Quantum Energy-Path Curvature (QEPC): A Geometric Framework for Path-Dependent Quantum Energy Costs
Abstract
Traditional quantum mechanics and thermodynamics predominantly treat energy as a state function, focusing on initial and final states while treating the intermediate trajectory as a secondary detail dictated solely by the equations of motion. In this paper, we challenge this paradigm by introducing the framework of Quantum Energy-Path Curvature (QEPC). We propose that between the exact same initial state and final state, different quantum evolution paths can incur fundamentally distinct effective energy costs. By defining a rigorous path energy functional and a corresponding path curvature metric over the control and state manifold, QEPC reveals a deeper geometric structure underlying quantum transitions. We formulate the mathematical foundations of path energy and curvature, derive key properties governing quantum evolution under geometric constraints, and present testable predictions where two distinct paths connecting identical initial and final states yield differing energy expenditures (E_Gamma_1 != E_Gamma_2). This framework broadens the conceptual boundaries of energy state functions and offers profound implications for quantum control, quantum thermodynamics, and fundamental quantum mechanics.
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