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, $⟨e^{-rt}⟩$ and $⟨e^{i\mathbf{q}\cdot \mathbf{r}}⟩$, 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.