一类非交换（超）空间上的n=3幂零微分

非对易时空中的几何、随机微积分和量子场

Geometry, stochastic calculus and quantum fields in a non-commutative space-time

非相对论和经典力学的代数是不稳定的代数结构。它们向稳定结构的变形分别导致了相对论和量子力学。同样，组合相对论性量子力学代数也是不稳定的。它的稳定性要求时空坐标的非交换性和基本长度常数的存在。新的相对论性量子力学代数对时空几何、量子随机微积分和量子场的构造有重要影响。本文研究了其中的一些影响。

Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

Non-Commutative Worlds and Classical Constraints

This paper reviews results about discrete physics and non-commutative worlds and explores further the structure and consequences of constraints linking classical calculus and discrete calculus formulated via commutators. In particular we review how the formalism of generalized non-commutative electromagnetism follows from a first order constraint and how, via the Kilmister equation, relationships with general relativity follow from a second order constraint. It is remarkable that a second order constraint, based on interlacing the commutative and non-commutative worlds, leads to an equivalent …

Clifford分析中的幂零李群：五个研究方向

Nilpotent Lie Groups in Clifford Analysis: Five Directions for Research

本文的目的是推广幂零李群（特别是海森堡群等）的Clifford分析和数学物理的相关模型的背景下。有人认为，这些群体的研究不足，与其他经典分支的分析。我们列出了五个大的方向，这似乎是有前途的进一步研究。保留字：Clifford分析，海森堡群，幂零李群，Segal-Bargmann空间，Toeplitz算子，奇异积分算子，伪微分算子，泛函微积分，联合谱，量子力学，旋量场。

非交换空间上的一个新微分

A new differential calculus on noncommutative spaces

在二维非对易量子空间上建立了一个$GL_{qp}(2)$不变微分方程。在这里，外量子平面的坐标空间被本质上是交换的（玻色子）微分所跨越。

Symbol calculus of square-integrable operator-valued maps

We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method we show that it behaves well with respect to the infinite tensor products. This construction subsumes examples coming from the study of magnetic Weyl calculus, the magnetic pseudo-differential Weyl calculus, the metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of so…

A Deformation Quantization Theory for Non-Commutative Quantum Mechanics

We show that the deformation quantization of non-commutative quantum mechanics previously considered by Dias and Prata can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined, and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef.

A Loomis-Sikorski theorem and functional calculus for a generalized Hermitian algebra

A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $ξ_a$ on the $σ$-orthomodular lattice of projections in $A$ and that $ξ_a$ determines the spectral…