. The basic ingredients of this formulation are the \textit{selective measurements}, whose algebraic composition rules define a mathematical structure called groupoid, which is associated with any physical system. After the introduction of the basic axioms of a groupoid, the concepts of observables and states, statistical interpretation and evolution are derived. An example is finally introduced to support the theoretical description of this approach.

2020-02-21math-phmath.MPquant-ph

Groupoids and Coherent states

Schwinger's algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmon…

2019-11-04math-phmath.MPquant-ph

更多相关论文收起更多论文

Noncommutative integration, quantum mechanics, Tannaka's theorem for compact groupoids and examples

We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We exhibit explicit examples and one of them will be the so-called quantum ratchet. This is related to Schwinger's algebra of selective measurements. Here we consider $\mathcal{G}$-kernels, transverse functions, modular functions, and quasi-invariant measures for Haar systems. Later we present our main result which is a version of Tannaka's theorem for Hausdorff com…

2023-03-21math-phmath.FAmath.MP

一种用于量子比特的拉格朗日路径积分方法

A Lagrangian path integral approach to the qubit

该研究基于广群概念推广施温格量子力学图像，提出量子比特的拉格朗日路径积分方法，给出历史求和计算方式，揭示传播子幺正性带来的拉格朗日量量子化条件及特例。

2024-01-24math-phmath.MPquant-ph

Schwinger's Picture of Quantum Mechanics II: Algebras and Observables

The kinematical foundations of Schwinger's algebra of selective measurements were discussed in a previous paper (arXiv:1905.12274) and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analysed. For that, the algebra generated by the observables, as well as the notion of state, are dicussed, and the structure of the transition functions, that plays an instrumental role in Schwinger's picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers …

2019-07-08hep-thmath-phmath.MP

无限量子自旋链的广群与代数

Groupoid and algebra of the infinite quantum spin chain

针对标准 Hilbert 空间无法描述有限密度热力学系统等问题，本文展示了无限自旋链量子力学的 Schwinger 描述中自然产生的一般代数，利用 DFS 态和 Tomita-Takesaki 模理论引入动力学，并应用于 Ising 模型。

2023-02-02quant-ph

Schwinger's Picture of Quantum Mechanics III: The statistical interpretation

Schwinger's algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the theory, was presented in two previous papers (arXiv:1905.12274 and arXiv:1907.03883). In this paper, a closer look to the statistical interpretation of the theory is taken and it is found that an interpretation in terms of Sorkin's quantum measure emerges naturally. It is proven that a suitable class of states of the algebra of virtual transitions of the theory allows to define quantum measures by me…

2019-09-16hep-thmath-phmath.MP