We present a N-dimensional quantization a la Berezin-Klauder or frame quantization of the complex plane based on overcomplete families of states (coherent states) generated by the N first harmonic oscillator eigenstates. The spectra of position and momentum operators are finite and eigenvalues are equal, up to a factor, to the zeros of Hermite polynomials. From numerical and theoretical studies of the large $N$ behavior of the product $λ\_m(N) λ\_M(N)$ of non null smallest positive and largest eigenvalues, we infer the inequality $δ\_N(Q) Δ\_N(Q) = σ\_N \overset{<}{\underset{N \to \infty}{\to}} 2 π$ (resp. $δ\_N(P) Δ\_N(P) = σ\_N \overset{<}{\underset{N \to \infty}{\to}} 2 π$) involving, in suitable units, the minimal ($δ\_N(Q)$) and maximal ($Δ\_N(Q)$) sizes of regions of space (resp. momentum) which are accessible to exploration within this finite-dimensional quantum framework. Interesting issues on the measurement process and connections with the finite Chern-Simons matrix model for the Quantum Hall effect are discussed.
Notions and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, is identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight we provide a rigorous approach to generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between …
We explore the method of old quantization as applied to states with nonzero angular momentum, and show that it leads to qualitatively and quantitatively useful information about systems with spherically symmetric potentials. We begin by reviewing the traditional application of this model to hydrogen, and discuss the way Einstein-Brillouin-Keller quantization resolves a mismatch between old quantization states and true quantum mechanical states. We then analyze systems with logarithmic and Yukawa potentials, and compare the results of old quantization to those from solving Schrodinger's equatio…
提出了量子化的概念，它独立于光子气体中能量分布的原始统计解释或氢原子中角动量的量子化。这样一个过程意味着有限的时空四间隔的存在，任何相对论的准备和测量的物理事件需要。有限四区间也是量子力学不确定关系的认识论根源。此外，光子传播过程中螺旋度的守恒是一个相对论不变量，它是Bell不等式[2，3，4]中非定域相互作用佯谬“出现”的根源，如A. Aspect et all [5] experiment.光子的自旋总是和它的动量相关。
It is clarified that Heisenberg quantization was proposed in empty space. Based on established experiments, the generalized Heisenberg quantization in physical space is obtained. Physical space quantization includes important new physics: Proving that physical space is noncommutative space; Exploring the existence of a non-zero minimal length scale, which leads to new space structures and the existence of the space minimal finite volume; Finding a new correlativity of the property of space with the motion status of the system: space non-commutativity is determined by the momentum non-commutati…
We propose a way to encode acceleration directly into quantum fields, establishing a new class of fields. Accelerated quantum fields, as we have named them, have some very interesting properties. The most important is that they provide a mathematically consistent way to quantize space-time in the same way that energy and momentum are quantized in standard quantum field theories.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.
We perform the momentum-space quantization of a spin-less particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar…
New approach to quantization of the relativistic Majorana field is presented. It is based on expansion of the field into eigenfunctions of the axial momentum -- a novel observable introduced recently. Relativistic invariance is used as the main guiding principle instead of canonical formalism. Hidden structure of the quantized Majorana field in the form of real Clifford algebra of Hermitian fermionic operators is unveiled. Generators of the Poincaré transformations in the Fock space are found as solutions of certain operator equations, without invoking the principle of correspondence with clas…
For a particle moving on a half-line or in an interval the operator $\hat p = - i \partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\hat p$ fails. Based upon a new concept for a self-adjoint momentum operator $\hat p_R$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\hat H$ and the momentum operator $\hat p_R$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\hat H)$ and $D(\hat p_R)$ in the H…
Physical problems for which the existence of non-trivial topological Pauli phase (i.e. fractional quantization of angular orbital angular momenta that is possible in 2D case) is essential are discussed within the framework of two-dimensional Helmholtz, Schroedinger and Dirac equations. As examples in classical field theory we consider a "wedge problem" -- a description of a field generated by a point charge between two conducting half-planes -- and a Fresnel diffraction from knife-edge. In few-electron circular quantum dots the choice between integer and half-integer quantization of orbital an…
Quantization of the canonically conjugate pair angle and orbital angular momentum
在量子力学的早期，如何计算一个以角度φ为基本正则变量的经典系统的问题一直是有争议的。问题是角度在相应的相空间上是一个多值或不连续的变量。补救办法是用光滑的周期函数cos phi和sin phi代替phi。在正则对（phi，l），l：轨道角动量（OAM）的情形下，相空间S_（phi，l）={phi in R mod 2 pi，l in R}具有圆柱的整体结构S^1 x R，在该圆柱上3个函数cos phi，sin phi和l的泊松括号服从平面欧氏群E（2）的李代数。这个性质提供了根据群E（2）或其覆盖群的不可约酉表示来量子化系统的基础。一个关键点是-由于子群SO（2）= S^1是多连通的-这些表示允许分数OAM l = n + c，c在[0，1）中。在Aharonov-Bohm和分数量子霍尔效应等情况下已经观察到了这种c不为0，它们对应于理想晶体中布洛赫波的准动量。本论文的主要目的是在外加磁场中寻找与拉盖尔-高斯激光模式量子光学有关的分数OAM。用E（2）的幺正表示描述的相空间S_（phi，l）的量子理论允许两种类型的“相干”态，详细讨论了它们的性质：非全纯最小测不准态和与Bargmann-Segal Hilbert空间有关的全纯最小测不准态。
