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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, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.2299/jsp.30.191

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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.
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.
In quantum theory, a wave function expresses the state of matter waves and is not considered to physically exist in general. Equations that are satisfied by a wave function are called wave equations. In classical physics, wave equations are well known and are derived from the Maxwell equations. However, the Maxwell equations do not define important physical properties, such as the lossless property of circuits. The lossless property is represented by the cascade matrix defined in analog circuit theory. The definition of the lossless property in wave equations became possible owing to the telegrapher's equations introduced by Heaviside. The cascade matrix can be determined from uniform transmission lines that can be obtained from the telegrapher's equations and those extended using the Riccati differential equation.
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.
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.
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 this lecture series, we attempt to construct a theoretical system of quantum theory that emphasizes the wave nature (in other words, that is based on circuit theory) and assumes quanta to be circuit elements. Circuit equations given by voltage and current have steady-state and transient responses. The steady-state response is expressed by complex numbers. For lossless circuits, the active and reactive powers can be distinguished, and the active power is expressed by a conserved quantity. In physics, physical phenomena are considered to occur as a result of some motion, suggesting the importance of motion. In circuit theory, a transient response is generated when a steady state transits to another steady state. In the transient response, the active and reactive powers cannot be distinguished. Therefore, the transient response is considered to be a physical phenomenon without a conserved quantity and to require special techniques involving difficult problems, such as renormalization.