Geometry of Quasi-Free States of CCR Algebras

Symmetry Reduction of Quasi-Free States

Given a group-invariant quasi-free state on the algebra of canonical commutation relations (CCR), we show how group averaging techniques can be used to obtain a symmetry reduced CCR algebra and reduced quasi-free state. When the group is compact this method of symmetry reduction leads to standard results which can be obtained using other methods. When the group is non-compact the group averaging prescription relies upon technically favorable conditions which we delineate. As an example, we consider symmetry reduction of the usual vacuum state for a Klein-Gordon field on Minkowski spacetime by …

投射谱定理、准自由态与点过程

The projection spectral theorem, quasi-free states and point processes

本文通过严格定义的CAR/CCR表示的联合谱测度，揭示了几类点过程如行列式点过程和permanental点过程的生成机制。

二次量子化代数的拟自由同构与模理论

Quasi-free isomorphisms of second quantisation algebras and modular theory

本文利用 Araki-Yamagami 对 CCR 拟自由表示拟等价性的刻画，提出基于单粒子模算子的抽象准则，判定 Bogolubov 变换诱导的二次量子化局部 von Neumann 代数同构的存在性，并探讨其对不同质量 Klein-Gordon 场真空局部正规性问题的应用。

Extension of representations in quasi *-algebras

Let $(A, A_o)$ be a topological quasi *-algebra, which means in particular that $A_o$ is a topological *-algebra, dense in $A$. Let $π^o$ be a *-representation of $A_o$ in some pre-Hilbert space ${\cal D} \subset {\cal H}$. Then we present several ways of extending $π^o$, by closure, to some larger quasi *-algebra contained in $A$, either by Hilbert space operators, or by sesquilinear forms on ${\cal D}$. Explicit examples are discussed, both abelian and nonabelian, including the CCR algebra.

Variable Planck's constant and scaling properties of states on Weyl algebra

We consider the possible quantum effect for infinite systems produced by variations of the Planck's constant. Using the algebraic formulation of quantum theory we study behaviour of states $ω$ defined as positive, normalized functionals on the canonical commutation relations algebra (CCR-algebra) under the changes of the defining relations of the CCR. These defining relations of the multiplication in the CCR-algebra depend explicitly on the value of the Planck's constant. We analyse to what extend changes of the $\hbar$ preserve the original state space (this gives restrictions on the admissib…

An indefinite metric model for interacting quantum fields with non-stationary background gravitation

We consider a relativistic Ansatz for the vacuum expectation values (VEVs) of a quantum field on a globally hyperbolic space-time which is motivated by certain Euclidean field theories. The Yang-Feldman asymptotic condition w.r.t. a "in"-field in a quasi-free representation of the canonic commutation relations (CCR) leads to a solution of this Ansatz for the VEVs. A GNS-like construction on a non-degenerate inner product space then gives local, covariant quantum fields with indefinite metric on a globally hyperbolic space-time. The non-trivial scattering behavior of quantum fields is analyzed …

The algebra of Grassmann canonical anti-commutation relations (GAR) and its applications to fermionic systems

We present an approach to a non-commutative-like phase space which allows to analyze quasi-free states on the CAR algebra in analogy to quasi-free states on the CCR algebra. The used mathematical tools are based on a new algebraic structure the "Grassmann algebra of canonical anti-commutation relations" (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasi-free fermionic states which is needed for the study of entanglement distillation within …

高斯态的拟等价性与模哈密顿量函数的能量估计

Quasi-equivalence of Gaussian states and energy estimates for functions of modular Hamiltonians

本文比较自由标量QFT的两个高斯态，研究GNS表示拟等价性、总能量差异及模哈密顿量函数差异三个视角的定量关系，提出以能量差异估计模哈密顿量函数差异的方法。

磁单极子的几何与准经典量子化

Geometry and quasi-classical quantization of magnetic monopoles

本文梳理了磁荷概念的物理与数学起源，构造了黎曼流形上磁单极子的磁拉普拉斯算子，并给出了其本征截面的准经典近似结果。

（拟）哈密顿系统与不可分解泊松几何

(Quasi) Hamiltonian Systems and Non-Decomposable Poisson Geometry

本文在不可分解广义泊松几何框架下系统研究哈密顿与拟哈密顿系统，证明满足李对称或守恒量不变性条件的动力系统可构造为拟哈密顿或哈密顿系统，提供统一几何框架。

Quasi-plane waves for a particle with spin 1/2 on the background of Lobachevsky geometry: simulating of a special medium

In the paper complete systems of exact solutions for Dirac and Weyl equations in the Lobachevsky space are constructed on the base of the method of separation of the variables in quasi-cartesian coordinates. An extended helicity operator is introduced. It is shown that solution constructed when translating to the limit of vanishing curvature coincide with common plane wave solutions on Minkowski space going in opposite z-directions. It is shown the problem posed in Lobachevsky space simulates a situation in the flat space for a quantum-mechanical particle of spin 1/2 in a 2-dimensional potenti…

The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction

We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted case). Poisson--quasi--Nijenhuis geometry is more general than generalized complex geometry and comprises it as a particular case. Next, we show how gauging and reduction can be implemented in the Hitchin model. We find that the geometry resulting form the BV master equation is c…

平面上准周期函数的几何

Geometry of quasiperiodic functions on the plane

本文综述了Novikov问题的最新进展，聚焦于平面上准周期函数水平线的几何，重点阐述三准周期函数的新结果及其在金属输运理论中的应用，并探讨了该问题在数学物理中的广泛意义。

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulatio…

Quantum (Matrix) Geometry and Quasi-Coherent States

A general framework is described which associates geometrical structures to any set of $D$ finite-dimensional hermitian matrices $X^a, \ a=1,...,D$. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and allows to extract the underlying classical space without requiring the limit of large matrices or representation theory. The approach is based on the previously introduced concept of quasi-coherent states. In particular, a concept of quantum Kähler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere $S^2…

SL(3|N) Wigner quantum oscillators: examples of ferromagnetic-like oscillators with noncommutative, square-commutative geometry

A system of $N$ non-canonical dynamically free 3D harmonic oscillators is studied. The position and the momentum operators (PM-operators) of the system do not satisfy the canonical commutation relations (CCRs). Instead they obey the weaker postulates for the oscillator to be a Wigner quantum system. In particular the PM-operators fulfil the main postulate, which is due to Wigner: they satisfy the equations of motion (the Hamiltonian's equations) and the Heisenberg equations. One of the relevant features is that the coordinate (the momentum) operators do not commute, but instead their squares d…

An Ocean Drum: quasi-geostrophic energetics from a Riemann geometry perspective

We revisit the discussion of the energetics of quasi-geostrophic flows from a geometric perspective based on the introduction of an effective metric, built in terms of the flow stratification and the Coriolis parameter. In particular, an appropriate notion of normal modes is defined through a spectral geometry problem in the ocean basin (a compact manifold with boundary) for the associated Laplace-Beltrami scalar operator. This spectral problem can be used to systematically encode non-local aspects of stratification and topography. As examples of applications we revisit the isotropy assumption…

狄拉克几何、变形理论与移位辛几何

Dirac geometry, deformation theory and shifted symplectic geometry

该研究发展狄拉克变形理论，在扭曲狄拉克几何与泊松几何间插值，证明其与狄拉克几何结构运算兼容，以紧致李群相关结构变形为例，还得到统一变形理论及约化空间的光滑变形。

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras A and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of A-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras A the full subcategory of symmetric A-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures…

托勒密泵与多滑动准周期晶格中的通用几何诱导漂移速度

Thouless pumps and universal geometry-induced drift velocity in multi-sliding quasi-periodic lattices

本文提出了一种通用框架，用于研究多滑动准周期晶格中的托勒密泵，揭示了拓扑漂移与准布里渊区几何之间的普遍关系，并通过模拟验证了其有效性。