A Direct Algebraic Approach to Normal Ordering of Exponential Bosonic Operators with Applications to Two-Dimensional Excitonic Form Factors
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.