Hamiltonian formulation of field–transmission line coupling and its classical–quantum correspondence

Abstract

A Hamiltonian formulation of electromagnetic field coupling to transmission-line systems is presented, providing a unified framework that connects classical field–conductor interaction models with their quantum mechanical description. Starting from the distributed circuit representation, the electromagnetic coupling is incorporated through canonical variables, allowing the total system to be expressed in terms of a well-defined Hamiltonian. The resulting equations of motion reproduce the established classical field-to-line coupling relations in the appropriate limit, while simultaneously enabling a consistent quantization of the transmission-line dynamics. This approach clarifies the physical meaning of the coupling terms, establishes the correspondence between classical field excitation and quantum operators, and provides access to quantum features, such as vacuum fluctuations and single-mode excitations. The formulation does not rely on phenomenological assumptions, making it applicable to a wide range of transmission-line configurations and frequency regimes. These results offer a physically transparent bridge between classical electromagnetic theory and quantum-enabled transmission-line systems, with potential relevance to quantum circuits, waveguide-based devices, and electromagnetic interaction modeling at the quantum–classical interface.