Traversable wormhole with GUP corrected Casimir effect in f(R,Lm) gravity

Traversable wormholes require exotic matter for stability, challenging their existence. Quantum mechanics offers a potential solution via the Casimir effect, which generates negative energy densities. In this study, we examine this interaction using two maximally localized quantum state models: the Kempf, Mangano, and Mann (KMM) model and the Detournay, Gabriel, and Spindel (DGS) model, incorporating Generalized Uncertainty Principle (GUP) corrections. We start by deriving the field equations for a generic $f(R,L_m)$ function, assuming a static and spherically symmetric Morris-Thorne wormhole metric. We then consider two specific gravity models: a linear model $f(R,L_m)=\frac{R}{2}+\alpha L_m$ and a nonlinear model $f(R,L_m)=\frac{R}{2}+L_m^n$, where $\alpha$ and $n$ are free parameters. Using the GUP-corrected Casimir effect, we derive the shape functions for these wormholes and investigate their existence. Next, we analyze the obtained wormhole solutions for each scenario, assessing the energy conditions at the wormhole throat with radius $r_0$. Our findings indicate that, for some arbitrary quantities, classical energy conditions are violated at the wormhole throat, highlighting the significant influence of GUP parameters on the geometry and physical properties of wormholes. Additionally, we explore the behavior of the equation of state (EoS) for each model. We further investigate the stability of the KMM and DGS wormhole solutions by applying the Tolman–Oppenheimer–Volkoff (TOV) equation. Finally, we use the volume integral quantifier to determine the amount of exotic matter required near the wormhole throat for both models, providing a comprehensive understanding of the exotic matter distribution necessary for maintaining wormhole stability.