Derivation of Hamiltonian and Symplectic Structure from Tellegen's Theorem as an Approach to the Quantum Quantization of Electrical Circuits

Abstract

In this research, we present a new theoretical framework that links the topology of electrical circuits, represented by Tellegen’ Theorem, which relies exclusively on Kirchhoff’s laws and the topology of the circuit, without regard to the nature of the elements, with Hamiltonian mechanics, which relies on the existence of a coherent structure that links the generalized coordinates to the accompanying moments in phase space, represented by the Hamiltonian.. We demonstrate that Tellegen's theorem is not merely a conservation of energy, but rather the topological constraint that generates the symplectic structure of the system. We derive the net Hamiltonian of the storage elements from the time-integration of Tellegen's theorem and prove that zero in Tellegen's relation is the topological expression of the least-action principle and is equivalent to the time-derivative of the Hamiltonian, dH/dt = 0. We demonstrate how the graph incidence matrix includes the symplectic matrix, which leads to phase space constancy, and that the graph is the true architect of the quantum phase space. We show that the symplectic structure is the classical formulation of Heisenberg's quantum uncertainty principle; ensuring the symplectic structure of the system necessarily entails the fulfillment of the uncertainty principle. We attempt to propose a quantization of Tellegen's theorem for use in quantizing electrical circuits used to fabricate qubits in superconducting processors such as Transmon and others