A Direct Algebraic Approach to Normal Ordering of Exponential Bosonic Operators with Applications to Two-Dimensional Excitonic Form Factors
Abstract
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, ⟨ e−rt⟩ and ⟨ eiq·r⟩, 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.
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BibTeX
@misc{nguyen2026direct,
title = {{A Direct Algebraic Approach to Normal Ordering of Exponential Bosonic Operators with Applications to Two-Dimensional Excitonic Form Factors}},
author = {Duy-Anh P. Nguyen and Ngoc-Tram D. Hoang and Dang-Khoa D. Le and Van-Hoang Le},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13884},
eprint = {2609.13884},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13884},
}
RIS
TY - UNPB
TI - A Direct Algebraic Approach to Normal Ordering of Exponential Bosonic Operators with Applications to Two-Dimensional Excitonic Form Factors
AU - Duy-Anh P. Nguyen
AU - Ngoc-Tram D. Hoang
AU - Dang-Khoa D. Le
AU - Van-Hoang Le
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13884
UR - https://arxiv.org/abs/2609.13884
AB - 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, ⟨ e^−rt⟩ and ⟨ e^iq·r⟩, 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.
ER -