H. -D. Doebner, P. Nattermann
In this contribution we review results on the kinematics of a quantum system localized on a connected configuration manifold and compatible dynamics for the quantum system including external fields and leading to non-linear Schrödinger equations for pure states.
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The results, different aspects and applications of our method of quantisation on configuration manifolds - called Borel Quantisation - were presented at meetings of the series `Symmetries in Science' and can be found in the published proceedings. The developments with numerous coauthors, on Borel quantisation and the related family of nonlinear Schrödinger equations called Doebner-Goldin equations, are reviewed and commented here.
Within the Hamiltonian framework, the propositions about a classical physical system are described in the Borel σ-algebra of a symplectic manifold (the phase space) where logical connectives are the standard set operations. Considering the geometric formulation of quantum mechanics we give a description of quantum propositions in terms of fuzzy events in a complex projective space equipped with Kähler structure (the quantum phase space) obtaining a quantized version of a fuzzy logic by deformation of the product t-norm.
Arbitrarily small changes in the commutation relations suffice to transform the usual singular quantum theories into regular quantum theories. This process is an extension of canonical quantization that we call general quantization. Here we apply general quantization to the time-independent linear harmonic oscillator. The unstable Heisenberg group becomes the stable group SO(3). This freezes out the zero-point energy of very soft or very hard oscillators, like those responsible for the infrared or ultraviolet divergencies of usual field theories, without much changing the medium oscillators. I…
Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the non-Abelian dihedral group D_n - is taken as its symmetry group. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.
A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle…
An anharmonic alliance: exact WKB meets EPT
本文研究多项式量子力学系统的谱，证明在精确 WKB 框架内，任何多项式势均可 $\hbar_0$-形变使得精确量子化条件简化为 $a_γ=-1$，从而为所有能量本征值导出 Borel 可求和的 EPT 级数。
We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S^1 including the Aharonov-Bohm type quantum description. The coherent states are constructed by Perelomov's method as group related coherent states generated by Weyl operators on the quantum phase space Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with…
It is shown that the well-known relativistic correction of quantum Hamiltonian that is present in textbooks appears after quantization of oversimplified relativistic kinetic energy decomposition. Using the proper expression one obtains the kinetic energy operator which relation with the ususal one is consistent with corresponding relation of kinetic energies in special relativity theory. The Schrödinger equation with this operator gives new interesting possibilities for quantum phenomena description.
Quantum mechanics predicts correlation between spacelike separated events which is widely argued to violate the principle of Local Causality. By contrast, here we shall show that the Schrödinger equation with Born's statistical interpretation of wave function and uncertainty relation can be derived from a statistical model of microscopic stochastic deviation from classical mechanics which is selected uniquely, up to a free parameter, by the principle of Local Causality. Quantization is thus argued to be physical and Planck constant acquires an interpretation as the average stochastic deviation…
The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the…
Guiding center quantization of a quantum Hall analog of Hawking radiation
该研究通过引导中心理论量子化，揭示量子霍尔类比霍金辐射实为类比安鲁辐射，其源于非对易平面上半平面电子的运动学约束。
The problem of time emerging in the canonical quantization procedure of gravity signals a necessity to properly define a relational time parameter. Previous approaches, which are here briefly discussed, make use of the dependence of the quantum system on semiclassical gravitational variables in order to define time. We show that such paths, despite the following studies, lead to a non-unitary evolution. We propose a different model for the quantization of the gravity-matter system, where the time parameter is defined via an additional term, i.e. the kinematical action, which acts as a clock fo…
Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac's Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied. In the second part of the paper we have considered the quantum theory of a charge moving in a torus kno…
The Hamiltonian formulation of scalar-tensor theories of gravity is derived from their Lagrangian formulation by Hamiltonian analysis. The Hamiltonian formalism marks off two sectors of the theories by the coupling parameter $ω(ϕ)$. In the sector of $ω(ϕ)=-3/2$, the feasible theories are restricted and a new primary constraint generating conformal transformations of spacetime is obtained, while in the other sector of $ω(ϕ)\neq-3/2$, the canonical structure and constraint algebra of the theories are similar to those of general relativity coupled with a scalar field. By canonical transformations…
Exact WKB in all sectors I: Potentials with degenerate saddles
本文提出一种对能量参数 $u$ 进行复化的新精确 WKB 方法，解决了多扇区局部谐和势的精确量子化问题，揭示了量子作用量中的 resurgence 结构与 $S$-对偶性。
A unified approach to quantum de Finetti theorems and SoS rounding via geometric quantization
本文通过几何量子化统一了量子 de Finetti 定理与 HSoS 舍入算法，提出量子化-舍入及逆过程的新框架，并给出新证明、应用及误差界。
Exact WKB analysis for ${\cal PT}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
本文通过精确WKB分析研究一类PT对称量子力学，构造能量量子化条件及其transseries解，并利用alien微积分与Borel重求和验证Ai-Bender-Sarkar猜想，揭示该猜想在ω>0与ω=0时的有效性差异及替代形式。
Several quantum gravity approaches and field theory on an evolving lattice involve a discretization changing dynamics generated by evolution moves. Local evolution moves in variational discrete systems (1) are a generalization of the Pachner evolution moves of simplicial gravity models, (2) update only a small subset of the dynamical data, (3) change the number of kinematical and physical degrees of freedom, and (4) generate a dynamical coarse graining or refining of the underlying discretization. To systematically explore such local moves and their implications in the quantum theory, this art…
Geometric Phase in Quantum Synchronization
本文研究自旋系统中量子极限环振荡器的几何相位，发现其随信号强度和失谐的结构与同步 Arnold 舌对应，并在同步阻塞区共同消失，同时给出慢旋转弱信号极限下的解析表达式。
The Schrödinger equation in the complex plane and quantum entanglement
通过将薛定谔方程解析延拓到复平面，定义复动量并分析其奇点结构，揭示了波函数零点与动量极点、量子化条件及纠缠的关联。
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