Quantum Thermodynamic Transformation Optics: A Unified Framework for Energy and Entropy With Application to the Casimir Force in Dissipative Metamaterials

A novel concept, quantum thermodynamic transformation optics (QTTO), is introduced as a theoretical framework integrating transformation optics with thermodynamics of quantum dissipative systems. Beyond conventional coordinate transformations, QTTO describes how physically implemented transformation-optical media with dispersion, loss, reservoir noise, and boundary conditions reorganize the statistical distribution of quantum energy and entropy-related field fluctuations. Using thermofield dynamics, we establish transformation rules showing how local fluctuation related energy and entropy densities are controlled by the Jacobian determinant of the mapping. Geometric compression enhances local statistical weight of quantum and thermal fluctuations, whereas expansion produces the opposite effect, with global conservation laws and the second law of thermodynamics preserved. This framework provides a thermodynamic design rule for predicting quantum-thermal field correlations in dissipative metamaterials through coordinate-induced optical geometry. As a practical test, we reformulate the Casimir effect, obtaining a reduced finite-temperature pressure law that connects quantum vacuum and classical-thermal regimes via a thermal weighting function. This relation shows how geometry and temperature jointly organize quantum pressure in an effective asymptotic description. Numerical results for Drude-Lorentz metamaterials support the analytical predictions and reproduce the relevant asymptotic behavior of the finite-temperature Lifshitz-Matsubara formulation while capturing the overall crossover structure. To our knowledge, QTTO provides one of the first analytical foundations for a metric-based thermodynamic framework of quantum fields in dissipative and curved optical media.