关于R^{2n}的Groenewold-Van Hove问题

Bypassing the Groenewold-van Hove Obstruction: A New Argument in Favor of Born-Jordan Quantization

There are known obstructions to a full quantization in the spirit of Dirac's approach, the most known being the Groenewold-van Hove no-go result. We show, following a suggestion of S. K. Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type T(p)+V(q). This reformulation leads to a non-injective quantization of all observables which, when restricted to po…

Born-Jordan Pseudodifferential Operators and the Dirac Correspondence: Beyond the Groenewold-van Hove Theorem

Quantization procedures play an essential role in microlocal analysis, time-frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, $a(x,ξ)$ an operator $\mathrm{Op}(a)$, according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol $a(x,ξ)=f(x)$ depends only on $x$ or $a(x,ξ)=g(ξ)$ depends only on $ξ$: \[ \mathrm{Op}(f\otimes1)u=fu,\quad\mathrm{Op}(1\otimes g)u=\mathcal{F}% ^{-1}(g\mathcal{F}u) \] where $\mathc…

量化障碍

Obstructions to Quantization

量子化并不是一个直截了当的命题，正如格罗内沃尔德和范霍夫在50多年前发现量子化的“障碍”所表明的那样。他们的“不通过定理”断言，不可能以物理上有意义的方式一致地证明相空间R^{2n}上可观察到的每个经典多项式。对于S^2和T*S^1也发现了类似的障碍，这支持了一个共同的信念，即不通过定理应该具有某种普遍性。令人惊讶的是，事实并非如此--已经证明，对T^2或T*R_+进行量子化没有任何障碍。在本文中，我们努力描绘的情况下，这种障碍会出现，并了解产生它们的机制。我们的目标是猜想-并在某些情况下证明-广义Groenewold-Van Hove定理，并确定最大的李子代数的观测，可以一贯量化。这需要研究泊松代数的结构及其表示。为了达到这些目的，我们包括了预量子化（在扩展的意义上）和量子化理论的阐述，这里用“可观测量的基本代数”来表述。然后，我们详细回顾了R^{2n}，S^2，T*S^1，T^2和T*R_+的已知结果，以及最近的理论工作。

The dynamics of the Koopman-van Hove angular and linear momentum operators

A recent formalism capturing the classical-quantum coupling in a Hamiltonian theory for probabilistic classical mechanics has been proposed: the Koopman-van Hove formulation. The aims of this report are twofolds. First, we rigourously expose this new framework for a non-expert audience and present a construction for the operatorial analogues to the classical angular and linear momentum operators. We then investigate the group actions generating their average as momentum maps as well as their associated dynamics. Finally, we illustrate the theory by deriving the Koopman-van Hove formulation to …

广义相对论序问题的Gröenewald Van Hove式

A Gröenewald Van Hove like formulation of the ordering problems of General Relativity

一个简单的正式重铸众所周知的论点有关的排序问题的广义相对论允许获得在这样一个背景下Gröenewald范霍夫一样的定理。

量子化：历史与问题

Quantization: History and Problems

本文梳理了量子化从经典相空间函数到量子算符映射的历史，重点分析狄拉克方案的缺陷、格罗内沃尔德-范霍夫定理的意义，并介绍外尔量子化与几何量子化等替代方案及其与狄拉克纲领的关系。