论文标题
量子普通化的Calogero-Moser系统来自免费的汉密尔顿还原
Quantum generalized Calogero-Moser systems from free Hamiltonian reduction
论文作者
论文摘要
具有$ 1/x^2 $排斥潜力的颗粒的一维颗粒系统称为Calogero-Moser系统。可以通过将耦合常数替换为额外的自由度来概括其经典版本，这些自由度跨越了$ \ mathfrak {so}（n）$或$ \ mathfrak {su}（n）$代数在Poisson Brackets方面。我们介绍了此广义模型的量子版本。由于经典的概括是通过自由系统的符合性降低获得的，因此我们提出了一种沿相似线路获得量子系统的方法。自由量子系统的还原导致了哈密顿量，该系统保留了经典系统动力学的差异，具体取决于基础，正交或统一的对称组。 The orthogonal system is known to be less repulsive than the unitary one, and the reduced free quantum Hamiltonian manifests this trait through an additional attractive term $\sum_{i<j}\frac{-\hbar^2}{(x_i-x_j)^2}$, which is absent when one performs the straightforward Dirac quantization of the considered system.我们提出了广义量子calogero-moser hamiltonian的详细而严格的推导，我们发现了粒子数量$ n = 2,3 $的光谱和波形，我们将汉密尔顿人的一般价值分别为$ n $。
The one-dimensional system of particles with a $1/x^2$ repulsive potential is known as the Calogero-Moser system. Its classical version can be generalised by substituting the coupling constants with additional degrees of freedom, which span the $\mathfrak{so}(N)$ or $\mathfrak{su}(N)$ algebra with respect to Poisson brackets. We present the quantum version of this generalized model. As the classical generalization is obtained by a symplectic reduction of a free system, we present a method of obtaining a quantum system along similar lines. The reduction of a free quantum system results in a Hamiltonian, which preserves the differences in dynamics of the classical system depending on the underlying, orthogonal or unitary, symmetry group. The orthogonal system is known to be less repulsive than the unitary one, and the reduced free quantum Hamiltonian manifests this trait through an additional attractive term $\sum_{i<j}\frac{-\hbar^2}{(x_i-x_j)^2}$, which is absent when one performs the straightforward Dirac quantization of the considered system. We present a detailed and rigorous derivation of the generalized quantum Calogero-Moser Hamiltonian, we find the spectra and wavefunctions for the number of particles $N=2,3$, and we diagonalize the Hamiltonian partially for a general value of $N$.
