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

A novel idea, Quantum Thermodynamic Transformation Optics (QTTO), is introduced in this article. This theoretical framework integrates the geometric formalism of transformation optics with the thermodynamic principles found in quantum dissipative systems. This concept goes beyond traditional coordinate transformations by affecting the distribution of quantum energy and entropy in a coherent thermodynamic manner as well as reshaping electromagnetic fields. By employing the thermofield dynamics approach, we establish new rules that show how local energy and entropy densities are influenced by the Jacobian determinant of the mapping. This indicates that when the geometry is compressed, it increases the generation of quantum energy and entropy density, while expansion has the opposite effect, all while adhering to the laws of conservation and the second law of thermodynamics. As a practical test, we reformulate the Casimir effect within this framework, yielding a continuous pressure law that connects the quantum and classical realms via a thermal weighting function. This relationship illustrates how both geometry and temperature jointly determine quantum pressure. Additionally, our numerical results for Drude-Lorentz metamaterials support our analytical predictions and align closely with the comprehensive Lifshitz-Matsubara formulation. In fact, QTTO offers a powerful and coherent platform for exploring energy, entropy, and quantum pressure in real-world dissipative media. Beyond our immediate findings, it opens up a systematic pathway for managing quantum thermal processes and controlling field fluctuations in metamaterials and curved optical environments.