Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group
Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group
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来自量子洛伦兹群几何的晶格上的刘维尔量子力学
DOI：
10.1088/0305-4470/27/13/040
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
V. Rogov
中科院分区：
文献类型：
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作者：
M. Olshanetsky;V. Rogov
我们考虑量子Lobachevsky空间Lq3，它被定义为Hopf代数Aq（SL2(C)）的一个子代数。Podles和Woronowicz引入的Aq（SL2(C)）的Iwasawa分解允许我们考虑Lq3上的星象层坐标的量子模拟。在这些坐标系中，属于Aq量子群对偶的Casimir元Uq（SL2(C)）作用于Lq3中的某一子空间导致了无限一维晶格上的二阶差分算子。在连续极限q到1时，它被转换成薛定谔哈密顿量，它将零模式描述为刘维尔场论（刘维尔量子力学）。我们计算了这个算子的谱（布里渊区）和特征函数。它们是q连续埃尔米特多项式，是麦克唐纳或罗杰斯-阿斯基-伊斯梅尔多项式的特殊情况。当各向异性参数γ和N趋近于无穷大或（γ，N）趋近于0时，该问题中的散射对应于ZN模型中第一级两能级盛装激发在非常特殊的极限下的散射。
We consider the quantum Lobachevsky space Lq3, which is defined as a subalgebra of the Hopf algebra Aq(SL2(C)). The Iwasawa decomposition of Aq(SL2(C)) introduced by Podles and Woronowicz allows us to consider the quantum analogue of the horospheric coordinates on Lq3. The action of the Casimir element, which belongs to the dual to Aq quantum group Uq(SL2(C)), on some subspace in Lq3 in these coordinates leads to a second order difference operator on the infinite one-dimensional lattice. In the continuous limit q to 1 it is transformed into the Schrodinger Hamiltonian, which describes zero modes into the Liouville field theory (the Liouville quantum mechanics). We calculate the spectrum (Brillouin zones) and the eigenfunctions of this operator. They are q-continuous Hermite polynomials, which are particular cases of the Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem corresponds to the scattering of the first two-level dressed excitations in the ZN model in the very peculiar limit when the anisotropy parameter gamma and N to infinity , or, equivalently, ( gamma ,N) to 0.