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The quantization of systems composed of transmission lines connected tolumped circuits poses significant challenges, arising from the interplaybetween continuous and discrete degrees of freedom. A widely adopted strategy,based on the pioneering work of Yurke and Denker, entails representing thelumped circuit contributions using Lagrangian densities that incorporate Dirac$\delta$-functions. However, this approach introduces complications, ashighlighted in the recent literature, including divergent momentum densities,necessitating the use of regularization techniques. In this work, we introducea $\delta$-free Lagrangian formulation for a transmission line coupled to alumped circuit without the need for a discretization of the transmission lineor mode expansions. This is achieved by explicitly enforcing boundaryconditions at the line ends in the principle of least action. In thisframework, the quantization and the derivation of the Heisenberg equations ofthe network are straightforward. We apply our approach to an analyticallysolvable network consisting of a semi-infinite transmission line capacitivelycoupled to a LC circuit.

A $\delta$-free approach to quantization of transmission lines connected to lumped circuits