指数玻色子算符正规排序的直接代数方法及其在二维激子形状因子中的应用
A Direct Algebraic Approach to Normal Ordering of Exponential Bosonic Operators with Applications to Two-Dimensional Excitonic Form Factors
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中文总结 AI 辅助
本文提出基于Wei-Norman分解的代数方法，将指数玻色子算符正规排序化为常微分方程，并用于推导二维激子形状因子，获得解析表达式。
中文摘要 AI 辅助
我们基于Wei-Norman分解方法，发展了一种系统性的代数方法，用于指数玻色子算符的正规排序，并将其应用于推导二维半导体材料中的解析激子形状因子。通过引入一个辅助参数，正规排序问题被简化为由底层闭李代数的对易关系确定的一组常微分方程。该方法首先针对与Heisenberg-Weyl代数和su(1,1)代数相关的指数算符进行说明，然后扩展到涉及su(2)以及一个包含两个耦合su(1,1)子代数的六生成元闭代数的双模玻色子算符。对于激子应用，Levi-Civita变换将二维激子问题映射到振子表示，从而提供了用玻色子产生和湮灭算符表示的自然表述。结合Rytova-Keldysh势的Laplace和Fourier表示，该表述将相互作用矩阵元简化为指数玻色子形状因子的计算。各向同性问题由三生成元su(1,1)代数控制，而各向异性情况则需要完整的六生成元代数以及额外的su(2)分解。我们获得了两个形状因子⟨e^{-rt}⟩和⟨e^{i𝐪·𝐫}⟩的显式解析表达式，为二维激子系统中以及可能涉及指数玻色子算符的其他量子问题中的矩阵元计算提供了有用的构建模块。
英文摘要
We develop a systematic algebraic approach, based on the Wei--Norman factorization method, to the normal ordering of exponential bosonic operators and apply it to derive analytical excitonic form factors in two-dimensional semiconducting materials. By introducing an auxiliary parameter, the normal-ordering problem is reduced to a system of ordinary differential equations determined by the commutation relations of the underlying closed Lie algebra. The approach is first illustrated for exponential operators associated with the Heisenberg--Weyl and $su(1,1)$ algebras, and is then extended to two-mode bosonic operators involving $su(2)$ and a six-generator closed algebra that contains two coupled $su(1,1)$ subalgebras. For the excitonic application, the Levi--Civita transformation maps the two-dimensional exciton problem onto an oscillator representation, providing a natural formulation in terms of bosonic creation and annihilation operators. Combined with the Laplace and Fourier representations of the Rytova--Keldysh potential, this formulation reduces the interaction matrix elements to the evaluation of exponential bosonic form factors. The isotropic problem is governed by a three-generator $su(1,1)$ algebra, whereas the anisotropic case requires the full six-generator algebra together with an additional $su(2)$ factorization. Explicit analytical expressions for both form factors, $\langle e^{-rt}\rangle$ and $\langle e^{i\mathbf q\cdot\mathbf r}\rangle$, are obtained, providing useful building blocks for matrix-element calculations in two-dimensional excitonic systems and potentially in other quantum problems involving exponential bosonic operators.
发表机构
- Thu Dau Mot University（胡志明市胡德茂大学）
- Ho Chi Minh City University of Education（胡志明市教育大学）
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