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Journal of Signal Processing

Circuit Theory Based on New Concepts and Its Application to Quantum Theory

Nobuo Nagai, Hirofumi Sanada, Takashi Yahagi

Published Aug 31, 2025• 0 citations
Publisher

Abstract

In the electromagnetic field formed by coaxial cables, electromagnetic waves of different frequencies are shown to propagate at the same speed (the speed of light c). Therefore, the variational method, which is based on the fact that electromagnetic waves of different frequencies have different propagation speeds, cannot be applied. In this session, the characteristics of light propagating through an electromagnetic field of glass, suitable for applying the variational method and for constructing lenses, are determined. Although the electromagnetic field is derived from the properties of dielectrics, it has been demonstrated to be equivalent to a circuit that represents the Schrödinger equation used in quantum mechanics. In other words, it may be possible to apply electromagnetic fields from classical physics (circuit theory) to quantum mechanics. Quaternions and wave functions that are considered to be related to complex functions will be the topic of future studies because the Schrödinger equation is a complex function.

DOI
10.2299/jsp.29.161

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In the electromagnetic field formed by coaxial cables, electromagnetic waves of different frequencies are shown to propagate at the same speed (the speed of light c). Therefore, the variational method, which is based on the fact that electromagnetic waves of different frequencies have different propagation speeds, cannot be applied. In this session, the characteristics of light propagating through an electromagnetic field of glass, suitable for applying the variational method and for constructing lenses, are determined. Although the electromagnetic field is derived from the properties of dielectrics, it has been demonstrated to be equivalent to a circuit that represents the Schrödinger equation used in quantum mechanics. In other words, it may be possible to apply electromagnetic fields from classical physics (circuit theory) to quantum mechanics. Quaternions and wave functions that are considered to be related to complex functions will be the topic of future studies because the Schrödinger equation is a complex function.
The electromagnetic waves derived from the Maxwell equations are characterized by being lossless. It is known that such lossless electromagnetic waves are obtained in the electromagnetic fields of coaxial cables and the electromagnetic fields generated using coils and capacitors. Because the physical behavior and effects of electromagnetic waves differ depending on frequency, the electromagnetic waves propagating through the electromagnetic fields of coaxial cables and the electromagnetic fields generated using coils and capacitors are limited to very high frequency (VHF) and ultrahigh frequency (UHF). Electromagnetic waves of other frequencies are left as problems for future research. Also, as stated in Section 8 of this article, a "lossless circuit" is also derived from the equations of motion for molecules (Newtonian mechanics). Namely, because the oscillations of a periodic system consisting of N molecules are expressed in the representation equivalent to the circuit representation of lossless electromagnetic waves, it has been demonstrated that a lossless circuit also arises from the motion of molecules with mass derived from Newtonian mechanics. This is the key physical phenomenon that links quantum mechanics to mathematics. We will leave this point as an important problem for future research.
Heaviside proposed the telegrapher's equations as an alternative to the Maxwell equations. The telegrapher's equations demonstrated that the electromagnetic waves propagate in coaxial cables at the velocity of light c. The electromagnetic waves are used in Fourier analysis, Laplace transformation, and z transformation. Electromagnetic waves exhibit the physical phenomenon of simultaneous “partial reflection and refraction”. Feynman found it tremendously important from the viewpoint of the physical phenomenon and advanced to the studies of quantum electrodynamics and quantum mechanics. In this session, we demonstrate that the physical phenomenon of partial reflection and refraction is an expression of the “law of conservation of energy” of electromagnetic waves, which falls into the category of classical physics.
The Schrödinger equation is derived from the concept that particles of matter such as electrons have a wave nature, and it is used to express phenomena specific to quantum mechanics, such as the advancing wave and the quantum tunneling effect. In this session, we apply circuit theory to the Schrödinger equation and define voltage and current using wave functions that satisfy the Schrödinger equation. By this method, we show that a lossless transmission line or circuit can be obtained. Thus, a circuit that satisfies the Schrödinger equation is lossless and is not directly related to heat, but the reactive power is related to the regulation of energy. Therefore, such a circuit can be discussed in a theoretical system that does not rely on probability theory.
Heaviside proposed the telegrapher’s equations as an alternative to the Maxwell equations and obtained the solution of the telegrapher’s equations by the operator method. The electromagnetic waves that satisfy the telegrapher's equations are obtained in the electromagnetic fields of coaxial cables and those of Lecher wires. Since electrons are considered to enable the physical behavior of electromagnetic waves in Lecher wires, in this session, we will obtain the wave functions that represent electrons as electromagnetic waves. Also, Heaviside derived Heaviside’s extension using the operator method and continued fractions, thus obtaining an LC ladder circuit having a periodic system. The LC ladder circuit is a future research topic. The nature of loading-in-lumps systems as electromagnetic waves and their mass will be reexamined in the next session.
Circuit theory, which explains the behavior of electromagnetic waves, concerns reflection waves caused by the difference in impedance. In this session, we demonstrate that electromagnetic waves with the angular frequency of nπ or nπ + π/2 exhibit interesting behavior in circuits with a high impedance ratio and a large amount of reflection. The physical phenomenon of increasing the amount of reflection is considered to be the same as the tunneling effect in quantum mechanics. This demonstration shows that the application of the telegrapher's equations expressing the electromagnetic fields and waves of the Maxwell equations to quantum mechanics may provide a clue to seeing the light at the end of the mysterious tunnel in quantum mechanics.
In Session 8, we demonstrated that circuit theory can be applied to the Schrödinger equation, a basic equation in quantum mechanics. In this session, we attempt to apply circuit theory to the Dirac equation using the Riccati differential equation and demonstrate that physical phenomena such as spin 1/2 can be explained on the basis of circuit theory. To this end, we obtain the one-dimensional Dirac equation for the complex equivalent circuit for a physical phenomenon. We show that the Dirac equation can be regarded as a type of low-energy effective field theory in quantum theory and that circuit theory can be applied to the Dirac equation.