Quantum many-body theory for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-deformed fermions and bosons

We present a comprehensive quantum many-body theory for $q$-deformed fermions and bosons, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters $k$ and $q$, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick theorem and extended Feynman diagrammatics tailored to $kq$ particles, allowing us to calculate two types of Green's functions. Explicit expressions for these Green's functions are derived in both direct and momentum spaces, providing key insights into the collective properties of $q$-fermionic and bosonic systems. Using a random-phase approximation, we estimate the dielectric function for $q$-fermion gas ($k=\ensuremath{-}1$) and analyze the Friedel oscillations, the plasmon excitations, and the energy-loss function. Our results demonstrate that the effective interaction is tuned by the value of $q$ so that a noninteracting limit is obtained as $q\ensuremath{\rightarrow}0$, where the Friedel and the plasma oscillations disappear. An optimal value of $q$, namely, ${q}^{*}$, the plasma frequency, and the energy-loss function show an absolute maximum and the effective interaction changes behavior.