Phase space quantization and Loop Quantum Cosmology: A Wigner function for the Bohr-compactified real line

Hydrogen atom in space with a compactified extra dimension and potential defined by Gauss' law

We investigate the consequences of one extra spatial dimension for the stability and energy spectrum of the non-relativistic hydrogen atom with a potential defined by Gauss' law, i.e. proportional to $1/|x|^2$. The additional spatial dimension is considered to be either infinite or curled-up in a circle of radius $R$. In both cases, the energy spectrum is bounded from below for charges smaller than the same critical value and unbounded from below otherwise. As a consequence of compactification, negative energy eigenstates appear: if $R$ is smaller than a quarter of the Bohr radius, the corresp…

Symmetries in exact Bohrification

The `Bohrification" program in the foundations of quantum mechanics implements Bohr's doctrine of classical concepts through an interplay between commutative and non-commutative operator algebras. Following a brief conceptual and mathematical review of this program, we focus on one half of it, called "exact" Bohrification, where a (typically noncommutative) unital C*-algebra A is studied through its commutative unital C*-subalgebras, organized into a poset C(A). This poset turns out to be a rich invariant of A. To set the stage, we first give a general review of symmetries in elementary quantu…

Polymer Quantum Mechanics as a Deformation Quantization

We analyze the polymer representation of quantum mechanics within the deformation quantization formalism. In particular, we construct the Wigner function and the star-product for the polymer representation as a distributional limit of the Schrödinger representation for the Weyl algebra in a Gaussian weighted measure, and we observe that the quasi-probability distribution limit of this Schrödinger representation agrees with the Wigner function for Loop Quantum Cosmology. Further, the introduced polymer star-product fulfills Bohr's correspondence principle even though not all the operators are w…

Wigner crystallization of electrons in a one-dimensional lattice: a condensation in the space of states

We study the ground state of a system of spinless electrons interacting through a screened Coulomb potential in a lattice ring. By using analytical arguments, we show that, when the effective interaction compares with the kinetic energy, the system forms a Wigner crystal undergoing a first-order quantum phase transition. This transition is a condensation in the space of the states and belongs to the class of quantum phase transitions discussed in J. Phys.~A \textbf{54}, 055005 (2021). The transition takes place at a critical value ${r_s}_{c}$ of the usual dimensionless parameter $r_s$ (radius …

相干环路态与角动量

Coherent loop states and angular momentum

本文在SU(2)不可约表示下研究Bohr-Sommerfeld态，证明其恢复角动量本征态基，自洽证明相干环路态内积渐近公式，并推导Wigner矩阵元几何公式。

Contextuality与量子力学基本定理

Contextuality and the fundamental theorems of quantum mechanics

本研究拓展拓扑斯量子力学视角，证明Wigner定理、Gleason定理等四大量子力学基本定理均与语境性相关，并用语境预层语言重述定理，给出适用于冯·诺依曼代数的通用版本，揭示语境性对量子理论结构的核心编码作用。